information entropy and number of principal components in shell...

11 June 1998

Ž .Physics Letters B 429 1998 1–6

Information entropy and number of principal componentsin shell model transition strength distributions

V.K.B. Kota a, R. Sahu a,b

a Physical Research Laboratory, Ahmedabad 380 009, Indiab Physics Department, Berhampur UniÕersity, Berhampur 760 007, India

Received 3 March 1998; revised 26 March 1998Editor: W. Haxton

Abstract

ŽThe bivariate Gaussian form for smoothed strength densities given by statistical spectroscopy equivalently by the.embedded Gaussian orthogonal ensemble EGOE is used to derive formulas for information entropy and number of principal

components, which are measures of complexity and choas, in transition strength distributions. They describe, in terms of theŽ .bivariate correlation coefficient z , shell model results and reduce to GOE results for zs0. q 1998 Elsevier Science B.V.

All rights reserved.

PACS: 05.45.qb; 21.60.Cs; 24.60.LzKeywords: Chaos; Shell model; Statistical spectroscopy; GOE; EGOE; Information entropy; Number of principal components; Strengthdistributions

w xRecently Zelevinsky, Brown and others analyzed 1,2 eigenvector amplitudes from large shell modelŽ .matrices in terms of, among other things, information entropy hereafter referred as S and number of principal

Ž .components NPC which are measures of complexity and chaos in many-body systems. Compared with theŽ .standard GOE Gaussian orthogonal ensemble of random matrices results, the shell model results show strong

Ž . Ž .secular energy variation; GOE gives Nr3 for NPC and ln 0.48 N for S where N is matrix dimension. On theŽ . w xother hand it is known from the developments in statistical spectroscopy SS 3–10 that embedded Gaussian

Ž Ž ..ensembles with kybody interactions EGOE k are more appropriate for interacting many-body systems suchas atomic nuclei; in the last few years with the developments in the subject of ‘quantum chaos’, there is new

w x w xinterest in developing further and applying SS and EGOE to atoms 11 , molecules and solids 12 , mesoscopicw xsystems 13 etc. Recognizing that analysis of eigenvector amplitudes is equivalent to dealing with transition

strengths and that transition strengths are observables, we derived in this letter, using the general results of SSw x Ž .9 for the smoothed with respect to energy bivariate strength densities, EGOE formulas for NPC and S for

Ž .ms 6, Js2,Ts0transition strength distributions. They are tested against shell model results in 2 s1d space and theyw xalso describe the shell model results in 1,2 .

Two important results of statistical spectroscopy, relevant for the present paper, are that in stronglyŽ . Ž . w x Ž .interacting shell model spaces essentially in 0"v spaces i the state densities take Gaussian form 4 and ii

0370-2693r98r$19.00 q 1998 Elsevier Science B.V. All rights reserved.Ž .PII: S0370-2693 98 00461-4

( )V.K.B. Kota, R. SahurPhysics Letters B 429 1998 1–62

w xthe bivariate strength densities take bivariate Gaussian form 9 . These results have their basis in the EGOEŽ .representation of the hamiltonian H which is in general one plus two-body in nuclear case and transition

w x Ž . Žoperators OO 4,9 . The EGOE k is defined in m-particle spaces i.e. in the space generated by distributing m.fermions over NN single particle states with a GOE representation in k-particle space for k-body operators.

Before going further it should be mentioned that from now on SS and EGOE are used interchangeably. TheŽ . Ž . w xeigenvalue density I E or its normalized version r E is defined by 4

²² :: ² :I E s d HyE sd d HyE sd r E ;Ž . Ž . Ž . Ž .21 1 EyeEGOE

r E ™ r E sr E s expy . 1Ž . Ž . Ž . Ž .GG ž /' 2 s2p s

Ž . Ž . Ž 2² : ² :² :In 1 PPP denotes trace similarly PPP denotes average , the e , s and d are centroid, width s is22. ² :variance and dimensionality respectively. Note that es H , s s Hye , ‘GG’ stands for Gaussian and² :Ž .

Ž . Ž .the bar over r E indicates ensemble average smoothening with respect to EGOE. In the shell modelexamples described ahead, the traces correspond to traces over fixed-JT spaces and in practice often the so

w x Ž . Ž X.called Edgeworth corrections 14 are added to the Gaussian form in 1 . The strength R E, E generated by aŽ X. X 2² :transition operator OO in the H-diagonal basis is R E, E s N E NOONE N . Correspondingly the bivariate

Ž X. Ž X. w xstrength density I E, E or r E, E which is positive definite and normalized to unity is defined by 9biv; OO biv; OO

X † X X X ² X : 2 † X² :² :² :I E, E s OO d HyE OOd HyE s I E N E NOONE N I E s OO OO r E, E² :Ž . Ž . Ž . Ž . Ž . Ž .biv; OO biv; OO

EGOEX X X

r E, E ™ r E, E sr E, EŽ . Ž . Ž .biv; OO biv; OO b iÕyGG ; OO

1s

2(2ps s 1yz1 2

=

2 2X X1 Eye Eye E ye E ye1 1 2 2exp y y2z q . 2Ž .2 ž / ž / ž / ž /½ 5s s s s2 1yzŽ . 1 1 2 2

Ž . 2 2 Ž .In 2 e and e are the centroids and s and s are the variances of the marginal densities r E and1 2 1 2 1; OO

Ž X .r E respectively. The bivariate reduced central moments of r are m s2 ; OO b iv ; OO p qq p

Hy e H y e† †2 1 ² :OO OO OO OO and zsm is the bivariate correlation coefficient. Though the EGOE forms¦ ;Ž . Ž . 11s s2 1

Ž . Ž .in 1 , 2 are derived by evaluating the averages over fixed-m spaces, in large number of numerical shell modelw xexamples it is verified that 2–8 they apply equally well in fixed- m, mT and mJT spaces. In practice, just as in

Ž .the case of state densities, bivariate Edgeworth corrections are added to the bivariate Gaussian form in 2 . ItŽ . Xshould be remarked that in general the mJT values for the E and E spaces need not be same. The Gaussian

Ž . Ž .forms in 1 , 2 give compact formulas for NPC and S in transition strength distributions incorporating theŽinformation that the hamiltonian and transition operators are of lower particle rank i.e. k,t<m where k and t

.are maximum particle ranks of H and OO respectively . To this end first we define NPC and S for transitionstrengths and then the ensemble average with respect to EGOE is carried out.

Ž .Let us introduce normalized strength RR, average smoothed normalized strength RR and locally renormalizedˆstrength R where

y1y1X X X X† 2 † 2² : ² :² : ² :RR E, E s ENOO OONE N E NOONE N , RR E, E s ENOO OONE N E NOONE N ,� 4Ž . Ž . � 4y1

X X X2 2ˆ ² : ² :R E, E s N E NOONE N N E NOONE N . 3Ž . Ž .� 4

( )V.K.B. Kota, R. SahurPhysics Letters B 429 1998 1–6 3

Then the measures NPC and entropy S for strength distributions are

y12X X XNPC s RR E, E , S sy RR E, E ln RR E, E . 4� 4Ž . Ž . Ž . Ž . Ž . Ž .Ý ÝE E½ 5

X XE E

2ˆThe EGOE expression for NPC is derived by writing NPC in terms of R and RR and using the fact thatŽ . Ž .EX X X 2ˆŽ . Ž . w x ² : ² :R E, E is Porter-Thomas P-T 6,15 ; i.e. the locally renormalized amplitudes E NOONE r N E NOONE N 1r2� 4

are Gaussian distributed with zero center and unit variance. It is not out of place to mention that in a recentw x Ž w x.study of strength fluctuations 16 and in many other similar investigations see for example 17 the local

X 2 w x² :averages N E NOONE N are obtained via a numerical smoothening procedure while in 6,15 SS forms are used;w x w x w x Ž .in 6 double polynomial expansion given in 5 is used and in 15 bivariate Gaussian form 2 is employed. The

ˆ w xP-T law for R gives 18 ,

2ˆ ˆ ˆ ˆR s1, R s3, Rln R s ln 0.48 . 5Ž . Ž .Ž . Ž . Ž .Ž . Ž . Ž .Eqs. 3 – 5 generate a compact formula for NPC ,E

y1 y1EGOE 2 2 2X X XˆNPC ™ R E, E RR E, E s 3 RR E, E sd E r3� 4 � 4Ž . Ž . Ž . Ž . Ž .� 4Ý ÝE eff½ 5½ 5

X XE E

y1 22X ˆ ˆr E, E s z EqDŽ .Ž . ˆ2 biv; OO 2 2X X X 2(s d r3 r E dE s d r3 s 1yz X expy ;Ž . Ž . Ž .Ž . ˆH1; OO 2X X ž /Xr E ž /Ž .

1r22X X X 2ˆ ˆs ss rs , D s e ye rs , Es Eye rs , Xs 2y s 1yz . 6Ž . Ž . Ž . Ž .Ž .ˆ ˆ2 2 2 2 1 1 2

Ž . Ž . Ž . Ž .In 6 the first equality follows directly from the definitions in 3 , 4 , the second equality from 5 and theŽ .third equality defines effective dimension d E which depends on the energy E. Rewriting d in terms ofeff eff

Ž . Ž . Ž .the bivariate strength densities and state densities defined in 1 , 2 gives the fourth equality in 6 and the finalŽ .formula follows from the smoothed SS or EGOE Gaussian forms for the two densities. In deriving the final

X Ž . X XŽ X.result the sum over E in 6 is replaced by integral over E with weight factor I E ; the centroid and width ofXŽ X. X X Ž . Ž . w xI E are s and e . The expression for d E given by 6 is first reported in 19 . Following the sameeff

Ž .arguments that gave 6 , the EGOE expression for the entropy S is derived,

EGOEX X X X X Xˆ ˆ ˆS ™ y RR E, E R E, E ln R E, E y R E, E RR E, E lnRR E, EŽ . Ž . Ž . Ž . Ž . Ž . Ž .� 4 Ž .Ý ÝE

X XE E

Xr E, EŽ .b iÕ ; OOX X Xs ln 0.48d y dE r E NE lnŽ . Ž .H OO X X

r E r EŽ . Ž .1; OO

2° ¶2 2 ˆ ˆs z EqD1ys 1yzŽ . ˆˆ ž /2 22X 2~ •(s ln 0.48d s 1yz exp expy2 2 2¢ ß2

2 2 ˆ ˆs z EqD1ys 1yzŽ . ˆˆ ž /2 22X 2(´exp S s0.48d s 1yz exp expy . 7Ž . Ž .Ž . ˆE 2 2 2

Ž . Ž . Ž . Ž .In going from the first equality to the second in 7 , results in 5 and 1 , 2 are used; note thatŽ X . Ž X. Ž .r E NE sr E, E rr E is a conditional density and it takes a Gaussian form with r and rOO b iÕ; OO 1; OO biv; OO 1; OO

Ž . Ž . Ž .taking Gaussian forms. The third equality in 7 is obtained by substituting the Gaussian forms in 1 , 2 for the


Ž .densities in the second equality and carrying out the integrations. In the dilute limit, with EGOE k for H andŽ . Žan independent EGOE t for OO in m-particle space i.e. in the situation, as it is the case with the numerical

Ž . Ž X.examples discussed ahead, that the initial E and final E spaces connected by the transition operator OO are. w x X Xsame and the H and OO operators are representable by EGOE , it is seen that 4,9,18 dsd , s ss ssss ,1 2

e se seseX and1 2

y1m my t

zs . 8Ž .ž / ž /k k

Ž . Ž .Then the basic forms of NPC and S are determined only by the correlation coefficient z ,E E

2 ˆ2d z E Eye4 ˆ(NPC s 1yz expy ; Es ;Ž . E 23 s1qz

2 2 ˆ2z z E2(exp S s0.48d 1yz exp expy . 9Ž . Ž .Ž .E 2 2

w x Ž .For GOE obviously zs0 9 and then the equations in 9 reduce to the well known GOE results stated in thebegining.

Ž .ms 6, Js2,Ts0Shell model calculations in 307 dimensional 2 s1d space are carried out, using the Rochester–Ž . Ž . Ž . Ž .OakRidge shell model code, for testing the EGOE results given by 6 , 7 . The hamiltonian Hsh 1 qV 2 is

w x Ž Ž .. 17 Ž Ž .defined by Kuo’s 20 two-body matrix elements V 2 and O single particle energies h 1 me sy4.15d5r2

. w xMeV, e s0.93 MeV, e sy3.28 MeV . The transition operator is chosen to be, basically as in 9 , thed s3r2 1r2

two-body part of H without the configuration–isospin centroid producing part. Thus OO in our example is aw x ² :two-body operator. For reasons explained in detail in 9,18 the diagonal matrix elements ENOONE of OO in H

diagonal representation are put to zero. With these choices it is seen that eseX sy32.78 MeV, sss

X s10.24ˆMeV, e se sy29.88 MeV, s ss s10.67MeV, D s0.28, s s1.04 and Xs1.25. More importantlyˆ1 2 1 2 2 2

Ž Ž .zs0.55 the EGOE estimate from 8 is 0.67 for one-body H and 0.4 for two-body H as ms6 and the. Ž . Ž .transition operator rank is ts2 in our example . Using these parameter values in 6 , 7 the EGOE curves for

Ž .NPC and exp S are constructed and compared in Fig. 1 with exact shell model results. The theoreticaldescription given by EGOE is in excellent agreement with the shell model results and they show clear

Ž .ms 12, Js2,Ts0departures from GOE results just as seen in the 3276 dimensional 2 s1d shell model results inw x Ž . Ž .1,2 . The EGOE results 9 give also a formula for the ratio exp S rNPC,

z 2

2 2 2exp ˆz 1yz EŽ .2exp S rNPCs 1.44 exp . 10Ž . Ž . Ž .22 2 1qzŽ .(1qz

Ž . Ž .Eq. 10 shows that exp S rNPC increases as the energy is away from the center and this behaviour is clearlyw x Žseen in Fig. 1. Let us now consider the shell model results given in 1,2 from now on referred as ZHB and

. Ž < < 2² :ZBFH respectively where squares of eigenfunction amplitudes C NC with C denoting shell modelE K K.basis states are analyzed.

w xIn the shell model analysis presented in ZHB and ZBFH, the correlation coefficient z can be estimated 8 inŽ .terms of the average width s of the individual shell model mean field basis states and the width s of thek E

2(eigenvalue distribution; zf 1y s rs . Using the s and s values given in ZBFH for theŽ .k E k EŽ .ms 12, Js2,Ts0 Ž .2 s1d example, it is seen that z;0.65 and comparing the results in Fig. 32 of ZBFH with 9

Ž . Žgives z;0.7. The Gaussian behaviour of NPC and exp S vs E seen in the examples given in ZBFH in ZHBŽ . . Ž .the NPC and exp S are plotted against the eigenstate number instead of E is immediately explained by 6 ,

Ž . Ž . Ž .7 , 9 and they clearly show that as z gets closer to unity the departures from GOE will be maximum. Eq. 9Ž . Ž .gives for the example in Fig. 1, the peak values of NPC and exp S to be 0.95, 0.97 of the GOE values while

( )V.K.B. Kota, R. SahurPhysics Letters B 429 1998 1–6 5

Ž . Ž . Ž .Fig. 1. Number of principal components NPC and information entropy S versus energy E for a strength distribution in six particleŽ .2 s1d shell space with Js2,T s0. The hamiltonian and the transition operator are defined in the text. Shown in the figure is also the ratio

Ž .exp S rNPC versus E. The exact shell model results are compared with the GOE and EGOE predictions; the EGOE predictions are givenŽ . Ž .by 6 , 7 .

ˆ ˆ ˆŽ . Ž .the exact values of s and D give 0.97, 0.98 ; note that the maximum values occur for EsyD r zs andˆ ˆ2 2 2 2

here the EGOE differ from GOE only by 2–3% as zs0.55 for our example. However for the example in Fig.Ž Ž ..32 of ZBFH the peak value estimates obtained using 9 differ from GOE values by 13% and 9% for NPC and

Ž .exp S respectively. These deviations are indeed observed by ZBFH and now it is clear that they are due to theŽ . Ž . Ž .fact that here z;0.7. In addition, the variation in NPC and S with correlation coefficient as given by 9E E

Ž .explains many other results reported by ZHB and ZBFH. For example with HsV 2 , they found the results tobe closer to GOE over much wider energy range. In this example s ;s and therefore zf0. This thenk E

Ž . Ž .explains the results for HsV 2 reported in ZHB and ZBFH. Using 9 it is seen that for zs0.1, 0.2 and 0.3ˆŽ .the EGOE values for exp S differ from GOE value by 3%, 12% and 25% respectively at Es2.5.

Ž . Ž .In conclusion statistical spectroscopy EGOE expressions for the measures NPC and exp S in transitionstrength distributions are derived in this letter. The bivariate correlation coefficient z that characterizes thestrength distributions determines the energy variation of the measures as seen in shell model results. The


Ž . Ž .agreement seen in Fig. 1 between the exact shell model results and the EGOE forms 6 , 7 confirm that thehamiltonian and the transition operator in our numerical example are well represented by EGOE. Thus EGOEŽ .and SS considerations are essential for dealing with questions related to complexity and chaos in finitequantum systems of interacting particles such as atomic nuclei. For example to study the region of onset of

w x w x w xchaos 13 , chaos and thermalization 21,13 , nature of chaos near yrast line at high-spins 22 etc. it is necessaryŽ . w x w xto go beyond the simple EGOE and SS and consider interpolating 13 and partitioned 23 EGOE’s just as it is

w xdone before for the Gaussian ensembles 24 . Some of these more general EGOE ensembles are beingw xinvestigated by using the large body of results available in statistical spectroscopy 3–10,23 and by further

extending them. Results of these investigations will be reported elsewhere.

Acknowledgements

Ž .One of the authors V.K.B.K. thanks S. Tomsovic for many valuable discussions and O. Bohigas for hisŽ .interest in the work. RS is thankful to DST Government of India for financial support.

References

w x Ž .1 V. Zelevinsky, M. Horoi, B.A. Brown, Phys. Lett. B 350 1995 141.w x Ž .2 V. Zelevinsky, B.A. Brown, N. Frazier, M. Horoi, Phys. Rep. 276 1996 85.w x Ž . Ž . Ž . Ž .3 J.B. French, K.F. Ratcliff, Phys. Rev. C 3 1971 94; C 3 1971 117; F.S. Chang, J.B. French, T.H. Thio, Ann. Phys. NY 66 1971

137.w x Ž . Ž .4 K.K. Mon, J.B. French, Ann. Phys. NY 95 1975 90.w x Ž . Ž .5 J.P. Draayer, J.B. French, S.S.M. Wong, Ann. Phys. NY 106 1977 472, 503.w x Ž .6 T.A. Brody, J. Flores, J.B. French, P.A. Mello, A. Pandey, S.S.M. Wong, Rev. Mod. Phys. 53 1981 385.w x7 S.S.M. Wong, Nuclear Statistical Spectroscopy, Oxford University Press, NY, 1986.w x Ž . Ž .8 J.B. French, V.K.B. Kota, Ann. Rev. Nucl. Part. Sci. 32 1982 35; V.K.B. Kota, K. Kar, Pramana - J. Phys. 32 1989 647.w x Ž . Ž .9 J.B. French, V.K.B. Kota, A. Pandey, S. Tomsovic, Ann. Phys. NY 181 1988 235.

w x Ž . Ž .10 V.K.B. Kota, D. Majumdar, Z. Phys. A 351 1995 365, 377; Nucl. Phys. A 604 1996 129.w x Ž .11 V.V. Flambaum, A.A. Gribakina, G.F. Gribakin, M.G. Kozlov, Phys. Rev. A 50 1994 267.w x Ž . Ž .12 J. Karwowski, Int. J. Quantum Chem. 51 1994 425; D.B. Waz, J. Karwowski, Phys. Rev. A 52 1995 1067; J. Planelles, F. Rajadell,

Ž .J. Karwowski, J. Phys. A 30 1997 2181.w x Ž . Ž .13 V.V. Flambaum, G.F. Gribakin, F.M. Izrailev, Phys. Rev. E 53 1996 5729; V.V. Flambaum, F.M. Izrailev, Phys. Rev. E 56 1997

Ž .5144; Ph. Jacquod, D.L. Shepelyansky, Phys. Rev. Lett. 79 1997 1837.w x14 M.G. Kendall, A. Stuart, Advanced Theory of Statistics, vol. 1, 3rd ed., Hafner Publishing Company, N.Y., 1969.w x15 V.K.B. Kota, R. Sahu, to be published, 1998.w x Ž .16 A.A. Adams, G.E. Mitchell, W.E. Ormand, J.F. Shriner Jr., Phys. Lett. B 392 1997 1.w x Ž .17 D.C. Medrith, Phys. Rev. E 47 1993 2405.w x18 S. Tomsovic, Ph.D. thesis, University of Rochester, 1986, unpublished.w x Ž . Ž .19 V.K.B. Kota, in: Proc. Symp. Nucl. Phys. 39 A 1997 208 published by Library and Information Services, B.A.R.C., Bombay, India .w x Ž .20 T.T.S. Kuo, Nucl. Phys. A 103 1967 71.w x Ž .21 M. Horoi, V. Zelevinsky, B.A. Brown, Phys. Rev. Lett. 74 1995 5194.w x Ž .22 M. Matsuo, T. Dossing, E. Vigezzi, S. Aberg, Nucl. Phys. A 620 1997 296.w x Ž .23 J.B. French, V.K.B. Kota, Phys. Rev. Lett. 51 1983 2183.w x Ž . Ž . Ž .24 J.B. French, V.K.B. Kota, A. Pandey, S. Tomsovic, Ann. Phys. NY 181 1988 198; D.M. Leitner, Phys. Rev. E 48 1993 2536; A.

Ž . Ž . Ž .Pandey, Chaos, Solitons and Fractals 5 1995 1275; T. Ghur, Ann. Phys. NY 250 1996 145.

11 June1998


CP-violation tests with polarized resonance neutrons

V.E. Bunakov 1, I.S. NovikovPetersburg Nuclear Physics Institute, 188350 Gatchina, Russia

Received 26 November 1997; revised 30 March 1998Editor: W. Haxton

Abstract

The enhancements of CP-violating effects in resonance neutron transmission through polarized targets are studied for 2possible versions of experiment. The importance is stressed of error analysis and of pseudomagnetic effects’ compensation.q 1998 Published by Elsevier Science B.V. All rights reserved.

PACS: 11.30.Er; 13.88.qe; 14.20.DhKeywords: CP-violation; Resonance neutrons

1. Introduction

w xIt was shown 1–3 about 15 years ago that CP-violation effects in transmission of polarized neutronsthrough the polarized target might be enhanced in the vicinity of p-resonances by 5–6 orders of magnitude.

Ž .Originally it was suggested to measure the difference in transmission of neutrons with spins parallel N andqŽ . Ž .antiparallel N to the vector k =I k and I are the neutron momentum and the target spin :q n n

N yN s ysq y q yh s f2 1Ž .T N qN s qsq y q y

Here N and N are the numbers of neutrons with the corresponding helicities transmitted through theq ypolarized target sample, s and s are the corresponding total neutron cross-sections. However, it wasq y

w xpointed 4 that without the special precautions the nuclear pseudo-magnetic precession of neutron spin togetherwith the precession induced by the P-violating interactions would give rise to numerous effects camouflaging

w xthe CP-violating ones. As a possible remedy of this nuisance it was suggested 4 to compensate the nuclearpseudo-magnetic field by the external magnetic field in order to nullify the neutron spin rotation angle f.

Ž y4However, in order to measure the CP-violating interaction with the reasonable accuracy about 10 of the. y5 w xP-violating one it was necessary to check the spin rotation angle with the precision of at least 10 rad 4 .

1 E-mail: Vadim. [email protected].

0370-2693r98r$19.00 q 1998 Published by Elsevier Science B.V. All rights reserved.Ž .PII: S0370-2693 98 00462-6

( )V.E. BunakoÕ, I.S. NoÕikoÕrPhysics Letters B 429 1998 7–148

w xIn order to circumvent the above difficulties Stodolsky 5 suggested to measure the difference N yN ,qy yqwhere N is the number of neutrons transmitted through the target and the subscript indices mean the neutronhelicity before and after the transmission. Consider the polarized neutron scattering amplitude of the form

w xfsAqp BP s PI qCP s Pk qp DPs P k =I 2Ž . Ž . Ž .t n n n t n n

where s is neutron spin, p is the target degree of polarization, A and B are the spin-independent andn t

spin-dependent parts of the strong interaction amplitude, D is the P- and CP-violating interaction amplitude,Žrespectively. The term C contains contributions from both weak P-violating and strong interaction from the

Ž .Ž . w x.term of the type s Pk k PI in scattering amplitude – see e.g. 6 .n n n

Stodolsky demonstrated that the difference

N yN ; Im DB) 3Ž . Ž .qy yq

is free from the above camouflaging effects. It is well-known that in order to improve the accuracy it ispreferable to measure the relative values, i.e. to normalize the above difference. Although Stodolsky neverbothered to introduce this normalization, it seems natural to consider the ratio

N yNqy yqTs 4Ž .

N qNqy yq

w xA few years later Serebrov 7 suggested to measure the quantity

N yN q N yNŽ . Ž .qq yy qy yqXs 5Ž .

N yN y N yNŽ . Ž .qq yy qy yq

One can easily see that

N yNqy yqXs1q2 6Ž .


˜The actual CP-violating effect causes the deviation of X from unity. Therefore the actually measured quantity X

N yNqy yqXs 7Ž .


is simply the one suggested by Stodolsky, but normalized in a rather odd manner.The main point is that up to now nobody cared to do the analysis of the energy dependence of the quantities

˜ w xT or X in the manner it was done for the originally considered CP-violating quantity h in Refs. 1–3 . It wasT

taken for granted that all these quantities would be enhanced in exactly the same way as h . However, all theT˜quantities in the numerators and denominators of T and X contain various combinations of real and imaginary

Ž . Ž .parts of all the four amplitudes A, B, C and D in Eq. 2 . Most of them show a rather complicated energyŽ w x.dependence see e.g. 3,6,8 in the resonance region. Some of them not only vary in magnitude, but even

˜change their sign. This means that up to now one does not know whether the suggested values T and X arereally enhanced and what is the magnitude of this enhancement, if any. Investigation of these problems is themain point of our present publication. For the time being we are not going to consider the false effects arisingfrom the difference of the polarizing and the analyzing power of polarizer and analyzer. We shall also restrictourselves with cases of ‘‘ideal geometry’’ when the incident beam polarization is either parallel or anti-parallelto the neutron momentum.

( )V.E. BunakoÕ, I.S. NoÕikoÕrPhysics Letters B 429 1998 7–14 9

2. Analysis of T

In order to obtain the expressions for the relative quantities of interest in terms of the energy-dependentŽ . w xcomplex amplitudes A, B, C and D of Eq. 2 , one might use the method developed in Ref. 9 . Introducing the

w xspin density matrix and the evolution operators of Ref. 9 , one obtains the expression for T :

Im DB)Ž .Ts2 8Ž .2 2< < < <D q B

Ž . Ž . w x ŽThe expressions for complex amplitudes D E and B E are obtained using the methods of Ref. 3 see alsow x.6,8 . The main contribution to the T-noninvariant amplitude D in the vicinity of the p -resonance comesIq1r2

from the term coupling this resonance with the corresponding s -resonance:Iq1r2

g s g pIq1r2 Iq1r2

Df V 9Ž .Ts pEyE q ıG r2 EyE q ıG r2Ž . Ž .Iq1r2 s Iq1r2 p

Ž .In the optimal cases like La target these s and p resonances contribute equally to the strongIq1r2 Iq1r2

amplitude B in this energy region. Taking into account all the other resonances would only lead to somenumerical changes, while the general qualitative picture would be the same. Therefore we consider:

g s g s g p g pIq1r2 Iq1r2 Iq1r2 Iq1r2

Bf q 10Ž .s pEyE q ıG r2 EyE q ıG r2Ž . Ž .Iq1r2 s Iq1r2 p

Ž . Ž .Inserting these expressions into Eq. 8 we see that the quantity T E in the vicinity of the p-wave resonanceenergy E is:p

g p V PGIq1r2 T pT E fy2 P 11Ž . Ž .s 2 2g EyE qG r4Iq1r2 Ž .p p

Here G stands for the p-resonance total width, while V is the matrix element of CP-violating interactionp T

causing the transition between the p- and s-resonance states. Further on in our numerical calculations we shally4 Ž y4 .assume the ratio of the CP-violating interaction strength to the P-violating one to be 10 i.e. V rV s10 .T P

The quantities g s, p stand for the neutron width amplitudes of the s- and p-resonances with spin Js Iq1r2.J

The sign of the effect is defined by the signs of g ’s and V . For the sake of simplicity we shall choose them inT

our numerical calculations so that the net effect is positive.Ž .We observe in Eq. 11 the resonance enhancement of the effect typical for all the symmetry-breaking effects

Ž w x.in nuclear reactions see 3,8 . In order to see explicitly the ‘‘dynamical enhancement’’, which is also typicalfor these effects, one might cast the value of T in this maximum in the following form:

g p V dIq1r2 TpT EsE f P P 12Ž .Ž .Iq1r2 sg d GIq1r2

Here d and G stands for the average resonance spacing and total width. One can see here the presence of the3 Ždynamical enhancement factors V rdfF P10 F is the strength of the CP-violating interaction relative toT T T

. 3the strong interaction one and of the resonance enhancement factors drGf10 coming from the fact that theŽ .effect is proportional to the time t; 1rG spent by the incident neutron in the CP-violating field of the target.

Ž w x. p s y3We also see the presence of the ‘‘entrance channel hindrance’’ factor see 3,8 g rg f10 typical for allthe low energy scattering experiments with P-violation. However, the resonance enhancement factor enters thequantity h quadratically, while T contains it only linearly. Therefore the net enhancement of the T quantity isT

only by a factor of 103 instead of the 105–106 factor in h .T


Ž .These conclusions are illustrated in Fig. 1a, where the energy behavior of the quantity T E is shown for theparticular case of the famous La p-resonance at E s0.75 eV.p

Consider now a very important problem of the optimal choice of the target sample thickness. One shouldmind that in the case of h value, likewise in the case of the longitudinal polarization P caused by theT

P-violating weak interaction, the correct expression for the experimentally measured ratio can be written asŽ w x.follows see e.g. 10,3,11 :

N yN s ysq y q yf PxPr 13Ž .

N qN 2q y

where x is the target sample thickness and r is the density of nuclei in this sample. Since the experimentallyobserved effect is linear in target thickness, it seems that one should choose the thickest target possible.However, the neutron countings N decrease exponentially with x. Therefore the statistical relative error of"

measuring each N value

dN 1 1x rs r2s s e 14Ž .'N NN ( 0

Ž .also increases exponentially with x N here stands for the number of polarized neutrons incident on the target .0

In order to find the optimal target thickness x one should estimate the relative error s rh of the quantity in0 h TT

Ž . Ž .the l.h.s. of Eq. 13 and define its minimum by equating the x-derivative of the relative error to zero . In this

Ž . Ž .Fig. 1. a Energy dependence of T in the vicinity of p -resonance. b Relative error of T as a function of target thickness withoutIq1r2Ž . Ž .compensation. c Energy dependence of T with full compensation. d Relative error of T as a function of target thickness with full

compensation.


Žway one obtains that the optimal thickness in the case of h quantity is 2ls2rsr here l stands for the meanT.free path of the neutron in the target sample . It is only by choosing the optimal x that one obtains the last line0

Ž .in Eq. 1 .The relative error of the quantity T looks more complicated. One can easily see that the main contribution to

it comes from the relative error of the numerator in T :

Ž .Im A x

Ž .Im f l 2 2< < < <s e q 1 D q BTf P P P 15Ž .) 2 )T sin u Im DBŽ . Ž .2 N( Im q x Re q xŽ . Ž .0 2 2ch ycos( ž / ž /Im f l Im f lŽ . Ž .

Ž .Here l is the neutron mean free path and the complex quantity q is defined as

21 1 12 2 2 2(qs sin u B q sin u D q Cqcos u B 16Ž . Ž . Ž . Ž .Ž .4 4 4

Ž .The angle between the target polarization and neutron momentum vectors is denoted as u . The sin u behaviourŽ . Ž . Ž .of Eqs. 15 reflects the fact that the CP-violating term in the amplitude 2 is proportional to sin u . Therefore

irrespective of the value of D the CP-violating effects disappear for uf0 and the relative error goes to infinity.Ž . 2The dependence of Eq. 15 on the target thickness x is complicated by the periodic cos oscillations. The

w xphysical origin of those oscillations is the pseudo-magnetic neutron spin rotation, discussed in Ref. 4 – theneutron spin performs about a hundred rotations per mean free path in the target sample. The explicit

Ž .dependence of the relative error 15 on the target thickness is shown in Fig. 1b for the case of the samep-resonance in La. The total number of the polarized neutrons N incident on the target was somewhat arbitrary0

chosen to be 1018.One can see from Fig. 1b that the first minimum of the relative error is located about xf10y2l. However, a

slight change of x increases the relative error by orders of magnitude, which makes the analysis of theexperimental results practically impossible.

w xThis forces us to return to our initial idea 4 of compensating the pseudomagnetic precession by the externalŽ . Ž .magnetic field. This field can be formally taken into account by substituting Re B in the initial Eq. 2 by

Re BX sRe B yH 17Ž . Ž . Ž .Ž .Here H stands for the value of the external magnetic field. Since the ’’pseudo-magnetic’’ amplitude B E is

XŽ .energy-dependent, we can do the compensation by, say, putting ReB E s0 at EsE qG r2. Fig. 1d showsp p

the dependence of relative error on x with this compensation. As expected, all the oscillations of Fig. 1bdisappear and the relative error shows a minimum at around xf2.5l.

Ž . Ž .However, the effect T itself depends on the value of ReB E – see Eq. 8 . Without the compensationŽ . Ž . Ž .Re B 4 Im B approximately by 3 orders of magnitude and the dominant contribution to the denominators

Ž . Ž . Ž .and numerators of Eq. 8 comes from it. If we do the above compensation, then Im B RRe B and the effectŽ < < .in the vicinity of p-resonance EyE FG can be expressed asp p

Re DŽ .XT fy2 18Ž .

Im BŽ .

Taking into account the energy dependence of the amplitudes, we get

g p V d EyEŽ .T pXT f4 P P 19Ž .s 2 2g G EyE qG r4s Ž .p


Therefore the effect now changes sign at around the resonance energy E and reaches at the points EfE "Gp p p

its maximal value:2p pg V d g V dT TXT f P P s P P 20Ž .s s ž /g G G g d G

Ž . Ž .Comparing these results with Eqs. 11 , 12 , we see that the compensating magnetic field, besides removing theoscillations of the relative error, also produced a important increase of the value of T itself, giving an extraresonance enhancement factor drG;103. It also radically changed the energy-dependence of the effect. By

Ž . Ž .comparing Eqs. 8 and 15 we see that the relative error in the presence of compensation decreases by thesame 3 orders of magnitude.

These conclusions are confirmed by the results of calculating the effect under conditions of completeŽ XŽ ..compensation Re B E qG r2 s0 – see Fig. 1c.p p

w xThus we see, that our initial idea 4 of compensation the pseudomagnetism turns out still to be quiteproductive. The only remaining point is to estimate the practically necessary accuracy of this compensation.

w xFollowing 4 , we still think that the practical way of controlling this accuracy is by measuring the neutron spinŽ . Ž .rotation angle fs2Re B rIm f Pxrl around I after its transmission through the target sample. Fig. 2a

Ž . Žshows the dependence of the effect T EsE qG r2 on the spin rotation angle which serves as a measure ofp p.the applied compensating magnetic field .

Fig. 2b shows the same dependence for the relative error. We see that both the effect and its relative error areŽ .optimal for practically complete compensation ff0 . However, the relative error changes only by a factor of

2–3 when the rotation angle varies from 08 to 2008. Thus the limitations on the accuracy of compensation arequite moderate from this point.

ŽA more essential limitation might come from the fact that the energy dependence of the effect and, to.somewhat less extend, its maximal value changes rapidly with increasing f. In order to see this, one might

Ž .compare the curves in Fig. 1c corresponding to fs08 and Fig. 1a, calculated without compensation.Therefore we decided to formulate the problem of the compensation accuracy in a slightly different way: We

assume that a reasonable value for the experimental energy resolution is DEf10y2 eV and consider theŽpractically reasonable accuracy Df of measuring f as a free parameter. Then the rotation angle f and thus the

. Ž .compensating field H should be chosen in such a way that energy maximum of the effect T E should beshifted by less than DE while varying the rotation angle in the interval from fyDf to fqDf. Onperforming a good deal of ‘‘computer experiments’’ we can state, that the accuracy Dfs58 is quite sufficientfrom this point of view.

Ž . Ž . Ž .Fig. 2. a The value of T E qG r2 as a function of neutron spin rotation angle f. b Relative error of T as a function of neutron spinp p

rotation angle f.


˜Fig. 3. Energy dependence of X for target thickness xf2.5l.

Thus we see, that the limitations on the accuracy of measuring the rotation angle in order to check thecompensation of pseudo-magnetic rotation are quite tolerable.

˜3. Analysis of X

˜Consider now the quantity X. As already mentioned, it differs from T only by the normalization factor.Therefore it is also enhanced in the vicinity of the p-wave resonance. However the new normalization makes the

Ž .effect itself and not only its relative error dependent both on the angle u and on the target thickness x.Moreover, the rapid energy oscillations are superimposed on the resonance behaviour of the effect. Thecharacter of these oscillations depend on the target thickness x in a very complicated way. For the sake of

˜illustration we show in Fig. 3 the energy dependence of X, calculated for xf2.5l. As we see, besides the˜oscillations, the value X without compensation roughly equals T and, likewise T , lacks the extra resonance

s sX Tenhancement factor. Our calculations also show that s . Therefore all our conclusions concerning theX T˜necessity of compensation and its consequences apply fully to the case of X.

˜However, the above analysis holds only for values of X which are sufficiently far from 908. On the other˜w xhand Serebrov 7 suggested to measure X in the vicinity of us908. This can be easily understood by˜considering the approximate expression for X:

sin2 u Im DB)Ž . Ž .Xfy 21Ž .2 ) ) )sin u Im DB q2Re q C qcos u BŽ . Ž . Ž .Ž .Ž .

˜We see that the denominator of X changes sign, passing through zero in the vicinity of us908. Thissmallness of the denominator leads to the seeming amplification of the whole effect, and Serebrov proposed to

s Xmake use of this amplification. However, the analysis of the relative error shows that it increases in theXpuncontrolled way and rapidly exceeds unity when u approaches .2

4. Summary

We can draw the following conclusions:Analysis of the CP-violating effect’s relative error is by no means less essential than analyzing the effect

Ž .itself. One can always normalize the CP-noninvariant difference 3 dividing it by a very small quantity.However such a normalization would not increase the accuracy of the measurement.


The necessity to compensate the pseudomagnetic precession is caused essentially by the fact that withoutsuch a compensation the accuracy of measurement varies with target thickness in a practically uncontrollableway.

The compensation of the pseudomagnetic precession increases by 3 orders of magnitude not only the effectitself but also the accuracy of its measurement. The net enhancement in the vicinity of p-wave resonance withcompensation reaches 6 orders of magnitude. The energy dependence of the effect changes drastically in thepresence of compensation.

w xAs a practical way to control the degree of compensation we suggest, following 3 , to measure the rotationangle f of neutron polarization around the target polarization vector. It seems sufficient to fix f with theaccuracy about 58.

w xThe CP-noninvariant quantity X suggested for measurement in Ref. 7 is shown to differ from Stodolsky’sŽ .CP-noninvariant difference 3 practically only by the choice of normalization factor. This factor becomes zero

in the vicinity of uspr2. Although the value of thus normalized effect tends to infinity, its relative error alsotends to infinity in this range of u values.

Acknowledgements

Ž .We acknowledge the support of the Russian Fund of Fundamental Studies grant No 97-02-16803 .

References

w x Ž .1 V.E. Bunakov, V.P. Gudkov, JETP Lett. 36 1982 38.w x Ž .2 V.E. Bunakov, V.P. Gudkov, Z. Phys. 308 1982 363.w x Ž .3 V.E. Bunakov, V.P. Gudkov, Nucl. Phys. A 401 1983 93.w x Ž . Ž .4 V.E. Bunakov, V.P. Gudkov, J. Phys. Paris 45 C3 1984 77.w x Ž .5 L. Stodolsky, Phys. Lett. B 172 1986 5.w x Ž .6 A.L. Barabanov, Nucl. Phys. A 614 1997 1.w x Ž .7 A.P. Serebrov, JETP Lett. 58 1993 14.w x Ž .8 V.E. Bunakov, Elementary Particles and Nuclear Physics 26 1995 287.w x Ž .9 S.K. Lamoreaux, R. Golub, Phys. Rev. D 50 1994 5632.

w x Ž .10 V.P. Alfimenkov, Nucl. Phys. 383 1983 93.w x Ž .11 V.E. Bunakov, L.B. Pikelner, Prog. Part. Nucl. Phys. 39 1997 387.

11 June 1998


Radius evolution of sodium isotopes in mean-fieldand generator coordinate methods

F. Naulin a,1, Jing-Ye Zhang b, H. Flocard c, D. Vautherin c, P.H. Heenen d,P. Bonche e

a DiÕision de Recherches Experimentales, Institut de Physique Nucleaire, F-91406 Orsay Cedex, France´ ´b Physics Department, UniÕersity of Tennessee, KnoxÕille, TN 37996-1200, USA

c DiÕision de Physique Theorique 2, Institut de Physique Nucleaire, F-91406 Orsay Cedex, France´ ´d Physique Nucleaire Theorique, U.L.B., CP129, 1050 Brussels, Belgium´ ´

e SPhT, CEA Saclay, F-91191 Gif sur YÕette Cedex, France

Received 1 December 1997Editor: J.-P. Blaizot

Abstract

Radii of sodium isotopes have been calculated by using the Hartree-Fock-BCS model and the Generator CoordinateŽ .Method GCM with different forces. It is found that Hartree-Fock-BCS results present a jump in both neutron and proton

radii from 22Na to 23Na. However, configuration mixing calculations performed with the GCM result in a smooth increaseof the neutron radius and an almost constant proton radius for the sodium isotopes. We analyze and discuss our results in thelight of recent experimental data. q 1998 Elsevier Science B.V. All rights reserved.

PACS: 21.10.Gv; 21.60.JzKeywords: Radii; Sodium; Mean-Field; Generator-Coordinate-Method; SIII; SLy4

Based on a systematic of measurements of theinteraction cross sections s , the mass radius evolu-I

Ž .tion for sodium isotopes As20 to 32 has beenw xreported 1 . Using independent information on pro-w xton radii 2 , Suzuki and his coworkers have deduced

the neutron rms radii. A remarkable finding of theiranalysis is an irregularity of the evolution of theneutron radius as a function of growing mass number

1 E-mail: [email protected] Unite de Recherche des Universites Paris XI et Paris VI´ ´

associee au C.N.R.S.´

A: it displays a decrease in 22 Na followed by asignificant increase in 23 Na. This contrasts with theevolution of the proton radius which stays almostconstant along the entire series of isotopes with only

w xa small decrease for As25 2 .This phenomenon has not yet attracted theoretical

attention although several calculations of the evolu-tion of proton and neutron radii of the ground stateof sodium isotopes have already been effected. Morethan twenty years ago a deformed Hartree-Fock cal-culation of proton radii using an early Skyrme phe-nomenological interaction was performed by Campi

w xet al. 3 . More recently results from a deformed


( )F. Naulin et al.rPhysics Letters B 429 1998 15–1916

Ž . w xRelativistic Mean Field RMF 1 and from severalspherical Hartree-Fock calculations with different

w xdensity-dependent Skyrme interactions 4 have be-come available. These calculations discuss the over-all increase of the neutron skin along the isotopechain that they describe rather well. On the otherhand, none adresses the specific issue of the reporteddrop and jump of the neutron radius at 22 Na and23 Na.

Among the reasons which could explain the be-havior of the neutron radius in 22 Na and 23 Na the

w xauthors of Ref. 1 remark that it might be due to themixture of the isomer state in 22 Na. However theisomer is expected to have a larger radius than theground state, so that a mixture would produce anincrease of the radius instead of a decrease. Also nodrop-jump is observed in the corresponding protonradius. We have chosen to adress this question, bymeans of a two-step microscopic approach whichtakes into account static and dynamic collective cor-relations. First, we perform a deformed Hartree-Fockcalculation to compute the quadrupole deformationenergy curves. This allows us to study the energyand deformation of the mean-field ground state andof any possible shape isomer. Second, we determinethe quantum collective wave function of the groundstate by means of a configuration mixing calculationperformed according to the Generator Coordinate

Ž .Method GCM .A detailed account of our way of solving the

constrained Hartree-Fock-BCS equations is given inw xRef. 5,6 . To achieve high precision in the numeri-

cal solution we use a 0.8 fm mesh and a 12 = 12 =

12 lattice. In order to investigate the possible influ-ence of the effective nucleon-nucleon interactions,two Skyrme forces have been used: the well tested

w xSIII set of parameters 7 and the more recentw xparametrization SLy4 8 . The correlations in the

pairing channel are derived from a zero range pairingw xforce 9 . The k-dependence of the particle-particle

channel is simulated by a density dependence whichgenerates an increased attraction in regions of lownuclear densities. It can be written in the form

V r ,s ;r ,sŽ .1 1 2 2

1ys .s r yr1 2 1 2sV d r yr f . 1Ž . Ž .0 1 2 ž /4 2

The density-dependence is determined by the func-Ž . Ž .tion f r s1yr r rr . As in earlier calculationsc

of high spin properties of medium and heavy nucleiw x y39 , we take a density scale r s0.16 fm and ac

y3 Žstrength V s1000 MeV= fm for both neutrons0. y3and protons for SIII and V s1150 MeV= fm0

for SLy4. In the numerical solution of the BCSequations we use a cutoff. It is 5 MeV above theFermi level for SIII and 5 MeV above and below theFermi level for SLy4. As all other previous analysis,the present calculation does not take into accounttime-odd terms of the Skyrme energy functional.This is a restriction for the odd and odd-odd nucleithat we consider here. Still we believe that thebreaking of time reversal symmetry associated to theunpaired nucleons is not large enough to invalidateour conclusions.

The deformation energy curves have been ob-tained with a quadratic constraining operator. For theSLy4 Skyrme force, the curves for sodium isotopeswith 20FAF25 are displayed in Fig. 1. For thelightest isotopes, we find a spherical minimum and asignificant stiffness with respect to prolate and oblatedeformations. When A increases, the deformationenergy curves become shallower. Eventually, a de-formed minimum appears for As23. For evenhigher mass numbers the lowest minimum is againspherical. The same qualitative features are foundwith the SIII force. Comparing our results with theMg study performed with similar conditions of cal-

w xculation in Ref. 10 , we see that results are quitesimilar except for Ns12. In this case, the 24 Mgenergy curve shows a very well defined minimum atlarge deformations while only a very shallow mini-mum is obtained for 23 Na. This confirms that defor-mation properties of these light nuclei depend cru-

w xcially on the number of particles. Kitagawa et al. 11have obtained a well pronounced deformed mini-mum for 20 Na in a deformed HF study of theAs20 isobars. The difference with our results isprobably due to the neglect by Kitagawa et al. ofpairing interactions and to the use of the SGII Skyrmeparametrization which has a very small symmetryenergy. The single particle level scheme obtained inour calculation is very similar to the standard Nils-son scheme. As a result our predictions for groundstate spins of even isotopes agree with the standardones.

( )F. Naulin et al.rPhysics Letters B 429 1998 15–19 17

Fig. 1. Potential energy curves calculated with interaction SLy4w x8 for sodium isotopes from mass number 20 to 25 in theconstrained mean-field approximation.

In Fig. 2a and 2b the evolution of neutron andproton radii is shown for the SIII and SLy4 force.For both interactions, one observes first a slightdecrease of the proton radius followed by a suddenincrease between masses 22 and 23. For neutrons theradius grows steadily. It also displays a jump be-tween masses 22 and 23. From the energy curves,one can relate the jumps in radii to the transitionfrom a spherical to a deformed minimum. We recallthat the proton radii extracted from accurate electron

w xscattering experiments 2 do not display an irregu-larity over the whole range of masses considered.

Fig. 1 shows that the deformed minimum of theenergy curves when it becomes lowest is not muchmore bound than the spherical configuration. In sucha situation, it is questionable whether the groundstate can be well described by means of a single

w xHartree-Fock-BCS state 12–14 . To analyze the ef-fect of configuration mixing we employ the GCMwhich, given a collective coordinate Q associated

with a one-parameter family of non orthonormalŽ .states C Q , diagonalizes the Hamiltonian H within

the subspace generated by this family. This leads tow xthe Hill-Wheeler equation 15 :

² < < :dQ C Q H C Q f QŽ . Ž . Ž .H 2 1 2 2

² < :sE dQ C Q C Q f Q , 2Ž . Ž . Ž . Ž .H 2 1 2 2

which can be reduced to a standard matrix diagonal-ization by introducing the collective wave function

y1r2 Ž . ² Ž . < Ž .:gsNN f , where NN Q ,Q s C Q C Q is1 2 1 2

the overlap matrix.To solve numerically the Hill-Wheeler equation,

we use a discrete mesh in the collective variable Qwith the points reported in Fig. 1. A detailed study ofthe GCM for axial quadrupole deformations alongwith the numerical method applied in this work can

w xbe found in Ref. 16 . In particular this reference'explains how one can define the operator 1r NN to

take care of the accumulation of eigenvalues of NN

Ž . Ž .Fig. 2. Neutron bold circles and proton crosses radii forŽ .sodium isotopes from mass number 20 to 26: a calculated with

w x Ž .HFqBCS and interaction SIII 7 , b calculated with HFqBCSw x Ž .and interaction SLy4 8 , c calculated with HFqBCSqGCMŽ . w xand interaction SLy4, d experimental values from 1

( )F. Naulin et al.rPhysics Letters B 429 1998 15–1918

near the value zero when the number of mesh pointsin the lattice goes to infinity. This involves theintroduction of a cut-off. The stability of the presentresults with respect to the cutoff has been carefullychecked. The matrix elements which appear in theHill-Wheeler equation involve quasiparticle statesrather than simple Slater determinants. The calcula-tions take explicitly into account the fact that twodistinct deformations correspond to different bases ofcreation and annihilation operators. Since BCS statesdo not have well defined neutron and proton num-bers it is also necessary to extend the GCM formal-ism by enforcing the average neutron and protonnumbers of the collective wave functions. This is

ˆ ˆdone by constraining ya Nya P. For each nu-n p

cleus an adjustment of a and a has been maden p

which ensures that the GCM ground state collectivewave functions have the correct neutron and protonnumbers. We have found that in general the valuesof a and a resulting from the constraining proce-n p

dure came out close to the neutron an proton Fermilevels respectively. When it was not the case, wehave checked that it happened in situations where theaverage particle numbers were correct irrespective ofthe value of the constraint.

Fig. 2c shows that the evolution of radii calcu-lated with the GCM does not display the irregularbehavior at As22 and 23 observable in the mean-field results. The dynamically determined superposi-tion of states with different deformations within theground state wave functions associated with the GCMhas smoothened the transition from spherical to de-formed shape. The proton radii first increase veryslightly and then decrease. Besides, the neutron radiiincreases regularly as a function of A. For the SLy4force the absolute values of radii for both neutronsand protons are slightly larger than the measuredones and also larger than values calculated with theSIII interaction. Squares of collective wave functionsare displayed in Fig. 3 for mass number 20, 21, 22,23, 24, 25. An important shape mixing is observed.Indeed the wave functions generally spread over arange of quadrupole moments of the order of 100fm2. The gradual evolution of the nuclear deforma-tion from As20 to As25 is visible in Fig. 3.

To summarize our results, we note that the jumpin neutron and proton radii between mass 22 and 23found in mean-field calculations disappears when

Fig. 3. Squares of collective GCM ground state wave functionsw xcalculated with interaction SLy4 8 for sodium isotopes from

mass number 20 to 25.

improved wave functions including superpositions ofvarious shapes are constructed by solving the HillWheeler equation. Although for a given nucleus weimpose that the wave functions have the correctneutron and proton numbers only in average, we donot believe that this is the reason for the smearing inthe radii evolution. Indeed we have checked that theparticle number fluctuation is very similar in the

Žmean-field solutions which shows the irregular ra-. Ždius behavior and in the GCM ground state which

.does not show it . We thus believe that refinedcalculations involving configuration mixing wouldpredict no jump of nuclear radii even if exact projec-tion on nucleon number was performed. This conclu-sion, which agrees with the results of accurate mea-surements of proton radii, is at variance with thevalues inferred recently for neutron radii. In additionwith the restriction related to possible isomer-groundstate mixing in the data, we note that the extractionof neutron radii requires a model whose foundationis not as solid as that which is used to extract charge

( )F. Naulin et al.rPhysics Letters B 429 1998 15–19 19

radii from electron scattering experiment. Let us alsonote that 22 Na is a nucleus in which shell modelcalculations encounter special difficulties in explain-

w xing data 17 . Our results for the evolution of neutronradii also suggest the growth of a neutron skin withincreasing mass in agreement with earlier calcula-

w xtions and experimental observations 1,4 .

Acknowledgements

Ž .Two of us J.Y.Z. and D.V. wish to thank theYukawa Institute for Theoretical Physics of the Ky-oto University where part of this work was per-formed. J.Y.Z. would like to thank I. Tanihata forcalling his attention on this issue. D.V. thanks to theOak Ridge National Laboratory where part of thiswork was proceeded. J.Y.Z. thanks to the Divisionde Physique Theorique de l’IPN for the hospitality´extended to him. We are grateful to the Institut duDeveloppement et des Ressources en Informatique´

Ž .Scientifique IDRIS for extended computing facili-ties. This work has been partly supported by thebelgian Ministery of Science Policy under the grantPAI-P3-043.

References

w x Ž .1 T. Suzuki et al., Phys. Rev. Lett. 75 1995 3241.w x Ž .2 G. Huber et al., Phys. Rev. C 18 1978 2342.w x3 X. Campi, H. Flocard, A.K. Kerman, S.E. Koonin, Nucl.

Ž .Phys. A 251 1975 193.w x Ž .4 B.A. Brown, W.A. Richter, Phys. Rev. C 54 1996 673.w x5 P. Bonche, H. Flocard, P.H. Heenen, S.J. Krieger, M.S.

Ž .Weiss, Nucl. Phys. A 443 1985 39.w x6 N. Tajima, P. Bonche, H. Flocard, P.H. Heenen, M.S. Weiss,

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Ž .Phys. A 621 1997 706.w x Ž .11 H. Kitogawa, N. Tajima, H. Sagawa, Z. Phys. A 358 1997

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11 June 1998


The physical origin of the Wigner termand its persistence in heavy nuclei

Nissan ZeldesThe Racah Institute of Physics, The Hebrew UniÕersity of Jerusalem, Jerusalem 91904, Israel

Received 17 February 1998; revised 1 April 1998Editor: W. Haxton

Abstract

The physical origin of the Wigner term in terms of correlated nucleon pairs is pointed out. Empirical evidence is shownfor the distribution of the Wigner term energy throughout a shell and for its persistence to heavy nuclei. The data indicatethat Zs126 might well be the next proton spherical major magic number after Zs82. q 1998 Elsevier Science B.V. Allrights reserved.

PACS: 21.10.Dr; 21.30.Fe; 21.60.Cs

Recent progress in the production of heavier pro-w xton-rich NfZ nuclei 1,2 has aroused renewed

interest in the Wigner term in nuclear masses,presently directly observable up to As60, and itsinterpretation in terms of neutron-proton correlationsw x w x3–9 . The question has also been raised 3,5,10 ifthe Wigner cusp discontinuity of the mass surface atNsZ persists to heavier as yet unknown nuclei,with strong Coulomb interaction and different orbitoccupancies for neutrons and protons.

The present Letter addresses these points. We firstclarify pedagogically the interrelations of correlatednucleon pairs and the Wigner term by straightfor-wardly rederiving the well known single-j shell low-est-seniority energy expression as the ground stateŽ . Žg.s. energy of correlated nucleon pairs see also

w x.Ref. 11 . Then we consider visually some empiricalevidence from masses for the Wigner term energydistribution throughout a shell and its persistence in

unknown heavy p-rich nuclei, and present relatedquantitative results.

We start by noting the properties of the empiricalw xeffective shell model interaction V 12–14 which

are essential for the following:

0,n pV -0 , V -0 , V )0 , I -0 , 1Ž .0 1 2

2where V is the two-body matrix element V , V0 j Js0 1Ž .and V denote the respective 2 Jq1 -averages of2

Ž2the matrix elements V for Ts0,Õs2 odd Jj T J. Ž .values and for Ts1,Õs2 even J/0 values ,

w xwhere Õ is the seniority quantum number 15,16 ,0, n p Ž . 2and I is the overall 2 Jq1 -average of the V .j J

Ž .The relations 1 reflect the overall short-range at-tractive nature of the nuclear interaction.

In the lowest-seniority model with isospin theŽ . anuclear g.s. energy of even-even e-e and odd-a j

Ž .configurations, and the center of mass c.m. energyof g.s. J-multiplets with given seniority Õ and re-


( )N. ZeldesrPhysics Letters B 429 1998 20–26 21

Ž .duced isospin t of odd-odd o-o configurations, canbe intuitively written as

nucl aW j sN V qN V q N yN yN VŽ . Ž .0 0 1 1 t 0 1 2

sN V yV qN V yV qN V ,Ž . Ž .0 0 2 1 1 2 t 2

2Ž .

Ž .where V ,V and V are as in Eq. 1 , N is the total0 1 2 t1 Ž .number of nucleon pairs, a ay1 , N is the num-02

ber of Js0 nucleon pairs, and N is the number of1Ž .Ts0 nucleon pairs. Eq. 2 is a partly J-indepen-

dent approximation, where all V 2 are set equal toj Jo d d

2their average value V , and all V are set1 j J / 0e Õ en

equal to V .2

The number N is measured by the expectation1

value of the two-body isospin projection operator

1s p i ,k s y t P t . 3Ž . Ž .Ł Ý ÝTs0 i kž /4Ts0 i-k i-k

In a state with well defined total isospin T it is givenby

1 a ay1 1 3Ž .aN j T s y T Tq1 y a .Ž . Ž .1 4 2 2 4

4Ž .

Similarly, the number N is measured by the0

expectation value of the two-body projection opera-tor

1s p i ,k s q i ,k , 5Ž . Ž . Ž .Ł Ý ÝJs0 2 jq1Js0 i-k i-k

where q is the seniority or pairing operator. In thew xseniority scheme it has the eigenvalues 15,16

1 1aN j ÕtT s ayÕ 4 jq8yayÕŽ . Ž . Ž .0 2 jq1 4

yT Tq1 q t tq1 . 6Ž . Ž . Ž .

Ž . Ž .Due to the relations 1 , Eq. 2 is minimized bymaximizing the pair numbers N and N , which0 1

corresponds to lowest T and Õ and highest compati-ble t. For e-e and odd-a configurations this corre-

< < Ž .sponds to T s T and Õ,t equal respectively tog.s. zŽ . Ž .0,0 and 1,1r2 . On the other hand, for an o-o

< <configuration the state Ts T ,Õs0 is forbidden byz

the Pauli Principle, and there is competition between< < < <Õs0,Ts T q1 and Õs2,Ts T . Moreover, inz z

the latter case both ts1, even-J and ts0 odd-Jvalues are allowed when the respective neutron andproton numbers n and p are different, whereas for

Žnsp the ts1, even-J possibility is excluded. TheŽ .competition between pairing Js0 and isopairing

Ž .Ts0 correlations in o-o nuclei is considered inw x Ž w x w x. .Ref. 17 see also Refs. 18 and 19 .

< <Altogether one has for T s T :g.s. z

1 1aN j T s a 4 jq8yaŽ . Ž .0 g . s . 2 jq1 4

yT Tq1 qdN Õt 7Ž . Ž . Ž .0

jq 1Ž .with dN Õt equal respectively to 0,y ,y10 2 j q 12 j q 3and y for e-e, odd-a, o-o n/p and o-o nsp2 j q 1

configurations. For heavier o-o nsp configurationswith T s1,Õs0 the N is the same as for theirg.s. 0 g . s.

e-e isobaric analogs.Ž Ž .With the g.s. values of N and N Eqs. 4 and1 0

Ž . < <. Ž .7 with Ts T , Eq. 2 becomes the well knownzw x Ž .result 15,16 for the J-T scheme lowest-seniority

energies

1nucl aW j Ts T s a ay1 bŽ .Ž .g .s . z 2

3 1q T Tq1 y a ´q apŽ .

4 2

a1y y1 1Ž .

q yž /2 2

n p1y y1Ž .

y2

2 jq1 2 jq3P 1yd q d p 8Ž . Ž .I ,0 I ,0ž /2 jq2 2 jq2

where Isnyp is the neutron excess, and

1 1bsV y V yV q V yV , 9Ž .Ž .Ž .2 2 1 2 04 4 jq2

1 1´s V yV q V yV , 10Ž .Ž .Ž .2 1 2 02 2 jq1

( )N. ZeldesrPhysics Letters B 429 1998 20–2622

2 jq2psy V yV . 11Ž .Ž .2 02 jq1

The parameter p is the pairing energy, and ´ is theŽ .shell model symmetry energy parameter.

For comparison we write down as well the nu-clear g.s. energy of e-e and odd-a jnqp configura-tions, and the c.m. energy of g.s. J-multiplets of o-oconfigurations, in the n-p scheme, where each of theneutron and proton groups is separately in a welldefined state of lowest seniority, having the highestnumber of Js0 pairs 1:

1nucl n pW j j s nqp pŽ . Ž .g .s . ž /2

n ny1 p py1Ž . Ž .0,n pq q dqnpI

2 2

nqp n p1y y1 1 1y y1Ž . Ž .

q y y p ,ž /2 2 2

12Ž .0, n p Ž .with I as in Eq. 1 and

1dsV q V yV . 13Ž .Ž .2 2 02 jq1

Ž .From the definitions of the coefficients of Eqs. 8Ž .and 12 in terms of the two-body effective interac-

Ž .tions satisfying the relations 1 one obtains:

bQ0 , I 0,n p ,p-0 , ´ ,d)0 . 14Ž .Ž . Ž .With the relations 14 Eq. 12 represents three

parallel quadratic surfaces looking like a valley run-ning along the nsp line, like the macroscopic partof the Liquid Drop mass equation, with the odd-asurface lying midway between the lower e-e and thehigher o-o ones. On the boundaries of the shell

Ž .region, where either n or p equals 0 or 2 jq1 , theŽ .n-p and J-T schemes coincide, and Eq. 8 coincides

1 This is a valid approximation to the g.s. for a non-diagonalw xshell region in the N-Z plane 18,20 , with different neutron and

proton valence subshells, but not for non-semimagic nuclei in adiagonal region, where they are the same.

Ž .with 12 . On the other hand, inside the region Eq.Ž .8 gives more binding. The difference

nucl a nucl n pW j Ts T yW j jŽ .Ž .g .s . z g .s .

1 1 1s nyp y nqp q np ´Ž .

2 2 2 jq1n p

1y y1Ž .q

2

=1 1

1yd y d p 15Ž . Ž .I ,0 I ,02 jq2 2 jq2

comprises a negative part depending on ´ , which iscommon to all nuclei, and a smaller part dependingon p for o-o nuclei, which is negative when n/pand positive for nsp.

Fig. 1 shows g.s. binding energies B snucl aW j of e-e nuclei in the 1 f shell, calcu-Ž .g .s . 7r2

Ž .Fig. 1. Binding energy B for As48 isobars above and for Is0Ž .isodiapheres below in the n-p and J-T schemes. Calculated with

w xthe empirical interaction of Ref. 14 .


w xlated with the empirical interaction of Ref. 14 , forŽ Ž .. Ž Ž ..both J-T Eq. 8 and n-p Eq. 12 couplings. The

continuous build-up of the additional binding in theJ-T scheme towards the middle of the shell, and theisobaric cusp at nsp, are obvious.

1The cusp is due to the term T´s nyp ´ in2

Ž .Eq. 8 , known as the Wigner term in nuclear massequations. The physical reason for it is the need that

< <T attains its lowest possible value T , in order tog.s. zŽ Ž .. Ž Ž ..maximize both N Eq. 4 and N Eq. 6 , in1 0

order to take full advantage of the strongly attractiveŽ Ž ..nature of V and V Eq. 1 .1 0

Recent more comprehensive calculations in thew x w x1d2 s 3,7 and 1 f 2 p 6,7 major shells, with varying

relative strengths of the Ts0 and Ts1 interactions,demonstrate disappearance of the Wigner term cuspwhen the Ts0 part of the interaction approacheszero.

There is a Wigner term and splitting according toparity type also in binding energies calculated for l a

configurations in LS coupling with SU symmetry,4w x w xboth without pairing 16,21 and with it 16,22 .

These energies are derivable similarly to our treat-ment of the single-j shell, by respectively maximiz-ing the number of nucleon pairs in space-symmetricŽ .L states, and the numbers of nucleon pairs ineÕ en

Ž .space-symmetric unpaired L /0 states and ineÕ enŽ .the paired Ls0 state. The maximized pair num-

< <Ž < < .bers comprise a term T T q4 , and the ensuingz zŽ .symmetry energy varies like T Tq4 rather than

Ž . Ž .T Tq1 , Eq. 8 . There has been recent renewedw xinterest in the LS coupling SU mass equation 5,23 .4

Ž .Adding to Eq. 8 the energy E of the A0 0

nucleons in the core, the sum of valence nucleonsingle-particle energies ac, and the Coulomb energy,one obtains the total g.s. energy:

1E A qa,Ts T sE qa cq pŽ .g .s . 0 z 0 ž /2

a ay1 3Ž .q bq T Tq1 y a ´Ž .

2 4a

1y y1 1Ž .q y pž /2 2

n p1y y1Ž .

q 1yd kqd lŽ .I ,0 I ,02

qECoulomb . 16Ž .

The pairing parameters k and l include the addi-tional g.s. binding of o-o nuclei as compared to the

Ž .c.m. of g.s. multiplets in the last line of Eq. 8 .When there are several simultaneously filling

Ž .mixed valence subshells in a major shell Eq. 16 is auseful approximation near shell region boundaries 2,with a,n and p denoting the total respective num-bers of valence nucleons, and the parameters repre-senting average major shell values. With this mean-ing, and including as well the sum of nucleon mass

Ž .excesses, NDM qZDM , Eq. 16 is the Approxi-n HŽ .mate Major Shell Lowest Seniority AMSLS mass

w xequation 18 in a diagonal shell region.We address first the question whether neutron-

proton Ts0 correlations and the resulting Wignerterm are a localized effect limited to the close neigh-

w xbourhood of NsZ as is sometimes 6,7,26 stated,or the effect is distributed throughout the shell, as

Ž .comes out from our derivation of Eq. 8 and illus-trated in Fig. 1.

The question is best approached by studying longcomplete isobaric chains extending over both sidesof the NsZ line in a diagonal shell region. Suchcomplete chains are obtainable from known IG0isobars by subtracting from the experimental masses

1 Žthe neutron-proton mass differences DM yn2

.DM I and the Coulomb energy, remaining with theHŽ .purely nuclear part of Eq. 16 , and using charge

symmetry.Fig. 2 shows plots of such masses, and corre-

sponding double beta decay energies, for isobarswith A s 72 and 73 in the 28-50 shell region. Themass plots are highly charge-symmetric, with a cuspat NsZ. There is a discontinuous decrease of Q y yb b

associated with crossing the cusp towards the protonrich side. On each side of the discontinuity theQ y y plot is an approximately straight line, and theb b

slopes of the two lines are equal. This is all inŽ .qualitative agreement with the AMSLS Eq. 16 .

2 Far inside shell regions, where there are large deformations,Ž .Eq. 16 gets distorted by configuration interaction with seniority

w xviolation 18,24 , even though its basic features survive. In partic-w x 2ular 25 , a generalized two-parameter symmetry energy ´T qz T

Ž .with an augmented Wigner term z )´ agrees better withŽ .isobaric masses than the single-parameter expression ´T T q1

Ž . Ž .of Eq. 16 . See also the concluding discussion of Table 1.


Ž . y yFig. 2. Isobaric mass parabolas above and corresponding Qb b

Ž . Ž . Ž .values below for As72 left and As73 right isobars in the1nuclear Ž28-50 shell region. For I G 0 DM s DM y DM yn2

. Coulom b nuclearŽ .D M I y E , and for I - 0 D M A , I sHnuclearŽ . w x CoulombDM A,y I . DM data are taken from Ref. 27 and E

w xcalculated from Ref. 28 .

For a quadratic mass parabola without a cusp theŽ .E D M

y yQ f4 plot is a single straight line. Fromb bE I

the figure it is obvious, that by eliminating a fewmasses around the cusp the remaining Q y y plotb b

would still consist of two parallel straight lines witha relative vertical displacement, rather than a singlestraight line. Thus the effects of the Wigner termcannot be eliminated by excluding few nuclei aroundNsZ, and they are effective throughout the shell.

We shall use the same method of extrapolationbased on charge symmetry to address the possibleexistence of a Wigner cusp in the higher diagonalshell regions, 50-82 and 82-126. In these regions theavailable isobaric chains are incomplete. They com-prise experimentally known neutron-rich isobars witha considerable neutron excess, and their charge-sym-metrically extrapolated proton-rich mirror nuclei. In-termediate masses around NsZ are lacking.

Fig. 3 illustrates the situation in the 50-82 shellregion by showing plots of masses and double betadecay energies for As131 and 132 isobars, andFig. 4 shows such plots for A s 207 and 208 in the82-126 region. Like in Fig. 2 neutron-proton massdifferences and Coulomb energies have been sub-tracted from the experimental masses. The plots looklike those of Fig. 2, with the central parts missing: anisobaric Q y y line connecting proton-rich nuclei isb b

a straight line with the same slope as the line throughthe neutron-rich mirror isobars, and it is displaceddownwards with respect to the latter. This indicates acusp intersection of the two quadratic parabolic arcsthrough the masses when they are extended towardseach other all the way to Is0.

Thus, to the extent that extrapolation of isobaricparabolas to Is0 can be justified, Figs. 3 and 4

Fig. 3. The same as Fig. 2, for As131 and 132 isobars in the50-82 shell region. The thin Q y y lines are from a Leastb b

Ž . Ž .Squares fit of Eq. 17 see text .


Fig. 4. The same as Fig. 3, for As207 and 208 isobars in the82-126 shell region.

indicate the persistence of the Wigner term up toisobars with neutron and proton numbers between 82and 126.

This finding has a bearing on the location of thenext proton magic number beyond Zs82. The

< <Wigner term was derived above as a result of Ts Tz

isospin coupling of neutrons and protons in the sameŽ .mixed valence subshell. Its occurrence indicates theoccurrence of the same valence subshells, with thesame magic numbers, for both neutrons and protons.Thus Fig. 4 points out Zs126 as the next protonmagic number after Zs82, similarly to the neutron

w xcase, rather than the traditional Zs114 29–31 .This is in conformity with results of recent self-con-

w xsistent mean field calculations 32 , and with a recentw x Ž .suggestion 33 based on BE 2 systematics.

In conclusion we present some quantitative re-sults. For a generalized symmetry energy ´T 2 qz TŽ .second footnote , and equal pairing parameters k

Žand l which is the situation beyond the 1 f shell7r2w x w x.18 , 24 , one has:

Qnucleary y A , I s2´ Iy2 qd Q y y I , 17Ž . Ž . Ž . Ž .b b b b

where Qnucleary y is the purely nuclear part of Q y y,b b b b

Ž .y yshown in Figs. 2–4, and d Q I equal respec-b b

tively to 2z ,z ,0,yz and y2z for IG4, Is3, IsŽ .2, Is1 and IF0. For IG4 and IF0 Eq. 17

describes two parallel straight lines with a slopegiven by 2´ and a relative vertical displacement of4z , like in the lower parts of Figs. 2-4.

Numerical values of the coefficients ´ and z andtheir statistical errors, obtained by Least Squares

Ž . y yadjustments of Eq. 17 to the isobaric Q datab b

shown in Figs. 2-4, are given in Table 1. The tableshows as well the numerical values of ´ adjusted

Žwhen assuming zs0 no Wigner term, like in theŽ .n-p scheme Eq. 12 , and in the traditional Liquid

. ŽDrop model and when assuming zs´ like in the

Table 1Ž . Ž . Ž .Adjusted values in MeV of the symmetry energy coefficients ´ and z and of the mean errors m for Eqs. 17 , 16 and 12

Parameters Mass Number Ain MeV

72 73 131 132 207 208

( )´ and z independent, Eq. 17´ 0.966"0.004 0.906"0.011 0.481"0.003 0.459"0.007 0.312"0.007 0.345"0.013z 1.456"0.036 1.750"0.080 2.712"0.059 3.156"0.175 2.815"0.256 1.497"0.500m 0.117 0.207 0.120 0.164 0.085 0.162

( )zs´ , Eq. 16´ 1.014"0.007 1.001"0.013 0.577"0.006 0.561"0.004 0.376"0.002 0.374"0.001m 0.451 0.693 1.056 0.751 0.317 0.203

( )zs0, Eq. 12´ 1.120"0.025 1.123"0.029 0.603"0.008 0.583"0.005 0.385"0.002 0.383"0.001m 1.444 1.385 1.338 0.908 0.363 0.240


Ž ..AMSLS Eq. 16 . The mean error m of each adjust-ment is given as well. It is defined by m

2s Sd r Nym 1r2 , where d are the individ-Ž .Ž .i i

ual deviations, N the number of data participating inthe fit, and m the number of independent adjustablecoefficients.

One observes the following:1. The relative statistical errors of ´ and of z are

always small, indicating the statistical signifi-cance of both coefficients. The significance of anindependent z is further pointed out by the smaller

Ž .mean errors m of Eq. 17 as compared to Eq.Ž . Ž Ž .12 and also to Eq. 16 with its restricted

.Wigner term .2. The values of ´ decrease when A increases,

roughly like the Ay1 dependence of the symme-try energy of the Liquid Drop model.

3. The ratios zr´ are larger than 1. As a rule theyincrease with A in agreement with the findings of

w xRef. 25 , indicating stronger effects of configura-tion interaction with seniority violation in heavier

Ž .nuclei see second footnote .

References

w x Ž .1 W. Mittig et al., Nucl. Phys. A 616 1997 329c.w x Ž .2 C. Baktash, Prog. Part. Nucl. Phys. 38 1997 291.w x3 D.S. Brenner, C. Wesselborg, R.F. Casten, D.D. Warner,

Ž .J.-Y. Zhang, Phys. Lett. B 243 1990 1.w x Ž .4 N.V. Zamfir, R.F. Casten, Phys. Rev. C 43 1991 2879.w x5 P. Van Isacker, D.D. Warner, D.S. Brenner, Phys. Rev. Lett.

Ž .74 1995 4607.w x Ž .6 W. Satula, R. Wyss, Phys. Lett. B 393 1997 1.w x7 W. Satula, D.J. Dean, J. Gary, S. Mizutori, W. Nazarewicz,

Ž .Phys. Lett. B 407 1997 103.w x Ž .8 W.D. Myers, W.J. Swiatecki, Nucl. Phys. A 601 1996 141.w x Ž .9 W.D. Myers, W.J. Swiatecki, Nucl. Phys. A 612 1997 249.

w x10 D.S. Brenner, D.D. Warner, N.V. Zamfir, R.F. Casten, B.D.Ž .Foy, in: C.A. Bertulani, L.F. Canto and M.S. Hussein Eds. ,

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Scientific, Singapore, in press.w x Ž . Ž .12 J.P. Schiffer, Ann. Phys. NY 66 1971 798.w x Ž .13 J.P. Schiffer, W.W. True, Rev. Mod. Phys. 48 1976 191.w x Ž .14 W.W. Daehnick, Phys. Rep. 96 1983 317.w x15 A. de-Shalit, I. Talmi, Nuclear Shell Theory, Academic

Press, New York, 1963.w x16 I. Talmi, Simple Models of Complex Nuclei, Harwood Aca-

demic Publishers, Chur, 1993.w x Ž .17 N. Zeldes, S. Liran, Phys. Lett. B 62 1976 12.w x18 N. Zeldes, in: Handbook of Nuclear Properties, D.N. Poenaru

Ž .and W. Greiner Eds. , Clarendon Press, Oxford, 1996, p. 12.w x Ž .19 R. Rudolph et al., Phys. Rev. Lett. 76 1996 376.w x20 M.G. Mayer, J.H.D. Jensen, Elementary Theory of Nuclear

Shell Structure, Wiley, New York, 1955.w x Ž .21 E.P. Wigner, E. Feenberg, Rep. Prog. Phys. 8 1941 274.w x22 G. Racah, in: L. Farkas Memorial Volume, A. Farkas and

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11 June 1998


Matrix theory on non-orientable surfaces

Gysbert Zwart 1

Institute for Theoretical Physics, UniÕersity of Utrecht, Princetonplein 5, 3584 CC Utrecht, The Netherlands

Received 5 January 1998; revised 5 March 1998Editor: P.V. Landshoff

Abstract

We construct the Matrix theory descriptions of M-theory on the Mobius strip and the Klein bottle. In a limit, these¨provide the matrix string theories for the CHL string and an orbifold of type IIA string theory. q 1998 Elsevier Science B.V.All rights reserved.

1. Introduction

Different string theories have been claimed to berelated via one unifying theory, M-theory. This the-ory is supposed to reproduce the various weak cou-pling string theories in certain limits of its modulispace, and to give the correct interpolation in be-

Ž w x.tween, at finite values of the coupling see e.g. 1 .The proposal of matrix theory as a definition of

w xM-theory in the infinite momentum frame 2,3 hasallowed many of the claims to be verified. Thematrix formulations of type II and heterotic strings in

w xten dimensions were constructed in 4 , and compact-ifications to lower dimensions on tori and orbifolds

Ž w xwere investigated as well see 3 and references.therein .

Here we would like to concentrate on compactifi-cations of M-theory to nine dimensions on non-orientable surfaces: the Mobius strip and the Klein¨bottle. Type II compactifications on these manifolds,in the form of certain orientifold models, were con-

w xsidered in 5,6 . These authors also argued which

1 E-mail address: [email protected].

nine dimensional string theories would appear aslimits of M-theory on these surfaces: the Mobius¨

w xstrip yields the nine-dimensional CHL string 7 ,Žwith gauge group E arising as the twisted sector8

Ž .living on the single boundary of the strip, followingw x.8 , and the Klein bottle represents, in a suitablelimit, a type IIA string in nine dimensions moddedout by half a shift over the circle accompanied by the

Ž .FLoperation y1 .We use the orientifold models to obtain the matrix

description of these theories. The models are con-structed as torus compactifications of matrix theory,modded out by an appropriate symmetry group con-sisting of a reflection and shifts over half a period ofthe circles. We first consider the Mobius strip. We¨construct the 2q1 dimensional gauge theory de-scribing the dynamics of zero-branes on the Mobius¨strip. The base manifold of the gauge theory is itselfagain a Mobius strip. The construction is similar to¨

w xthat of heterotic matrix theory 9 . We show how inthe limit of weak string coupling we indeed recovera string theory with chiral fermions producing oneE gauge group, the CHL string.8

Then we turn to the Klein bottle compactification.In this case the T-duality that we have to perform to


( )G. ZwartrPhysics Letters B 429 1998 27–3428

construct the 2q1 dimensional theory describingthe dynamics is less straightforward, we have to use

w xthe original construction of 10,11 to find the gaugetheory. The base manifold of the gauge theory is nota Klein bottle. Rather, the geometric structure of theKlein bottle is reflected in the structure of the gaugefields; different modes of the fields turn out tosatisfy different conditions. Finally we again find thetype IIA string theory, modulo the required symme-try, as a limit of this three-dimensional model.

After the preprint of this paper was made publicw xon the archives, 12 appeared, which also studies the

matrix descriptions of M-theory on the Mobius strip¨and the Klein bottle. These authors find similarresults to ours in the case of the Mobius strip. For¨the Klein bottle, however, they argue that the gaugetheory base space is a Klein bottle, and not a cylin-der as we found. A possible resolution to this contra-

w xdiction was suggested in 13 . There it was found,using non-commutative geometry techniques, thatthe topology of the base space depends on the valueof the background anti-symmetric tensor field Bthrough the original Klein bottle. In the absence ofthis background the result exactly coincides with ourconclusions, i.e. the gauge theory base space is acylinder with the same field content as presented inthis paper. If a half-integral B-field Wilson surface isswitched on, however, the result is a Klein bottle.

w xPresumably in the work of 12 there is such aB-field, although it is not obvious to us where itenters their argument.

2. M-theory on a Mobius strip¨

Ž .In constructing the M atrix -theory compactifiedon a Mobius strip we will start from a related type¨

w xIA theory studied by 5,6 , and consider the dynam-ics of D0-branes in that model.

The IA theory in question is given by the type IIAtheory compactified on a two torus, with radii R ,1,2

divided out by the symmetries V II and SS SS .1 1 2

Here V is the world sheet orientation reversal, II1

inverts the first coordinate, making the first circleinto a line segment, and the SS are shifts in thei

compact directions by half a period:

X ™X qp R , X ™X qp R .1 1 1 2 2 2

In the M-theory point of view V II will become the1

inversion of the eleventh direction, and the resultingcompactification manifold is the Mobius strip.¨

M-theory on a circle divided by the inversion ofthe circle was demonstrated to be equivalent to

w xheterotic E =E string theory 8 . One E factor8 8 8

lives on each ten-dimensional boundary. Weak cou-pling corresponds to shrinking the compact direction.In the present case we divide out one more symme-

Ž .try, which exchanges the two boundaries SS and1

rotates by half a period in an extra compact directionŽ .SS . In the heterotic string, this is exactly the2

operation producing a nine-dimensional CHL stringw x7 , so we expect our matrix model to produce matrixCHL string theory in the limit of vanishing R .1

We will first review Dabholkar and Park’s analy-sis of the orientifold model underlying the matrixdescription. On closed string states with momentumnumbers n ,n , the action of SS SS is simply1 2 1 2Ž .n1qn 2y1 . The untwisted closed string spectrum istherefore that of the usual type IA theory in eightdimensions, except that those states with odd n qn1 2

Žare projected out note that this does not affect the. Ž .massless spectrum . In addition there are massive

twisted states, having half integer winding numbers.The open string spectrum is obtained by calculat-

ing the Klein bottle contribution to the RR-tadpoles.The amplitude with V II in the trace gives the1

usual 32 D8-branes, aligned along the 2-direction.Their X coordinates have to be compatible with1

Ž .both II and SS ; the maximal symmetry is SO 16 ,1 1

obtained when sixteen D8-branes lie on the orien-Žtifold plane X s0 and the other sixteen, by SS1 1

.symmetry, on X sp R . At strong coupling the1 1

symmetry is expected to become E , similarly as in8

the regular IA theory. The V II SS SS in the trace1 1 2

gives a contribution that vanishes in the long tubelimit. Finally, the twisted channels vanish altogether,since they necessarily have non-vanishing windingnumber in the 2-direction, incompatible with V inthe trace. The resulting massless spectrum is thesame as that of the CHL model in eight dimensions,and the two models were in fact claimed to be dual

w xby 5,6 .We will now analyse the dynamics of D0-branes

Ž .in this background, and study the M atrix -theorylimit in which the model is assumed to be lifted toeleven dimensions. As is well known the full dynam-

( )G. ZwartrPhysics Letters B 429 1998 27–34 29

ics on the compact space is described by a gaugetheory in 2q1 dimensions, which is obtained by

w xT-dualising the D0-branes 10 . Since the T-duals ofthe shift symmetries are not obvious to us, we find itconvenient to describe the system slightly differ-ently. Make a change of coordinates to

X X2 1X s " . 2.1Ž ." R R2 1

If we include into the orbifold group the translationsthat make the R2 into a torus, the group is generatedby

SS SS , SSy1 SS , and V II .1 2 1 2 1

The first two elements shift X resp. X by oneq yunit, creating an ordinary torus out of the X X -q yplane. The effect of II is to exchange the two1

coordinates:

X Xq yII s ,1 ž / ž /X Xy q

and V again exchanges left and right movers. Interms of these coordinates we therefore have a type

ŽIIA theory on a torus, divided out by one unconven-.tional symmetry. The operation is drawn in Fig. 1;

taking the shaded region as fundamental domain,instead of one of the triangles, one easily verifiesthat this indeed represents a Mobius strip.¨

Fig. 1. On the left, the torus is drawn, with the symmetry to bedivided out. The result is a Mobius strip. This can be most easily¨seen in the right-hand figure, where instead of the triangle we takethe shaded region as fundamental domain; the arrows indicate theidentification of the two sides.

The metric on the torus is, in terms of the originalradii R ,1,2

1 2 2 2 2R qR R yR2 1 2 1Gs . 2.2Ž .

2 2 2 2ž /4 R yR R qR2 1 2 1

In particular, when the two radii of the original torusR˜Ž .are equal R , the new radii are Rs .'2

The orientifold plane is now really only oneplane, situated along the diagonal X sX . Thisq ydiagonal is the boundary of the Mobius strip . We¨again compute the open string spectrum to verifythat this is indeed the correct model. In the Kleinbottle calculation, the oscillator contributions are thesame as before; furthermore we have a momentumsum with p sp , and a winding mode sum withq yw syw . There are no twisted sectors, since theq yorientifold group has only the one element V II .1

The resulting Klein bottle amplitude is

8V8 2AA sy dl 16 . 2.3Ž . Ž .HKB 4X28p aŽ .

To find the normalisation we also compute the cylin-Žder diagram for convenience we take the case where

. ŽR sR . Here we have diagonal momenta along1 2n. Žthe eight-branes ps , and winding transverse

˜'2 R

R.to the eight-branes over a multiple of . The'2

amplitude is

8V8 2AA sy dl Trg . 2.4Ž . Ž .HCyl 14X28p aŽ .

We clearly need only sixteen D8-branes, distributedsymmetrically around the orientifold plane. The fac-tor of two difference with the usual result comesfrom the difference between winding and momentumsums in the Klein bottle and cylinder. In this repre-sentation we therefore obtain a somewhat simplerpicture, where again the maximal symmetry is

Ž .SO 16 .Ž .Now we go to the M atrix -theory description.

We have to consider the quantum mechanics of N1D0-branes, with masses , on the torus modded outR11

by the orientifold group. In the limit where N™`

this will describe M-theory on the Mobius strip in¨the infinite momentum frame.


To construct the theory we first take the modeldescribing zero-branes of type IIA on a torus, andthen divide out the symmetry. Until further notice wetake R sR sR. The dynamics is described by a1 2

Ž .2q1 U N SYM theory with sixteen supersymme-tries, whose action is the dimensional reduction fromthe ten-dimensional NNs1 gauge theory:

12 2 2S sy Tr F q2 g D XH žIIA mn YM m i24 gYM

24yg X , X q fermions . 2.5Ž ./YM i j

The theory contains seven scalars and eight three-di-mensional fermions, all in the adjoint representation.The two-dimensional space is the T-dual of the torusthe zero-branes moved on; its radii are

3'2 ll 11rs

R R11

Ž .ll is the eleven-dimensional Planck length , and11

the coupling is

2 R112g s .YM 2R

We now have to factor out the required symmetryV II . Under T-duality of the X and X coordi-1 q ynates this symmetry is converted into yV II . Its1

fixed planes are therefore the lines X syX , per-q ypendicular to the eight-branes before T-dualising, asexpected. The action on the fields is as follows:

A x , x ™yAT yx ,yxŽ . Ž .0 q y 0 y q

A x , x ™AT yx ,yxŽ . Ž .q q y y y q

A x , x ™AT yx ,yxŽ . Ž .y q y q y q

X x , x ™X T yx ,yxŽ . Ž .i q y i y q

1 T'c x , x ™ 2 g 1qg c yx ,yx .Ž . Ž . Ž .I q y q 0 I y q2

2.6Ž .

The transformation rule of the fermions is motivatedby the fact that the action of yII can be obtained1

by first rotating the X X plane over ypr2 andq ythen reflecting in the line X s0. Choosing three-di-ymensional gamma-matrices

g s is , g ss , g ss ,0 2 q 1 y 3

we find the rotation is represented by

1 1' 'p i 2 22 2expi g g sq y 1 1ž / ' '2 2 ž /y 2 22 2

1's 2 1q is ,Ž .22

while the reflection is implemented by multiplicationby g .q

These transformations relate the fields on bothŽsides of the fixed plane. Note that these sides are

.actually connected . In particular they impose condi-tions on the fields living on the fixed plane, breaking

Ž .the U N symmetry to a subgroup. Effectively thegauge theory itself lives on a Mobius strip, with¨specific conditions on the fields on the boundary.

Let us then determine the two-dimensional spec-trum living on the fixed line, obtained by restrictingthe three-dimensional fields to the boundary. First ofall we have a two-dimensional vector, consisting ofthe three-dimensional vectors tangent to the line

1' Ž .X syX : A and 2 A yA . From the trans-q y 0 q y2

Ž .formation rules 2.6 we see that these have to beantisymmetric, so that the gauge group in two di-

Ž .mensions is broken to O N . The remaining bosonic1' Ž .fields, 2 A qA and X , are in the symmetricq y i2

representation of this group. The three-dimensionalfermions are split up in two sets of two-dimensionalfermions of definite chirality. The chirality operatoron the fixed line is

1 1' 'g s 2 g g yg sy 2 g 1qg ,Ž . Ž .3 0 q y q 02 2

so that the spinors of negative chirality are in thesymmetric, and those of positive chirality in theadjoint representation.

As in similar models where symmetries with fixedpoints are divided out, we expect extra twisted mat-ter on the fixed line. In the type IA description theseextra states arise from the quantisation of stringsconnecting the D2-brane to the D8-branes; eachtwo-eight string gives rise to one massless fermion,in the fundamental representations of the D2 and D8gauge groups. We expect therefore sixteen Majo-

Ž .rana-Weyl spinors in the fundamental of SO N .Ž .The M atrix -theory motivation for the extra mat-

ter is that the two-dimensional theory as it stands is


anomalous, due to the different representations of theleft- and right-handed spinors. In fact the anomalycan be cancelled by adding 32 positive chiralityfermions in the fundamental representation.

There seems to be a discrepancy between the twoways of counting twisted fermions. This can beresolved by making the anomaly argument a bitmore precise. The gauge anomaly is an anomaly ofthe three-dimensional gauge theory, supported at theboundary. To calculate the actual coefficient of the

w xanomaly, following 8 , we perform a gauge transfor-mation whose gauge parameter L is constant alongthe direction perpendicular to the fixed plane:

E qE Ls0; 2.7Ž . Ž .q y

then we will find the two-dimensional anomaly. ButŽ .now note that 2.7 implies that L can be indepen-

dently defined only along half of the fixed line. TheŽline X qX sconstant intersects the fixed line orq y

.its copies under translation in two different points.We conclude that, by symmetry, half of the anomalyis supported on the lower component of the bound-ary, the other half on the upper component. There-fore, to cancel the anomaly we indeed only have toadd sixteen chiral fermions in the fundamental repre-sentation.

We now wish to go to the limit in moduli spacewhere the CHL string is weakly coupled, and find

w xthe underlying matrix string theory 4 . The couplingconstant of the string is related to the length of theline segment in the X direction,1

3R 21

l s .CHL ž /ll 11

In the weak coupling limit we therefore have to sendR ™0. Let us see what this means in terms of the1

gauge theory. The torus of the gauge theory is theT-dual of the X X torus, which had the metricq yŽ .2.2 . T-dualising means inverting the metric andmultiplying by a

X 2 s ll 6 rR2 , which yields1111

Ry2 qRy2 Ry2 yRy22 1 2 16 2Gs ll rR . 2.8Ž .1111 y2 y2 y2 y2ž /R yR R qR2 1 2 1

So we see that in the weak coupling limit the radii ofthe torus remain equal, and go to infinity, but the

angle between the two periodic coordinates goes toyp .

Ž .The fundamental domain of the torus thereforedegenerates from a two-dimensional diamond to aone-dimensional line, which will become the string

( ˆŽ .Fig. 2 . At the same time its surface, detG be-Ž .comes infinite. In this IR limit the dimensionful

gauge coupling constant also goes to infinity.The fields living on the line are first of all the

untwisted ones. The surviving ones are those that areindependent of the coordinate transverse to the diag-onal. We have eight scalars and eight right movingfermions, all in the symmetric representation ofŽ .O N , and furthermore the gauge fields and adjoint

spinors which are left movers. Then there are the leftmoving twisted fermions in the fundamental ofŽ .O N . There are sixteen of these, but their periods

are twice those of the untwisted fields.In the infrared limit the commutator of the X-fields

is required to vanish, so the coordinate fields can besimultaneously diagonalised. At the same time thegauge multiplet decouples. The eigenvalues of theX ’s may be permuted by a Weyl reflection uponcircling the string; one so obtains the long strings astwisted sectors. The left moving fundamental spinorsare also exchanged by this Weyl reflection upon

Žcompleting one cycle which has double the length.of a right moving cycle . Furthermore, the element

Ž .y1 of O N acts non-trivially on the fundamentalspinors; we will have to project on the subspace ofstates invariant under this element. It is then clearhow to obtain the CHL spectrum: in the sector withlong strings of length n, the left moving bosons have

1moding . Since the fermions’ periods are twicen

those of the bosons, they have half the normalŽ .moding: in the anti-periodic sector A they have

1 mmodes q , while in the P-sector their modes are4n 2 n

Fig. 2. In the weak string coupling limit, the strip collapses to aline.


m 1. The left moving vacuum energy is y in the2n 2 n

A-sector, and 0 in the P-sector. States of zero massŽ .therefore have SO 16 adjoint quantum numbers in

the A-sector, while in the P-sector they group to-Ž .gether in a spinor and an anti-spinor of SO 16 . One

of the latter two is projected out by the y1 elementŽ .of O N , so that we are left with the adjoint repre-

sentation of E , as expected for the CHL-string.8

3. M-theory on a Klein bottle

For the construction of the matrix model compact-ified on a Klein bottle we start from IIA theorycompactified on a two-torus, with radii R and R ,1 2

w xand then divide out the symmetry V II SS 5,6 .1 2

When we lift this to eleven dimensions, the operationV II is the inversion of the eleventh coordinate. The1

topology of the M-theory compactification manifoldis then indeed that of a Klein bottle. In terms of thetype IIA theory, the inversion of the eleventh dimen-

Ž .FLsion is interpreted as the operation y1 , with FL

the left moving space-time fermion number. This canbe checked by comparing the action of both opera-tions on the massless fields: they multiply all RR-fields by y1. In the limit of small eleventh dimen-sion we therefore expect to recover a weakly coupledtype IIA string, modded out by the symmetry

Ž .FLSS y1 .2

Let us first again review the orientifold model. InŽ .n2the closed string sector, the shift SS acts as y1 ,2

with n the momentum number along the second2

circle. If we are interested in the massless spectrum,we therefore have to find those states invariant underV II . This is easily done by performing a T-duality1

along the first circle. Then V II is converted to V ,1

and we obtain, in the massless sector, the closedstring spectrum of the type I string. There are noclosed string twisted sectors, since the only symme-try we divide out contains V . Then there could be

Ž .open strings, but upon calculating the world sheetKlein Bottle contribution to the RR-tadpoles, onefinds that this vanishes. The reason is that the mo-mentum factor in the amplitude is of the form

ta X n22Rn y 2Ž .Ý y1 e . Poisson resummation shows that this

vanishes in the t™0 limit. In conclusion, the theoryis consistent without D-branes. This is consistent

with the observation that the symmetry we dividedout has no fixed points.

We now want to introduce zero-branes in thisset-up. Their dynamics will be described by a certain2q1-dimensional gauge theory, which in the limitN™` should capture the dynamics of the wholetheory. The usual strategy to find the appropriategauge theory is to use T-duality, as we did in theprevious section. There we could circumvent theproblem of T-dualising the half shift SS by adopt-2

ing suitable new coordinates. In the present casehowever, we found no such obvious mechanism toT-dualise.

Instead, we will construct the corresponding two-dimensional gauge theory following the original

w x mstrategy of 10,11 . In the matrices X for N zero-branes on a compact space we include entries for theimages of the zero-branes under translation over the

Ž .circles and strings wrapping the circles , so that theX m are infinite-dimensional. The invariance underthe translations then poses restrictions on the varioussub-matrices. In the case of zero-branes on a torus,with no further symmetries divided out, one findsthat in the compact directions the X m-matrices havethe structure of a covariant derivative, acting on a

Ž .field in the fundamental representation of U N . Thevarious N=N blocks in the infinite matrices thencorrespond to the Fourier components of the gaugefields.

In the case at hand we also want to divide out thesymmetry V II SS . The shift SS has the effect of1 2 2

adding one to the indices denoting the image alongthe two-direction, plus increasing the diagonal entryof X by p R . V transposes the matrices, while II2 2 1

multiplies X , and the indices denoting the image1

along the one-direction, by y1.We demand the matrices to be invariant under this

transformation, which places restrictions on the vari-ous blocks. It turns out that we can again representthem as covariant derivatives, acting on two fields inthe fundamental representation, one periodic along

1the x direction with radius , and the other an-2 R2

tiperiodic:

fq n ,n e2p iŽn1 x1 R1qn 2 x 2 R 2 .Ž .1 2Fs .1

. .y 2p iŽn x R qŽn q x R1 1 1 2 2 2� 0f n ,n eŽ . 21 2


The X can be identified as the Fourier decomposi-1,2

tion of the covariant derivatives

A x , x B x , xŽ . Ž .1 1 2 1 1 2X syiE q ,1 1 † Tž /B x , x yA x ,yxŽ . Ž .1 1 2 1 1 2

A x , x B x , xŽ . Ž .2 1 2 2 1 2X syiE q ,2 2 † Tž /B x , x A x ,yxŽ . Ž .2 1 2 2 1 2

3.1Ž .

with B antiperiodic in the x direction, and satisfy-m 2Ž . T Ž .ing B x , x s"B x ,yx , with the y signm 1 2 m 1 2

for ms1, and the q for ms2. The other, non-com-pact, directions have the same structure as X , but2

lack of course the derivative.In this representation, the translations along the

torus are represented by the unitary transformations

e2p i x1 R1 0U s ,1 2p i x Rž /1 10 e

e2p i x 2 R 2 0U s ,2 2p i x Rž /2 20 e

while the operation SS is given by2

0 ep i x 2 R 2

U s ,SS p i x R2 ž /2 2e 0

which duly squares to U .2

The fermions behave similarly. We have 16Ž .fermionic coordinates matrices S . The symmetriesa

are the same, except that II acts as the ten-dimen-1

sional gamma matrix g on the ten-dimensional1Ž .chiral spinor index. If we then choose a basis inwhich g is diagonal, we find that half of the1

Ž .fermions have the structure of X the gauginos and1Ž .half of them are in the other representation matter .

The gauge theory describing M-theory on a KleinBottle is quite strange. Whereas in the case of theMobius strip we found a gauge theory whose base¨manifold was again a Mobius strip, in the Klein¨Bottle case this is definitely not so. Rather, we have

Ž .gauge fields having some part the A-field living ona torus, without any further restriction beside beinghermitian. The off-diagonal blocks, B, B†, the an-

Žtiperiodic modes of the gauge field or the odd.modes when going to a circle of double the radius ,

only have independent components on half of thetorus; effectively they live on a cylinder, and are

forced to be symmetric on one boundary and anti-symmetric on the other.

We now still want to identify the limit of thetheory in which it describes a weakly coupled typeIIA string. This limit corresponds to R ™0. In the1

gauge theory we see that this implies that x ’s1

period goes to infinity. So again, as expected, wefind that the dimensionless gauge coupling diverges,and that only the zero modes in the x direction2

survive. This means that we effectively obtain atwo-dimensional gauge theory, which we interpret asthe world sheet theory of a string. Since the B fieldsare anti-periodic in the x direction, they do not2

have zero modes, so their masses go to infinity andthey can be ignored. The massless fields that remainon the world sheet are therefore the regular unitarycomponents A. The world sheet theory is thereforethat of the type IIA string, as we expected.

Then we have to show that this theory indeed isinvariant under the extra symmetry. First of all it isclearly invariant under the large gauge transforma-tion U , identifying X with X q2p R , so that the2 2 2 2

theory indeed lives on a circle. To see that the otherŽ .FLsymmetry indeed reduces to SS y1 , note that in2

the strong coupling gauge theory we are considering,the vanishing of the potential implies that all thefields can be simultaneously diagonalised. Thereforethe matter fields are invariant under U , sending XSS 22

to X qp R . The gauge field and the gauginos, on2 2

the other hand, get a y1, so to have a full symmetrywe need to add to SS a transformation multiplying2

these by an extra y1. In the strong coupling limitwe are considering, the gauge field drops out of theaction, so basically we have to check that thefermions with g equal to y1 are precisely the1

fermions of definite chirality in the dimensionallyreduced gauge theory.

This can be easily verified by writing the zero-brane lagrangian and inserting the solutions for theX and S . In a basis for the ten-dimensionalm a

gamma-matrices withG s1m is , G sg ms ,0 2 i i 1

with g a set of nine-dimensional gamma-matrices,i

we have that the ten-dimensional chiral spinors areŽ .of the form S ,0 . The interaction term in the matrixa

model lagrangian is then of the formT i w xTr yS g X ,S . 3.2Ž .Ž .i


Inserting the representations we found above, the1q1-dimensional fermion derivative terms reduceto

dx Tr yiST E yg 1E S , 3.3Ž .Ž .Ž .H 1 0 1

so that indeed g 1 determines the two-dimensionalchirality of the spinor. In the weak coupling limit thesymmetry therefore can be correctly identified as

Ž .FLSS y1 .2

4. Conclusion

We have derived the gauge theory models de-scribing matrix theory on two non-orientable sur-faces, the Mobius strip and the Klein bottle. The¨Mobius gauge theory lives on a Mobius strip itself.¨ ¨

Ž .In the bulk we have a U N 2q1-dimensionalgauge theory. On the boundary the symmetry is

Ž .reduced to O N , with a matter multiplet in thesymmetric representation. Anomaly cancellation re-quires extra twisted chiral fields on the boundary. Inthe limit of a small line segment the chiral fermionsrepresent the level two E current algebra living on8

the CHL string.For the Klein bottle compactification, the T-

dualisation necessary for finding the gauge theorywas less straightforward; we had to find a covariantderivative representation for the zero-brane coordi-nate matrices ‘‘by hand’’. The base manifold of thegauge theory turns out to be a hybrid of a torus forthe periodic modes of the fields and a cylinder forthe anti-periodic modes in the x -direction. Alterna-2

tively one might describe the theory as living on acylinder, with two gauge fields that are identified onthe boundaries, and another field that on one bound-ary is symmetric, on the other anti-symmetric. In thelimit of weak type IIA string coupling, the anti-peri-odic modes do not survive the dimensional reduc-tion, and we are left with a type IIA matrix string

Ž .FLinvariant under the extra symmetry SS y1 .2

Acknowledgements

The author is indebted to Erik Verlinde for adviceand discussions, and to FOM for financial support.

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11 June 1998


On non-linear superfield versions of the vector-tensor multiplet

E. Ivanov a,1, E. Sokatchev b,2

a BogoliuboÕ Laboratory of Theoretical Physics, Joint Institute for Nuclear Research, 141980 Dubna, Russiab Laboratoire de Physique Theorique ENSLAPP 3 Groupe d’Annecy: LAPP, Chemin de BelleÕue BP 110, F-74941 Annecy-le-Vieux Cedex,´

France

Received 5 February 1998; revised 27 February 1998Editor: L. Alvarez-Gaume

Abstract

We propose a harmonic superspace description of the non-linear vector-tensor Ns2 multiplet. We show that there existtwo inequivalent version: the old one in which one of the vectors is the field-strength of a gauge two-form, and a new one inwhich this vector satisfies a non-linear constraint and cannot be expressed in terms of a potential. In this the new versionresembles the non-linear Ns2 multiplet. We perform the dualization of both non-linear versions in terms of a vector gaugemultiplet and discuss the resulting holomorphic potentials. Finally, we couple the non-linear vector-tensor multiplet to anabelian background gauge multiplet. q 1998 Published by Elsevier Science B.V. All rights reserved.

1. Introduction

Ž . w xRecently, there was a revival of interest in the Ns2 vector-tensor VT multiplet 1 , mainly due to the factthat it describes the axionrdilaton complex in heterotic Ns2 four-dimensional supersymmetric string vacuaw x2 . The VT multiplet is a variant representation of the Ns2 vector multiplet, such that one of the physical

Ž .scalars of the latter is traded for a gauge antisymmetric tensor notoph off shell. The known off-shellŽ .formulation of the VT multiplet 8q8 components necessarily implies the presence of a central charge in the

Ns2 superalgebra. It is real and acts on the component fields in a highly non-trivial way. As was observed inw x3,4 , there exist at least two different versions of the VT multiplet. Their basic difference is in the coupling ofthe tensor and vector gauge fields: in the so-called ‘‘non-linear’’ version the tensor field couples to the

Ž .Chern–Simons CS form of the vector one, while no such CS coupling is present in the ‘‘linear’’ version.Ns2 supersymmetry is realized in these two cases in essentially different ways: non-linearly in the first caseand linearly in the second one. The two versions also radically differ in what concerns couplings to background

w xNs2 vector multiplets and Ns2 supergravity 4,5 . Note that the central charge transformations can be global

1 E-mail address: [email protected] E-mail address: [email protected] URA 14-36 du CNRS, associee a l’Ecole Normale Superieure de Lyon et a l’Universite de Savoie.´ ` ´ ` ´


( )E. IÕanoÕ, E. SokatcheÕrPhysics Letters B 429 1998 35–4736

or local. In the latter case an extra vector multiplet gauging the central charge should be introduced from thebeginning. When coupling the VT multiplet to supergravity, the central charge is necessarily gauged.

An exhaustive component analysis of the two versions of the VT multiplet together with their couplings tow xbackground vector multiplets was given in 3,4 , assuming that the central charge transformations are local. As

Žfor the superfield formulations of the VT multiplet which are most natural when dealing with off-shell.supermultiplets , until recently they were constructed only for the linear version, both in the free case and in the

w x w xpresence of couplings to background vector multiplets 6–9 . There exist formulations in the standard 6–8 asw x 4well as in harmonic 9 Ns2 superspaces .

Our purpose in this letter is to give such a formulation for the non-linear version, both for the pure VTmultiplet and for the case when CS couplings to the background vector multiplets are switched on. We make use

Ž . w xof the harmonic superspace HSS approach 11–13 as most adequate to Ns2 supersymmetry and demonstratethe existence of two inequiÕalent non-linear Õersions of the VT multiplet. The first one is just the version

w xdiscovered in 3,4 , while the second is essentially new: it cannot be reduced to the ‘‘old’’ one by any fieldredefinition. Its most characteristic feature is the modification of the r.h.s. of the Bianchi identity for the

Ž .three-form the field strength of the tensor gauge field by terms quadratic in the latter and in the auxiliaryŽfields. As a result, the Bianchi identity has no local solution in terms of a tensor gauge potential note that one

w x .of the primary assumptions of 3,4 is the existence of such a potential . We show that the bosonic action of thisnew version of the VT multiplet vanishes as a consequence of the modified Bianchi identity for the three-form.Nevertheless, a non-trivial action is obtained upon dualization, i.e. after implementing this identity in the actionwith a scalar Lagrange multiplier. In this aspect, the situation is quite similar to the case of non-linear Ns2

w xmultiplet 14 . The dual action exhibits all the features of special Kahler geometry typical of the actions of¨Ns2 vector multiplets and is fully specified by a non-polynomial holomorphic potential. We propose amanifestly supersymmetric version of the dualization procedure.

Here we restrict our study to the rigid case, postponing the discussion of the gauged central charge and, moregenerally, Ns2 supergravity to a future publication. Also, when discussing the superfield CS couplings tobackground vector multiplets, for the sake of simplicity we consider one abelian multiplet coupled to the ‘‘old’’non-linear version of the VT multiplet. Further generalizations will be presented elsewhere.

2. Preliminaries

Let us first briefly recall some facts about the HSS description of the linear version of the VT multiplet, to aw xlarge extent following Ref. 9 .

The basic object of such a description is the real harmonic superfield

aa 5 " a " a " i˙ ˙LsL x , x ,u ,u ,uŽ .subject to the constraints

22q qD Ls D Ls0 , 2.1Ž . Ž . Ž .q qD D Ls0 , 2.2Ž .a a

DqqLs0 . 2.3Ž .

4 w xWhen this work was near completion, we became aware of the parallel work 10 where a harmonic superspace formulation of theŽnon-linear version of the VT multiplet was given at the level of rigid central charge and without considering CS couplings to extra vector

.multiplets .

( )E. IÕanoÕ, E. SokatcheÕrPhysics Letters B 429 1998 35–47 37

Here

E Eqi y " a a i " " a a i " q q˙ ˙u u s1 , u su u , u su u , D s 'E , D s 'E ,i i i a ya a ya˙ ˙ya yaEu Eu

E 22qq qi qa qa q q qa qa˙ ˙D su y2 iu u E q i u y u E qu E qu E 2.4Ž . Ž . Ž .ž /aa 5 ya ya˙ ˙yiE u

are the basic quantities of the central-charge extended HSS in the analytic basis. This basis is chosen so that thecovariant spinor derivatives Dq, Dq are ‘‘short’’ and the coordinate setsa a

5 a a 5 qa qa " i˙ ˙z ' x , x ,u ,u ,u 2.5� 4 Ž .

and

aa qa qa " i˙ ˙� 4z' x ,u ,u ,u 2.6Ž .

are closed under the Ns2 supersymmetry transformations. They are called analytic subspaces. The harmonicqq q qderivative D commutes with D , D , and so it preserves analyticity. In what follows we will always use thea a

Ž . Ž .analytic basis. The subspaces 2.5 , 2.6 are real, i.e. closed under some generalized conjugation. Ourw xconventions are those of Ref. 11 .

Ž . Ž .The set of constraints 2.1 – 2.3 reduces the infinite component content of L to that of the off-shell linearVT multiplet when formulated via the field strengths of the notoph and vector gauge potentials 5.

It is convenient to represent L by its analytic components, i.e. by the functions on the subspace z 5 whichya ya Ž . Ž .appear in the decomposition of L in powers of u , u . The constraints 2.1 , 2.2 imply

5 y q 5 y q 5Ls l z qu f z qu f z 2.7Ž .Ž . Ž . Ž .a aŽ . Ž .here and in what follows the spinor indices are contracted according to the rule , , while 2.3 leads to thea a

two harmonic constraints

qq q q q qD lqu f qu f s0 , 2.8Ž .

qq q qq qD f sD f s0 . 2.9Ž .

Thus an equivalent description of the VT multiplet is given in terms of the analytic scalar and spinorq q qŽ . Ž .superfunctions l, f . f subject to the harmonic constraints 2.8 , 2.9 . Note that f is transformed undera a a˙

Ns2 supersymmetry as a standard analytic superfield while l has unusual transformation properties:

q i y q i y qd f s0 , d lsye u f ye u fa i i

i iwhere e , e are infinitesimal transformation parameters.a a

5 Ž .At present it is unclear, even at this simplest linear level, what could be if existing! the HSS description of the VT multiplet in termsof superfield potentials.


Ž . Ž . 5Eqs. 2.8 , 2.9 fully determine the action of the central charge generator ErE x on the component fields inl, fq. In what follows it will be more convenient to define its action directly on the analytic quantities l, fq.

Ž .This can be done using the following trick. As a consequence of the harmonic condition 2.3 we have

DyyLs0 , 2.10Ž .yy Ž . qqwhere D is the harmonic derivative conjugate in the usual sense to D

E 22yy yi ya ya y y ya ya˙ ˙D su y2 iu u E q i u y u E qu E qu E 2.11Ž . Ž . Ž .ž /aa 5 qa qa˙ ˙qiE u

Ž . qq Ž .it does not preserve analyticity! . Together with D they form the SU 2 algebra:

w qq yyx 0 w 0 ""x ""D , D sD , D , D s"2 D , 2.12Ž .0 Ž . Ž 0 0 q q. Ž .where D is the operator counting the harmonic U 1 charge D ls0, D f s f . Substituting 2.7 into

Ž .2.10 , we find the set of constraints:

Eyyls0 , 2.13Ž .yy q yy qE lqE f s0 , E lyE f s0 , 2.14Ž .qa a qa a˙ ˙

iq qE lq E f qE f s0 , 2.15Ž .˙ ž /ab qa a qa a˙ ˙2

i iqa qaE ly E f s0 , E lq E f s0 , 2.16Ž .5 qa 5 qa2 2

q qa q qa˙E f yE f s0 , E f qE f s0 , 2.17Ž .5 a a a 5 a a a˙ ˙ ˙yy yi qi Ž .where E su ErE u . An important corollary of Eqs. 2.16 is the reality condition

qa qaE f qE f s0 . 2.18Ž .qa qa

Introducing

y q yy y q yyD s D , D , D s D , D , 2.19Ž .a a a a˙ ˙

Ž .it is easy to show that another form of 2.18 is

ya q y qaD f sD f , 2.20Ž .a a

w xwhich is just the reality condition of Ref. 9 .Ž . Ž . Ž . Ž .Combining relations 2.13 – 2.17 with Eqs. 2.8 , 2.9 , it is easy to show that the irreducible field content

of l, fq coincides with that of the linear version of the VT multiplet:

< 5 5 q < i 5 ql sf x , x , E f'G x , x , f s f x , x u ,Ž . Ž . Ž .5 a a i

q< 5 5 q < 5 5E f sF x , x q ie G x , x , E f sh x , x q iE f x , x 2.21Ž . Ž . Ž . Ž . Ž .qb a Ž ba . ba qa b ba ba˙ ˙ ˙

<where means the lowest component of a given superfield. After simple algebraic manipulations involving theŽ .above constraints, all other components, including those obtained by acting on 2.21 with E , are expressed as5

Ž .x-derivatives of the basic quantities 2.21 . For instance,1 aaE Gs b E f . 2.22Ž .5 a a2 ˙

The Bianchi identities for F , h also directly follow from the constraints. For instance, acting by E on the˙Ža b . a b 5Ž . Ž . Ž .reality condition 2.18 and on Eq. 2.15 , and making use of Eqs. 2.17 afterwards, one gets, respectively,

EPhs0 , 2.23Ž .


and

b b .E F yE F s0 , 2.24Ž .˙ab Ž ba Ža˙a

which are the Bianchi identities for the notoph and vector gauge field strengths.w xIn Ref. 9 there was proposed a nice general recipe of constructing HSS actions for super-multiplets with a

non-trivial realization of the central charge, such that they are still given by integrals over the standard analyticŽ . 6subspace 2.6 . The action is given by

22y4 q q qqSs dz i u y u LL 2.25Ž . Ž . Ž .Hwhere dzy4 'dud4 xd4uq. The real Lagrangian density LLqq should be:1. analytic:

q qq q qqD LL sD LL s0 . 2.26Ž .a a

2. harmonically ‘‘short’’:

DqqLLqqs0 . 2.27Ž .The second condition immediately leads to the important property that the x 5 derivative of the integrand in

Ž . Ž Ž .. Ž .2.25 is a total x and u-derivative recall 2.4 and so disappears upon integration. As a result the action 2.25does not depend on x 5 or, to put it differently, is invariant under central charge transformations. The Ns2

Ž . Ž w x.supersymmetry of 2.25 is not manifest, but can be easily checked see 9 .w xIn the case under consideration two Lagrangian densities of this sort exist 9 :

qq q q q q q q q qa LL s i D LD LyD LD L s i f f y f f , 2.28Ž . Ž .Ž . Ž .1

qq q q q q q q q qb LL s D LD LqD LD L s f f q f f . 2.29Ž . Ž .Ž . Ž .2

The first density gives the free action of the linear VT multiplet. The second one is a total x-derivative, i.e.gives a topological invariant. Both of them can be generalized to include CS couplings to external Ns2 vector

w xgauge multiplets 9 .

3. Non-linear VT multiplets

As was already mentioned, a characteristic feature of the non-linear version of the VT multiplet discovered inw x3,4 is the presence of CS coupling-induced terms of the vector gauge field in the Bianchi identity for thenotoph gauge field strength. The simplest way to obtain such terms in the HSS description is to modify thelinear VT multiplet constraints as follows:

2q q q q qD Lsa L D LD Lqb L D LD L , 3.1Ž . Ž . Ž . Ž .q q q qD D Lsg L D LD L , 3.2Ž . Ž .a a a a˙ ˙

DqqLs0 , 3.3Ž .Ž . Ž . Ž . Ž .with a L , b L being complex and g L sg L real functions of L, arbitrary for the moment. Note that

Ž . Ž . Ž . Ž .3.1 , 3.2 provide the most general deformation of the linear constraints 2.21 , 2.22 consistent with the

6 w xNote that a similar HSS action is used to describe the massive central-charged hypermultiplet 11,13 .


Ž . Ž .preservation of the harmonic U 1 charge and the harmonic condition 3.3 . It is worth mentioning that inq qprinciple the latter can also be deformed by adding appropriate bilinears of D , D into its r.h.s. We do nota a

consider such non-minimal possibilities here.Ž . Ž .The constraints 3.1 , 3.2 should satisfy the evident self-consistency conditions

2 2q q q q qa q qD D Ls0 , D D LsD D D L , 3.4Ž . Ž . Ž .Ž .a a a a˙ ˙

which amount to the following set of differential equations for the coefficients:X

gya s ayg gybb , 3.5Ž . Ž . Ž .b

X s ay2g b . 3.6Ž . Ž .Thus we have four real differential equations for five real functions. However, we are actually dealing with fourunknowns due to the reparametrization freedom

˜ ˜L™L , LsL L 3.7Ž .Ž .Ž . Ž .in 3.5 , 3.6 . Under such reparametrizations the coefficients transform as follows:

X XX X X X X˜a™asLay ln L , b™bsLb , g™gsLgy ln L . 3.8Ž . Ž . Ž .˜ ˜Ž . Ž . Ž .We can choose different gauges with respect to 3.8 in order to simplify the set 3.5 , 3.6 . A very

convenient gauge amounts to choosing

gs0 , 3.9Ž .which implies

X Xa sbb , b sab . 3.10Ž .

Ž . Ž .In this gauge the constraints 3.1 – 3.2 become simpler:

2q q q q q q qD LsaD LD LqbD LD L , D D Ls0 . 3.11Ž . Ž .a a

y yŽ .The main advantage of the constraints in the form 3.11 is that there appear no mixed terms in the u , ua a

Ž . Ž Ž . .expansion of L. Indeed, the solution to the second of Eqs. 3.11 is cf 2.7 in the linear case :

2 2 22 2 21 1y q y q y q q y q qLs lqu f qu f y u a f qb f y u a f qb f . 3.12Ž . Ž . Ž . Ž . Ž .Ž . Ž .4 4

Ž .It is easy to find the general solution to the Eqs. 3.10 , but before doing this, we point out that additionalŽ .restrictions on the coefficient functions a , b come from the harmonic condition 2.10 . Applying the reasoning

Ž . Ž . Ž . Ž .which lead to Eqs. 2.13 – 2.17 , one finds the analogs of the latter for the non-linear case. Eqs. 2.13 , 2.14Ž .preserve their form, while those from 2.15 on are modified by non-linear terms:

Eyyls0 , 3.13Ž .yy q yy qE f qE ls0 , E f yE ls0 , 3.14Ž .a qa a qa˙ ˙

iq qE lq E f qE f s0 , 3.15Ž .Ž .aa qa a qa a˙ ˙ ˙2

iqa yy q q yy q qE ly E f yaE f f ybE f f s0 , and c.c. 3.16Ž .Ž .5 qa2

i i 22˙q qb q q q q yy q q qE f yE f q aE f f qbE f f q abE f f q f s0 and c.c. 3.17Ž . Ž .Ž .˙ Ž .5 a a b qa qa a2 4


A new phenomenon in the non-linear case is the appearance of a new self-consistency condition as a result ofy 2 y 2Ž . Ž . Ž .equating to zero the coefficient of the monomial u u in 2.10 . It reads

22q qE ayb f q bya f s0 . 3.18Ž . Ž . Ž .Ž . Ž .5

q q Ž . Ž .Working out the derivative E and expressing E l, E f , E f from Eqs. 3.16 , 3.17 , we see that there appear5 5 5 a 5 a

unacceptable algebraic constraints on the fermions fq, unless we demand

asb . 3.19Ž .Ž .This new constraint, together with 3.10 , imply

Xa saa . 3.20Ž .

Ž .Putting asaq ib in 3.20 , we find

bsconstand

aX sa2 qb2 . 3.21Ž .Ž .The solution to the differential Eq. 3.21 depends on the value of the constant b. If b/0, one obtains

asb tan Lqc q i , 3.22Ž . Ž .where c is a new integration constant; if bs0, the solution is

1asy . 3.23Ž .

Lqc

Ž .Note that after choosing the gauge 3.9 we still have the freedom of global rescalings and shifts of L. UsingŽ . Ž . Ž .this, we can fix the constants b, c in 3.22 or 3.23 , for example, bs1, cs0. Thus, in the gauge 3.9 we

obtain two distinct solutions:

1i as tan Lq i ; ii asy . 3.24Ž . Ž . Ž .

L

ŽThey give rise to two inequiÕalent Õersions of the non-linear VT multiplet remember that we have already.exhausted the freedom of redefinition of L .

The principle difference between these two versions is in the following. It is easy to deduce the analogs ofŽ . Ž . Ž .the Bianchi identities 2.23 , 2.24 for both non-linear versions. Eq. 2.24 does not change, implying that

F , F are still expressed in the standard way through the vector gauge potential. At the same time, the˙Ža b . Ža b .˙Ž .identity 2.23 is drastically modified:

i i 2 12 2 2 2EPhq aF yaF q aya h y Ef y2G y aqa EfPhs0 , 3.25Ž . Ž . Ž . Ž .Ž . 24 4

where

˙2 a b 2 a b˙F sF F , F sF F ˙ab a b˙

Ž . Ž . Ž .and asa f . For the second solution ii in 3.24 asasy1rf, so after the redefinition

aa aa aa˜h ™h sfh 3.26Ž .˜one gets the standard CS-term-modified Bianchi identity for h

i2 2˜EPhy F yF s0 . 3.27Ž . Ž .

4


aa˜Ž .It can still be solved through the antisymmetric gauge field notoph after an appropriate shift of h by the CSŽ . Ž .one-form. This means that the solution ii in 3.24 corresponds just to the non-linear version of the VT

w x Ž . Ž .multiplet discovered in 3,4 . At the same time, there is no way to reduce 3.25 to 3.27 in the new caseŽ . Ž . Ž .corresponding to the solution i in 3.24 . There it is impossible to solve the identity 3.25 through a notoph

Ž .potential at least, locally , though we still end up with 8q8 off-shell degrees of freedom. Thus we encounteran essentially new version of the VT multiplet in this case 7.

Ž . Ž .It is easy to find the analogs of the free actions 2.28 , 2.29 for both non-linear versions at hand. One startsfrom the Ansatz

qq q q q qLL sA L D LD LqA L D LD L 3.28Ž . Ž . Ž .Ž . Ž .and solves the differential equations for A, A following from the analyticity constraint 2.26 . In both cases i ,

Ž . Ž . Ž .ii in 3.24 we get in this way two-parameter solutions for A L :

1i A L sd tan Lq i qd 1qL tan Lq i ; ii A L sg q ig L , 3.29Ž . Ž . Ž . Ž . Ž . Ž . Ž .1 2 1 2L

where d , g are arbitrary real constants.1,2 1,2Ž .The explicit form of the superfield Lagrangian density 3.28 is the most interesting case of the new solution

Ž .i is2 2qq q q q q q qLL sd D Lqd D LD LqD LD LqL D L . 3.30Ž . Ž . Ž .1 2

Ž .It is instructive to work out the component bosonic Lagrangian corresponding to 3.30 . As a preparatory step itaais convenient to redefine h as follows:

1aa a a˙ ˙˜h s h , 3.31Ž .

cosf

1 21 1if 2 if 2 2 2˜ ˜EPhq e F qe F q h y cosf Ef q2G s0 . 3.32Ž . Ž . Ž .4 22cosf

ŽThen a straightforward computation yields up to an overall normalization constant, modulo a total x-derivative.and after putting the auxiliary field Gs0

i 1 22 1 2 2 2 2 2˜ ˜LL sÕ f Ef y F qF y F yF tanfy h y EPh , 3.33Ž . Ž . Ž . Ž . Ž .bos 2 2½ 52 cosfcos f

where

Õ f 'd qd f .Ž . 1 2

Ž .Substituting 3.32 into this expression, we find the surprising result

LL s0! 3.34Ž .bos

Nevertheless, one can obtain a non-vanishing action after dualizing the notoph covariant field strength. Thispoint is discussed in the next Section.

7 The relation between these two non-linear versions of the VT multiplet resembles that between the two well-known multiplets of Ns2w x w xsupersymmetry without central charge, the tensor 15 and non-linear 14 ones. Both of them have the same number of off-shell degrees of

freedom and in both cases there is a constraint on the vector component. In the case of the tensor multiplet this constraint is of the notophŽ .type 3.3 and it can be locally solved through the notoph potential. In the case of the non-linear multiplet the constraint is modified and

Ž . Ž .resembles 3.25 it also contains terms bilinear in the vector field strength in its r.h.s. . No local solution to this modified constraint in termsof a gauge potential can be given.


Ž .As the last topic of this Section we present an alternative to the gauge 3.9 :

bsbeic , bsconst/0 3.35Ž .which yields

X X2gya s ayg gy2b , ibc sb ay2g . 3.36Ž . Ž . Ž . Ž .Ž .This time the analog of 3.18 implies the additional constraint

aybygs0 . 3.37Ž .Ž . Ž .Eqs. 3.36 , 3.37 have two different solutions:

asb 2cos2lq isin2l , gsbcos2l , bsbey2 i l , lsarctanaeyb L , 3.38Ž . Ž .as2b , gsbsb , 3.39Ž .

Ž .a, b being integration constants. The solution 3.38 corresponds to the ‘‘new’’ version of non-linear VTŽ . Ž . Ž .multiplet, while its as0 contraction 3.39 yields the ‘‘old’’ version. The constraints 3.1 , 3.2 at as0

exhibit an invariance under the shift L™Lqconst, so the relevant actions are scale-invariant. The cases a/0and as0 cannot be related by any field redefinition, since we have already fixed the reparametrization freedom

Ž .while choosing 3.35 .Ž . Ž .The most general solution 3.38 was obtained in the gauge 3.35 , and it has the advantage of being

non-singular in the two important limits as0 and bs0 which lead, respectively, to the scale-invariantŽ .non-linear version 3.39 and to the linear version. However, when constructing the invariant actions and

inspecting the deformations of the Bianchi identities in the general a/0, b/0 case, it is more convenient toŽ .stay in the gauge 3.9 . The precise relation between the two gauges is as follows:

22q q q q q q˜ ˜ ˜ ˜ ˜ ˜D Ls D Ls2c ba cot2lq i D LD Lq cot2ly i D LD L ,Ž . Ž . Ž . Ž .1

1q q ˜ ˜D D Ls0 , Lyc y 2l . 3.40Ž .a a 2˙ 2c ba1

˜Here, c , c are arbitrary integration constants reflecting the residual freedom of shifting and rescaling L. They1 2Ž .can always be chosen so as to guarantee the limits as0 andror bs0 to be non-singular in the gauge 3.9 too.

Ž . Ž .Finally, we note that it is rather straightforward to check that in the case ii in 3.29 the invariants enteringŽ .with constants g and g take, respectively, the following form in the gauge 3.35 :1 2

yb L q q q q y3bL q q q q;e D LD LqD LD L and ; ie D LD LyD LD L . 3.41Ž .Ž . Ž .qq qq Ž Ž . Ž ..Thus they generalize the Lagrangians LL and LL of the linear case Eqs. 2.29 , 2.28 . Note that the first2 1

Ž . w xLagrangian in 3.41 is reduced to a total derivative 10 .

4. Dual versions of the VT actions

The dual form of the above actions is obtained by implementing the notoph constraint in the Lagrangian withŽ .the help of a Lagrange multiplier. In the case of the constraint 3.32 this leads to the action

1 2X 1 1if 2 yif 2 2˜ ˜LL syl EPhq e F qe F q h y cosf Ef . 4.1Ž . Ž . Ž .bos 4 2ž /2cosf

aa˜Now h is unconstrained, and one can integrate it out by its algebraic equation of motionaaE l

aah scosf . 4.2Ž .l


After that we get a typical sigma-model action

l l 2 2X if 2 yif 2LL sy e F qe F q cosf Ef q E lnl . 4.3Ž . Ž . Ž . Ž .bos 4 2

Let us make once more an analogy with the non-linear Ns2 multiplet. There one cannot write down aŽ Ž . . w xnon-vanishing and SU 2 invariant action for this multiplet itself 16–18 , but the dual action obtained by

implementing the defining constraint with the help of a Lagrange multiplier yields a non-trivial sigma-modelaction in its bosonic sector.

Ž . Ž .No such subtleties occur in the case of the ‘‘old’’ non-linear version corresponding to the solution ii 3.29 .Ž . Ž .The only effect of substituting the constraint 3.27 into the appropriate analog of the Lagrangian 3.33 is the

cancellation of the terms proportional to g , in accord with the previous statement that the invariant proportional1Ž .to g is a total derivative. In this case we have the following bosonic Lagrangian before dualization :1

12 1 2 2 2˜LL sg f Ef y f F qF y h . 4.4Ž . Ž . Ž .bos 2 2f

Ž .The analog of the dual Lagrangian 4.3 reads

1 i i2 2X 1 12 2LL sf g Ef q El y g fq l F y g fy l F . 4.5Ž . Ž . Ž .bos 2 2 22 2ž / ž /4 g 2 22

Ž . Ž .Both actions 4.3 and 4.5 can be recast in the generic form of the bosonic part of the action of an Ns2gauge multiplet:

iX X X XX XX2 2LL s EFFE zyE zEFF qFF F yFF F . 4.6Ž .Ž .bos 2

Ž . Ž .The holomorphic potential FF z for the action 4.3 is

iyi zFF z s e , zsfq ilnl 4.7Ž . Ž .

2Ž .and for the action 4.5

g i2 3FF z syi z , zsfy l . 4.8Ž . Ž .6 2 g2

Ž . w xThe potential 4.8 can be obtained from that of Ref. 3,4 , by freezing the Ns2 vector multiplet which gaugesŽ .the central charge. The potential 4.7 is new and it would be interesting to study whether it may occur in a

stringy context.The dualization procedure described above can be carried out in a fully off-shell supersymmetric way. For

simplicity we explain this on the example of the linear version of the VT multiplet. We take the superspaceŽ . Ž . Ž . Ž .action 2.25 , 2.28 and add to it the harmonic constraints 2.8 , 2.9 with analytic superfield Lagrange

multipliers:

22y4 q q q q q q q qq q q qq q qq qq q q q qSs dz u y u f f y f f qH D f qH D f qG D lqu f qu f .Ž . Ž . Ž . Ž .H ½ 54.9Ž .

Note that the Lagrange multiplier Hqa has a non-standard supersymmetry transformation law in order toq qcompensate for the variation of the first term. We assume that the central charge is still realized on f , f as in

Ž .2.17 , whereas on l it acts as follows:

iqa qaE ls E f yE f 4.10Ž .Ž .5 qa qa4


Ž Ž .the reality condition 2.18 is not imposed at this stage, it appears only as a result of the variation w.r.t. some.Lagrange multiplier . This realization of the central charge is compatible with supersymmetry. The first term in

Ž .4.9 is invariant under central charge transformations on its own. The requirement of central charge invarianceof the rest of the action determines the central charge transformation properties of the Lagrange multipliers:

iq qa qq qq˙E H syb H y b G , E G s0 . 4.11Ž .5 a a a qa 5˙ 4

Ž .It can be shown that upon elimination of the infinite set of auxiliary fields from 4.9 we are left in the bosonicsector with two scalars and an abelian gauge vector field, which belong to an on-shell Ns2 vector multipletdual to the original VT one. More details and the treatment of the non-linear versions will be given elsewhere.

5. Coupling to an external vector multiplet

Ž .Here we shall deform the non-linear superfield constraints their ‘‘old’’ version to switch on the CScoupling to one external abelian vector multiplet.

Ž . Ž .We choose the gauge 3.35 and the simplest constant solution to Eqs. 3.36 :

as2gs2b , bsbeic , bsconst , csconst . 5.1Ž .w xThus our starting point is the set of constraints describing the nonlinear version of 3,4

2q q q ic q q q q q qD Ls2bD LD Lqbe D LD L , D D LsbD LD L . 5.2Ž . Ž .a a a a˙ ˙

As we saw before, some additional self-consistency conditions require for the given solution

eic s1 . 5.3Ž .This phase factor can be non-trivial in the presence of extra vector multiplets.

The abelian vector multiplet is represented by its superfield strength W which does not depend on theharmonics and obeys the chirality condition and the Bianchi identity

22qq " " q qD Ws0 , D WsD Ws0 , D Ws D W . 5.4Ž . Ž . Ž .a a

Ž . Ž .In order to find an appropriate self-consistent deformation of 5.2 , such that it is reduced to 5.2 afterŽ . Ž .switching off W, we start from the most general form of such a deformation of the r.h.s. of Eqs. 3.1 , 3.2

Ž . Ž .consistent with the harmonic U 1 charge q2 of the l.h.s. and the constraint 3.3 . All the coefficients,including a , b and g , are assumed to be arbitrary functions of L, W and W, with proper reality conditions

Ž .imposed. Next, we exploit the integrability conditions 3.4 . They lead to a huge number of equations on theŽ . Ž .coefficients. Among them we still have Eqs. 3.5 , 3.6 . To simplify the set of self-consistency conditions we

utilize, like in the pure L case, the reparametrization freedom

˜L™L L,W ,W . 5.5Ž .Ž .Ž . Ž . Ž .We can still impose the gauges 3.5 or 3.9 on the coefficients a , b , g . We choose 3.5 , with b having no

dependence on L, W and W,

bsconst . 5.6Ž .There still remains the freedom of shifting L by a real function of W, W. It can be used to further restrict ther.h.s. of the deformed constraints.

Ž .Even after fixing the gauges we are still left with a considerable set of equations. We first solve the Eqs. 3.6Ž .for a , b , g . As was stated above, for simplicity we choose the solution 5.1 , where c is still independent of L


but now depends on W, W. This dependence has to be specified by solving the rest of the consistencyŽ .conditions. Fortunately, the latter is greatly simplified under the choice 5.6 .

Ž .As a result, we find the following self-consistent deformation of the constraints 5.2 :

2 2q q q ic q q q q q q qD Ls2bD LD Lqbe D LD LqlD LD Wqv D WD Wqn D WŽ . Ž .q q ic q qqE n D WD WqE n e D WD W , 5.7Ž .W W

q q q q q q q q q qD D LsbD LD Lqs D LD Wys D LD Wyv D WD W . 5.8Ž .a a a a a a a a a a˙ ˙ ˙ ˙ ˙

Ž .Here all the coefficients, except for n , are expressed through the function c W,W which satisfies the followingremarkable equations:

icE cse b c , E E cs0 . 5.9Ž .W W W W

Ž Ž . .The general solution of this system respecting Eq. 5.3 in the limit Ws0 is given by

cs i ln 1y ikW y ln 1q ikW . 5.10Ž . Ž . Ž .Ž .Ž . Ž .Here k is a real integration constant. Explicitly, the coefficients in 5.7 , 5.8 are as follows:

ik i k 1 k 2

ls , ss , vsy . 5.11Ž .1y ikW 2 4b1q ikW 1y ikW 1q ikWŽ . Ž .

Ž .It remains to specify the coefficient n in 5.7 . It is given by the following expression:

ic 2 bLns e a W qa W e , 5.12Ž . Ž . Ž .Ž .a W being an arbitrary holomorphic function.

Ž . Ž . Ž . Ž .The constraints 5.7 , 5.8 with the coefficients given by Eqs. 5.11 and 5.12 describe the most generalŽ .deformation of the ‘‘old’’ nonlinear VT constraints 5.2 in the presence of one extra vector multiplet. It should

be pointed out that the deformation presented here does not distinguish an external vector multiplet from onethat gauges the central charge. Indeed, the above derivation relied merely upon the anticommutativity of Dq,a

q Ž .D and the constraints 5.4 . These properties are valid irrespectively of whether W is some external gaugea

superfield strength or it is the strength of a superfield gauging the central charge.Ž .An additional selection rule results from enforcing a self-consistency condition like 3.18 . It leads to

w xdrastically different consequences for the cases of rigid and gauged central charges 19 . In the rigid case we areŽ Ž . . Ž . 8dealing with when W is treated as an external U 1 superfield gauge strength it still requires 5.3 that entails

ks0. As a result, in this case the deformation above is fully specified by the choice of the holomorphicaaŽ . Ž . Ž .function a W in 5.12 . The standard CS modification of the Bianchi identity for h arises for a W scW, c

being the appropriate CS coupling constant. However, all the self-consistency conditions are still fulfilled by anŽ .arbitrary a W . Though the modified Bianchi identity has no local solution in the general case, we expect that

the ‘‘dualization’’ of this identity with the help of a Lagrange multiplier vector multiplet may yield anacceptable local theory.

As our last topic we give here the relevant invariant action. The analytic Lagrangian density LLqq for theW-deformed case can be constructed by the method of undetermined coefficients, like we proceeded in the

8 Our special thanks are due to S. Kuzenko for bringing up this point to us.


Ž . qqprevious Section. The analyticity constraint 2.26 fixed LL up to three integration constants, thus yieldingthree independent invariants

1 2qq ybL q q bL qLL ;e D LD Lq e y1 D G W qc.c. , 5.13Ž . Ž . Ž . Ž .Ž1. b2 12 2qq y3bL q q q ybL ybL qLL ; i e D LD Lq D e y1 G W q 1ye D G W yc.c. ,Ž . Ž . Ž . Ž . Ž . Ž .Ž2. ½ 5b b

5.14Ž .1 2 2qq ybL q q ybLLL ; e y1 D Wq2 D 1ye W qc.c. , 5.15Ž . Ž . Ž . Ž . Ž .½ 5Ž3. 2b

Ž . Ž .where G W is a ‘‘potential’’ for a W ,

a W sE G W .Ž . Ž .W

Ž .The first two densities extend the invariants 3.41 , while the third one is new, since it vanishes when Ws0. ItŽ .still reduces to the first Lagrangian in 3.41 under the choice Wsconst. Note that all these invariants were

chosen to be well-defined in the limit bs0 by extracting some pure W densities22q q; D FF W q D FF W 5.16Ž . Ž . Ž . Ž . Ž .

with some appropriate FF. They can be omitted without loss of generality.

Acknowledgements

We are indebted to B. de Wit, N. Dragon, R. Grimm, M. Hasler, S. Kuzenko and B. Zupnik for valuablediscussions. The work of E.I. was partially supported by the grants RFBR 96-02-17634, RFBR-DFG 96-02-00180, INTAS-93-127ext and INTAS-96-038. He thanks D. Lust, C. Preitschopf and P. Sorba for their kind¨hospitality at the Humboldt University, Berlin and at LAPP, Annecy, where a part of this work was carried out.

References

w x Ž .1 M.F. Sohnius, K.S. Stelle, P.C. West, Phys. Lett. B 92 1980 123.w x Ž .2 B. de Wit, V. Kaplunovsky, J. Louis, D. Lust, Nucl. Phys. B 451 1995 53.¨w x Ž .3 P. Claus, B. de Wit, M. Faux, B. Kleijn, R. Siebelink, P. Termonia, Phys. Lett. B 373 1996 81.w x Ž .4 P. Claus, P. Termonia, B. de Wit, M. Faux, Nucl. Phys. B 491 1997 201.w x5 P. Claus, B. de Wit, M. Faux, B. Kleijn, R. Siebelink, P. Termonia, Ns2 Supergravity Lagrangians with Vector-Tensor Multiplets,

Preprint KUL-TF-97r24, THU-97r26. JUB-EP-97r73, ULB-TH-97r18, ITP-SB-97-63, DFTT-62r97, hep-thr9710212.w x Ž .6 A. Hindawi, B.A. Ovrut, D. Waldram, Phys. Lett. B 392 1997 85.w x Ž .7 I. Buchbinder, A. Hindawi, B.A. Ovrut, Phys. Lett. B 413 1997 79.w x8 R. Grimm, M. Hasler, C. Herrmann, The Ns2 vector-tensor multiplet, central charge superspace and Chern–Simons couplings,

Preprint CPT-97rP.3499, hep-thr9706108.w x9 N. Dragon, S.M. Kuzenko, U. Theis, The Vector-Tensor Multiplet in Harmonic Superspace, Preprint ITP-uh-20r97, hep-thr9706169.

w x10 N. Dragon, S.M. Kuzenko, Self-interacting vector-tensor multiplet, Preprint ITP-UH-24r97, hep-thr9709088.w x Ž .11 A. Galperin, E. Ivanov, S. Kalitzin, V. Ogievetsky, E. Sokatche, Class. Quant. Grav. 1 1984 469.w x Ž .12 A. Galperin, E. Ivanov, V. Ogievetsky, E. Sokatchev, Class. Quant. Grav. 2 1985 601, 617.w x13 A. Galperin, E. Ivanov, V. Ogievetsky, E. Sokatchev, Harmonic Superspace, the review paper, still in preparation.w x Ž .14 B. de Wit, R. Philippe, A. Van Proeyen, Nucl. Phys. B 219 1983 143; B. de Wit, P.G. Lauwers, A. Van Proeyen, Nucl. Phys. B 255

Ž .1985 569.w x Ž . Ž .15 J. Wess, Acta Phys. Austr. 41 1975 409; B. de Wit, A. Van Holten, Nucl. Phys. B 155 1979 530.w x Ž .16 A. Galperin, E. Ivanov, V. Ogievetsky, E. Sokatchev, Class. Quant. Grav. 4 1987 1255.w x Ž .17 U. Lindstrom, B. Kim, M. Rocek, Phys. Lett. B 342 1995 99.¨ ˇw x Ž .18 E. Ivanov, A. Sutlin, Class. Quant. Grav. 14 1997 843.w x19 N. Dragon, E. Ivanov, S. Kuzenko, E. Sokatchev, U. Theis, Ns2 Rigid Supersymmetry with Gauged Central Charge, in preparation.

11 June 1998


Double-photon decays of B - and B -mesons in the MSSMs d

G.G. Devidze 1, G.R. Jibuti 2

High Energy Physics Institute, Tbilisi State UniÕersity, UniÕersity St.9, Tbilisi 86, Georgia

Received 5 December 1997; revised 2 April 1998Editor: P.V. Landshoff

Abstract

B-mesons double radiative decays B ™gg , B ™gg are investigated in frame of supersymmetric extension of thes d

standard model. The branching ratios are calculated. q 1998 Published by Elsevier Science B.V. All rights reserved.

Prior to their direct detection on colliders, super-symmetry effects may manifest themselves implicitlyin rare processes. Supersymmetry gives elegant solu-tions to the problems of the standard model and

Ž .predicts minimal supersymmetry the masses of su-persymmetry particles in the range 100 GeVyŽ . ŽO 1 TeV with the exception of the lightest super-

.symmetry particle . In the present paper we considerthe B -meson’s rare double-photon radiative decayss,d

B ™gg in the framework of the minimal super-s,dŽ .symmetric standard model MSSM .

The investigation of the B-meson’s rare decays isof great interest in order to test the standard modeland beyond standard model physics. It seems thatexperimental investigation of beauty physics will beone of the major topics at the world facilities. The

Ž . w xL3 collaboration CERN established that 1

Br B ™gg -14.8P10y5Ž .s

Br B ™gg -3.95P10y5 1Ž . Ž .d

The experimental interest stimulates theoretical in-

1 E-mail: [email protected] E-mail: [email protected].

w xvestigations 2–5 . All group of the authors haveŽ .obtained in the frame of the standard model the

same values for the branching ratios in the leadingorder 1rM 2 :W

Br B ™gg s 3.0"1.0 P10y7Ž . Ž .s

Br B ™gg s 1.2"1.1 P10y8 2Ž . Ž . Ž .d

One can write down the amplitude for the decaysB ™gg in the following form, which is corrects,d

after gauge fixing for final photons:

T B ™gg se m k e n kŽ . Ž . Ž .s ,d 1 1 2 2

= a bAg q iBe k k 3Ž .mn mna b 1 2

mŽ . n Ž .where e k and e k are the polarization vec-1 1 2 2

tors of final photons with momenta k and k re-1 2

spectively. The diagrams contributing into A and BŽ .parts of amplitude 2 are presented in Fig. 1. In the

framework of supersymmetric extension of the stan-dard model one has following set of diagrams which

Žcontribute into decays B ™gg , B ™gg the dia-s d

grams are classified by particles which appear in the. . Ž .loops : a charged gauge fermions chargino and up

.scalar quarks, b charged Higgs particles and up. Ž .quarks, c neutral gauge fermions neutralino and

0370-2693r98r$19.00 q 1998 Published by Elsevier Science B.V. All rights reserved.Ž .PII: S0370-2693 98 00447-X

( )G.G. DeÕidze, G.R. JibutirPhysics Letters B 429 1998 48–50 49

Fig. 1. B mesons double radiative decays: B ™gg , B ™gg ,s d) ˜ ˜ ˜ ˜ ) ) 0 0Ž .X su ,u , H ,d ,d ,d ,d ;Y s x ,x ,u,x ,x , g, g .˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜L R L R L R

.down scalar quarks, d gluino and down scalarŽ .quarks. Using Eq. 3 we find that the branching

ratio of the decays B ™gg can be represented ass,d

Br B ™ggŽ .s ,d

1 12 4 2s 4NAN q M NBN . 4Ž .B32p M G 2B to t

Let us mention that our calculations are performed inthe frame of Feynman-t’Hooft gauge and we usedimension regularization technique for divergentFeynman diagrams. Only one particle reducible dia-grams contain divergent parts. The divergent partsmutually cancel in the sum of amplitudes and due to

w xthe GIM mechanism 6 . The higgs particles contri-bution into A and B have the following formsŽcontributions of other supersymmetric particles are

.significant small :

'2 mb 2 2 )As i G f eQ M U U f xŽ . Ž .ÝF B D B bi l i 1 i2 ½16p ml

m m 1l b2 2qcot b f x y tan b f x ,Ž . Ž .2 i 2 i2 5xM HŽ . i

'2 mb 2)Bs i G f eQ U U f xŽ . Ž .ÝF B D bi l i 1 i2 ½8p ml

m m 1l b2 2qcot b f x q tan b f x ,Ž . Ž .2 i 2 i2 5xM HŽ . i

5Ž .

where f is the B-meson decay constant, U beingB i j

elements of Cabbibo-Kobayashi-Maskawa matrix, lss,d, Q sy1r3 is charge of down quarks, Md Bs

Žfm qm , M fm qm ; tanbsÕ rÕ , Õb s B b d 2 1 1,2d

. w xbeing vacuum expectation of the Higgs bosons 7and we have introduced following notation:

m2 uŽ .ix s ,i 2M HŽ .

y3 xq8 x 2 y5x 3 q 6 x 2 y4 x ln xŽ .f x s ,Ž .1 32 1y xŽ .

2 3 4 Ž 2 .31 xy84 x q69 x y16 x q6 x x y6 xq4 ln xŽ .f x s ,2 4Ž .12 1y x

6Ž .Ž . Ž .Using formulae 2 – 5 one can estimate the branch-

ing ratio of the decays B ™gg , B ™gg . Thes dŽ .dependence of the Br B ™gg on the parameterss,d

. .Fig. 2. Branching ratios of the decays a B ™gg , b B ™gg .s d

( )G.G. DeÕidze, G.R. JibutirPhysics Letters B 429 1998 48–5050

Ž .tanb and M H are presented on Fig. 2. We haveŽ .used the following set of parameters: G B s5Pt o t s

10y4 eV, m s 174 GeV, m s 4.8 GeV, m st b sw x0.5 GeV, m s0.3 GeV, f s f s200 MeV 8 . Ford B Bs d

Ž Ž .small value of the Higgs particles masses M H f.100 GeV and large tanb the supersymmetric contri-

bution into branching ratio of the decay B ™gg ares

significant large than standard model estimateŽ Ž . y6 Ž . y7 .Br B ™gg ;10 )Br B ™gg ;10 .susy s SM s

In the wide range of supersymmetric parametersŽ Ž . .1 - tanb - 50, 100 GeV - M H - 350 GeV thesupersymmetric contribution on the same level or

Ž Ž .more as standard model estimate Br B ™ggSM sy7 . y6 Ž;10 . The value of the branching ratio 10 and

y7 .10 too will be measurable in the foreseen future.

The authors express their deep gratitude to N.Amaglobeli, T. Kopaleishvili, A. Tavkhelidze, Z.

Garuchava, T. Sakhelashili and M. Xaindrava forsupport and interesting discussions.

References

w x1 L3 Collaboration, preprint CERN-PPE-95-136; L3 Collabora-tion, contributed paper to the EPS Conference, EPS-0093-2,Brussels, 1995.

w x Ž .2 G.-L. Lin, J. Liu, Y.-P. Yao, Phys. Rev. D 42 1990 2314.w x Ž .3 H. Simma, D. Wyler, Nucl. Phys. B 344 1990 283.w x Ž .4 S. Herrlich, J. Kalinowski, Nucl. Phys. B 381 1993 1176.w x5 G.G. Devidze, G.R. Jibuti, A.G. Liparteliani, Nucl. Phys. B

Ž .468 1996 241.w x6 S.L. Glashow, J. Illiopoulos, L. Maiani, Phys. Rev. D 10

Ž .1970 897.w x Ž .7 H.E. Haber, G.L. Kane, Phys. Rep. 117 1985 75.w x Ž .8 Review of Particle properties, Phys. Rev. D 54 1996 .

11 June 1998


Scaling violation through squark and light gluino production

L. Clavelli 1, I. Terekhov 2

Department of Physics and Astronomy, UniÕersity of Alabama, Tuscaloosa, AL 35487, USA

Received 27 August 1997; revised 12 March 1998Editor: H. Georgi

Abstract

In the light gluino scenario, squarks in the 100 GeV mass region can be copiously produced at the Tevatron without asecond heavy particle. Their subsequent dijet decay into quark plus gluino leads to non-scaling structure in the inclusive jetX distribution. The expected behavior is similar to recent observations. q 1998 Published by Elsevier Science B.V. AllT

rights reserved.

PACS: 11.30.Pb; 14.80.Ly

Recent anomalies in the production of jets in ppannihilation have stimulated significant interest as apossible sign of physics beyond the standard model.A case in point is the inclusive jet transverse energycross section which was reported by CDF as exhibit-ing a dip at low E and a rise at high E relative toT T

w xthe standard model predictions 1 . This behaviorwas cited as a possible indication of quark substruc-ture or of various other non-standard-model effectsw x2 . Among these latter was the suggestion that thejet E distribution could be due, in the light gluinoT

scenario, to extra jet activity from production ofgluino pairs and to the expected slower running of

w xa 3,4 . In addition to these two effects, the possibles

production of a squark in association with a lightgluino could explain one of the several possible

Žbumps visible in the CDF data. For a discussion ofw xother indications of a light gluino see 5 and for a


discussion of direct phenomenological signals in fu-w x .ture searches see 6 . On the other hand it was also

w xfound possible 7 to fit the data, apart from the lowŽ .-50 GeV E values, by readjusting the gluonT

distribution function in the proton in a way stillconsistent with other data or by changing the renor-

w xmalization scheme 8 . Thus, whether or not newphysics is contained in the Fermilab data must awaitfurther analysis. It is significant that the angulardistributions of the jets in various dijet mass bins areconsistent with that expected from the standard modelw x9 . This would seem to rule out many non-standard-model explanations of the data. However, it has beenshown that the light gluino hypothesis would lead todijet angular distributions in practice indistinguish-able from the standard model expectations except indijet mass bins containing an up squark or down

w xsquark 13,14 . This is due to the fact that thestructure of light gluino production amplitudes isquite similar to that of other light partons dominatedby massless particle exchanges in the t and s chan-nels. However a squark, once produced, will, in the


( )L. ClaÕelli, I. TerekhoÕrPhysics Letters B 429 1998 51–5452

light gluino case, decay into dijets with an isotropicangular distribution in its rest frame.

The light gluino hypothesis, therefore, seems vi-able only if the valence squarks are below about 150GeV or above 650 GeV since currently analyzeddata does not constrain the lower or higher mass

w xregions 15,4,14 . The possible bump at about 550GeV seen by CDF and discussed in the light gluino

w xcase in 15 might, therefore, be a statistical fluctua-tion. This interpretation is supported by the failure of

w xthe later analysis 16,17 to confirm a bump in thisregion. On the other hand the angular distributionshave not been published in the vicinity of a possible

w xparticle near 100 GeV suggested by the E data. 3 .T

Furthermore D0 has not published data spanning therelevant low E region. Assuming it is relatively lessT

attractive to have squarks above 650 GeV, the studyof jet angular distributions in the 100 GeV to 150GeV dijet mass region could therefore be crucial tothe light gluino hypothesis.

Ž .The parton distribution functions pdf’s in thestandard model must presently be treated as theoreti-cally arbitrary functions; the primary constraints arefrom data on deep inelastic scattering and fromdirect photon production. This freedom eliminatesthe necessity of, although not the possibility of, newphysics in explaining the Fermilab results for the jettransverse energy cross sections at high E . It doesT

not at present, however, allow for an understandingof the behavior below 50 GeV E . In addition, it isT

possible to study suitably defined scaling distribu-tions whose ratio at two different Fermilab energiesis relatively insensitive to modifications of the partondistributions. Such a quantity is the scaled inclusivejet transverse energy cross section, which is pre-dicted to have the form

ds L m2s sa m F X , , 1Ž . Ž .s Tž /' 'dX s sT

with

2 ETX s . 2Ž .T 's

Here m is conventionally taken to be E r2, L is theT

QCD dimensional transmutation parameter, and mrepresents any particle mass appearing in the theory.Since the lowest order cross sections in QCD are

proportional to a 2, this has been factored out, al-s

though it also could be written as a function of thefirst two arguments of the scaling function F. If all

'masses in the theory are negligible compared to s ,F depends on s only through its second argumentcoming from scaling deviations in the parton distri-bution functions and from logarithmic scaling viola-tions due to higher order terms in a both of whichs

effects are expected to be small. The ratio of theŽ .scaling distribution at two different high energies is

therefore expected to be approximately constant inŽ .X for X R0.05 independent of small modifica-T T

w xtions of the pdf’s. The CDF collaboration 18 haspresented preliminary results for the ratio

'sdsrdX s s630Ž .Tr X s . 3Ž . Ž .T 'sdsrdX s s1800Ž .T

Ž .In the standard model r X departs from unity dueTŽ .to the running of the coupling constant in 1 and

due to logarithmic scaling violations in the pdf’s.This leads to a predicted value for r near 1.8 approx-imately independent of X . The data show a system-T

atic tendency to be below this prediction. In the lightgluino case a runs more slowly than in the standards

model leading to a value for r near 1.6 againapproximately constant if the squarks are too high inmass to be copiously produced. These values dependsomewhat on the assumed scale msE r2 whichT

seems to be preferred by the CDF study of theseparate E distributions.T

The current data for the ratio of the transverseenergy distributions at two energies are preliminary.Although systematic errors are still under study andcould raise the overall normalization of the scalingcurve, it is difficult to imagine systematic errorsseriously affecting the point-to-point errors in such away as to produce the observed structure. The re-

Ž .ported structure in r X is, therefore, a tantalizingT

suggestion of the existence of strongly interactingparticles whose masses are not negligible comparedto 630 GeV and which then lead to a strong effect

Ž .from the third argument in the F of 1 . Weaklyinteracting particles, such as the W and Z bosons, donot have a sufficiently high production cross sectioncompared to QCD jet production to affect the rparameter significantly. Similarly the top quark or

( )L. ClaÕelli, I. TerekhoÕrPhysics Letters B 429 1998 51–54 53

the prevalent supersymmetry hypothesis with pairproduced heavy squarks and gluinos have productioncross sections too low to be helpful in the currentcontext. In addition, such states are not expected tohave prominent dijet decay modes.

On the other hand, in the light gluino scenario,which can be obtained in the context of the con-strained supergravity related SUSY breaking modelby setting the universal gaugino mass m to zero, a1r2

single heavy squark can be produced in associationwith a light gluino leading to greatly enhanced squark

w xproduction cross sections as discussed in 15,3 . Inthe light gluino case, but not in the standard SUSYpicture, a squark will have a predominantly dijetdecay into quark plus gluino. Such a squark would

Žproduce a dip in the r ratio before smearing due to.experimental resolution and hadronization at ap-

Ž .proximately X sm GeV r1800 followed by a peakTŽ .at m GeV r630. There is in fact some indication in

the data for such a low X dip followed by a peak atT

roughly three times higher X . In this letter weT

explore this basic signature which should be pre-served independent of details such as the choice ofrenormalization scale, assumptions about squark de-generacy splitting, hadronization smearing effects,etc. These latter effects will be studied in a fulllength paper to follow. Our present purpose is not topresent a definitive fit but to illustrate the size of thedip-bump structure that might appear for simplechoices of parton distribution functions and variouspreliminary assumptions about renormalization scaleand squark width enhancement. In the current workwe adopt the light gluino hypothesis and consider as

w xin 15,3,14 the lowest order standard model pro-cesses together with the effect of sparticle productionprocesses

˜ ˜GG™GG 4Ž .

˜ ˜QQ™GG 5Ž .

˜ ˜QG™QGG. 6Ž .

Ž . Ž .Processes 4 and 5 , while increasing the jet activ-ity by some 6% do not have a significant effect onthe r ratio. The possibility of a squark intermediate

Ž .state in process 6 leads however to structure in r asdiscussed above. The effect is shown in Fig. 1 in the

.Fig. 1. CDF data on the X scaling distribution compared to aTŽ . .the standard model prediction dashed line , b light gluino with

Ž . Ždecoupled effectively infinitely massive squarks dot-dashed. .line , and c light gluino plus 133–135 GeV valence squarks

Ž .solid lines . See text.

higher solid curve for a squark mass of 135 GeV.The structure shown in the r parameter theory as afunction of X is due to an intermediate squark inT

Ž .the process of 6 . This process has a collinearlogarithmic mass singularity as the gluino mass ap-proaches zero. Although this feature is not problem-atic at present energies, the summation of thesesingularities would be expected to build an intrinsicgluino distribution in the proton due to gluon split-ting. Such a gluino distribution would decrease thegluon distribution in the proton. Several fits to pdf’sincluding light gluino effects have been publishedw x10,11 . As an example illustrating the effect ofintrinsic gluinos, in Fig. 1 we also plot for compari-son the scaling curve resulting from the Ruckl-Vogt¨pdf set where the 2 to 3 process of 6 is replaced by

˜ ˜ Ž .the 2 to 2 process QG™QG. Other non-resonantgluino initiated processes are also included here. The

Žprocess of 6 would of course still occur at a reduced. Ž 3.level together with other OO a effects which wills

partially cancel in the scaling ratio. A complete studyof these higher order effects awaits further calcula-

( )L. ClaÕelli, I. TerekhoÕrPhysics Letters B 429 1998 51–5454

tion. In Fig. 1the lower solid line corresponds to theintrinsic gluino pdf set with a squark mass of 133GeV and the renormalization scale chosen as theparton CM energy. The higher solid line corresponds

w xto the standard 12 CTEQ3 pdf set with scale set asE r2.T

In lowest order QCD, the squark width is pre-dicted to be

G s2 M a r3. 7Ž .˜ ˜Q Q s

This width would be increased somewhat by elec-troweak decays of the squark and by QCD correc-tions to the hadronic decays. In addition the four

Žvalence squark states would be split in mass bysome 10 to 20 GeV in the simple Supergravity

.related model To roughly exemplify these effects, inthe upper solid curve of Fig. 1 we have increased thesquark width by a factor of 1.3. The width of theobserved structure is not grossly inconsistent withthe expected squark width. A more detailed study,including the effects on the separate E distributionsT

will be part of the later paper which will also providespace for detailed consideration of resolution andhadronization smearing effects. Since the peaks, ifthey exist, sit on a steeply falling background, theeffect of resolution might be to move the observedpeak upward. Given the preliminary nature of thedata and uncertainties in the actual amount of smear-ing present in the data we would not consider anysquark mass between 100 GeV and 140 GeV asdefinitively counter-indicated at present. The possi-ble bump in the data near X ;0.28 could be fit by aT

squark of mass 180 GeV but such a mass is disfa-vored by dijet angular distribution measurementsw x13,14 .

Our main conclusion at this point is that the lightgluino hypothesis together with valence squarks inthe 100 GeV to 140 GeV region is in qualitativeagreement with current experimental indications asexemplified in Fig. 1. An attempt to present a precisefit must await further understanding of systematicuncertainties in both the experiment and the theory.It seems unlikely that the current magnitude of theobserved structure could be fit in any model involv-ing pair production of two heavy particles withcoupling strength a or less or in any model withs

additional gauge bosons in the electroweak sector.

There has been a recent attempt to understand theanomalies in the scaling data from a Regge point of

w xview 19 . Current fits from this point of view dosucceed in reducing the theoretical scaling curve bysome 10% from the standard QCD prediction in theregion X -0.1. This is still far from observationsT

and does not, at present, predict the observed struc-ture.

If one ignores the structure apparent in the currentexperimental analysis, the light gluino fit withsquarks far above the 100–200 GeV mass range,shown in the dot-dashed line of Fig. 1 seems to bealso preferred by the data over the standard modelfit.

This work was supported in part by the Depart-ment of Energy under grant DE-FG02-96ER40967.

References

w x Ž .1 F. Abe et al., CDF Collaboration, Phys. Rev. Lett. 77 1996438.

w x2 R.S. Chivukula, A.G. Cohen, E.H. Simmons, Phys. Lett. BŽ .380 1996 92.

w x Ž .3 L. Clavelli, I. Terekhov, Phys. Rev. Lett. 77 1996 1941.w x Ž .4 Z. Bern, A.K. Grant, A.G. Morgan, Phys. Lett. B 387 1996

804.w x5 L. Clavelli, in: Proceedings of the Workshop on the Physics

of the Top Quark, IITAP, Iowa State Univ., Ames Ia, 1995,World Scientific Press.

w x Ž .6 G. Farrar, Phys. Rev. D 51 1995 3904; Phys. Rev. Lett. 76Ž .1996 4115.

w x7 H.L. Lai, J. Huston, S. Kuhlmann, F. Olness, J. Owens, D.Ž .Soper, W.K. Tung, H. Weerts, Phys. Rev. D 55 1997 1280.

w x Ž .8 M. Klasen, G. Kramer, Phys. Lett. B 386 1996 384.w x Ž .9 F. Abe et al., CDF Collaboration, Phys. Rev. Lett. 77 1996

5336.w x Ž .10 R.G. Roberts, W.J. Stirling, Phys. Lett. B 313 1993 453.w x Ž .11 R. Ruckl, A. Vogt, Z. Phys. C 64 1994 431.¨w x Ž .12 H.L. Lai et al., CTEQ Collaboration, Phys. Rev. D 51 1993

4763.w x Ž .13 J. Hewett, T. Rizzo, M. Doncheski, Phys. Rev. D 56 1997

5703, hep-phr9612377.w x Ž .14 I. Terekhov, Phys. Lett. B 412 1997 86, hep-phr9702301.w x Ž .15 I. Terekhov, L. Clavelli, Phys. Lett. B 385 1996 139.w x16 D0 Collaboration, S. Abachi et al., Phys. Rev. Lett. 75

Ž .1995 618.w x17 CDF Collaboration, F. Abe et al., Fermilab-PUB-97r023-E.w x18 A. Bhatti, Fermilab-Conf-96r352-E, presented at the DPF

conference, Minneapolis, August 1996.w x19 V.T. Kim, G.B. Pivovarov, J.P. Vary, hep-phr9709303; V.T.

Kim, G.B. Pivovarov, hep-phr9709304.

11 June 1998


The interaction of dyons in the mean field approximation

B.V. MartemyanovInstitute of Theoretical & Experimental Physics, 117259, B. Cheremushkinskaya 25, Moscow, Russia

Received 4 March 1998; revised 2 April 1998Editor: P.V. Landshoff

Abstract

The interaction of dyons in the mean field approximation is considered. The result of interaction is the mass term fordyonic field in the effective Lagrangian. Due to the mass term the profile function of dyon falls off exponentially at largedistances. q 1998 Published by Elsevier Science B.V. All rights reserved.

1. Introduction

There is a hope that dyons, euclidean solutions ofgauge theory, can be the fluctuations of vacuum

w xfields that are responsible for confinement 1–4 . IfŽwe consider the gas of dyons the simple superposi-

.tion of dyonic solutions , neglect the interaction ofdyons and omit their possible interference in thecontribution to the Wilson loop we will get the

w xphenomenon of ‘‘superconfinement’’ 2–4 . The‘‘superconfinement’’ means that the Wilson loop

Ž .average W follows not the area low confinement

W;exp yCr 2Ž .where C is some constant, r is the radius of the loopŽ .the circle, for example , but falls off like

Ž X 3. Ž X .exp yC r C is the constant . It was put forwardw xthe idea 2–4 that due to the interaction of dyons

something like ‘‘Debye screening’’ should appear inthe dyonic gas transforming the ‘‘superconfinement’’to confinement. In this paper we will consider theinteraction of dyons in the mean field approximationand argue that effectively at large distances thedyonic field is exponentially damped. Such a damp-ing is known for a long time in the case of instanton

w xgas model of vacuum 5 , where the damping wasobtained from the Feynman variational principle. Wewill consider the connection of the mean field ap-proximation to the Feynman variational principle.Now we are not in a position to say whether thisdamping is the desired ‘‘Debye screening’’ or not. Inthis sense the problem needs further investigation.

2. Mean field approximation

Let us consider the gas of dyons: N dyons in thevolume V. Dyons are described by their degrees of

Žfreedom: positions, color orientations we considerŽ . .SU 2 gauge theory , ‘‘velocities’’. We consider one

isolated dyon in the field of other dyons and averagethe total action over their degrees of freedom. In thisway we can obtain the effective action for the dyon.Taking the minimum of the effective action we canfind the modified dyonic solution. And at last we canuse this modified solution to calculate some parame-ters of the effective action.

The described procedure is usually called themean field approximation. We will show further the


( )B.V. MartemyanoÕrPhysics Letters B 429 1998 55–5756

connection of this procedure with some form of theFeynman variational principle. And now let us de-scribe the details.

Ž . Ž . Ž .The total gauge field A x sa x qB x ,m m m

Ž .where a x is the gauge field of the isolated dyonm

Ž .and B x is the field created by other dyons. Them

total action is equal to

214 a a int ,aSs d x f qF q f , 1Ž .Ž .H mn mn mn4

int,a abcŽ b c b c .where f sge a B qB a , f and F aremn m n m n mn mn

field tensors for potentials a and B respectively.m m

The averaging of S over the field B is rather simplem

Ž .and as a result we get the effective action S foreff

the isolated dyon

m214 2 2S s d x f q a qS , 2Ž .Heff mn m B4ž /2

g 2 12 2 2² : ² :where m s B ,S s F . The constant Sm B mn B2 4

is inessential for the calculation of the modifiedone-dyon solution.

The effective action is no more guage invariantand this is the reflection of the fact that the interac-

Žtion of dyons depend in the model simple superposi-.tion of individual dyons on the gauge used for

w xindividual dyonic field 2 . The problem here is thatthe superposition of dyon’s potentials is not only thesolution of Yang-Mills equations but also gives theresulting field tensor depending on the gauge we are

Žusing for the dyon solution the sum of potentials inone gauge is not connected to that in another gauge

.by any gauge transformation . The gauge should betaken in such a form as to get a minimal interaction

Ž .of dyons at large distances, at least . The interactionof dyons at large distances depends crucially on theasymptotic behaviour of dyonic potential and thelatter depends on the class of the gauge, the differentclasses being connected by singular gauge transfor-

w xmations. In Ref. 2 it was pointed out that theinteraction of two dyons vanishing at infinity can beobtained in the so called ’t Hooft gauge. We will usethis gauge here also. We use the ’t Hooft gaugeŽ .E a s0 for the individual dyon solution in them m

superposition ansatz also for the following reasons.First, the topological charges of the dyons are

Ž .summed in the considered singular gauge. Second,Žthe interaction of dyons is minimal in this gauge at

.least at infinitesimal gauge transformation . AndŽ .third, because of the screened due to the mass term

dyon solution satisfies the condition E a s0 auto-m m

matically and transforms to the unscreened solutionin the m2 ™0 limit, the unscreened solution shouldsatisfy the condition E a s0 also.m m

In the ’t Hooft gauge the unscreened dyon solu-tion is equal to

gaa sye n f r ,t qd ag r ,t ,Ž . Ž .i i ab b i

gaa synaf r ,t ,Ž .0

1 sinhg r coshg rf r ,t s qg y ,Ž . ž /r coshg rycosg t sinhg r

sing tg r ,t sg . 3Ž . Ž .

coshg rycosg t

Here g is the inverse size of the dyon. The massŽ .term in the effective action 2 modifies the solution

Ž . Ž .3 . If m<g the limit of pointlike dyons qualita-tively we can say that the internal part of the dyon is

Žmainly unchanged the function g and the last twoŽ . .terms of function f in 3 are unchanged and the

1Ž .first term of function f Coulomb-like tail trans-rŽ .exp ymr 1forms to at r4 . So, we have obtainedr m

the exponential damping of the dyonic solution atlarge distances. Obviously, the problem needs furthernumerical consideration at this point. The parameter

Ž .m of the effective action 2 can be now calculatedusing the modified one-dyon solution.

Ž . Ž .According to Eqs. 2 , 3

12 2 2² :m s Ny1 g aŽ . m2

Ny1Ž .3 2 2 2s 4p r dr f qg . 4Ž .Ž .H2V

In the limit of pointlike dyon we have approximately

3 12m fn4p , m<g , 5Ž . Ž .

2 2mNwhere ns is 3-d density of dyons.V

For the unscreened solution the integral in formu-Ž . Ž .lar 4 is divergent. Eq. 5 has the selfconsistent

solution

3'ms 3p n . 6Ž .

( )B.V. MartemyanoÕrPhysics Letters B 429 1998 55–57 57

3. The connection to the Feynman variationalprinciple.

The modification of the dyon solution can be alsofound by Feynman variational principle. Let us as-sume that we are calculating the partition function Zfor N dyons:

Zs exp yS , 7Ž . Ž .Hwhere the integral goes over the dyons degrees offreedom. According to Feynman variational principle

² :ZGZ exp y SyS , 8Ž .Ž .1 1

where S and Z are the simplified action and the1 1

partition function. If we choose the action S as a1Ž .sum of N independent no interaction terms: S s1

Ns then1

² :Z s exp yNs sAexp yNs , S sNs ,Ž . Ž .H1 1 1 1 1

9Ž .

where A is the volume of the space of dyons degreesof freedom. So, we get

² :ZGAexp y S . 10Ž . Ž .² :For N dyons S is equal to

1 N Ny1Ž .14 2 2 2 4 2² : ² :S sN d x f q g a d xa .H Hmn m m4 4 2

11Ž .² : ŽThe variation of S over a we are looking for them

² :.maximum of Z and minimum of S gives thesame equation as the variation of S over a . Theeff m

mass parameter m2 is equal to12 2 2² :m s Ny1 g a ,Ž . m2

Ž .in agreement with the result 4 obtained in the meanfield approximation. We have applied the mean field

Žapproximation to the case of instantons with fixed

.size r for example . The mass term for the instantonfield calculated in such a way coincides with that of

w xRef. 5 where the same problem was considered inthe context of Feynman variational principle.

4. Conclusion

We have considered the problem of the dyonsinteraction in the mean field approximation. In thisapproximation the interaction is effectively takeninto account in the form of mass term for the dyonic

Ž .field. The mass parameter m is selfconsistentlyŽ Ž ..determined by the density of dyons see Eq. 5 . If

Ž .other fluctuations not of dyonic type are present inthe vacuum they also contribute to m2. So, m is

3'larger than 3p n . The effect of the considered massterm is the exponential damping of the dyonic fieldat large distances.

Acknowledgements

This work is partly supported by the RFFI grants96-02-00088G and 97-02-17491. The author wouldlike to thank Yu.A. Simonov for stimulating discus-sions.

References

w x Ž .1 Yu.A. Simonov, Sov. J. Yad. Fiz. 43 1985 557.w x2 B.V. Martemyanov, S.V. Molodtsov, Yu.A. Simonov, A.I.

Ž .Veselov, Pis’ma Zh. Eksp. Teor. Fiz. 62 1995 695.w x3 B.V. Martemyanov, S.V. Molodtsov, Sov. J. Yad. Fiz. 59

Ž .1996 766.w x4 B.V. Martemyanov, S.V. Molodtsov, Yu.A. Simonov, A.I.

Veselov, Sov. J. Yad. Fiz., 1997, to be published.w x Ž .5 D.I. Diakonov, V.Yu. Petrov, Nucl. Phys. B 245 1984 259.

11 June 1998


The massive Schwinger model- a Hamiltonian lattice study in a fast moving frame

Helmut Kroger 1, Norbert Scheu 2¨Departement de Physique, UniÕersite LaÕal, Quebec, Que. G1K 7P4, Canada´ ´ ´

Received 25 February 1998; revised 6 April 1998Editor: H. Georgi

Abstract

We present a non-perturbative study of the massive Schwinger model. We use a Hamiltonian approach, based on amomentum lattice corresponding to a fast moving reference frame, and equal time quantization. We present numericalresults for the mass spectrum of the vector and scalar particle. We find good agreement with chiral perturbation theory in the

Ž w Ž . xstrong coupling regime and also with other non-perturbative studies Hamer et al. Phys. Rev. D 56 1997 55 , Mo andw Ž . x.Perry J. Comput. Phys. 108 1993 159 in the non-relativistic regime. The most important new result is the study of the

u-action, and computation of vector and scalar masses as a function of the u-angle. We find excellent agreement with chiralperturbation theory. Finally, we give results for the distribution functions. We compare our results with Bergknoff’svariational study from the infinite momentum frame in the chiral region. q 1998 Published by Elsevier Science B.V. Allrights reserved.

1. Introduction

In order to study QCD non-perturbatively, latticegauge theory has been most successful. However,there are some observables, where computationalprogress in lattice gauge theory has been slow. Ex-amples are: Higher excited states of hadrons and

Žmesons, finite density thermodynamics quark-gloun.plasma , dynamical scattering calculations of cross

sections or phase-shifts and hadron structure func-tions, in particular in the region of small Q2 and


Ž y2 y5.small x 10 to 10 . In our opinion a non-per-B

turbative Hamiltonian approach may be a viablealternative. In deep inelastic lepton-hadron scatter-ing, the success of the parton model suggests thephysical idea to use a fast moving frame also for acomputational study of those processes. The partonmodel can be justified using the operator productexpansion. In equal-time quantization, the Breit-frame is the most convenient choice of frame in

w xorder to interpret structure functions. In Ref. 1 theauthors have proposed such a scheme based onequal-time quantization, using a lattice Hamiltonianon a momentum lattice corresponding to a fast mov-

Ž .ing frame Breit-frame . It has been applied to thescalar f 4 theory in 3q1 dimensions. Distribution


( )H. Kroger, N. ScheurPhysics Letters B 429 1998 58–63¨ 59

functions and the mass spectrum in the close neigh-bourhood of the critical line of the second orderphase transition have been computed.

Here we study QED , the so-called massive1q1

Schwinger model, in this framework. This model isphysically interesting due to the following proper-

Ž . Ž .ties: i It is super-renormalizable. ii It has a limitwhere the model is analytically soluble: the massless

w x Ž . Ž . Ž .limit 2 . iii It has an axial anomaly. iv One canstudy the topological properties associated with u-ac-tions and u-vacua. Most of the analytic work on this

w xmodel has been presented in Refs. 3–9 This modelis a widely studied benchmark problem for newmethods. It has been studied by a variational method

w xin the infinite-momentum frame 10 , in discretizedw xlight-cone quantization 12 , on the light-cone using

w xorthogonal functions 13 , on a finite lattice in equalw xtime quantization 11 and by strong coupling series

w xexpansion 14 .Ž .The purpose of this work is to show: a The use

of a fast moving frame, such that Õ-c, in conjunc-tion with equal-time quantization works well also inQED . It is not necessary to go to the infinite1q1

momentum frame or to quantize on the light-cone.We obtain quite precise results in the ultra-relativis-tic region mrg™0. It has been claimed that lowenergy states of the massive Schwinger model arenot well aproximated in equal-time field theory. Weshow here that the problem is not equal-time quanti-zation, but the use of the rest-frame, which we avoid

Ž .to do here. b We consider as most important newresults of this work the non-perturbative results ofthe dependence of the vector and scalar mass on theu-angle.

2. Method

Starting from the Lagrangian, we use the axialgauge A3 s0 to obtain the Hamiltonian

L 3 3Hs dx cg iE cqccŽ .H 3yL

g 2 1L 3 † †q dx c c c c . 1Ž .Ž . Ž .H 22 yEyL 3

One introduces a space-time lattice given by spacing2 La with Ns lattice nodes. Via discrete Fouriera

transformation one goes over to a momentum lattice,p pwith cut-off Ls and resolution Dps . Moti-a L

vated by the parton picture we make the assumptionthat left-moving particles are not dynamically impor-tant, if physical particles are considered from areference frame characterized by a velocity Õ

Ps which is close to the velocity of light. We thusE

consider a momentum lattice where 0Fp0, p3 FP.In order to minimize the number of virtual particlepairs created from the vacuum, we choose a smalllattice size. The reason for this is that the number ofvacuum pairs is roughly proportional to a vacuumdensity times the lattice size. On the other hand, afast-moving physical particle is Lorentz contracted;

Žthus it fits in a small lattice volume when compared.to the rest-frame .

For the purpose of computing the mass spectrum,we need to determine the vacuum energy. Becausethe vacuum has the quantum number Ps0, the

Ž .vacuum energy and only this is computed in therest frame. Because the model is super-renormaliz-able, one can perform the continuum limit a™0.The only renormalization necessary is the subtractionof the vacuum energy. On a space-time lattice, one

Ž .has to satisfy a physical condition scaling windowa-j a-L, where j is the correlation length indimensionless units, related to the physical mass of

1the ground state by Ms . In a strongly relativisticj a

system, M<P, thus when a™0 the scaling win-dow is replaced by 1rP-1rM-L. For more de-

w xtails compare with Ref. 1 .

3. Numerical results

3.1. Mass spectrum

We diagonalize the Hamiltonian in a sector withmomentum Ps0 to obtain the vacuum energy E .Õac

Then we diagonalize the Hamiltonian in a sectorP/0 corresponding to a relativistic velocity. Thisyields an energy spectrum EX . The physical energiesn

are obtained from E sEX yE . The mass spec-n n Õac2 2(trum is then given by M s E yP . The lown n

lying states of the massive Schwinger model are aw xvector state and next a scalar state 13,14 . The

behaviour in the ultra-relativistic region mrg™0

( )H. Kroger, N. ScheurPhysics Letters B 429 1998 58–63¨60

has been computed in chiral perturbation theory upw x w xto second order by Adam 16 and Vary et al. 17 .

The vector particle mass behaves as

2M mÕs1qAcos uŽ .Ž0. Ž0.ž /M M

2mq ByCcos 2uŽ . Ž0.ž /M

3mqO ,Ž0.ž /ž /M

As2 eg f3.5621, Bf5.4807, Cf2.0933,2Ž .

Ž0. 'where M sgr p is the boson mass of the mass-less Schwinger model. The mass of the vector bosonin the chiral region is shown in Fig. 1. The vector

Ž .particle is almost entirely a fermion-antifermion qqbound state. In our lattice results presented in Fig. 1only this qq sector has been taken into account. Wefind good agreement with chiral perturbation theoryw x16,17 and with finite lattice results by Hamer et al.w x14 . Fitting our data to a second order polynomial inmrg yields the coefficients As0.56, Bs1.74 andCs0.22. Results for the binding energy M y2mÕ

of the vector boson in the non-relativistic region are

Fig. 1. The dimensionless mass of the vector boson versus mr gw xin the chiral region. Comparison with Hamer et al. 14 and chiral

w xperturbation theory 16,17 .

Fig. 2. The dimensionless binding energy of the vector bosonŽ .versus log mr g in the non-relativistic region. Comparison with2

w x w xHamer et al. 14 and Mo and Perry 13 .

shown in Fig. 2. We compare our results with finitew xlattice results 14 and with results by Mo and Perry

w x13 using light-cone quantization with an expansionin orthogonal functions. The corresponding resultsfor the binding energy M y2m of the scalar bosons

are shown in Fig. 3. Our results correspond to the qqsector. For small masses, i.e. mrgf1r2 to 1r8, theqqqq sector becomes important. Thus our resultshave a larger error in that region. Results on thescalar mass in the ultra-relativistic region mrg--1

Fig. 3. Same as Fig. 2, but for scalar boson.


Fig. 4. Dependence of the dimensionless mass of vector and scalarparticle on the u-angle before proper subtraction of u-vacuum

Ž .energy raw data . Comparison with chiral perturbation theoryw x16,17 .

taking into account the qqqq sector are shown inFigs. 4 and 5

3.2. Dependence on u-angle

The massive Schwinger model has u-vacua andone can study its u-action. The u-physics of themassive Schwinger model on a circle has been dis-

w xcussed by Manton 9 . The wavefunction is invariant

Fig. 5. Dependence of the dimensionless mass of vector and scalarparticle on the u-angle after subtraction of u-vacuum energy.

w xComparison with chiral perturbation theory 16,17 .

under local gauge transformations. The u-angle is aparameter, which characterizes the behavior of thewavefunction under global gauge transformations

w x i nu w xC A ™e C A , where ns0,"1,"2, PPP . Onecan also transform Hilbert space such that this com-mon phase factor disappears and the wavefunctionbecomes completely invariant under local as well asglobal gauge transformations. This corresponds to

w x w x w xintroducing a u-action S A,u s S A y u q A ,w xwhere q A is the topological charge, or an equiva-

lent u-Hamiltonian. The mass spectrum of low-lyingstates as a function of the u-angle is shown in Figs. 4

Ž .and 5. Fig. 4 shows results raw data , where theŽ .usual vacuum energy E us0 has been sub-Õac

tracted in all u-sectors, which is unphysical, e.g., forusp . It shows, however, the behavior of the vectormass for usp when increasing the lattice fromNs6 to 24. One observes a cusp in the curves,becoming more pronounced when N increases. Thisbehavior is due to level crossings with higher lying

Ž .states not shown in the figure . Fig. 5 shows theŽ .same results when the vacuum energy E u hasÕac

been subtracted in the corresponding u-sector. Oneobserves agreement with first order chiral perturba-

Žtion theory in the ultra-relativistic regime mrg-.0.04 for Ns6 and mrg-0.01 for Ns24 . The

slope of those curves is related to the expectationvalue of the chiral condensate. The Feynman-Hellmann theorem relates the fermion condensate-cc) to the derivative of the vacuum energy with

w xrespect to the fermion mass. Adam 15 has com-puted the condensate in lowest order of chiral pertur-bation theory,

-cc) msacos u q byccos 2uŽ . Ž .Ž0. Ž0.M M

2mqO .Ž0.ž /ž /M

eg

as f0.2835, bf0.7825, cf0.7163. 3Ž .2p

Ž . Ž .Eqs. 2 , 3 imply

Ž0.M E-cc)s M , 4Ž .Õ2p E m ms 0

( )H. Kroger, N. ScheurPhysics Letters B 429 1998 58–63¨62

which in analogy to the Feynman-Hellmann theoremrelates the condensate to the vector mass. When weextract the slope E M rE m from Fig. 5, we obtainÕ

Ž .-cc)rgs0.16cos u , compared to the exact re-sult of the massless Schwinger model given by

g-cc) es cos u f0.1599cos u . 5Ž . Ž . Ž .'g 2p p

The u-dependence and global gauge transformationshave been treated in standard lattice gauge theory

Ž .based on compact link variables U x by Hamer etm

w xal. 18 . Our results are in agreement with theirs.

3.3. Distribution functions

In the Hamiltonian approach it is easy to computethe wave function of a low-lying state. From thewave function one can obtain information on itsparton structure, i.e., the number of partons and theirmomentum distribution. The distribution function ofthe vector boson is given by

1† †f x s-C P N b b qd d NC P ) ,Ž . Ž . Ž .B Õ k k k k Õ2

6Ž .

where x skrP is the fraction of momentum of theB

vector boson carried by the parton, i.e., fermion. InFig. 6 we display our lattice results for mrgs0 to0.28. In the massless limit the distribution functionhas the shape of a box. Our results are compared

Fig. 6. Distribution function of the vector boson for differentmr g in the chiral region.

Fig. 7. Distribution function of the vector boson for mr g s' w x0.1r p . Comparison with results by Mo and Perry 13 and

w xBergknoff 10 .

with variational calculations using the infinite-w xmomentum frame by Bergknoff 10 and also with

w xthose by Mo and Perry 13 using the light-cone. Themost sensitive region is the ultra-relativistic region.We find agreement in shape with Mo and Perry’sresults and very good agreement with Bergknoff’sresults as shown in Fig. 7. The agreement with Moand Perry’s results becomes even better than theagreement with Bergknoff if their distribution func-tion is normalized to one. In Fig. 8 we also displaythe convergence behavior of the vector distributionfunction for mrgs32, i.e., in the non-relativistic

Fig. 8. Distribution function of the vector boson for mr g s32 inthe non-relativistic region.


region. It can be seen that a reasonably large latticeŽ . ŽNf200 is needed to resolve this function which

.has a sharp peak at x s1r2 when mrg™` .B

4. Summary

In this work we have applied a Hamiltonian lat-tice approach in a fast moving reference frame tostudy the massive Schwinger model. We find that themethod works well for the computation of thelow-lying mass spectrum, also in the presence of theu-action, and distribution functions. Here we haveinvestigated only us0,p . It is interesting to furtherstudy the dependence on other u-angles andhigher-lying states. More u-angles can be studied,e.g. by adding one negative momentum state to thebasis. Those investigations are feasible, but wouldrequire a higher computational effort.

Acknowledgements

This work has been supported by NSERC Canada.N.S. acknowledges support by an AUFE fellowshipfrom DAAD Germany.

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w x Ž .10 H. Bergknoff, Nucl. Phys. B 122 1977 215.w x Ž .11 D.P. Crewther, C.J. Hamer, Nucl. Phys. B 170 1980 353.w x Ž .12 T. Eller, H.C. Pauli, S.J. Brodsky, Phys. Rev. D 35 1987

1493.w x Ž .13 Y. Mo, R.J. Perry, J. Comput. Phys. 108 1993 159.w x Ž .14 C.J. Hamer, Z. Weihong, J. Oitmaa, Phys. Rev. D 56 1997

55.w x Ž .15 C. Adam, Phys. Lett. B 363 1995 79.w x Ž .16 C. Adam, Phys. Lett. B 382 1996 383.w x Ž .17 J.P. Vary, T.J. Fields, H.J. Pirner, Phys. Rev. D 53 1996

7231.w x18 C.J. Hamer, J. Kogut, D.P. Crewther, M.M. Mazzolini, Nucl.

Ž .Phys. B 208 1982 413.

11 June 1998


Pion-nucleon coupling at finite temperature

C.A. Dominguez a, C. van Gend a, M. Loewe b

a Institute of Theoretical Physics and Astrophysics, UniÕersity of Cape Town, Rondebosch 7700, South Africab Facultad de Fisica, Pontificia UniÕersidad Catolica de Chile, Casilla 306, Santiago 22, Chile

Received 24 March 1998Editor: R. Gatto

Abstract

Ž . Ž .The pion nucleon vertex function at finite temperature is studied in the framework of: a the thermal linear sigmaŽ . Ž .model to leading one-loop order, and b a thermal QCD-Finite Energy Sum Rule. Results from both methods indicate that

the strength of the pion-nucleon coupling decreases with increasing T , vanishing at a critical temperature. The associatedmean-square radius is a monotonically increasing function of T , diverging at the critical temperature. This is interpreted asŽ .analytical evidence for the quark-gluon deconfinement phase transition. q 1998 Published by Elsevier Science B.V. Allrights reserved.

The temperature behaviour of hadronic Green’s functions, and their associated parameters such as masses,widths, couplings, etc., has received considerable attention lately, given its impact on the search for the

w xquark-gluon plasma 1 . Two successful theoretical frameworks for these studies are the thermal sigma modelw x w x2–4 , and QCD sum rules 5,6 . The former technique provides information on the T-dependence of pion andnucleon masses and widths associated, respectively, with the real and imaginary parts of their two-point Green’s

w xfunctions. While these masses show no appreciable variation with temperature, 2,3 their widths exhibit aw xdramatic increase with increasing T 2,4 . This result is in line with the expectation that hadronic widths,

w xinterpreted as absorption coefficients in the thermal bath, should diverge at some critical temperature 7 . Thisprovides a signal or phenomenological order parameter for the quark-gluon deconfinement phase transition.

Ž .Another such signal is the thermal behaviour of hadronic couplings and form factors three-point functions ,which should vanish at a critical temperature, where the associated mean square radii should diverge. This has

w x w xbeen explicitly confirmed for the electromagnetic form factor of the pion 8 , and for the rho-pi-pi coupling 9 .Ž .In this note we study the p NN vertex function at finite temperature using the thermal linear sigma model, as

Ž .well as QCD sum rules. The purpose is to obtain additional confirming analytical evidence for thedeconfinement phase transition, as well as information on this vertex function, which should be of use in hadrongas models at finite temperature.

Ž .We begin with the linear sigma model, and consider the p NN vertexX2 2G q sV q u p g t u p 1Ž . Ž . Ž .Ž . Ž . f 5 a i

2 Ž X .2where the nucleons are on-mass shell, and the pion has virtual mass q s p yp . The renormalization of theŽ . w x Žsigma model at Ts0 is discussed e.g. in 10 , and the renormalization of the p NN vertex before the


( )C.A. Dominguez et al.rPhysics Letters B 429 1998 64–71 65

. w xinvention of dimensional regularization and the MS scheme may be found in 11 . At finite temperature wew xshall use the Dolan-Jackiw real time propagators 12 , together with the fact that thermal corrections do not

induce any new kind of ultraviolet corrections. Hence, the thermal theory may be renormalized as at Ts0. WeŽ .have done this using dimensional regularization and the MS scheme. To leading one-loop order, the relevant

Ž . 2diagrams are shown in Fig. 1 a-e . At the kinematical point q s0, and in the chiral limit, the expression forŽ 2 .the irreducible vertex V q is given by

V 0 sg 1qb 0 g 2 2Ž . Ž . Ž .Ž .where

2 2 2 21 5 M M 5 M Ms s s sb 0 s y q 3y lnŽ . 2 2 2 2 2ž /3 616p M M M MN N N N

2 2(4M yM1 M N ss2 2 2 2(q 5M y8 M 4M yM arctan 3Ž .Ž .s N N s43 MM sN

Ž .with M and M being the nucleon and sigma meson masses, respectively. Eq. 2 may be regarded as anN s

Ž .Fig. 1. Leading one-loop diagrams contributing to g in the linear sigma model.p NN

( )C.A. Dominguez et al.rPhysics Letters B 429 1998 64–7166

Ž .expression for the effective coupling constant g in the chiral limit. It is equal to g if b 0 vanishes. Thisp NNŽ .happens for M ,1300 MeV; in fact, b 0 is small and negative if M is bigger than this value. One shoulds s

Ž . w xrecall that in the linear sigma model g s1, so that the Goldberger-Treiman relation GTR 13 becomes:Aw xM sgf . Using the chiral symmetry limit values 14 : M ,800 MeV, and f ,80 MeV, one finds: g, 10,N p N p

not far from the experimental value g , 13. In any case, here we are only interested in the temperaturep NNŽ 2 . Ž 2 .behaviour of the ratio V q ,T rV q ,0 ; particularly in the possibility that this ratio vanishes at a critical

temperature, and that the mean square radius diverges there. This will turn out to be largely independent of theŽ . Ž .particular value assumed by g or g and b 0 or equivalently M , although the specific value of thep NN s

critical temperature does depend on the latter.Ž .Turning to the temperature corrections to the graphs shown in Fig. 1, we need only consider Fig. 1 b and

Ž . Ž Ž .. Ž .Fig. 1 d identical to Fig. 1 e , as Fig. 1 c is Boltzmann suppressed on account of M ,M ))m . TheN s p

Ž .thermal correction to the graph Fig. 1 b is given by

d4k k 2 n k d k 2 ym2Ž . Ž .B 0 pX2 3G q ,T syg u p g t u p 4Ž . Ž . Ž .Ž . Hf 5 a i 4 2 2X 2 22p p yk yM pyk yMŽ . Ž . Ž .N N

Ž . Ž z r T .y1where n is the Bose thermal factor: n z s e y1 . Hence, this contribution vanishes in the chiralB BŽ .limit. That of Fig. 1 d is found to be

22 2 4 2M ym d k 2m yku 2pd k n kŽ . Ž .Ž .Ž .s p N B 0X2 3G q ,T sg u p g t u p 5Ž . Ž . Ž .Ž . Hf 5 a i2 4 2 22 22 f M 2p kqq yM pyk yMŽ . Ž . Ž .p N s N

We choose for convenience a Lorentz frame in which the incoming nucleon is at rest with respect to the heatŽ .bath ps0 . We have checked that the final results are largely independent of the choice of frame. The thermal

effective p NN coupling in the chiral limit, and at q2 s0, may then be written as

V 0,T g 2T 2Ž .s1y 6Ž .2 2V 0,0 12 M 1qg b 0Ž . Ž .Ž .N

where the GTR has been used. Notice that an extrapolation in temperature of this result implies a criticaltemperature

2(T s 12 1qb 0 g M rg 7Ž . Ž .Ž .d N

Ž .which depends on the value of the sigma-meson mass through b 0 ; numerically, T ,150y300 MeV ifd

M ,1300y1600 MeV. One should not assign too much importance to the specific numerical values of thiss

critical temperature andror the sigma-meson mass, to wit. First, the relations among parameters in the sigmamodel are valid at the 25y30% level; e.g. g s1 instead of the experimental value g s 1.26, and g ,A A p NN

Ž . Ž . 210 from the GTR , as opposed to the experimental value g , 13, etc. Second, Eq. 6 and the Tp NN

dependence is a consequence of the one-loop approximation. Higher loop corrections will induce higher orderŽ .in T terms which will alter the numerical value of the critical temperature. These will be suppressed, though,by inverse powers of the nucleonrsigma-meson masses. A similar situation arises in chiral perturbation theory

Ž . 2 Ž . w xand the T-dependence of the pion decay constant f T . To order T , f T vanishes at T ,240 MeV 15 ,p p c

while higher order corrections bring down this value considerably. What we find important here, is that thep NN coupling at leading order in T decreases with increasing temperature.

Ž 2 .Next, we consider the mean-square radius associated with V q ,T , and defined as

2 2 2-r ) V 0,0 E V q ,T E V q ,0Ž . Ž . Ž .p NN Ts 8Ž .2 2 2V 0,T-r ) E q E qŽ .p NN 0 2q s0


We have calculated this ratio numerically, the result being shown in Fig. 2. An extrapolation in temperatureindicates quite clearly the divergence of this mean square radius. This may be interpreted as a signal fordeconfinement, to the extent that the size of the nucleon, as probed by a pion, increases with increasingtemperature, becoming infinite at TsT .d

Ž .We study next the same vertex function, but in the framework of QCD Finite Energy Sum Rules FESR .This will provide important independent support to the above result, especially since the QCD sum ruletechnique, unlike the sigma model, does not entail any expansion in powers of the temperature. To this end, webegin the analysis at zero temperature and introduce the three-point function

XX 2 4 4 iŽ p P xyqP y.² :P p , p ,q s i d x d y 0 T h x J y h 0 0 e 9Ž . Ž . Ž . Ž . Ž .Ž .HH 5

Ž . Ž .where the nucleon and pion interpolating currents, h x and J x , respectively, are chosen as5

a b m ch x se u x Cg u x g g d x 10Ž . Ž . Ž . Ž . Ž .abc m 5

J x s i u x g u x yd x g d x 11Ž . Ž . Ž . Ž . Ž . Ž .5 5 5

where C is the charge conjugation operator. The couplings of these currents to the nucleon and the pion aredefined as

² < < :0 h 0 N p ,s sl u p ,s 12Ž . Ž . Ž . Ž .N

2² < < :N p ,s J 0 N p ,s su p ,s g g q u p ,s 13Ž . Ž . Ž . Ž . Ž . Ž .Ž .2 2 5 1 1 2 2 5 P 1 1

where

f m2 gp p p NN2g q s 14Ž .Ž .P 2 2m q ymq p

and where m is the average of the up and down quark masses. In our normalization, the pion decay constant isqŽ .f ,93 MeV. The hadronic representation of the imaginary part of the vertex function Eq. 9 is obtained byp

inserting a complete set of hadronic states. After summing over spins, and making the standard nucleon-poleŽ .approximation thus including in the continuum all the radial excitations of the nucleon one obtainsX 2 < 2 2 2 X 2ImP s,s ,q sg l M ig qu p d syM d s yMŽ .Ž . Ž . Ž .HAD P N N 5 N N

X X X 2 <qu sys u s ys ImP s,s ,q 15Ž . Ž . Ž .Ž . QCD0 0

Since we are interested in the pion-nucleon coupling in the vicinity of q2 s0, we can safely neglect any q2

dependence in g . This dependence would arise from the contribution of the radial excitations of the pion,p NNXŽ . 2 2

Xp 1300 etc., which in the chiral limit is a correction of order q rM . As usual, the hadronic continuum withp

Ž .Fig. 2. Thermal behaviour of the p NN mean square radius, Eq. 8 , in the linear sigma model.


thresholds s and sX is modelled by the QCD spectral function. The leading order diagrams needed to compute0 0

the latter are shown in Fig. 3. In the chiral limit, the relevant structure to be sought is proportional to quq2. ItŽturns out that the diagram Fig. 3a does not have this behaviour, while that of Fig. 3b plus all other related

.diagrams gives

² :qq ig qu5X X2 <ImP s,s ,q s sqs 16Ž . Ž .Ž . QCD 22p q

where -uu),-dd)s-qq) has been used. By means of Cauchy theorem, and assuming globalŽ .quark-hadron duality, one obtains the lowest dimensional FESR

f s sX s qsXŽ .p 0 0 0 0g s 17Ž .p NN 3 28p l MN N

w xwhere use has been made of the Gell-Mann, Oakes and Renner relation 162 2 ² :f m sy2m qq 18Ž .p p q

Since the dispersion in p2 ss and pX 2 ssX refers to the two nucleonic legs, it is reasonable to assume s ,sX .0 0Ž . w xAn analysis of the two-point function involving the nucleonic current h x 17 in the framework of QCD FESR

yields3 ² :s qq02 2 2l s , l M sy s 19Ž .N N N 04 2192p 8p

Ž .which determines the nucleon mass in terms of s . Conversely, using M and -qq) as input, Eq. 19 fixes0 NŽ .l and s . In this fashion, Eq. 17 becomes: g s48p f rM ,15, not far from the experimental valueN 0 p NN p N

g ,13. This level of agreement is more than enough for our purpose here, which is to determine thep NN

Fig. 3. Leading order QCD diagrams entering the determination of the spectral function relevant to g .p NN


Ž . Ž .temperature behaviour of the p NN coupling, i.e. the ratio g T g 0 . To achieve this, we havep NN p NN

calculated the thermal corrections to the QCD spectral function with the result

1 ig qu5X X 2 X2² :ImP p , p ,q s qq p f p ,T qp f p ,T 20Ž . Ž . Ž . Ž .� 424p q

where

< < < < < < < <p y p x p q p x1 0 0f p ,T s dx 1yn yn 21Ž . Ž .H F Fž / ž /2 2y1

Ž X .and a similar expression for f p ,T . The FESR in this case becomes

1 f T XŽ . Ž . Ž .s T s Tp X X X0 0g T s ds ds sf p ,T qs f p ,T 22Ž . Ž . Ž . Ž .H Hp NN 3 28p M T l TŽ . Ž . 0 0N N

where use has been made of the thermal Gell-Mann, Oakes and Renner relation, which has been recently shownw xto be valid over a wide range of temperatures 18 . The temperature behaviour of f , valid up to the criticalp

w x Ž .temperature, has been obtained in 19 . The function s T has been determined from a FESR for the two-point0w x Ž . Ž . Ž Ž . Ž ..2function involving the axial-vector current 20,21 ; it scales as: s T rs 0 , f T rf 0 . The thermal0 0 p p

Ž . w xnucleonic coupling l T has been determined from a QCD-FESR in the nucleon channel, with the result 17N

3s TŽ .02 2l T sl 0 1qG T 23Ž . Ž . Ž . Ž .N N až /s 0Ž .0

where

576 vr2 vr2Ž .s T' 0G T s dv dx dy x vy2 xŽ . Ž .H H Ha 30 0 vr2yxs TŽ .Ž .0

= yn x yn y qn x n y qn vyxyy n x qn y y1 . 24� 4Ž . Ž . Ž . Ž . Ž . Ž . Ž . Ž .F F F F F F F

Ž . Ž yz r T .y1 Ž .and n z s 1qe is the Fermi factor. With all this information one can then solve Eq. 22 , afterF

choosing a particular Lorentz frame. Our choice is the rest frame of the incoming nucleon, i.e. ps0 and p0's s , however, the final results are quite insensitive to the choice of frame. The result for the ratioŽ . Ž .g T rg 0 as a function of TrT is shown in Fig. 4. One can clearly appreciate the vanishing of thisp NN p NN d

Ž .Fig. 4. Thermal behaviour of the p NN coupling determined from the QCD-FESR Eq. 22 .


Ž .Fig. 5. Thermal behaviour of the p NN mean square radius, Eq. 26 , according to the QCD-FESR.

Ž .coupling at the critical temperature. This is the temperature at which f T vanishes, i.e. the critical temperaturep

Ž .for the chiral-symmetry restoration phase transition, which is basically the same temperature at which s T0w xvanishes, i.e. the critical temperature for the quark-gluon deconfinement phase transition 20,21 .

Finally, the mean-square radius associated to the pion-nucleon vertex

E2 2

2² : <r s6 ln g q ,T 25Ž .Ž .T q s0p NN p NN2E q

can be easily calculated, with the resulty1

XŽ . Ž . 1s T s T X X0 02 '² :r s ds ds dx s 1y2n s r2 qs 1y2n zŽ .Ž .Ž .Ž .T H H Hp NN F F½ 50 0 y1

=

XX6 1 sqsŽ . Ž . 1s T s T X0 0 2 z r Tds ds dx n z e 1qx 26Ž . Ž .H H H XF' < <2T syss0 0 y1

whereX < X <sqs q sys x

zs 27Ž .'4 s

Ž .and the rest frame of the incoming nucleon has been used. A numerical evaluation of Eq. 26 gives the resultshown in Fig. 5. Notice that since we have made the pion-pole approximation, g at Ts0 is independent ofp NN

q2. The mean-square radius is non-vanishing only at finite temperature, where a q2 dependence appears throughthe Fermi factors.

In summary, the vanishing of the pion-nucleon coupling, and the divergence of the associated mean-squareradius, at a critical temperature has been shown to follow from the thermal linear sigma model at leadingŽ . Ž .one-loop order, as well as from a thermal QCD-FESR. This may be viewed as analytical evidence supportingthe existence of the quark-gluon deconfinement phase transition. As the critical temperature is approached, thestrength of the coupling of pions to nucleons is quenched, and at the same time, the size of the nucleon asprobed by the pion gets bigger. The qualitative agreement between the two methods lends further support to theextension of the QCD sum rule program to finite temperature. It should be noticed that potential non-diagonal

w x Ž .vacuum condensates 22 do not enter our FESR because of their higher dimensionality. We have emphasizedmany times in the past that QCD-FESR are far better than e.g. QCD-Laplace transform sum rules, to the extent

Ž .that the lowest dimensional thermal FESR do not involve unknown non-diagonal vacuuum condensates.

Ž .This work has been supported in part by the Foundation for Research Development South Africa , and byŽ .Fondecyt Chile under grant No. 1950797.


References

w x Ž . Ž .1 For reviews see e.g. F. Karsch, Nucl. Phys. A 590 1995 367; H. Meyer-Ortmanns, Rev. Mod. Phys. 68 1996 473.w x Ž .2 H. Leutwyler, A.V. Smilga, Nucl. Phys. B 342 1990 302.w x Ž . Ž .3 A. Larsen, Z. Phys. C 33 1986 291; C. Contreras, M. Loewe, Int. J. Mod. Phys. A 5 1990 2297.w x Ž .4 C.A. Dominguez, M. Loewe, J.C. Rojas, Phys. Lett. B 320 1994 377.w x Ž .5 A.I. Bochkarev, M.E. Shaposnikov, Nucl. Phys. B 268 1986 220.w x Ž . Ž .6 C.A. Dominguez, M. Loewe, Phys. Rev. D 52 1995 3143; Z. Phys. C 51 1991 69.w x Ž . Ž . Ž .7 R.D. Pisarski Phys. Lett. B 110 1982 155; C.A. Dominguez, Nucl. Phys. B Proc. Suppl. 15 1990 225; C.A. Dominguez, M.

Ž .Loewe, Z. Phys. C 49 1991 423.w x Ž .8 C.A. Dominguez, M. Loewe, J.S. Rozowsky, Phys. Lett. B 335 1994 506.w x Ž .9 C.A. Dominguez, M. Loewe, M.S. Fetea, Phys. Lett. B 406 1997 149.

w x10 B.W. Lee, Chiral Dynamics Gordon and Breach, New York, 1972.w x Ž .11 J.A. Mignaco, E. Remiddi, Nuovo Cimento 1 A 1971 376.w x Ž .12 L. Dolan, R. Jackiw, Phys. Rev. D 9 1974 3320.w x Ž .13 C.A. Dominguez, Rivista Nuovo Cimento 8 1985 1.w x Ž .14 J. Gasser, H. Leutwyler, Phys. Rep. 87 1982 77.w x Ž .15 J. Gasser, H. Leutwyler, Phys. Lett. B 184 1987 83.w x Ž .16 M. Gell-Mann, R. Oakes, B. Renner, Phys. Rev. 175 1968 2195.w x Ž .17 C.A. Dominguez, M. Loewe, Z. Phys. C 58 1993 273.w x Ž .18 C.A. Dominguez, M. Loewe, M.S. Fetea, Phys. Lett. B 387 1996 151.w x Ž .19 A. Barducci, R. Casalbuoni, S. de Curtis, R. Gatto, G. Pettini, Phys. Rev. D 46 1992 2203.w x Ž .20 C.A. Dominguez, M. Loewe, Phys. Lett. B 233 1989 201.w x Ž .21 A. Barducci, R. Casalbuoni, S. de Curtis, R. Gatto, G. Pettini, Phys. Lett. B 244 1990 311.w x Ž .22 T. Hatsuda, Y. Koike, S.H. Lee, Nucl. Phys. B 394 1993 221.

11 June 1998


Couplings of pions with heavy baryonsfrom light-cone QCD sum rules in the leading order of HQET

Shi-lin Zhu, Yuan-Ben DaiInstitute of Theoretical Physics, Academia Sinica, P.O. Box 2735, Beijing 100080, China

Received 27 January 1998; revised 31 March 1998Editor: H. Georgi

Abstract

Ž ) . Ž ) .The couplings of pions with heavy baryons g S , S and g S , L are studied with light-cone QCD sum rules in the2 3

leading order of heavy quark effective theory. Both sum rules are stable. Our results are g s1.56"0.3"0.3,2

g s0.94"0.06"0.2. q 1998 Elsevier Science B.V. All rights reserved.3

PACS: 12.39.Hg; 14.20.Lq; 13.75.Gx; 12.38.LgKeywords: HQET; Pion heavy baryon coupling; Light cone QCD sum rule

1. Introduction

Important progress has been achieved in the interpretation of heavy hadrons composed of a heavy quark withŽ . w xthe development of the heavy quark effective theory HQET 1 . HQET provides a systematic expansion of the

heavy hadron spectra and transition amplitude in terms of 1rm , where m is the heavy quark mass. Of courseQ Q

one has to employ some specific nonperturbative methods to arrive at the detailed predictions. Among thew xvarious nonpeturbative methods, QCD sum rules is useful to extract the low-lying hadron properties 2 .

w xThe couplings of the heavy mesons with pions has been analysed with QCD sum rules 3–13 . The couplingsw xof heavy baryons with soft pions are estimated from QCD sum rules in an external axial field 13 . In this

approach the mass difference D between the baryons in the initial and final states is approximately taken to bezero.

Ž .In this work we employ the light-cone QCD sum rules LCQSR in HQET to calculate the couplings g to2,3

the leading order of 1rm . The LCQSR is quite different from the conventional QCD sum rules, which is basedQŽ .on the short-distance operator product expansion OPE . The LCQSR is based on the OPE on the light cone,

which is the expansion over the twists of the operators. The main contribution comes from the lowest twistoperator. Matrix elements of nonlocal operators sandwiched between a hadronic state and the vacuum definesthe hadron wave functions. The LCQSR approach has the advantage that the double Borel transformation isused so that the the continuum contribution is treated in a way better than the external field approach. Moreover,

1Ž .the final sum rule depends only on the value of the wave function at a specific point like w u s , which isp 0 2

w xmuch better known than the whole wave function 10 .


( )S.-l. Zhu, Y.-B. DairPhysics Letters B 429 1998 72–78 73

2. Sum rules for the coupling constants

We first introduce the interpolating currents for the heavy baryons:aT b ch x se u x Cg d x h x , 1Ž . Ž . Ž . Ž . Ž .L abc 5 Õ

aT b m cqh x se u x Cg d x g g h x , 2Ž . Ž . Ž . Ž . Ž .S abc m t 5 Õ

1m aT b mn m n c

qq)h x se u x Cg u x yg q g g h x , 3Ž . Ž . Ž . Ž . Ž .S abc n t t t Õž /3

Ž . Ž . Ž .where a, b, c is the color index, u x , d x , h x is the up, down and heavy quark fields, T denotes theÕ

transpose, C is the charge conjugate matrix, g mn sg mn yÕ mÕn, g m sg yÕÕ m, and Õ m is the velocity of theˆt t m

heavy hadron.The overlap amplititudes of the interpolating currents with the heavy baryons is defined as:

² < < :0 h L s f u , 4Ž .L L L

² < < :0 h S s f u , 5Ž .S S S

f )Sm ) m² < < :) )0 h S s u , 6Ž .S S'3

m w x) )where u is the Rarita-Schwinger spinor in HQET. In the leading order of HQET, f s f 14 .S S S

w xWe adopt the same notations for g as in 15 . The coupling constants g and g are defined through the2,3 2 3

following amplitudes:g3

) m)M S ™L p s i u q u , 7Ž .Ž .c c L m Sc cfp

g2) s n r m

)M S ™S p s ie Õ q u g g u , 8Ž .Ž .c c nrsm t S t 5 Sc c'6 fpŽ .where f s132 MeV, q is the pion momentum. The process 8 is kinematically forbidden. It is veryp m

important to get a reliable estimate of the coupling g since it is not directly accessible experimentally.2

In order to derive the sum rules for the coupling constants we consider the correlators

1qÕ 1ˆ X4 yi kP x m mn m n m n r² < < :) )d x e p q T h 0 h x 0 s yg q g g ie q Õ g g G v ,v ,Ž . Ž . Ž . Ž .Ž .H S S t t t m r nn t t 5 S , Sž /2 3

9Ž .1qÕ 1ˆ X4 yi kP x m n mn m n² < < :

) )d x e p q T h 0 h x 0 s q yg q g g G v ,v , 10Ž . Ž . Ž . Ž . Ž .Ž .H S L t t t t S , Lž /2 3X t Ž . X X 2where k skyq, q sq y qPÕ Õ , vs2ÕPk, v s2ÕPk and q s0.m m m

Ž X. Ž .)Let us first consider the function G v,v in 10 . As a function of two variables, it has the followingS , L

pole terms from double dispersion relation

4 g f ) f c cX3 S L

q q , 11Ž .XX')3 f ) 2 L yv 2 L yv2 L yv 2 L yvŽ . Ž .p S LS L

Ž . Ž .) ) )where f etc are constants defined in 4 - 6 , L sm ym .S S S Q

Ž X.)Neglecting the four particle component of the pion wave function, the expression for G v,v with theS , L

tensor structure reads`

yi k x T² < < :4 dt dxe d yxyÕt Tr p q u 0 d x 0 g CiS yx Cg , 12Ž . Ž . Ž . Ž . Ž . Ž .� 4H H 5 n0

( )S.-l. Zhu, Y.-B. DairPhysics Letters B 429 1998 72–7874

Ž .where iS yx is the full light quark propagator with both perturbative term and contribution from vacuumfields.

² < < :iS x s 0 T q x ,q 0 0Ž . Ž . Ž .2² :x qq x 1 x xˆ ˆ1 m mn² :s i y y qg sPGq y ig du sPG ux y4 iu G ux g uŽ . Ž .Hs s n2 4 2 2 2½ 512 1922p x 16p x x0

q PPP . 13Ž .Ž X.

)Similarly for G v,v we have:S , S

`yi k x T² < < :4 dt dxe d yxyÕt Tr p q u 0 d x 0 g CiS yx Cg , 14Ž . Ž . Ž . Ž . Ž . Ž .½ 5H H r n

0

To the present approximation, we need the following OPE near the light cone for two- and three-particle pionw xwave functions 10 :

< <-p q d x g g u 0 0)Ž . Ž . Ž .m 5

x 2q1 1miuq x 2 4 iuq xsyif q du e w u qx g u qOO x q f x y du e g u , 15Ž . Ž . Ž . Ž . Ž .Ž .H Hp m p 1 p m 2ž /qx0 0

f m21p p iuq x< <-p q d x ig u 0 0)s du e w u , 16Ž . Ž . Ž . Ž . Ž .H5 Pm qm 0u d

f m21p p iuq x< <-p q d x s g u 0 0)s i q x yq x du e w u . 17Ž . Ž . Ž . Ž . Ž . Ž .Hmn 5 m n n m s6 m qmŽ . 0u d

< <-p q d x s g g G ux u 0 0)Ž . Ž . Ž . Ž .ab 5 s mn

i q xŽa qÕa .1 3s if q q g yq q g y q q g yq q g DDa w a e , 18Ž . Ž . Ž . Ž .H3p m a nb n a mb m b na n b ma i 3p i

< <-p q d x g g g G Õx u 0 0)Ž . Ž . Ž . Ž .m 5 s a b

x q x qa m b m i q xŽa qÕa .1 3s f q g y yq g y DDa w a eŽ .Hp b am a bm i H iž / ž /qPx qPx

qm i q xŽa qÕa .1 3q f q x yq x DDa w a e 19Ž . Ž . Ž .Hp a b b a i 5 iqPx

and

˜< <-p q d x g g G Õx u 0 0)Ž . Ž . Ž . Ž .m s a b

x q x qa m b m i q xŽa qÕa .1 3s if q g y yq g y DDa w a eŽ .˜Hp b am a bm i H iž / ž /qPx qPx

qm i q xŽa qÕa .1 3q if q x yq x DDa w a e . 20Ž . Ž . Ž .˜Hp a b b a i 5 iqPx1 dr˜ ˜ ŽThe operator G is the dual of G : G s e G ; DDa is defined as DDa sda da da d 1ya yaab a b a b a bdr i i 1 2 3 1 22

. m Ž . Ž 1 m Ž ..ya . Due to the choice of the gauge x A x s0, the path-ordered gauge factor Pexp ig H dux A ux has3 m s 0 m

been omitted.


Ž . Ž . Ž .The wave function w u is associated with the leading twist 2 operator, g u and g u correspond to twistp 1 2Ž . Ž .4 operators, and w u and w u to twist 3 ones. The function w is of twist three, while all the waveP s 3p

Ž . Ž . Ž . Žfunctions appearing in Eqs. 19 , 20 are of twist four. The wave functions w x ,m m is the renormalizationi. Ž .point describe the distribution in longitudinal momenta inside the pion, the parameters x Ý x s1i i i

representing the fractions of the longitudinal momentum carried by the quark, the antiquark and gluon.Ž . Ž . 1 Ž .The wave function normalizations immediately follow from the definitions 15 – 20 : H du w u s0 p

1 Ž . 1 Ž . 2 Ž . Ž . Ž . Ž .H du w u s 1, H du g u s d r12, HDDa w a sHDDa w a s 0, HDDa w a s yHDDa w a s˜ ˜0 s 0 1 i H i i 5 i i H i i 5 i2 a 2˜Ž . < <d r3, with the parameter d defined by the matrix element: -p q dg G g u 0)s id f q .s am p m

Ž . Ž .Expressing 12 and 14 with the pion wave functions, we arrive at:

v t vX t`i 1X iŽ1yu. iu

)G v ,v s f dt due eŽ . 2 2H HS , L p3 0 0

=

21 t it2² : ² :m w u y qq q qg sPGq w u q g u q t g uŽ . Ž . Ž . Ž .p s s p 2 12 2½ 5ž /16 qPÕp t

v t vX t2 dt 1 w Ž .x Ž .i 1y a yua i a yua1 3 1 3q f duu DDa e e qPÕ w a , 21Ž . Ž . Ž .2 2H H H3p i 3p i2 tp 0

Xv t v t`2 dt it1X iŽ1yu. iu 2

)G v ,v s f due e w u q g u q t g uŽ . Ž . Ž . Ž .2 2H HS , S p p 2 12 3 qPÕp t0 0

v t vX tf dt 1p w Ž .x Ž .i 1y a yua i a yua1 3 1 3q du DDa e e2 2H H H i2 tp 0

=1

w a yw a q yu w a , 22Ž . Ž . Ž . Ž .˜ ˜5 i H i 5 iž /2

3 2 2 2² : Ž . ² : ² :where m s1.76 GeV, f s132 MeV, qq sy 225 MeV , qg sPGq sm qq , m s0.8 GeV . Forp p s 0 0

large euclidean values of v and vX this integral is dominated by the region of small t, therefore it can be

approximated by the first a few terms.After Wick rotations and making double Borel transformation with the variables v and v

X the single-poleŽ .terms in 11 are eliminated. Subtracting the continuum contribution which is modeled by the dispersion integral

in the region v,vX Gv , we arrive at:c

)L qLS L2f v v 4 4p c c3T)g f f sy e m w u T f ya w u Tf y g u q G uŽ . Ž . Ž . Ž .3 S L p s 0 2 p 0 0 1 0 2 02 ž / ž /½' T T T T8 3 p

2 'am 4 4 f f 3 v0 p 3p cG 3q w u y g u q G u q I u T f , 23Ž . Ž . Ž . Ž . Ž .p 0 1 0 2 0 3 0 32 2 2 ž /54T TT T 4p

x kyx nŽ .where f x s1ye Ý is the factor used to subtract the continuum, v is the continuum threshold. un ks0 c 0k!T T T 2 G1 1 2 Ž . ² : Ž . Ž .s , T' , T , T are the Borel parameters asy 2p qq . The functions G u and I u are1 2 2 0 3 0T q T T q T1 2 1 2

defined as:

u0

G u s g u du , 24Ž . Ž . Ž .H2 0 20

1qu0w a ,1qu y2a ,a yu w a ,1ya ya ,aŽ . Ž .1ya3p 1 0 1 1 0 3p 1 1 3 312GI u s da y da . 25Ž . Ž .H H3 0 1 3 2a yu au 01 0 30


Ž . Ž .We have used integration by parts to absorb the factors qPÕ and 1r qPÕ . In this way we arrive at theŽ .simple form after double Borel transformation. In obtaining 23 we have used the Borel transformation

1T a vˆ Ž .formula: BB e sd ay .v T

Similarly we have:2 LS'3 2 v v vc c cT2 4 2 2

)g f f sy f e w u T f q4G u T f y4 g u T fŽ . Ž . Ž .2 S S p p 0 3 2 0 1 1 0 12 ž / ž / ž /½ T T T8p

2 'a am 3 2 v0 c2 G 2y m w u q m w u q f I u T f , 26Ž . Ž . Ž . Ž .p s 0 p s 0 p 4 0 12 2 ž /59 T36T 4p

GŽ .where the function I u is defined as:4 0

1qu0 da 1 a yu1ya 3 1 012GI u s da w a yw a q y w a . 27Ž . Ž . Ž . Ž . Ž .˜ ˜H H4 0 1 5 i H i 5 iž /a 2 au 0 3 30

Ž . Ž . w xFrom 23 and 26 we know that both g and g are negative using the notations in 15 . In the following2 3

we will always discuss the absolute values of g .2,3

3. Determination of the parameters

Ž . Ž .In order to obtain the coupling constants from 23 – 26 we need the mass parameters L’s and the couplingw xconstants f ’s of the corresponding interpolating currents as input. The results are 14

3L s0.8 GeV f s 0.018"0.002 GeV ,Ž .L L

3L s1.0 GeV f s 0.04"0.004 GeV . 28Ž . Ž .S S

Ž . Ž . Ž .For the sum rule 23 and 26 the continuum threshold is v s 2.5"0.1 GeV.cw xWe use the wave functions adopted in 10 to compute the coupling constants. Moreover, we choose to work

1 w xat the symmetric point T sT s2T , i.e., u s as traditionally done in literature 10 . Such a choice is very1 2 0 2

reasonable for the sum rules for g since S ) and S are degenerate in the leading order of HQET. The mass2 c c

difference between S ) and L is about 0.2 GeV. Due to the large values of T , T ;3.2 GeV4D usedc c 1 2

below, the choice of T sT is acceptable. We use the scale ms1.3 GeV, at which the values of the various1 21Ž . Ž . Ž . Ž . Ž . Ž .functions appearing in 23 – 26 at u s are: w u s1.22, w u s1.142, w u s1.463, g u s0.0340 p 0 P 0 s 0 1 02

2 Ž . 2 GŽ . GŽ . 2GeV , G u sy0.02 GeV , I u sy2.75 and I u sy0.24 GeV . We have used the asymptotic2 0 3 0 4 0Ž . Ž . Ž . Ž . Ž . GŽ . GŽ .forms for the wave functions w a , w a , w a , w a and w a to calculate I u and I u ,˜ ˜3p i H i 5 i H i 5 i 3 0 4 0

since these wave functions are not known very well. f s0.0035 GeV 2.3p

4. Numerical results and discussion

We now turn to the numerical evaluation of the sum rules for the coupling constants. Since the spectralŽ . Ž . Ž . 2 3density of the sum rule 23 – 26 r s is either proptional to s or s , the continuum has to be subtracted

carefully. We use the value of the continuum theshold v determined from the corresponding mass sum rule atcw xthe leading order of a and 1rm 14 .s Q

The lower limit of T is determined by the requirement that the terms of higher twists in the operatorexpansion is reasonably smaller than the leading twist, say F1r3 of the latter. This leads to T)1.3 GeV for

Ž . Ž .the sum rules 23 – 26 . In fact the twist-four terms contribute only a few percent to the sum rules. The upper


Fig. 1. Dependence of g on the Borel parameter T for different values of the continuum threshold v . From top to bottom the curves2 c

correspond to v s2.6,2.5,2.4 GeV.c

limit of T is constrained by the requirement that the continuum contribution is less than 50%. This correspondsto T-2.2 GeV.

The variation of g with the Borel parameter T and v is presented in Fig. 1 and Fig. 2. The curves2,3 cŽ .correspond to v s2.4,2.5,2.6 GeV from bottom to top respectively. Stability develops for the sum rules 23c

Ž .and 26 in the region 1.3 GeV -T-2.2 GeV, we get:

g f ) f s 2.5"0.4 =10y3 GeV6 , 29Ž . Ž .2 S S

g f ) f s 6.8"0.4 =10y4 GeV6 , 30Ž . Ž .3 S L

where the errors refers to the variations with T and v in this region. And the central value corresponds toc

Ts1.6 GeV and v s2.5 GeV.cŽ .Combining 28 we arrive at

g s1.56"0.3"0.3 , 31Ž .2

g s0.94"0.06"0.2 , 32Ž .3

where the second error takes into account the uncertainty in f’s.) 'w x Ž . w xThe recent CLEO measurement 16 of the S ™L p decays gives g s 3 0.57"0.1 17 , where thec c 3

)'factor 3 arises from the different notaions. The decay S ™S p is kinematically forbidden so the directc c

Fig. 2. Dependence of g on the Borel parameter T for different values of the continuum threshold v . From top to bottom the curves3 c

correspond to v s2.6,2.5,2.4 GeV.c


Ž .measurement of g is impossible. In the large-N limit of QCD N is the number of colors g is related to2 c c 2,3 'w x'the nucleon axial charge g s1.25, g s3r2 g , g s 3r2 g 18,19 . The quark model result is g s 3 =A 2 A 3 A 3Ž . w x w x0.61, g s1.5= 0.93"0.16 17 . The short-distance QCD sum rules with the external field method 13 yields2

Ž . Ž .'g s1.5= 0.4;0.7 , g s 3r2 = 0.4;0.7 . In this work we employ light-cone QCD sum rules to2 3

calculate the strong coupling constants g . Both sum rules for g and g is very stable with reasonable2,3 2 3

variations of the Borel parameter T and the continuum threshold v as can be seen from Fig. 1 and Fig. 2. It isc

interesting to note that the numerical values of g and g are consistent with both the experimental data and the2 3w x w xquark model result 17 . It is also interesting to notice the deviation from the large N limit prediction 18,19 .c

Moreover the result from the short-distance QCD sum rules is compatible with the present work with thelight-cone QCD sum rule approach if we use the same values of v and f ’s though the errors are quite large.c

Acknowledgements

S.-L. Zhu was supported by the National Postdoctoral Science Foundation of China and Y.D. was supportedby the National Natural Science Foundation of China.

References

w x Ž . Ž .1 B. Grinstein, Nucl. Phys. B 339 1990 253; E. Eichten, B. Hill, Phys. Lett. B 234 1990 511; A.F. Falk, H. Georgi, B. Grinstein,Ž . Ž .M.B. Wise, Nucl. Phys. B 343 1990 1; F. Hussain, J.G. Korner, K. Schilcher, G. Thompson, Y.L. Wu, Phys. Lett. B 249 1990 295;¨Ž . Ž .H. Georgi, Phys. Lett. B 240 1990 447; J.G. Korner, G. Thompson, Phys. Lett. B 264 1991 185.¨

w x Ž .2 M.A. Shifman, A.I. Vainshtein, V.I. Zakharov, Nucl. Phys. B 174 1979 385, 448, 519.w x3 Y.B. Dai et al., report BIhep-thr-97-005, hep-phr9705223.w x Ž .4 N. Di Bartolomeo, F. Feruglio, R. Gatto, G. Nardulli, Phys. Lett. B 347 1995 405.w x Ž .5 P. Colangelo, G. Nardulli, A. Deandrea, N. Di Bartolomeo, R. Gatto, F. Feruglio, Phys. Lett. B 339 1994 151.w x Ž .6 P. Colangelo, F. De Fazio, N. Di Bartolomeo, R. Gatto, G. Nardulli, Phys. Rev. D 52 1995 6422.w x Ž .7 V.L. Eletski, Y.I. Kogan, Z. Phys. C 28 1985 155.w x8 A.G. Grozin, O.I. Yakovlev, preprint BUDKERINP-94-3, hep-phr9401267.w x Ž .9 S. Narison, H.G. Dosch, Phys. Lett. B 368 1996 163.

w x Ž . Ž .10 V.M. Braun, I.E. Filyanov, Z. Phys. C 44 1989 157; V.M. Belyaev, V.M. Braun, A. Khodjamirian, R. Ruckl, Phys. Rev. D 51 1995¨6177.

w x Ž .11 T.M. Aliev, N.K. Pak, M. Savci, Phys. Lett. B 390 1997 335.w x12 P. Colangelo, F. de Fazio, BARI-THr97-250.w x13 A.G. Grozin, O.I. Yakovlev, WUErITP-97-016, hep-phr9706421.w x Ž . Ž .14 A.G. Grozin, O.I. Yakovlev, Phys. Lett. B 285 1992 254; S. Groote, J.G. Korner, O.I. Yakovlev, Phys. Rev. D 54 1996 3447; D 55

Ž .1997 3016.w x Ž .15 P. Cho, Phys. Lett. B 285 1992 145.w x Ž .16 CLEO Collaboration, G. Brandenburg et al., Preprint CLNS 96r1427, CLEO 96-13 1996 .w x Ž .17 H.-Y. Cheng, Phys. Lett. B 399 1997 281.w x Ž .18 Z. Guralnik, M. Luke, A.V. Manohar, Nucl. Phys. B 390 1993 474.w x Ž .19 E. Jenkins, Phys. Lett. B 315 1993 431.

11 June 1998


Determining the coupling of a Higgs boson to ZZat linear colliders

J.F. Gunion a, T. Han a,b, R. Sobey a

a DaÕis Institute for High Energy Physics, UniÕersity of California, DaÕis, CA 95616, USAb Department of Physics, 1150 UniÕersity AÕenue, UniÕersity of Wisconsin, Madison, WI 53706, USA

Received 27 January 1998; revised 1 April 1998Editor: H. Georgi

Abstract

We demonstrate that, at a 500 GeV eqey collider, inclusion of the ZZ-fusion process for production of a lightstandard-model-like Higgs boson can substantially increase the precision with which the ZZh coupling can be determinedŽ .using the model-independent recoil mass technique as compared to employing only Zh associated production. q 1998Published by Elsevier Science B.V. All rights reserved.

1. Introduction

Ž .Once a neutral Higgs boson h is discovered,Ž .determining its coupling to Z bosons g is ofZZh

fundamental importance. 1 It is this coupling whichmost directly reflects the role of the h in electroweaksymmetry breaking. In the minimal Standard ModelŽ .SM , where the Higgs sector consists of a singleHiggs doublet field, there is only one physical Higgsboson eigenstate, with coupling g sgm rcosu ,ZZh Z W

Ž .where g is the SU 2 coupling and u is the weakW

mixing angle. In contrast, in a theory with manyscalar doublets andror singlets, the ZZ couplings of

1 For recent reviews of Higgs boson phenomenology, see e.g.,w xRefs. 1,2 .

Ž .the individual neutral Higgs bosons h are gener-i

ally reduced in magnitude, but must obey the sumw x 2 w x2rule 3 Ý g s gm rcosu . The sum rule be-i ZZh Z Wi

comes still more complicated if triplet Higgs repre-sentations are present. Precise determination of gZZh

for each and every observed h will therefore becrucial to knowing whether or not we have found allthe Higgs bosons that participate in electroweaksymmetry breaking and to understanding the fullstructure of the Higgs sector.

In eqey collisions, the dominant Higgs bosonproduction diagrams involving the ZZh coupling areof two types:

eqey™Zh 1.1Ž .

eqey™eqeyZ )Z ) ™eqeyh . 1.2Ž .

Ž .There is constructive interference of the amplitudeq y q y Ž .for e e ™Zh™e e h with that for 1.2 . How-


( )J.F. Gunion et al.rPhysics Letters B 429 1998 79–8680

ever, it is desirable both for simplicity and in orderto maximize experimental accuracy for the gZZh

determination to impose cuts such that this interfer-Ž . Ž .ence is very small; the Zh 1.1 and ZZ-fusion 1.2

amplitudes can then be considered as leading toeffectively independent production processes. 2 Themain point of this paper is to demonstrate that, in thecase of a light Higgs boson, when the eqey collider

'Ž .is operated at full energy e.g. s s500 GeV , theZZ-fusion production mode will make a crucial con-tribution to the determination of the ZZh coupling.Indeed, by combining the ZZ-fusion and Zh produc-tion processes we find that the error for g achievedZZh'at s s500 GeV can be competitive with that which

Ž .is attained using Zh associated production alone at'lower s near the maximum in the Zh cross section.

Further, if the linear collider is run in the eyey

mode, the only source of h production, and onlymeans for determining the ZZh coupling is from

y y y y w xZZ-fusion e e ™e e h 5 .Ž . Ž .Processes 1.1 and 1.2 have quite different

characteristics. For a lighter Higgs boson and lower'Ž . Ž .center of mass energy s process 1.1 dominates

' 'Ž .with a maximal cross section at s ;m q 2 m ,Z h'while for a heavier Higgs boson or higher s ,Ž . Žprocess 1.2 becomes more important the cross

' .section increasing logarithmically with s . This isshown in Fig. 1, where we present the total cross

q y q y 'Ž .section for e e ™e e h solid as a function of sfor several Higgs boson masses, m s80,120,160h

and 200 GeV. Dashed curves present the contribu-tion only from eqey™Zh with Z™eqey. Notethat the ZZ fusion cross section becomes larger than

'that of Zh for s )300 GeV.In both the Zh and ZZ-fusion channels, the Higgs

'Ž .signal can be easily detected for m Q 0.7y0.8 sh

by reconstructing the Higgs mass peak via the hdecay products. However, in order to determine the

ŽZZh coupling in a model-independent manner i.e.

2 Nonetheless, our calculations will always employ the full SMw xmatrix elements 4 , including all interfering diagrams, for any

particular signal final state. Full matrix elements are also em-ployed for background processes contributing to any particularfinal state. However, interference between the signal and back-ground is neglected; this is an excellent approximation whenconsidering a very narrow light Higgs boson.

q y q y Ž .Fig. 1. Total cross section for e e ™ e e h solid as a'function of s for m s80,120,160 and 200 GeV. Dashed curvesh

present the contribution only from eq ey™ Zh with Z™ eq ey.

independent of the h’s branching ratio to any particu-.lar channel , it will be crucial to identify the Higgs

signal through the ‘‘recoil mass’’ variable,

2 2 ' q yM ssqM y2 s E qE , 1.3Ž . Ž .rec ll ll ll ll

where M is the invariant mass of the final statell ll

lepton pair and the E are the lepton energies in thell

c.m. frame; here, llse,m is possible for processŽ . 3 Ž .1.1 while only llse is relevant for process 1.2 .Due to detector resolution effects, M will displayrec

a peaked distribution of finite width, much broaderthan the physical width of a light Higgs boson,centered about m . If we can measure the inclusiveh

cross section associated with the M peak in such arec

way that there is small sensitivity to the h decay,then we can obtain a direct determination for theZZh coupling g .ZZh

3 One could also consider reconstructing an M peak usingrecŽ . Ž .the Z™ qq hadronic decays in Eq. 1.1 in order to increase the

signal statistics. However, the energyrmomentum resolution forjets is much worse than for leptons and the backgrounds in thehadronic decay channels are larger, implying a less sharp signalpeak above background. Thus, we will consider only the leptonicmodes.

( )J.F. Gunion et al.rPhysics Letters B 429 1998 79–86 81

Since we allow the Higgs to decay to anything,our background is composed of many processes. Forllse, for example, we must consider all processes ofthe type:

eqey™eqeyX , 1.4Ž .q y q ywith Xs ll ll ,t t ,nn and qq.

Ž .Many analyses of the process 1.1 in the inclu-sive M context have appeared in the literature; see,rec

w xfor example, 6–8,2,9 and references therein. How-Ž .ever, process 1.2 has received limited attention

w x2,10 . In particular, complete background computa-tions for the inclusive signal are given for the firsttime in the present paper.

The rest of the paper is organized as follows. InSection 2 we explore the relative importance of the

Ž . Ž . 2 Žprocesses 1.1 and 1.2 for determing g assum-ZZh. y ying a SM-like h , and comment on the e e ™

eyeyh ZZ-fusion production mode. Section 3 sum-marizes our results and their implications for the

Ž .relative importance for the g determination ofZZh' 'running at s s500 GeV vs. lowering s to a valuenear the maximum in the Zh cross section.

2. Determining the coupling g 2Z Z h

In order to reconstruct the signal peak in M viarecŽ .Eq. 1.3 , we must assure that the charged leptons

are detected. Thus, we impose the following ‘‘basic’’acceptance cuts

< <cosu -0.989,ll

p ll " )15 GeV, p ll qll y )30 GeV,Ž . Ž .T T

M )40 GeV, and M )70 GeV. 2.1Ž .ll ll rec

The polar angle cut roughly simulates the detectorw xacceptance for a beam hole of 150 mrad 8 , and the

cut on M is imposed in order to suppress eventsll ll

from photon conversion.The sharpness of the reconstructed M peak isrec

determined by the momentumrenergy resolution forŽthe charged leptons. The energy of an electron but

.not that of a muon can be measured in the electro-magnetic calorimeter. The momentum of either amuon or an electron can be determined from atracking measurement of its curvature in the mag-

netic field of the detector. We consider two possibili-ties for the energy resolution of the electromagneticcalorimeter:

'I: NLCrEM DErEs12%r E [1%;'II: CMSrEM DErEs2%r E [0.5%.

where [ denotes the sum in quadrature and E is inGeV. Case I is that currently discussed for the NLC

w xelectromagnetic calorimeter 8 ; case II is that of thew xCMS lead-tungstate crystal 11 . We also consider

two possibilities for the momentum resolution fromtracking:III: NLCrtracking

y4 y3 2'Dprps2=10 p[1.5=10 r p sin u ;IV: SJLCrtracking Dprps5=10y5 p[10y3,with p in GeV. Case III is that specified for the

w xtypical NLC detector in 8 and case IV is thatŽ .quoted for the ‘‘super-JLC’’ SJLC detector design

w x12 .It is important to note that for an electron the

tracking determination of its momentum is not statis-tically independent of the electromagnetic calorime-ter measurement of its energy. Thus, the two mea-surements cannot be combined; one should usewhichever measurement provides the best result. Inorder to provide a clean comparison of the differentpossibilities, we will present results in which weanalyze all events using either the calorimeter energymeasurement or the tracking measurement; i.e. wedo not choose the best measurement on an event-by-event basis.

It is useful to compare the fractional resolutions,Ž .r'DErE E;p for llse,m , for the different

cases to one another as a function ofenergyrmomentum. Using us908 in III, one finds

Žthat r )r for any E)70 GeV lower E for u inIII I.the forward or backward direction and that r )rI IV')r for E)100 GeV. At s s500 GeV, leptonII

energies above 100 GeV are typical for the lightHiggs masses studied here; then, NLCrtracking willnot be useful for an electron, whereas SJLCrtrack-ing might be, depending upon the EM calorimeter.Since the muon energyrmomentum can only bemeasured by tracking, NLCrtracking will result inthe Zh™mqmyh channel having a poorer signal tobackground ratio than either the Zh™eqeyh or theZZ-fusion eqeyh channel. On the other hand, if themachine energy is lower, e.g. near the peak in the Zhcross section for small m , electron energies areh


q y q y 'Fig. 2. Recoil mass distributions for e e ™e e X at s s500 GeV. The solid and dashed curves give the Zh signal for m s90 andh

120 GeV, respectively. The dotted line is the summed SM background. Results for the different energyrmomentum resolution, cases I-IV,Ž . Ž . Ž .are shown in the four different panels. The cuts of Eqs. 2.1 , 2.2 and 2.3 have been imposed.

smaller, and for the majority of events theNLCrtracking measurement of the electron energyis competitive with the NLCrEM measurement. Thiswill be apparent from the figures in the next section.

To suppress the SM background more effectivelyand for physics clarity, it is beneficial to divide thestudy into the two natural categories: the Zh associ-ated production and ZZ fusion processes.

q y q y 'Fig. 3. Recoil mass distributions for e e ™e e X at s s250 GeV. The solid and dashed curves give the Zh signal for m s90 andh

120 GeV, respectively. The dotted line is the summed SM background. Results for the different energyrmomentum resolution, cases I-IV,Ž . Ž .are shown in the four different panels. The cuts of Eqs. 2.1 and 2.2 have been imposed.


2.1. eqe y™Zh™eqe yh and mqmyh

'We first discuss a linear collider with s s500 GeV. In order to isolate the Zh class of eventswe require

< < < <M ym -10 GeV , cosu -0.8 . 2.2Ž .ll ll Z ll

The mass cut largely eliminates the leading back-ground from eqey™WqWy and the polar anglecut helps reduce the other large background fromeqey™ZZ. Since the Z boson in the signal is not

Ž 2only central, but also energetic with E s sym qZ h2 ' '.m r2 s f s r2, we can further reduce the back-Z

ground by imposing a cut of

p eqey )80 GeV . 2.3Ž . Ž .T

The signal to background ratio is improved after thecuts to the extent that the Higgs must be nearlydegenerate with the Z for the M peak to have anyrec

significant background. Fig. 2 presents the recoilmass distributions, dsrdM , for eqey™eqeyX atrec's s 500 GeV. Results for different energyrmomentum resolutions, cases I-IV, are shown in thefour different panels. The solid and dashed curvesgive the Zh signal for m s90 and 120 GeV, respec-h

tively. The dotted line is the summed SM back-ground.

The eqey™Zh cross section reaches a maximum' 'near s ;m q 2 m . A relevant question is howZ h

much improvement is possible by running the ma-chine at a lower energy nearer the maximum crosssection, as would be possible once m is knownhŽ .either from LHC or NLC data . To illustrate, we

'consider s s250 GeV. Fig. 3 shows the recoil massdistributions, dsrdM , similar to those in Fig. 2rec'but for s s250 GeV. We see not only that the crosssection rate is larger, but also that the signal peak ismuch sharper because of the better determination for

'lepton energyrmomentum at this lower s .We estimate the relative statistical error for the

cross section measurement as

'Rs SqB rS , 2.4Ž .where S and B are the numbers of signal andbackground events for a given luminosity; neglectingsystematic uncertainty from correcting for our cuts,this is also the error on the coupling g 2 . In Table 1ZZh

Ž .we compare the errors as a function of m in theh

Table 1Percentage accuracy for g 2 based on Zh channel eq ey™ZZh

eq ey h cross section measurements with Ls200fby1 assuming' 'Ž . Ž .a s s500GeV and b s s250GeV. Results for the four

different lepton energyrmomentum resolutions are shown. The'Ž . Ž .cuts of Eqs. 2.1 and 2.2 have been imposed for both s s

' '500GeV and s s250GeV. For s s500GeV we have imposedŽ .the additional cut 2.3 . Note that only one kind of lepton is

counted here.

' Ž . Ž .s GeV Resolution Mass bin m GeVh

80 90 100 120 140

Ž .500 I m "10 15% 15% 12% 8.3% 7.2%hŽ .II m "10 11% 12% 10% 6.2% 6.4%hŽ .III m "10 19% 21% 18% 15% 12%hŽ .IV m "10 11% 12% 10% 6.8% 6.5%hŽ .250 I m "10 5.7% 6.7% 6.3% 4.8% 6.6%hŽ .II m "10 5.8% 6.8% 6.7% 4.8% 6.5%hŽ .III m "10 5.6% 6.7% 6.7% 4.7% 6.7%hŽ .IV m "10 5.8% 6.7% 6.7% 4.7% 6.7%h

Zh mode, with Z™eqey, for different resolution'choices, taking s s500 GeV and 250 GeV and

assuming an integrated luminosity of Ls200 fby1.'At s s500 GeV, the error ranges from ;15% at

m s80 GeV to ;7% at m s140 GeV. In general,h h

the accuracy is not greatly affected by the detector.The exception is case III which yields poorer resultsthan the other cases. Results for the g 2 error atZZh' Žs s250 GeV are ;6% more or less independent

.of resolution choice for most values of m , worsen-h

ing to ;7% at the Z peak. Thus, by running themachine at an energy near the maximum in the Zhcross section one should be able to improve the

Table 22 wThe percentage accuracy for g obtained by combining via Eq.ZZh

Ž .x q y2.5 results for the Zh cross section measurement in the e eq y Ž . q y q y™ e e h NLCrEM channel and the e e ™ m m h

y1 'Ž .NLCrtracking channel, assuming Ls200 fb and s s' Ž . Ž .500GeV or s s250GeV. The cuts of of Eqs. 2.1 and 2.2

'have been imposed for both energies and for s s500GeV weŽ .impose the additional cut 2.3 . The mass bins of Table 1 have

been employed.

' Ž . Ž .s GeV m GeVh

80 90 100 120 140

500 12% 12% 10% 7.3% 6.2%250 4.0% 4.7% 4.6% 3.4% 4.7%


q y q y ' Ž . Ž .Fig. 4. Recoil mass distributions for e e ™e e X at s s500 GeV. Results for m s90 GeV solid , and 120 GeV dashed are shown.hŽ .Results for the different energyrmomentum resolution, cases I-IV, are shown in the four different panels. We impose the cuts of Eqs. 2.1

Ž .and 2.6 .

accuracy for this particular mode by about a factor of2.

We compute the net error for the g 2 determina-ZZh

tion in the Zh channel by including measurementsfor both the Z™eqey and the Z™mqmy final

Ž .states. The net error R is given bynet

Ry2 sRy2 qRy2 . 2.5Ž .net e m

In obtaining the error for the eqey final state, wecompare results found using the EM calorimeter tothose found using tracking and adopt the superiorchoice. Thus, for example, if we assume NLCrEMand NLCrtracking, this means using case I resultsfor the Z™eqey final state and case III results forthe Z™mqmy final state. The results are illustrated

'in Table 2. At s s250 GeV our results indicate thatan error for g 2 of less than 5% can be expected. 4

ZZh

4 This, and other results obtained here for the Zh mode arew xgenerally consistent with those of Refs. 6,7,2,9 for m Rh

110GeV, when the same electromagnetic-calorimeterrtrackingresolution assumptions are made. For lower m , we find higherh

w xbackgrounds as compared to the estimates made in Ref. 2 ,resulting in larger errors.

2.2. eqe y™eqe yh and eye y™eye yh Õia ZZ-fu-sion

The major advantage for the ZZ-fusion channelŽ . Ž .1.2 over the Zh channel 1.1 is that the crosssection increases logarithmically with energy; at's s500 GeV and for m s120 GeV it is about 10h

fb as compared to the Zh cross section of about 2.5fb. To remove the large ZZ background, we require,

Ž .in addition to the basic cuts of Eq. 2.1 ,

M )100 GeV . 2.6Ž .ee

The only penalty from the M cut is a 20% decreaseee

in the signal rate due to elimination of the construc-tive interference of the ZZ-fusion amplitude with theZh amplitude.

Fig. 4 presents signal and background curves afterŽ . Ž .the cuts of Eqs. 2.1 and 2.6 . Although some

background persists, 5 the much larger signal ratemakes a good measurement of the cross sectionfeasible. The corresponding results for the error of

5 The signal-to-background ratio can be further enhanced bystronger cuts, but the signal rate is also reduced and the error inthe determination of g 2 is not improved.ZZh


Table 3Percentage accuracy for g 2 based on the ZZ-fusion channelZZh

eq ey™ eq ey h cross section measurements with Ls200fby1

'at s s 500 GeV. Results for NLCrEM, CMSrEM andŽ .SJLCrtracking resolution cases I, II and IV are shown;

NLCrtracking yields much poorer results than NLCrEM. TheŽ . Ž .cuts of Eqs. 2.1 , and 2.6 have been imposed.

Ž .Resolution Mass bin m GeVh

80 90 100 120 140

Ž .I m "10 9.7% 10% 9.6% 6.9% 7.2%hŽ .II m "10 8.8% 9.7% 11% 6.3% 6.8%hŽ .III m "10 19% 18% 17% 15% 14%hŽ .IV m "10 9.1% 9.9% 9.2% 6.6% 7.0%h

the g 2 determination are presented in Table 3.ZZh

Using NLCrEM calorimetry, case I, the error rangesfrom 10% to 7%. As in the Zh channel, there is littledifference between results for the resolution cases I,II and IV, whereas case III yields substantially poorerresults.

2 'The ultimate accuracy for g at s s500 GeVZZh

is obtained by combining the Zh and ZZ-fusionchannel results. In Table 4, we present the error as afunction of m , assuming that NLCrEM energyh

resolution is employed for the eqeyh final states ofthe Zh and ZZ-fusion channels and that NLCrtrack-ing resolution is employed for the mqmyh final stateof the Zh channel. Note that the achievable accuracy

'is competitive with that obtained at s s250 GeVŽusing the Zh mode alone the ZZ-fusion mode being.not useful at this low energy , especially at the larger

m values.h

At an eyey collider, Higgs production is entirelyfrom the ZZ-fusion process, eyey™eyeyh . Theresults presented in Table 3 are essentially applica-ble, except that the background level is slightly

y y w xsmaller in the e e case 5 .

Table 4Combined percentage accuracy for g 2 as obtained by includingZZh

the three measurements: eq ey™ eq ey h for the Zh and ZZ-fu-Ž . q y q y Žsion NLCrEM plus e e ™m m h for the Zh NLCrtrack-

y1'.ing at s s500GeV and Ls200fb . Cuts as specified inTables 1 and 3 are imposed.

Ž .m GeVh

80 90 100 120 140

7.5% 7.7% 6.9% 5.0% 4.7%

3. Summary and conclusions

We have investigated the precision with which theŽ 2 .Higgs boson to ZZ coupling g can be deter-ZZh

mined by employing the recoil mass distribution inq y Ž .ll ll h llse,m final states coming from the Zh and

ZZ-fusion production channels at an eqey colliderŽ . y yor ZZ-fusion alone at an e e collider. From the

Ž .fully inclusive recoil mass distribution a directdetermination of g 2 , that is independent of anyZZh

assumptions regarding the branching ratio for theHiggs boson to decay into any particular channel,can be made. We considered a number of electro-magnetic calorimetry and tracking resolution optionsfor the detector. Results for the four options consid-ered are similar, except that the tracking specified forthe ‘typical’ NLC detector yields poor results at's s500 GeV. For the calorimetry and tracking res-olutions specified for this typical NLC detector, wefind the following results for the error of the g 2

ZZh

determination for m in the range 80y140 GeV:hq y 'Ø If the e e collider is run at s s500 GeV, the

ZZ-fusion production mode yields smaller statisti-Žcal error than does the Zh production mode even

after combining both eqeyh and mqmyh final.states in the latter case .

Ø Taking both Zh channel and ZZ-fusion channely1'measurements at s s500 GeV and 200 fb into

account, the combined accuracy ranges from 7.5%Žat m s80 GeV to 4.7% at m s140 GeV Tableh h

.4 .Ø Running at a lower energy near the Zh cross

section maximum is possibly fruitful. For accu-y1 'mulated luminosity of 200 fb at s s250 GeV,

Ž .the error found using the dominant Zh produc-tion mode is less than 5% for 80Fm F140 GeVhŽ .Table 2 .

Ø The statistical error of a measurement in theZZ-fusion eyey™ eyeyh channel would besomewhat better than that for a measurement in

q y q y Ž .the ZZ-fusion e e ™e e h channel Table 3 ,due to smaller backgrounds.

The most significant implication of our results isthat it may not be necessary, or even appropriate, torun at low energy in order to obtain the best possibleaccuracy for the g 2 determination. When runningZZh'at the full energy of s s500 GeV, if m R120 GeVh

Ž .then for NLCrEM and NLCrtracking the com-


bined Zh and ZZ-fusion error of Table 4 is very'close to that obtained when running at s s250 GeV

for the same integrated luminosity. Further, it seemslikely that it will prove desirable from the point ofview of other physics to accumulate more luminosity' 'at s s500 GeV than at s s250 GeV, and it is

w xalso possible 13 that the instantaneous luminosity atŽthe lower energy will be lower assuming that the

'interaction region is optimized initially for s s.500 GeV . In either case, the lower energy running

might not significantly improve the g 2 determina-ZZh

tion even if m -120 GeV.h

Thus, we conclude that use of the ZZ-fusion modeŽ .as well as the Zh mode should provide a veryvaluable increase in the accuracy that can be achievedfor the determination of the ZZ coupling of a SM-likeHiggs boson at a lepton collider operating at highenergy.

Acknowledgements

This work was supported in part by the DOEunder contracts No. DE-FG03-91ER40674 and No.DE-FG02-95ER40896, and in part by the Davis In-stitute for High Energy Physics.

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Ž .H. Haber, J. Siegrist Eds. , Electroweak Physics and Beyondthe Standard Model World Scientific Publishing Co., pp.23–145.

w x2 J.F. Gunion, L. Poggioli, R. Van Kooten, C. Kao, P. Row-Ž .son, in: D.G. Cassel, L.T. Gennari, R.H. Siemann Eds. ,

New Directions for High-Energy Physics, Proceedings of the1996 DPFrDPB Summer Study on High-Energy Physics,June 25—July 12, 1996, Snowmass, CO, Stanford LinearAccelerator Center, 1997, pp. 541–587.

w x Ž .3 J.F. Gunion, H. Haber, J. Wudka, Phys. Rev. D 43 1991904.

w x4 We have made use of the Helicity Amplitude package,MadGraph by T. Stelzer, W. Long, Comput. Phys. Commun.

Ž .81 1994 357.w x5 V. Barger, J. Beacom, K. Cheung, T. Han, Phys. Rev. D 50

Ž . Ž .1994 6704; T. Han, Int. Journ. Mod. Phys. A 11 19961541.

w x Ž .6 P. Janot, in: F. Harris, S. Olsen, S. Pakvasa, X. Tata Eds. ,Proceedings of the 2nd International Workshop on Physicsand Experiments with Linear eq ey Colliders, Waikoloa, HI,1993, World Scientific Publishing, p. 192.

w x7 K. Kawagoe, in: F. Harris, S. Olsen, S. Pakvasa, X. TataŽ .Eds. , Proceedings of the 2nd International Workshop onPhysics and Experiments with Linear eq ey Colliders,Waikoloa, HI, 1993, World Scientific Publishing, p. 660.

w x8 Physics and Technology of the Next Linear Collider: aReport Submitted to Snowmass 1996 BNL 52-502, FNAL-PUB-96r112, LBNL-PUB-5425, SLAC Report 485, UCRL-ID-124160.

w x q y9 Physics with e e Colliders by ECFArDESY LC PhysicsŽ .Working Group E. Accomando et al. , hep-phr9705442.

w x10 P. Minkowski, presentations at the U.C. Santa Cruz Work-shops on ey ey Colliders.

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tion with R. Van Kooten.w x13 J. Irwin, private communication.

11 June 1998


Inclusive decay rate for B™X qg in next-to-leadingd

logarithmic order and CP asymmetry in the standard model 1

A. Ali a, H. Asatrian b, C. Greub c

a Deutsches Elektronen-Synchrotron DESY, Hamburg, Germanyb YereÕan Physics Institute, Alikhanyan Br., 375036-YereÕan, Armenia

c Inst. f. Theor. Physik, UniÕ. Bern, Bern, Switzerland


Abstract

Ž .We compute the decay rate for the Cabibbo-Kobayashi-Maskawa CKM -suppressed electromagnetic penguin decay2Ž .B™ X qg and its charge conjugate in the next-to-leading order in QCD, including leading power corrections in 1rmd b

2 ² Ž .:and 1rm in the standard model. The average branching ratio BB B™X qg of the decay B™X qg and its chargec d dy6 y5² Ž .:conjugate B™ X qg is estimated to be in the range 6.0=10 F BB B™X qg F2.6=10 , obtained by varyingd d

the CKM-Wolfenstein parameters r and h in the range y0.1FrF0.4 and 0.2FhF0.46 and taking into account otherparametric dependence. In the NLL approximation and in the stated range of the CKM parameters, we find the ratioŽ . ² Ž .: ² Ž . Ž .R dgrsg ' BB B™X g r BB B™X g to lie in the range 0.017FR dgrsg F0.074. Theoretical uncertainties ind s

this ratio are estimated and found to be small. Hence, this ratio is well suited to provide independent constraints on the CKMŽ . Ž Ž . Ž .. Ž Žparameters. The CP-asymmetry in the decay rates, defined as a B™X g ' G B™X g yG B™ X g r G B™CP d d d

. Ž .. Ž .X g qG B™ X g , is found to be in the range 7y35 %. Both the decay rates and CP asymmetry are measurable ind d

forthcoming experiments at B factories and possibly at HERA-B. q 1998 Elsevier Science B.V. All rights reserved.

1. Introduction

Electromagnetic penguins were first measured bythe CLEO collaboration through the exclusive decay

w w xB™K qg 1 , followed by the measurement ofw xthe inclusive decay B™X qg 2 . The presents

w xCLEO measurements can be summarized as 3 :² : y4BB B™X qg s 2.32"0.57"0.35 =10 ,Ž . Ž .s

² w : y5BB B™K qg s 4.2"0.8"0.6 =10 .Ž . Ž .1Ž .

Very recently, the inclusive radiative decay has also

1 Work partially supported by Schweizerischer Nationalfonds.

been reported by the ALEPH collaboration with aŽ . w xpreliminary branching ratio 4 :

² :BB H ™X qgŽ .b s

s 3.29"0.71"0.68 =10y4 . 2Ž . Ž .² Ž .:The quantity BB B™X qg is the branchings

ratio averaged over the decays B™X qg and itss

charge conjugate B™ X qg . The branching ratio insŽ .2 involves a different weighted average of theB-mesons and L baryons produced in Z 0 decaysbŽ .hence the symbol H than the corresponding oneb

Ž .given in 1 , which has been measured in the decayq y 0 0Ž .F 4S ™B B , B B .


( )A. Ali et al.rPhysics Letters B 429 1998 87–9888

These measurements have stimulated an impres-sive theoretical activity, directed at improving theprecision of the decay rates and distributions in the

Ž .context of the standard model SM and beyond thestandard model, in particular supersymmetry. In theSM-context, the complete next-to-leading-logarith-

Ž .mic NLL contributions have been painstakinglyw xcompleted 5–14 and leading power corrections in

2 w x 2 w x1rm 15–17 and 1rm 18–20 have also beenb c

calculated for the decay rate in B™X qg . Thiss

theoretical work allows to calculate the branchingratios in the SM with an accuracy of about "9%,

² Ž .: Ž . y4yielding BB B™X qg s 3.50"0.32 =10s² Ž .: Ž . y4and BB H ™X qg s 3.76"0.30 =10 , inb s

Žreasonable agreement with the CLEO and pre-.liminary ALEPH measurements, respectively. The

Ž . Ž .decay rates in Eqs. 1 and 2 determine the ratio ofŽ . w xthe Cabibbo-Kobayashi-Maskawa CKM 22 ma-

< ) < < < < <trix elements V V rV . Since V and V havet s t b cb cb tb

been directly measured, these measurements can be< < w xcombined, yielding V s0.033"0.007 21 . Thet s

< <central value of V is somewhat lower than thet s< < < <corresponding value of V , V s 0.0393 "cb cb

0.0028, but within errors the two matrix elements arefound to be approximately equal, as expected fromthe CKM unitarity.

The interest in measuring the decay rate in B™Ž .X qg and its charge conjugate B™ X qg liesd d

in that it will determine the CKM-Wolfenstein pa-w xrameters r and h 23 in a theoretically reliable way.

Likewise, this decay will enable us to search for newphysics which may manifest itself through enhancedbdg andror bdg effective vertices. These verticesare CKM-suppressed in the standard model, but newphysics contributions may not follow the CKM pat-tern in flavor-changing-neutral-current transitions andhence new physics effects may become more easily

Ž .discernible in B™X qg and its charge conjugated

than in the corresponding CKM-allowed vertices bsgand bsg. Closely related to this is the question ofCP-violating asymmetry in the decay rates forB™ X qg and its charge conjugate B™X qg ,d d

which may provide us with the first measurements ofthe so-called direct CP violation in B physics. Withthe weak phase provided dominantly by the CKMmatrix elements V and V in the decay B™X qt d ub d

g , the perturbatively generated strong phases can becalculated by taking into account the charm and up

quark loops in the electromagnetic penguins, whichgenerate the necessary absorptive contributions. This

Žcalls for an improved theoretical estimate of BB B. Ž . Ž .™X qg and BB B™ X qg hence a in thed d C P

SM.In what follows, we shall discuss for the sake of

definiteness the decays of the b-quark b™sqŽ . Ž .g qg and b™dqg qg , whose hadronic tran-

scriptions are B™ X qg and B™ X qg , respec-s d

tively, but the discussion applies for the chargeconjugate decays B™X qg and B™X qg ass d

well with obvious changes. When it is essential todifferentiate between the decay of a B meson and itscharge conjugate B, we shall do so. The branchingratio for the CKM-suppressed decay B™ X qgd

was calculated several years ago in partial next-to-Ž . w xleading order by two of us A.A. and C.G. 24 .

Subsequent to that, the CP asymmetries in the decayŽ .rates in the leading logarithmic LL approximation

w xwere calculated in the SM 25,26 , and in somew xextensions of the SM 27,28 . Much of the theoreti-

cal improvements carried out in the context of thedecay B™ X qg mentioned above can be takens

over for the decay B™ X qg . Like the formerd

decay, the NLL-improved and power-corrected de-cay rate for B™ X qg rests on firmer theoreticald

ground and consequently has much reduced theoreti-cal uncertainty; in particular, the one arising from thescale-dependence which in the LL approximation isnotoriously large, becomes drastically reduced in thecomplete NLL order, presented here. Hence, theimproved theory for the decay rate would allow toextract more precisely the CKM parameters from themeasured branching ratio B™ X qg . Of particulard

theoretical interest is the ratio of the branching ra-tios, defined as

² :BB B™X qgŽ .dR dgrsg ' , 3Ž . Ž .² :BB B™X qgŽ .s

in which a good part of theoretical uncertaintiesŽ .such as from m , m , BB etc. cancel. Anticipatingt b sl

Ž . w xthis, the ratio R dgrsg was proposed in 24 as a< <good measure of V . We now calculate this ratio int d

the NLL accuracy, determine its CKM-parametricdependence precisely and estimate theoretical errors.

( )A. Ali et al.rPhysics Letters B 429 1998 87–98 89

The CP-asymmetry in the decay rates, defined as

a B™X qgŽ .C P d

G B™X qg yG B™ X qgŽ . Ž .d d' 4Ž .

G B™X qg qG B™ X qgŽ . Ž .d d

has not so far been calculated in the NLL precision.We recall that, as opposed to the decay ratesŽ . Ž .G B™ X qg and G B™ X qg , which receives d

contributions starting from the lowest order, i.e.,Ž nŽ . nŽ ..terms of the form a m ln m rm , the CP-odds b W bŽ .numerator in Eq. 4 is suppressed by an extra factor

Ž . Ž na , i.e., it starts with terms of the form a m as s b snŽ ..= log m rm . To simplify the language in theW b

following, we refer to this statement by saying thatthe CP-odd numerator starts at order a . This resultss

in a moderate scale dependence of a , arising fromC P

the Wilson coefficients which contain a term propor-Ž .tional to a ln m rm which is not compensated bys b b

the matrix elements in this order. We show thescale-dependence of a numerically by varying theC P

scale m in the range 2.5 GeVFm F10 GeV. Theb b

compensation of this scale dependence requires theŽ 2 .knowledge of the O a contributions in the matrixs

elements of the operators in the Wilson productexpansion, which is not yet available. However, it isnot unreasonable to anticipate that a judicious choiceof the scale m in B decays may reduce the NLLb

corrections. Since the results for the CP-even part,Ž .i.e., the denominator in Eq. 4 , are known in the LL

approximation, and with the help of the present worknow also in the NLL accuracy, this information canbe used to guess the optimal scale. We make thechoice m s2.5 GeV for which we show that theb

NLL corrections in the decay rates become minimalŽ .see Fig. 1 . Of course, one can not insist that thisfeature must necessarily also hold for a . NotC P

Ž 2 .having the benefit of the complete O a calcula-s

tion for a , this particular choice of m is anC P b

educated guess based on the inclusive decay ratespresented here.

Ž .The branching ratio BB B™ X qg and ad C P

depend on the parameters r and h and this depen-dence is the principal interest in measuring thesequantities. To estimate them, we vary these parame-ters in the range y0.1FrF0.4 and 0.2FhF0.46,which are the 95% C.L. ranges allowed by the

w xpresent fits of the CKM matrix 29 . In addition to

Fig. 1. Average branching ratio of the processes B™ X qg andd

B™X qg , plotted as a function of the scale m for the centrald b

values of the input parameters m , m rm , BB , m , a andb c b sl t emŽ . Ž . Ž .a m . The solid dashed curve shows the NLL LL result.s Z

this, there are other well-known parametric depen-dences inherent to the theoretical framework beingused here, such as a , m , m , m , a and BB .s t b c em sl

We estimate the resulting uncertainty in the branch-ing ratios for B™X qg and B™ X qg in and d

Ž .analogous way as has been done for BB B™ X qgsŽ .and its charge conjugate . The resulting average

² Ž .: Ž Žbranching ratio BB B™X qg s BB B™X qd d. Ž ..g qBB B™ X qg r2 and the CP rate asymmetryd

a are found to be in the range 6.0=10y6 FC P² Ž .: y5 ŽBB B™X qg F2.6=10 , and a s 7yd C P.35 %, with most of the dispersion arising due to the

CKM parametric dependence of these quantities. Forw xthe central values of the fit-parameters 29 rs

² Ž .:0.11,h s 0.33 one obtains BB B ™ X q g sdŽ . y5 Ž .1.61 " 0.12 = 10 and a s 11.5 y 16.5 %,C P

where the errors reflect the uncertainties stemmingw xfrom varying m g 2.5 GeV,10.0 GeV and the restb

Ž .of the parameters. The ratio R dgrsg , defined inŽ .Eq. 3 , is largely free of parametric uncertainties;

the residual theoretical error on this quantity is smallbut correlated with the values of r and h. Amus-ingly, the theoretical uncertainty on this ratio almostvanishes for the central values of the CKM-fit pa-

Ž .rameters! see Fig. 5 . However, as the presentlyallowed CKM-domain is large, one can take the

Ž . Ž .largest uncertainty DR dgrsg rR dgrsg s"7%,which is found for hs0.2 and rs0.4, as an upper

Ž .limit on this uncertainty. The ratio R dgrsg wouldthen provide theoretically the most robust constrainton the CKM parameters. For the ratio itself, varyingthe CKM parameters in the 95% C.L. range, we find


Ž .0.017FR dgrsg F0.074, with the central valuebeing 0.046. Hence, even a first measurement of thisratio will provide a rather stringent constraint on theŽ .r,h domain. We show this for three assumed

Ž .values, R dgrsg s0.017, 0.046, 0.074 taking intoŽ .account the theoretical errors see Fig. 6 .

Although the shape of the photon energy spectrain B™X qg and B™X qg is very similar, wes d

think that a measurement of the much rarer decayB™X qg should become feasible at future experi-d

ments like CLEO-III and B-factories, because thesefacilities will allow for a good Krp discrimination.

This paper is organized as follows: In Section 2,we discuss the theoretical framework and present thesalient features of the calculation for the decay ratesin B™ X qg and its charge conjugate process ands

the CP asymmetry in the decay rates. In Section 3,we work out the corresponding decay rates and CP

Ž .asymmetry for B™ X qg , and the ratio R dgrsg .d

Section 4 contains the numerical results and weconclude with a summary in Section 5.

2. Decay rates and CP asymmetry in B™X Hgs

and B™X Hgs

The appropriate framework to incorporate QCDcorrections is that of an effective theory obtained byintegrating out the heavy degrees of freedom, whichin the present context are the top quark and W "

bosons. The effective Hamiltonian depends on theŽunderlying theory and for the SM one has keeping

.operators up to dimension 6 ,

84GFHH b™sg qg sy l C m O m ,Ž . Ž . Ž .Ž . Ýe f f t i i'2 is1

5Ž .

where the operator basis and the corresponding Wil-Ž . w xson coefficients C m can be seen elsewhere 11 .i

The symbol l 'V V ) is the relevant CKM factort t b t s

and G is the Fermi coupling constant.F

The Wilson coefficients at the renormalizationŽ .scale m sO m are calculated with the help of theb b

renormalization group equation whose solution re-quires the knowledge of the anomalous dimensionmatrix in a given order in a and the matchings

Ž .conditions, i.e., the Wilson coefficients C msm ,i W

calculated in the complete theory to the commensu-rate order. The anomalous dimension matrix in the

w x w xLL 30 and the NLL approximation 11 are known.The NLL matching conditions have also been workedout in the meanwhile by several groups. Of these, thefirst six corresponding to the four-quark operators

w xhave been derived in 31 , and the remaining two,Ž . Ž .C msm and C msm , were worked out in7 W 8 W

w x w x8 and confirmed in 9,13,14 . In addition, the NLLcorrections to the matrix elements have also beencalculated. Of these, the Bremsstrahlung corrections

w x Žwere obtained in 5,24 in the truncated basis involv-.ing the operators O , O , and O and subsequently1 2 7

w xin the complete operator basis 6,7 . The NLL virtualw xcorrections were completed in 10 . This latter contri-

bution plays a key role in reducing the scale-depen-dence of the LL inclusive decay width. All of thesepieces have been combined to get the NLL decay

Ž .width G B™ X qg and the details are given insw xthe literature 11–13

We recall that the operator basis in HH is in facte f fŽ .larger than what is shown in Eq. 5 in which

operators multiplying the small CKM factor l 'u

V V ) have been neglected. If the interest is inub u s

calculating the CP asymmetry, then they have to beput back. Doing this, and using the unitarity relationl syl yl , the effective Hamiltonian readsc t u

HH b™sg qgŽ .Ž .e f f

4GFsy l C m O m qC m O m� Ž . Ž . Ž . Ž .t 7 7 8 8'2

qC m O m qC m O mŽ . Ž . Ž . Ž .1 1 2 2

yl C m O m yO mŽ . Ž . Ž .Ž .u 1 1u 1

qC m O m yO m q PPP . 6Ž . Ž . Ž . Ž .4Ž .2 2 u 2

In this equation terms proportional to the smallWilson coefficients C , . . . , C are dropped as indi-3 6

cated by the ellipses. The relevant operators aredefined as:

a m aO m s s g T c c g T b ,Ž . Ž .Ž .1 L m L L L

a m aO m s s g T u u g T b ,Ž . Ž .Ž .1u L m L L L

mO m s s g c c g b ,Ž . Ž .Ž .2 L m L L L

mO m s s g u u g b ,Ž . Ž .Ž .2 u L m L L L


emnO m s m m s s b F ,Ž . Ž . Ž .7 b L mn R216p

ga amnO m s m m s T s b G . 7Ž . Ž . Ž .Ž .8 b L mn R216p

Ž .Note that the Wilson coefficients in Eq. 6 areŽ .exactly the same as those in Eq. 5 . Moreover, the

< < < <matrix elements -sg O b) and -sg g O b)iu iu

of the additional operators O and O are obtained1u 2 u

from those of O and O by obvious replacements.1 2

For our intent and purpose, we write the ampli-tudes for the processes b™sg and b™sg g in aform where the dependence on the CKM matrixelements is manifest. The amplitude for the first

Ž .process including the virtual corrections can bewritten as

4GF< <A b™sg sy -sg O b) D b™sg ,Ž . Ž .7 t r ee'2

D b™sg sl At q iAt ql Au q iAu . 8Ž . Ž .Ž .Ž .t R I u R I

It is straightforward to construct the real functionsAt , At , Au and Au from the expressions for theR I R I

w xvirtual correction in Ref. 10 and the NLL Wilsonw xcoefficients in Ref. 11 . The amplitude of the charge

conjugate decay b™ sg decay is then:

4GF< <A b™ sg sy -sg O b) D b™sg ,Ž . Ž .7 t r ee'2

) t t ) u uD b™sg sl A q iA ql A q iA .Ž .Ž .Ž . t R I u R I

9Ž .

The decay rate for the process b™sg then reads 2

m5 G2 ab F em 2< <G b™sg s D b™sg ,Ž . Ž .432p

< < 2D b™sgŽ .2 22 t t< <s l A q AŽ . Ž .t R I

2 22 u u< <q l A q AŽ . Ž .u R I

) t u t uq2 Re l l A A qA AŽ .t u R R I I

) t u t uy2 Im l l A A yA A . 10Ž .Ž .t u R I I R

The order a 2 terms, which are generated whens

2 Ž . Ž .Note that we have absorbed the factor Fs1y 8a r 3p ,sw x tpresent in Ref. 10 , into the term A .R

inserting the explicit expressions for the functionsAt , At , Au , and Au, are understood to be discarded.R I R I

The corresponding expression for the b™ sg decaycan be obtained from the preceding equation bychanging the sign of the term proportional to

Ž ) .Im l l .t u

An analogous expression for the decay widthŽ .G b™sg g of the Bremsstrahlung process, where

the CKM dependence is explicit, is also easily ob-w xtained from the literature 5,24,6,7,10 . To get rid of

the infrared singularity for E ™0, we included theg

virtual photonic correction to the process b™sg, asdiscussed in these references. Another possibility,

w xwhich was suggested in Ref. 11 , is to define thebranching ratio in such a way that the photon energyE has to be larger than some minimal value Emin.g g

It is customary to express the branching ratioŽ .BB B™ X qg in terms of the measured semilep-s

Ž .tonic branching ratio BB B™X lln ,ll

BB B™ X qgŽ .s

G B™ X qgŽ .ss BB B™X lln . 11Ž .Ž .ll

Gsl

ŽThe expression for G including radiative correc-sl. w xtions can be seen in Refs. 33 .

In addition to the perturbative QCD improve-ments discussed above, also the leading power cor-rections, which start in 1rm2 , have been calculatedb

to the decay widths appearing in the numerator andŽ . w xdenominator of Eq. 11 15–17 . The power correc-

tions in the numerator have been obtained assumingthat the decay B™ X qg is dominated by thes

magnetic moment operator O . Writing this correc-7

tion in an obvious notation as

G B™ X qg dŽ .s bs1q , 12Ž .0 2G B™ X qg mŽ .s b

one obtains d s1r2l y9r2l , where l and lb 1 2 1 2

are, respectively, the kinetic energy and magneticmoment parameters of the theoretical frameworkbased on heavy quark expansion. Using l s1

y0.5 GeV 2 and l s0.12 GeV 2, one gets d rm2 ,2 b bŽ 2 .y4%. However, the leading order 1rm powerb

corrections in the heavy quark expansion propor-tional to l are identical in the inclusive decay rates1


Ž . Ž .G B™ X qg and G B™X lln . The correctionss ll

proportional to l differ only marginally. Thus, in-2

cluding or neglecting the 1rm2 corrections makes abŽ .difference of only 1% in BB B™ X qg .s

The power corrections proportional to 1rm2, re-cŽsulting from the interference of the operator O and2

.O with O in B™ X qg , have also been worked1 7 sw xout 18–20 . Expressing this symbolically as

G B™ X qg dŽ .s cs1q , 13Ž .0 2G B™ X qg mŽ .s c

2 w xone finds d rm ,q0.03 20 .c c

It is convenient to express the branching ratio forB™ X qg in a form where the dependence on thes

CKM matrix factors is manifest:

BB B™ X qgŽ .s

< < 2 < < 2l lt us D q Dt u2 2< < < <V Vcb cb

Re l)l Im l)lŽ . Ž .t u t uq D q D . 14Ž .r i2 2< < < <V Vcb cb

Ž .The quantities D as t,u,r,i , which depend ona

various input parameters such as m ,m ,m ,m andt b c b

a , are calculated numerically and listed in Table 1.s² Ž .:The averaged branching ratio BB B™X qg iss

obtained by discarding the last term on the rightŽ .hand side of Eq. 14 . The CP-violating rate asym-

metry has been defined earlier. In terms of thefunctions D it can be expressed as:a

a B™X qgŽ .C P s

Im l)l DŽ .t u isy . 15Ž .2 2

)< < < <l D q l D qRe l l DŽ .t t u u t u r

Ž .Since the function D in the numerator in Eq. 15i

only starts at order a , the complete NLL expressionsŽ 2 .for a requires D up to and including the O aC P i s

term which is not known. Hence, in the LL approxi-mation, a consistent definition of a is the one inC P

which only the LL result for the denominator isretained, i.e., in this approximation one should dropterms proportional to D and D and keep only theu r

LL result for D , which is denoted as DŽ0. in thet t

following. The expression for a in this approxima-C P

tion then reduces to

Im l)l DŽ .t u ia B™X qg sy . 16Ž . Ž .C P s 2 Ž0.< <l Dt t

This is what we shall use in the numerical estimatesof a . Note that DŽ0., which is also shown in TableC P t

1, is proportional to the square of the LL Wilson0, effŽ .coefficient C m . To be precise, this Wilson7 b

coefficient is obtained from the Wilson coefficientsŽ .C is1, . . . , 8 at the matching scale msm byi W

Ž .using the 1-loop expression for a m in the renor-s

malization group evolution. Moreover, notice thatthe power corrections, which are contained in thefunctions D and D in the NLL branching ratiot rŽ .14 , drop out in the LL expression for a .C P

w xIn the Wolfenstein parametrization 23 , whichwill be used in the numerical analysis, the CKMmatrix is determined in terms of the four parametersA,lssinu , r and h, and one can express theC

< < 2quantities l , l and V in the above equations ast u cbw x Ž Ž 6..32 neglecting terms of O l :

l sAl4 ry ih ,Ž .u

l22 2l syAl 1y ql ry ih ,Ž .t ž /2

l syl yl ,c u t

< < 2 2 4V sA l . 17Ž .cb

3. Decay rates and CP asymmetry in B™X Hgd

and B™X Hgd

In complete analogy with the B™ X qg cases

discussed earlier, the relevant set of dimension-6operators for the processes b™dg and b™dg gcan be written as

HH b™dg qgŽ .Ž .e f f

4GFsy j C m O m qC m O m� Ž . Ž . Ž . Ž .t 7 7 8 8'2

qC m O m qC m O mŽ . Ž . Ž . Ž .1 1 2 2

yj C m O m yO mŽ . Ž . Ž .Ž .u 1 1u 1

qC m O m yO m q PPP ,Ž . Ž . Ž . 4Ž .2 2 u 2

18Ž .

where j sV V ) with jsu,c,t. The operators arej jb jdŽ .the same as in Eq. 7 up to the obvious replacement

of the s-quark field by the d-quark field. Moreover,


Ž .the matching conditions C m and the solutions ofi WŽ .the RG equations, yielding C m , coincide withi b

Ž .those needed for the process b™sg qg . The power2 2 Žcorrections in 1rm and 1rm besides the CKMb c

. Ž .factors are also the same for G B™ X qg anddŽ .G B™ X qg . However, the so-called long-dis-s

tance contributions from the intermediate u-quark inthe penguin loops are different in the decays B™ Xs

qg and B™ X qg . These are suppressed in thed

decays B™ X qg due to the unfavorable CKMs

matrix elements. In B™ X qg , however, there isd

no CKM-suppression and one has to include thelong-distance intermediate u-quark contributions. Itmust be stressed that there is no spurious enhance-

Ž .ment of the form ln m rm in the perturbativeu b< <contribution to the matrix elements - X g O B)d iu

Ž . w xis1,2 as shown by the explicit calculation in 10w xand also discussed more recently in 34 . In other

words, the limit m ™0 can be taken safely. Theu

non-perturbative contribution generated by the u-quark loop can only be modeled at present. In thiscontext, we recall that estimates based on the vectormeson dominance indicate that these contributions

w xare small 35 . Estimates of the long-distance contri-butions in exclusive decays B™rg and B™vg inthe Light-Cone QCD sum rule approach put thecorresponding corrections somewhere aroundŽ . Ž ".O 15% for the charged B decays and much

Ž . w xsmaller O 5% for the neutral B decays 36,37 .Model estimates based on final state interactionslikewise give small long-distance contribution for the

w xexclusive radiative B decays 38 . To take this uncer-tainty into account, we add an error proportional toD in LL approximation, viz. "0.1 DŽ0., in the nu-t t

merical estimate of the function D when calculatingrŽ . w xthe branching ratio BB B™ X qg 35 .d

Ž .In analogy to Eq. 14 the branching ratioŽ .BB B™ X qg in the SM can be written asd

BB B™ X qgŽ .d

< < 2 < < 2j jt us D q Dt u2 2< < < <V Vcb cb

Re j )j Im j )jŽ . Ž .t u t uq D q D , 19Ž .r i2 2< < < <V Vcb cb

Ž .where the functions D as t,u,r,i are the same asaŽ . Žin Eq. 14 . While these functions or some combina-

.tions thereof were obtained in partial NLL approxi-w xmation some time ago 24 , the complete NLL results

are presented here for the first time. For numericalvalues of these functions we refer to Table 1. An

² Žexpression for the averaged branching ratio BB B.:™X qg is obtained by dropping the last term ond

Ž .the right hand side in Eq. 19 .² Ž .:As the branching ratio BB B™X qg is verys² Ž .:well approximated by BB B ™ X q g ss

< < 2 < < 2 Ž .l D r V , the ratio R dgrsg , defined in Eq.t t cbŽ .3 , can be expressed as follows:

< < 2 < < 2 )j D j D Re j jŽ .t u u r t uR dgrsg s q q .Ž . 2 2 2D D< < < < < <l l lt tt t t

20Ž .

< < 2 < < 2 Ž .The leading term j r l in Eq. 20 is obviouslyt t

independent of any dynamical uncertainties; the sub-leading terms proportional to D rD and D rD areu t r t

still uncertain by almost a factor 2, but numericallyŽ .small compared to unity see Table 1 . Also, as we

shall see in the next section, the allowed values ofthe CKM parameters provide a further suppressionof these terms. Hence, the overall uncertainty inŽ .R dgrsg is small.

Using again the LL expression for the denomina-Ž .tor in Eq. 4 , the CP rate asymmetry can be written

as

Im j )j DŽ .t u ia B™X qg sy , 21Ž . Ž .C P d 2 Ž0.< <j Dt t

where DŽ0. stands for the LL expression of D . Int t

the numerical analysis, we will use the followingŽ .expressions for the quantities j in Eqs. 19 andj

Ž . Ž Ž 7..21 neglecting terms of O l :

3 3j sA l ry ih , j sA l 1yrq ih ,Ž . Ž .u t

j syj yj , 22Ž .c u t

2 2Ž . Ž . w xwith rsr 1yl r2 and hsh 1yl r2 32 .Note that all three CKM-angle-dependent quantitiesj start at order l3. Inserting these expressions intoj

Ž .Eq. 21 , a simple form for the CP rate asymmetry isobtained:

D hia B™X qg s . 23Ž . Ž .C P d 2Ž0. 2D 1yr qhŽ .t


4. Numerical estimates of branching ratios andCP asymmetries

We now proceed to the numerical analysis of ourresults. Based on present measurements and theoreti-cal estimates, we take the following values for the

Ž .input parameters: a M s0.118"0.003, m ss Z b

4.8 " 0.15 GeV, m rm s 0.29 " 0.02, m 'c b tŽ . Ž . Žm pole s 175 " 6 GeV corresponding tot

Ž . . Ž .m m s 168"6 GeV , BB s 10.49"0.46 %,Ž .t t s ly1 Ž .a s 130.3"2.3 . For the CKM matrix elementsem

we note that the parameters A and l are rather welldetermined. The parameters r and h are constrainedfrom unitarity fits. The updated fits, taking intoaccount also the lower bound on the mixing-induced

Žmass difference ratio DM rDM )20.4 yield ats d. w x"1s 29 ,

As0.81"0.057, ls0.22

hs0.33"0.065, rs0.11q0 .14 , 24Ž .y0 .11

where l, being very accurately measured, was fixedto the value shown. Note that the allowed range of r

is now asymmetric with respect to rs0 due to thementioned bound on DM rDM , which removess d

large negative-r values. We also note that the recentw xCKM fits reported in 39 yield an identical range for

h but they find rs0.156"0.090, which is morerestrictive for the lower bound on r than the analy-

w xsis in 29 , that we use here.In Table 1, we give the values of the functions

D , D , D and D , evaluated for the central valuest u r iŽ .of the parameters a m , m , m , a and thes Z t b em

semileptonic branching ratio BB . The other twosl

parameters m rm and m are varied as indicated.c b b

We note that the renormalization scale dependenceof D is significantly reduced in the NLL comparedt

to the LL result DŽ0.. As D , D , and D start att u r i

order a only, their m dependence is more signifi-s b

cant, but their contribution to the branching ratio israther small.

For the values of the input parameters givenabove, the theoretical branching ratio for the decayB™ X qg in the SM is calculated by us ass

y4Ž . Ž .BB B™ X qg s 3.50"0.32 =10 . This is tos

Table 1Ž 4 .Values of the NLL functions D , D , D , D divided by l fort u r i

the indicated values of the scale parameter m and the quark massb

ratio m rm . Also tabulated are the values for the LL functionc b

DŽ0.rl4t

m s2.5 GeV m s5 GeV m s10GeV m rmb b b c b

Ž0. 4D rl 0.131 0.106 0.086 0.27tŽ0. 4D rl 0.142 0.114 0.093 0.29tŽ0. 4D rl 0.155 0.125 0.101 0.31t

4D rl 0.150 0.147 0.140 0.27t4D rl 0.155 0.154 0.147 0.29t4D rl 0.161 0.163 0.157 0.31t4D rl 0.015 0.011 0.009 0.27u4D rl 0.016 0.012 0.009 0.29u4D rl 0.016 0.012 0.009 0.31u4D rl y0.033 y0.021 y0.014 0.27r4D rl y0.043 y0.028 y0.019 0.29r4D rl y0.055 y0.036 y0.025 0.31r4D rl 0.056 0.039 0.028 0.27i4D rl 0.062 0.042 0.031 0.29i4D rl 0.068 0.047 0.034 0.31i

w xbe compared with the recent result in Ref. 40 ,where the central value 3.46=10y4 is quoted forthe case in which the factor 1rG is – like in thesl

present work – expanded in a . Taking into account,s< < 2that we use l rV s0.96 in the present CKMt cb

w xframework, whereas in Ref. 40 a value 0.95 wasused for the same quantity, the results here and inw x40 are in agreement. To calculate a in the decayC P

rates for B™X qg and its charge conjugatesŽ .B™ X qg , we shall use the LL approximation 16s

for a . For the central values of the parameters weC PŽ Ž 2 .. Žobtain neglecting corrections of O l : a B™C P. Ž .X qg sy 2.11 h%, which gives for hs0.33"s

0.13 the following prediction for the decay rateŽ . Ž .asymmetry: a B™X qg sy 0.70"0.28 %.C P s

Note that these numbers correspond to the preferredscale m s2.5 GeV. For ms5 GeV, the asymmetryb

Ž . Ž .would be a B ™ X q g s y 0.59 " 0.25 %.C P s

Thus, the direct CP asymmetry in B™ X qg in thes

standard model turns out to be too small to bemeasurable.

We now discuss the decay B™ X qg and thed

CP conjugated process B™X qg . The averagedd² Ž .:branching ratio BB B™X qg strongly dependsd

on the CKM parameters r, h. Taking the centralŽ .values of the parameters a M , m , m rm , m ,s Z b c b t


BB , a and m s2.5 GeV, one obtains the follow-sl em b

ing prediction:

² :BB B™X qgŽ .d

2 2s2.43 1yr qhŽ .y5y0.35 1yr q0.07 =10 ,Ž .

,1.61=10y5

= for r ,h s 0.11,0.33 ,Ž . Ž .or r ,h s 0.107,0.322 . 25Ž . Ž .Ž .

In comparison, the result in the LL approximation² Ž .:for BB B™X qg , for the same values of thed

² Ž .:p a ram e te rs is : BB B ™ X q g sd2 2 y5 ² Ž1.61 1yr qh =10 . This gives, BB B™Ž .

.: y5 Ž . Ž .X q g s 1.45 = 10 for r,h s 0.11,0.33 .d

The difference between the LL and NLL results is;10%, increasing the branching ratio in the NLL

Ž . Ž .case see Fig. 1 for m s2.5 GeV . The scale m -b b² Ž .:dependence of BB B™X qg in the LL andd

NLL accuracy is shown in Fig. 1, fixing all otherparameters to their central values.

In Fig. 2 we give the r dependence of the branch-² Ž .:ing ratio BB B™X qg for hs0.20,0.33 andd

0.46, using m s2.5 GeV and the central values forb

all other parameters. We note that the dependence onh is not very marked. The branching ratio is largest

Žfor the smallest allowed value of r taken here as. Žrsy0.10 and the largest allowed value of h as-

Fig. 2. The r dependence of the average branching ratio forB™ X qg and B™X qg is shown for different values of h:d d

Ž . Ž . Žhs0.46 dashed curve ; hs0.33 solid curve ; hs0.20 dash-.dotted curve . All three curves correspond to m s2.5 GeV and tob

the central values of the input parameters m , m rm , BB , m ,b c b sl tŽ .a and a m . The vertical lines show the "1s range for rem s Z

w xfrom the CKM fits 29 .

.sumed here as hs0.46 , and may reach a value ofy5 ² Ž2.6=10 . The minimum value of BB B™X qd

.: y6g in the SM is estimated as 6.0=10 .Ž . Ž .The ratio R dgrsg in Eq. 20 can be expressed

in terms of the CKM parameters r and h as followsŽ < < 2 .expanding 1r l in powers of l :t

2 2R dgrsg sl 1ql 1y2 rŽ . Ž .

=Du2 2 2 21yr qh q r qhŽ . Ž .Dt

Dr 2q r 1yr yhŽ .Ž .Dt

,0.046 for r ,h s 0.11,0.33 ,Ž . Ž .or r ,h s 0.107,0.322 . 26Ž . Ž .Ž .

Our prediction for the direct CP asymmetryŽ . Ž .a B™X qg , based on the LL result 21 andC P d

for the central values of the input parameters, is:

a B™X qg m s2.5 GeVŽ . Ž .C P d b

0.44hs 2 21yr qhŽ .,0.16 for r ,h s 0.11,0.33 ,Ž . Ž .

or r ,h s 0.107,0.322 . 27Ž . Ž .Ž .The scale dependence of this result is as follows:

Ž . Ž .a m s5 GeV ,0.13 and a m s10 GeV ,C P b C P b

0.12. As argued earlier, we prefer m s2.5 GeV tob

estimate a , as for this choice of the scale the NLLC P

corrections in the decay rates are small.In Fig. 3 we show the h dependence of the direct

CP rate asymmetry for the B™ X qg decay ford

rsy0.10,0.11,0.25 and 0.40, using again m s2.5b

GeV and the central values of all other parameters.Ž . ŽThe smallest value of a B™X qg is 7% forC P d

.rsy0.10 and hs0.20 and may reach as high aŽ .value as 35% for rs0.4 and hs0.46 , as can be

seen in Fig. 3. We want to stress that the r-depen-² Ž .:dence of the branching ratio BB B™X qg andd

Ž .both the r- and h-dependence of a B™X qgC P d

are very marked. Hence, their measurements willhelp to determine these parameters more precisely.

To that end, it is important to estimate the theoret-ical uncertainties in the branching ratio, the ratioŽ .R dgrsg , and direct CP asymmetry in B™X qgd


Ž .Fig. 3. h dependence of the CP rate asymmetry a B™ X qgC P dŽ .for different values of r: r sy0.1 dashed curve ; r s0.11

Ž . Ž . Žsolid curve ; r s0.25 dash-dotted curve , r s0.4 long-short.dashed curve . All four curves correspond to m s2.5 GeV and tob

the central values of the input parameters m , m rm , BB , m ,b c b sl tŽ .a and a m . The vertical lines show the "1s range for hem s Z

w xfrom the CKM fits 29 .

and B™ X qg for given values of h and r. Wed

estimate these theoretical uncertainties by varyingw xthe scale m g 2.5 GeV,10.0 GeV and the input pa-b

rameters in their respective "1s-ranges given ear-lier. The procedure adopted is as follows: Individual

Ž² Ž .:. Ž Ž ..errors D BB B™X qg , D R dgrsg andi d iŽ .D a are estimated by varying each parameter ati C P

a time and the resulting errors are then added inquadrature, much the same way as it has been donefor estimating the theoretical uncertainty in the

Ž .branching ratio BB B™ X qg . As mentioned ear-s

lier, we add an error of "0.1 DŽ0. in the numericalt

estimate of the function D in order to take intor

account long-distance effects generated by interme-diate u-quarks. The resulting theoretical uncertainty

² Ž .:on BB B™X qg from all the sources is shownd

in Fig. 4 as "1s bands for the central valuehs0.33 as a function of r. For a given value of r

Ž² Žand h, the theoretical uncertainty is: D BB B™Xd.:. ² Ž .: Ž .qg r BB B™X qg s" 6y10 % on thed

branching ratio. This is much smaller than the factor² Ž .:4 dispersion in BB B™X qg due to the r andd

h dependence, shown in Fig. 2.Ž .The uncertainty in the ratio R dgrsg is even

² Žsmaller, since the theoretical errors on BB B™Xs.: ² Ž .:qg and BB B™X qg tend to cancel. Thed

residual theoretical uncertainty DRrR is correlatedwith the value of r and h, which is not difficult to

Ž .see from the relation in Eq. 26 . For the central

Fig. 4. r dependence of the average branching ratio for B™ Xd

qg and B™X qg for fixed hs0.33. The solid curve corre-d

sponds to m s2.5 GeV and the central values of the inputb

parameters. The upper and lower dashed curves show the theoreti-cal dispersion due to the errors in the input parameters m ,b

Ž .m rm , BB , m , a m ,a and due to the variation of thec b sl t s Z emw xscale m g 2.5 GeV,10.0 GeV . The long-distance contribution dueb

Žto the u-quark loop is also included in estimating the errors see.text . The vertical lines show the "1s range for r from the

w xCKM fits 29 .

value of the CKM fits hs0.33, this is shown in Fig.Ž .5 where we plot R dgrsg as a function of r.

Interestingly, the theoretical uncertainties almostvanish for r in the proximity of the ’’best fit’’ valuer s 0.11. The largest theoretical uncertaintyŽ .R dgrsg in the 95% C.L. allowed CKM-domain is

Ž .for the point r s 0.4, h s 0.2 whereŽ . Ž .DR dgrsg rR dgrsg s "7%, as the ratio

Ž .R dgrsg is smallest there. This study suggests that

Ž . Ž .Fig. 5. The ratio R dg r sg in % as a function of the CKMparameter r for a fixed value of hs0.33. The bands show thetheoretical uncertainties following from the error estimates dis-cussed in text. The vertical lines show the "1s range for r from

w xthe CKM fits 29 .


Ž .the impact of the measurement of R dgrsg on theCKM parameters will be largely determined by ex-perimental errors.

It is interesting to see with which theoreticalaccuracy the Wolfenstein parameters r and h getconstrained assuming an ideal measurement of R. Toillustrate this, we study the fixed-R contours in theŽ .r,h plane. Our procedure is as follows: We choosethree hypothetical values R s 0.017,0.046,0.074,emerging from our NLL analysis. For each of these

Ž .values, we solve Eq. 26 for h. Fixing all inputŽ .parameters, except r, leads to a curve in the r,h

Ž .plane fixed-R contour . Varying then the input pa-Ž .rameters a m , m , m rm , m and the scale ms z b c b t b

Žone at a time followed by adding the individual. Ž .errors in quadrature , leads to a band in the r,h -

plane for each value of R. In Fig. 6 these bands areshown for the values of R indicated above. Theunitarity triangle corresponding to the ’’best fit’’

Ž .solution rs0.11,hs0.33 is also drawn for orien-tation. One sees again that the theoretical uncertain-

Ž .ties are minimal practically vanishing for the ’’bestfit’’ solution.

In Fig. 7, we show the uncertainty on a due toC P

the scale variation and due to the input parameters asŽ .a function of h with fixed rs0.11 . As mentioned,

the power corrections drop out in the LL approxima-tion. For given values of r and h, we find:Ž . ŽD a ra s"17%. Since the asymmetry a BC P C P C P

Ž .Fig. 6. Fixed-R contours in the r,h plane, obtained by varyingw xthe input parameters and the scale m g 2.5 GeV,10.0 GeV . Theb

Ž .three bands shown in the figure correspond to Rs0.017 bottom ,Ž . Ž .Rs0.046 middle and Rs0.074 top . The unitarity triangle

w xcorresponding to the ’’best fit’’ solution from the CKM fits 29 isalso shown.

Ž .Fig. 7. h dependence of the CP rate asymmetry a B™ X qgC P d

for fixed r s0.11. The solid curve corresponds to m s2.5 GeVb

and the central values of the input parameters. The upper andlower dashed curves show the theoretical dispersion due to the

Ž .errors in the parameters m , m rm , BB , m , a m , a andb c b sl t s Z emw xdue to the variation of the scale m g 2.5 GeV,10.0 GeV . Theb

w xvertical lines show the "1s range for h from the CKM fits 29 .

. Ž™X qg itself varies between 7% and 35% seed.Fig. 3 in the presently allowed range of the parame-

ters r and h, the residual theoretical uncertainty isnot a serious hindrance in testing the CKM paradigmfor CP violation in these decays. Of course, it will benice to complete the calculation for a in the NLLC P

approximation, which hopefully will reduce the theo-retical uncertainty on this quantity considerably.

5. Summary

To summarize, we have presented theoretical esti-² Ž .:mates of the branching ratio BB B™X qg andd

Ž .the ratio R dgrsg in the NLL approximation, andŽ .a B™X qg in the LL approximation in SM,C P d

working out also theoretical errors. Varying theCKM-Wolfenstein parameters r and h in the rangey0.1FrF0.4 and 0.2FhF0.46 and taking intoaccount other parametric dependences stated earlier,our numerical results can be summarized as follows:

y6 ² : y56.0=10 F BB B™X qg F2.6=10 ,Ž .d

0.017FR dgrsg F0.074 ,Ž .0.07Fa B™X qg F0.35 . 28Ž . Ž .C P d

The central values of these quantities correspondingŽ .to the ’’best fit’’ parameters rs0.11, hs0.33

y 5Ž . Ž .are: BB B™ X q g s 1.61 " 0.12 = 10 ,dŽ . Ž . ŽR dgrsg s0.046 and a B™X qg s 11.5yC P d

. Ž .16.5 %, with practically no error on R dgrsg . This


ratio is also otherwise found to be remarkably stableagainst variation in the input parameters, with them a x im u m u n c e r ta in ty e s t im a te d a s

Ž . Ž . ŽDR dgrsg rR dgrsg s"7% for rs0.4, hs.0.2 . These quantities are expected to be measurable

at the forthcoming high luminosity B facilities. TheŽ .CP-violating asymmetry a B™X qg in the SMC P s

is found to be too small to measure. We emphasizethe need to complete the NLL-improved calculation

Ž .for a B™X qg .C P d

Acknowledgements

We would like to thank Francesca Borzumati andDaniel Wyler for helpful discussion. This work waspartially supported by INTAS under the ContractINTAS-96-155.

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11 June 1998


Implications for B™hK and B™glueballqK modesfrom observed large B™h

XKqX

Xiao-Gang He a, Wei-Shu Hou b, C.-S. Huang c

a School of Physics, UniÕersity of Melbourne, Melbourne, Australiab Department of Physics, National Taiwan UniÕersity, Taipei, 10764 Taiwan, ROC

c Institute of Theoretical Physics, Academia Sinica, Beijing, China

Received 11 January 1998Editor: H. Georgi

Abstract

X Ž X .The unexpectedly large branching ratios for B™h K h X decays could be of gluonic origin. We study thesŽ . Ž .implications for B™hK hX and PK PX , where P is the pseudoscalar glueball. In the mechanism proposed bys s

Fritzsch, large branching ratios are predicted for these modes. The B™hK rate is barely within the experimental limit, andB™PK , PX could be at the 0.1% and 1% level, respectively. Smaller but less definite results are found for the mechanisms

of g )™hXg via the gluon anomaly. q 1998 Published by Elsevier Science B.V. All rights reserved.

Large branching ratios for exclusive B™hXK and semi-inclusive B™h

X X decays have been reportedsw xrecently, giving 1

Br B"™hXK " s 7.1q2 .5 "0.9 =10y5 ,Ž . Ž .y2 .1

Br B0 ™hXK 0 s 5.3q2 .8 "1.2 =10y5 ,Ž . Ž .y2 .2

Br B™hX X s 6.2"1.6"1.3 =10y4 2.0 - p - 2.7 GeV . 1Ž . Ž . Ž .Ž .s h ’

Ž .Factorization calculations of four quark operators in the Standard Model SM indicate that exclusive branchingw xratios could be accounted for by choice of form factors 2–4 . However, the four quark operators do not seem

w x w xsufficient for the semi-inclusive decay 2 . Several mechanisms have been proposed 5–10 to explain the latter,w xand ways to distinguish some of the mechanisms have also been proposed 11 . We wish to study here the

w ximplications for gluonic origins 5,6,8 of such enhancements. In particular, we concentrate on the mechanismw xproposed in Ref. 8 , which postulates an ad hoc effective Hamiltonian of the form

mn˜H sa a G s b G G . 2Ž .eff s F L R mn

w xIn the notation of Ref. 8 , a contains a factor of V . Similar effective Hamiltonians can be generatedt sw x Ž X .perturbatively at one loop level in the SM 12 . However, the resulting Br B™h X would be smaller bys

w x Ž .more than two orders of magnitude. It was therefore argued in Ref. 8 that Eq. 2 may arise fromnon-perturbative effects. It may also arise from new physics. It turns out that a single parameter a could account


( )X.-G. He et al.rPhysics Letters B 429 1998 99–105100

X Ž .for both inclusive and exclusive h rates of Eq. 1 . Although the semi-inclusive m spectrum seems to favorXs)

X w x3-body decay over 2-body, which supports the anomaly induced g ™h g mechanism 5,6 , some smearingw x Ž . Ž .effect cannot yet be ruled out 8 . Fitting Eq. 2 to Eq. 1 , we find that h and pseudoscalar glueball P modes

Ž .could then be quite large: Br B™hK is barely within the experimental limit, while B™PK , PX could be ats

the 0.1% and 1% level, respectively.X ˜ mn XŽ . Ž .To calculate Br B™h X from Eq. 2 , one has to project out the G G content in h , i.e. extract thes mn

˜ mn X² < < :matrix element 0 G G h . Using the definitionsmn

hsh cosuyh sinu , hX sh sinuqh cosu ,8 0 8 0

ug g uqdg g dy2 sg g sm 5 m 5 m 58² < < : ² < < :0 j h s 0 h s if p ,m 8 8 8 m'6

ug g uqdg g dqsg g sm 5 m 5 m 50² < < : ² < < :0 j h s 0 h s if p , 3Ž .m 0 0 0 m'3

w xwe obtain 13

1m 8 2² < < : ² < < :0 E j h s f cosu m s 0 i2m ug uq i2m dg dy i4m sg s h ,m 8 h u 5 d 5 s 5'6

1X Xm 8 2² < < : ² < < :X0 E j h s f sinu m s 0 i2m ug uq i2m dg dy i4m sg s h ,m 8 h u 5 d 5 s 5'6

1 3asm 0 2 mn˜² < < : ² < < :0 E j h syf sinu m s 0 i2m ug uq i2m dg dq i2m sg sq G G h ,m 0 h u 5 d 5 s 5 mn' 4p3

1 3a sX Xm 0 2 mn˜² < < : ² < < :X0 E j h s f cosu m s 0 i2m ug uq i2m dg dq i2m sg sq G G h . 4Ž .m 0 h u 5 d 5 s 5 mn' 4p3

Neglecting the small up and down quark masses, we arrive at

3as 3mn 2˜ '² < < :0 G G h s cosu f y 2 sinu f m ,( Ž .mn 8 0 h24p

3as X 3mn 2˜ '² < < : X0 G G h s sinu f q 2 cosu f m . 5( Ž .Ž .mn 8 0 h24p

Ž . Ž . XWe emphasize that Eqs. 3 – 5 do not make explicit assumptions of the quark and gluon content of h and h

Ž .mesons. The parameters u , f and f , however, are still not quite certain. In the following we shall use A8 0w x Ž . w xusy17.08, f rf , f rf s1.06 13 , and B usy22.08, f rf s1.38 and f rf s1.06 14,15 , to8 p 0 p 8 p 0 p

X Ž X .illustrate the sensitivity to h-h parameter values. To account for the observed central value of Br B™h Xsy4 Ž . y1s6.2=10 , one needs a f0.012 0.015 GeV , where the subscript stands for inclusive. The numbers arein

Ž .given for Set A Set B respectively, a notation which we shall employ from now on.Ž . Ž X .Assuming that Eq. 2 also dominates the large exclusive Br B™h K , one finds

2 24p m ymB KX 3y y 2 K 2' X XA B ™h K sa G sinu f q 2 cosu f m F m , 6(Ž . Ž .Ž . Ž .F 8 0 h 0 h23 2 m qmŽ .b s

K Ž 2 . w xXwhere the form factor F m is estimated to be 0.33 16 . With the a values obtained earlier, we find0 h in

( )X.-G. He et al.rPhysics Letters B 429 1998 99–105 101

Ž X . y5Br B™h K ,6.6=10 , which is consistent with experimental data. In turn, we could extract a se xŽ . y1 Ž " X ". Ž . y1 Ž X 0.0.012 0.016 GeV from Br B ™h K , and a s0.011 0.014 GeV from Br B ™h K , which aree x 0

consistent with a .inŽ Ž .. Ž . w xWe can easily obtain Br B™hK hX by using Eq. 5 . If the mechanism of Ref. 8 is the sole source fors

X Ž X . Ž Ž .. Ž X Ž X ..the large B™h K h X branching ratios, then Br B™hK hX is simply related to Br B™h K h Xs s s

by

Br B™hK hX sR2 F Br B™hXK h

X X , 7Ž . Ž . Ž .Ž . Ž .s s

where

2'A B™hK hX cosu f y 2 sinu f mŽ .Ž .s 8 0 hRs s , 8Ž .X X 2' XA B™h K h X mŽ . sinu f q 2 cosu fŽ .s h8 0

and F is a phase space correction factor. For inclusive decays,

2 22 21y m qm rm 1y m ym rmŽ . Ž .(ž / ž /h s b h s bFs , 9Ž .

2 22 2X X1y m qm rm 1y m ym rmŽ . Ž .(ž / ž /h s b h s b

while for exclusive decays

2 22 21y m qm rm 1y m ym rmŽ . Ž .(ž / ž /h K B h K BFs . 10Ž .

2 22 2X X1y m qm rm 1y m ym rmŽ . Ž .(ž / ž /h K B h K B

Ž . Ž . Ž .Note that Eqs. 5 and 8 give R, 0.42 0.69 . This means that the slight numerical change from Set A to SetB results in a factor of 2.7 difference in predictions for h modes. The branching ratio for inclusive B™hX iss

therefore

Br B™hX s1.1 3.0 =10y4 p )2.1 GeV , 11Ž . Ž . Ž .Ž .s h

which is still consistent with the experimental upper limit of 4.4=10y4 at the 90% C.L. for 2.1 GeV-p -h

2.7 GeV. For exclusive decays, using a from above givesin

Br By™hKy s1.2 3.2 =10y5 , 12Ž . Ž . Ž .y5 Ž .which is higher than the experimental limit of 0.8=10 90% C.L. . If Set B is the case, then the Fritzsch

Ž . Xscenario of Eq. 2 would be in grave difficulty. Scaling from exclusive B™h K decay gives similar results. ItŽ . Xis somewhat surprising that the one parameter model of Fritzsch, Eq. 2 , could account for both B™h and

B™h data. Our results underpin the importance of improved experimental measurements in probing thismechanism.

It is interesting to speculate that, since this mechanism invokes the large gluon content of hX, there should be

a large branching ratio for B™PK , where P is the pseudoscalar glueball. To obtain the branching ratio for˜ mn² < < :B™PK , we need to know the matrix element 0 a G G P . This can be obtained by using QCD sums mn

a ˜a 2w x Ž .² < < :rules, as in Refs. 17,18 . Early calculations give 3r4p 0 a G G P f0.87f m for glueball masss mn mn p PŽ Ž . .around 1.4 GeV i.e. taking h 1440 as glueball candidate . Recent lattice calculations suggest, however, that

w xglueball masses are heavier, i.e. m s2.3"0.2 GeV 19 , and the sum rule result could be quite different. WePw xwill follow the analysis in Ref. 18 , but include the h contribution to the low energy spectral density which was

previously neglected.


The basic idea of QCD sum rule analysis in the present case is to match the dispersion relation for theŽ .vacuum topological susceptibility T s with the result found by using the operator product expansion. The

definitions of T and the spectral density r are given by

3as4 i q x a a˜² < < :T s s i d xe 0 T j x j 0 0 , j s G G ,Ž . Ž . Ž .Ž .H ps ps ps mn mn4p

r s sr pole s qr cont s u sys ,Ž . Ž . Ž . Ž .1

23a 2 as spole 2 4 2 cont 2 2r s s f m d sym , r s , 1q5 s 'bs ,Ž . Ž .Ž .Ý i i i ž /ž /4p p pi

3as2 mn mn˜² < < :f m s 0 G G i , 13Ž .i i a a4p

where isP,hX,h, and s is the continuum threshold. To improve series convergence, and to remove subtraction1Ž . ` Ž . ys r M 2

dependence, one makes the following Borel transformations on T s : H Im T s e dsrs and0` Ž . ys r M 2 2H Im T s e ds, with M a free parameter. One gets0

`ys r M 2

epoler s dsŽ .H

s0

2 2ysr Ms e 3a D1 s 6conts r s dsqp D q q . . .Ž .H 4 2ž / ž /s 4p M0

=

2` s 3a 112 2 sysr M pole ysr M conte r s dss e r s dsqp D qO q . . . , 14Ž . Ž . Ž .H H 6 2ž / ž /ž /4p M0 0

a a abc a b c ˜ 2² < < : ² < < : Ž .where D s4 0 G G 0 and D s8 g f 0 G G G 0 . Numerically we use D 'p 3a r4p D4 mn mn 6 s ma a b bm 4,6 s 4,6Ž . y2 4 Ž . y2 6 w xs 1.44"0.61 =10 GeV , 0.25"0.11 =10 GeV 18 respectively. To further simplify, we follow

w x 2Ref. 18 and take M to infinity, leading to

bs2 bs31 12 2 2 2 2 2 2 4 2 4 2 4˜ ˜X X X Xf m q f m q f m s qD , f m q f m q f m s qD . 15Ž .p P h h h h 4 p P h h h h 62 3

Since f X and m X are known, f and m can be determined from knowing what to take for the continuumh,h h ,h P P

threshold s .1

Some information on s can be obtained from comparing with lattice studies, which give m s2.3"0.21 PŽGeV in the quenched approximation. The quenched approximation is equivalent to QCD without quarks but

.lattice studies incorporate running coupling with N s3 . Applying QCD sum rules, we obtain similar results asfŽ . Xin Eq. 15 , except that there are no h and h mesons. Also, in the absence of quarks, there are no light meson

states for glueball to decay into, hence the continuum threshold s in this case would be larger than s . Using0 12 w xthe lattice glueball mass of 2.3 GeV, s is determined to be 7.4 GeV 18 . For the lower bound on s , we0 1

require it to be larger than the glueball mass, since otherwise the sum rule approach breaks down. The glueballmass in QCD need not be the same as the lattice result in quenched approximation. To get an idea on how mP

and f can be determined, we take s to range from 7.4 GeV 2 to 6 GeV 2. Using the two sets of u and fP 1 0,8Ž .discussed earlier and a s0.33 at m scale , we gets P

m s2.42 GeV, f s1.15 1.16 f , for s s7.4 GeV 2 ,Ž .P P p 1

m s2.29 GeV, f s0.92 0.94 f , for s s6.0 GeV 2 . 16Ž . Ž .P P p 1


w xThe difference between Set A and B is small in the given range for s . In Ref. 18 the lower limit for s was1 1

taken to be as low as 3 GeV 2. We consider this to be too low for the following reason. The effect of including h

in the sum rule is not significant for s between 7.4 to 6.0 GeV 2, but cannot be ignored for s , 3 GeV 2. The1 12 Ž 2 . Ž .inclusion of h increases the value for s such that s is larger than 3.24 GeV 3.15 GeV for Set A Set B ,1 1

and below these values there is no physical solution to the sum rule equations. When s is close to the above1

lower bounds, the solution is in any case very sensitive to small changes.Ž .We are now ready to calculate the branching ratio for B™PK PX . Using the glueball masses and decays

Ž .constants in Eq. 16 , we obtain

°1.9 3.2 for m s1.44 GeVŽ . PBr B™PK PXŽ .Ž .s ~ 11 20 for m s2.29 GeVŽ ., 17Ž .PX XBr B™h K h XŽ .Ž .s ¢21 36 for m s2.42 GeVŽ . P

As argued, the low glueball mass value of 1.44 GeV is not plausible. The numerical value for this case is takenw xfrom Ref. 17 for sake of illustration. The branching ratio is very sensitive to m becauseP

˜ mn 2 X X² <Ž . < : Ž . Ž Ž ..0 3a r4p G G P Am . Hence, Br B™PX is at least twice as large as Br B™h K h X , and iss mn P s s

likely as large as 1%, which is truly huge. Since the perturbative estimate for the inclusive b™sg ) penguin isw xitself ,1% 20 , this indicates once again that the Fritzsch mechanism is probably not the sole source for the

X Ž X .large branching ratios for B™h K h X . However, it does suggest that one should search for the pseudoscalars

glueball in B decays, since the scenario is not yet completely ruled out. For exclusive decay, one finds" "Ž . Ž .Br B ™PK ; 0.1% or even higher for m ,2.3 GeV. Taking Br h ™KKp ,6.6% as a guide, oneP c

" Ž .should search for a fast K p ,2.1 GeV recoiling against a KKp system with mass ; 2.2–2.3 GeV.K

Background should be small since both B™D KqX and usual D™KKqX decays are suppressed. Such as

large rate for B™PK should be readily observable if G is not too broad, which is likely the case since thePqq Ž . w x2 glueball candidate, the j 2220 , has a width of only 20 MeV 21 .

Before we conclude, let us comment on the predictions for B™hX and PX for the mechanism proposeds sw x w x Xin Refs. 5,6 . It was proposed in Ref. 5 that the coupling of h to two gluons via the gluon anomaly may be

able to explain the semi-inclusive decay by the process b™sg ) ™sghX within the SM. However, it was

X w xsubsequently pointed out that, because the effective h -g-g coupling arises from the Wess-Zumino term 13,22 ,

N( F mn˜L sa h G G , 18Ž .eff s 0 mn4p f0

it is proportional to a . If this runs with the q2 of the virtual gluon g ) , the branching ratio is reduced by asw x w xfactor of 3 6 . Enhanced bsg coupling due to new physics is then needed 6 , with the tantalizing prospect of

Ž .potentially large CP violation effects. A critical question is the form factor behavior of Eq. 18 . Analogy with2 w xthe photon anomaly suggests 1rk suppression 9 , where k is the difference between the two gluon momenta.

w xHowever, the analogy fails precisely 6 because of binding effects in the two gluon channel, with theŽ . Xnormalization of Eq. 18 fixed by the h-h system at low energy. The study of the off-shell behavior of the

gluon anomaly has yet to begin.Turning back to phenomenology, since one of the gluons appear in the final state, one can no longer use Eq.

Ž . X Ž . Ž X.5 . Using naive h–h mixing as an estimate of the ratio A b™sgh rA b™sgh ; tanu , the inclusiveŽ .Br B™hX could well be below the experimental upper limit. For the exclusive process, as the actuals

X w xtransition is bq™h qsgq and there are no IR singularities, it was noted 6 that if the sgq system evolves intothe K meson when the gluon is soft, it could account for the exclusive B™h

XK rate. However, since it is


difficult to make quantitative predictions, the large exclusive B™hXK branching ratio could well be coming

from other mechanisms, such as from four quark operators in the SM with appropriate form factors.To evaluate the branching ratio for B™PX , we need to estimate the Pgg coupling. For sake of illustrations

w xwe use an Ansatz given in Ref. 23 ,

2cLmn˜L sy P G G , 19Ž .eff mn2k

where k is the difference of the two gluon momenta. For g ) ™Pg where P and g are both on-shell, one has2 2 2 2 Ž .k s2 q ym where q is the virtual gluon momentum. But in the small k limit, Eq. 19 could generateP

w xfinite glueball mass 23 at one loop level, giving

16c2L22m s 4ln2y1 . 20Ž . Ž .P 2p

Of course there should be other contributions to m2 , but saturating m2 with the above relation gives the upperP PŽ . Ž .bound c L s0.59 m , and Eq. 19 becomes1 max P

1.2 m2 PP mn˜L ,y G G . 21Ž .eff mn2 2 m2 q ym PP

Ž .Comparing with Eq. 18 , the most salient feature is the absence of an a factor. It also highlights our ignorancesŽ . X

Xof the form factor behavior of Eq. 18 . We emphasize that m does not arise from h -P mixing, nor from someh

Ž . w xAnsatz analogous to Eq. 19 , but from topological effects through the gluon anomaly 22 . Since the effectiveŽ . X Ž . Ž .P-g-g coupling of Eq. 21 is only slightly larger than the h -g-g coupling of Eq. 18 , we find Br B™PXs

Ž X .;) Br B™h X is likely. Though sizable, the branching ratios for B™PX and PK in the scenario ofs sw x w xRefs. 5,6 are not as large as in the case of Ref. 8 . However, more quantitative predictions in the present case

are difficult, especially for the exclusive case.ŽTo conclude, we have shown that the mechanism proposed by Fritzsch for explaining large Br B™

X Ž X ..h K h X gives large branching ratios for B decays with an h or a glueball in the final state. The predictedsŽ .branching ratio for B™hK is barely within the experimental upper limit, while Br B™PK could be as large

Ž .as 0.1%, with Br B™PX ten times larger. If this is the case, perhaps the pseudoscalar glueball P could bes

first discovered in B decays. Improved experiments will provide decisive information about this mechanism.

Acknowledgements

This work was supported in part by the ARC, the NSC, and the NNSF. X.G.H. acknowledges the support ofK.C. Wong Education Foundation, Hong Kong.

References

w x1 CLEO Collaboration, R. Godang et al., CLNS 97-1522, hep-exr9711010; P. Kim, talk presented at FCNC97, Santa Monica,California, February 1997; J. Alexander, B. Behrens at BCONF97, Honolulu, Hawaii, March 1997; D.H. Miller, talk presented atEPS-HEP97, Jerusalem, Israel, August 1997i; A. Anderson et al., CLEO CONF 97-221, 1997; D.M. Asner et al., CLEO CONF 97-13;J. Smith, talk presented at the 1997 Aspen Winter Physics Conference on Particle Physics, January 1997.

w x2 A. Datta, X.-G. He, S. Pakvasa, hep-phr9707259.w x3 A. Ali, C. Greub, hep-phr9707251.w x4 H.-Y. Cheng, B. Tseng, hep-phr9707316.w x Ž .5 D. Atwood, A. Soni, Phys. Lett. B 405 1997 150.w x6 W.-S. Hou, B. Tseng, hep-phr9705304, to appear in Phys. Rev. Lett.


w x7 I. Halperin, A. Zhitnitsky, hep-phr9705251.w x8 H. Fritzsch, hep-phr9708348.w x9 A. Kagan, A. Petrov, hep-phr9707354.

w x Ž .10 F. Yuan, K.-T. Chao, Phys. Rev. D 56 1997 2495; D.S. Du, C.S. Kim, Y. Yang, hep-phr9711428; M.R. Ahmady, E. Kou, A.Sugomoto, hep-phr9710509.

w x11 D.S. Du, M.Z. Yang, hep-phr9711272.w x Ž . Ž .12 X.-G. He et al., Phys. Rev. Lett. 64 1990 1003; Jiang Liu, Y.-P. Yao, Phys. Rev. D 41 1990 2147.w x Ž . Ž .13 R. Akhoury, J.-M. Frere, Phys. Lett. B 220 1989 258; P. Ball, J.-M. Frere, M. Tytgat, Phys. Lett. B 365 1996 367.` `w x14 E. Venugopal, B. Holstein, hep-phr9710382.w x Ž .15 Particle Data Group, Phys. Rev. D 54 1996 1.w x Ž . Ž .16 M. Bauer, B. Stech, Phys. Lett. B 152 1985 380; M. Bauer, B. Stech, M. Wirbel, Z. Phys. C 34 1987 103.w x Ž . Ž .17 V. Novikov et al., Phys. Lett. B 86 1979 347; K. Senba, M. Tanimoto, Phys. Lett. B 105 1981 297.w x18 G. Gabadadze, Preprint, hep-phr9711380.w x Ž .19 G.S. Bali et al., UKQCD Collaboration, Phys. Lett. B 309 1993 378.w x Ž . Ž .20 W.S. Hou, A. Soni, H. Steger, Phys. Rev. Lett. 59 1987 1521; W.S. Hou, Nucl. Phys. B 308 1988 561.w x Ž .21 J.Z. Bai et al., BES Collaboration, Phys. Rev. Lett. 76 1996 3502.w x Ž . Ž .22 G.M. Shore, G. Veneziano, Nucl. Phys. B 381 1992 2; P.H. Damgaard, H.B. Nielsen, R. Sollacher, Nucl. Phys. B 414 1994 541,

and references therein.w x Ž .23 G.J. Gounaris, Nucl. Phys. B 320 1989 253.

11 June 1998


ž /Final state interaction phases in B™Kp decay amplitudes

D. Delepine, J.-M. Gerard, J. Pestieau, J. Weyers´ ´Institut de Physique Theorique, UniÕersite catholique de LouÕain, B-1348 LouÕain-la-NeuÕe, Belgium´ ´

Received 11 March 1998Editor: P.V. Landshoff

Abstract

A simple Regge pole model for Kp scattering explains the large phase eid between isospin amplitudes which ispŽ . Ž .observed at the D meson mass df . It predicts df148y208 at the B mass. Implications for B™Kp decays and2

extensions of the model to other two-body decay channels are briefly discussed. q 1998 Elsevier Science B.V. All rightsreserved.

1. Introduction

With B-factories forthcoming, detailed checks ofthe precise CP-violation pattern predicted by thestandard model will become possible. However it isby no means trivial to extract reliable information onCP-violation parameters from various B-decaymodes. One of the problems is of course how toestimate ‘‘hadronic effects’’ such as final state inter-

Ž .action FSI phases. Although these phases are of noparticular interest by themselves, they do play animportant role for many potential signals of CPviolation in hadronic B-decays.

The relevant question concerning these FSI phasesis whether they are significantly different from 1 ornot. Clearly the answer to this question depends onthe hadronic channels considered. Here we will focus

Ž .our attention on Kp channels where experimentalw xdata also exist for B decays 1 . There are two

Žisospin invariant scattering amplitudes Is1r2 and.Is3r2 and the quantity one wants to estimate, as a

Ž . Ž . Ž .function of energy, is d s sd s yd s namely3 1

the difference between the S-wave phase shifts in theŽ . Ž .Is3r2 d and Is1r2 d amplitudes. As a3 1

Ž .matter of fact d s has been measured at the D massŽ 2 . w xssm where it is found 2 to be around pr2.D

Naively one does not expect such a huge FSI angleat ssm2 to become negligible at ssm2 but,D B

obviously, a more quantitative argument is calledfor.

The main purpose of this letter is to suggest aRegge model as a general strategy for determining

w xFSI angles 3 . Past experience with p N and KNscattering amplitudes strongly suggests that such amodel should work quite well for Kp scatteringover an energy range which includes the D and Bmeson masses.

The dominant Regge exchanges to consider inKp™Kp scattering are, respectively, the PomeronŽ .P and the exchange degenerate ry f trajectories2

in the t-channel and in the u-channel the exchangedegenerate K ) yK ) ) trajectories. In the next sec-tion we briefly recall a few properties of thesetrajectories and then proceed to show in Section 3


( )D. Delepine et al.rPhysics Letters B 429 1998 106–110´ 107

that with all parameters fixed phenomenologicallyour model automatically accounts for the observed

p2Ž .d m , . From the known energy dependence ofD 2

Ž 2 .Regge trajectories one then readily predicts d mB

close to 20 degrees, namely quite a sizeable FSIangle at the B mass, as naıvely expected. These are¨our main results.

To conclude this note we first comment on obvi-Ž .ous implications of our results for B™Kp decays

and then end with several general remarks on theŽ .parametrization of any quasi two-body decay am-

plitude of the B mesons.

( )2. A Regge model for Kp™Kp scattering am-plitudes

We take s,t,u to be the usual Mandelstam vari-Ž .ables. In the s-channel, Kp™Kp scattering am-

plitudes are linear combinations of the isospin invari-ant amplitudes As and As . In the t-channel1r2 3r2Ž .KK™pp , we have isospin invariant amplitudes

t Ž . t Ž .A isospin 0 and A isospin 1 and, similarly, in0 1Ž . uthe u-channel Kp™p K , we define A and1r2

Au . The relations between these amplitudes are3r2

given by the crossing matrices

11s tA ' A61r2 0

ss t1Až / ž /A3r2 1 1y� 02'6

Au1r21r3 4r3

s . 1Ž .už /y2r3 1r3 Až /3r2

In a Regge model, s-channel amplitudes at highŽ .energy large s are parametrized as sums over

Ž .Regge pole exchanges in the crossed channels: nearŽ .the forward direction t small , t-channel exchanges

Ždominate while near the backward direction t close.to ys or u small , it is the u-channel exchanges

which are relevant.The generic form, at large s, of a Regge pole

exchanged in the t-channel is given by

Ž .a tyip a Ž t .tqe syb t . 2Ž . Ž .ž /sinpa t sŽ . o

Ž . Ž .In Eq. 2 , b t is the residue function, t theŽ .signature ts"1 and

a t sa qaX t 3Ž . Ž .o

Ž .is the linear Regge trajectory with intercept a ando

slope aX; finally s is a scale factor usually taken aso

1 GeV 2. For a Regge pole exchanged in the u-chan-Ž .nel, the generic form is similar to Eq. 2 but with

the variable t replaced by u.Ž .The leading trajectory highest intercept is the

Ž .so-called Pomeron P . It has the quantum numbersŽ .of the vacuum Is0,tsq1 and its exchange de-

scribes ‘‘diffractive scattering’’. The Pomeron al-ways contributes to elastic scattering and describesquite well the bulk of hadronic differential cross-sec-tions over a wide energy range.

In the energy interval which is of interest to ushere, namely

3 GeV 2 QsQ35 GeV 2 4Ž .

a very simple but excellent phenomenologicalparametrization of the Pomeron trajectory and residuefunction is given by

a t s1 5Ž . Ž .P

and

b t sb 0 ebP t 6Ž . Ž . Ž .P P

with

2.5 GeVy2 Qb Q3 GeVy2 7Ž .P

w xobtained from fits 4 to elastic p p, pp and KpŽ .differential cross-sections using factorization . As a

result the Pomeron contribution to At now reads0Ž 2 .s s 1 GeVo

A s ib 0 ebP ts. 8Ž . Ž .P P

The next trajectories to consider are the ry f2

trajectories in the t-channel and the K ) yK ) ) tra-jectories in the u-channel. The r trajectory hasTs1,tsy1 while the f trajectory has Ts0,ts2

q1; similarly the K ) trajectory has Ts1r2,tsy1 while the K ) ) trajectory has the opposite sig-nature. Because of the absence of exotic resonancesŽ .no Kp resonances with Is3r2 , the r and f2

trajectories as well as the K ) yK ) ) ones must be

( )D. Delepine et al.rPhysics Letters B 429 1998 106–110´108

exchange degenerate. Specifically this means that ther and f trajectories coıncide¨2

1a t sa t ( q t 9Ž . Ž . Ž .r f 22

and that their residues are related i.e.

b t b tŽ . Ž .f r2 s . 10Ž .' 26) ) ) ŽSimilarly, for the K yK trajectories in the

Ž . .SU 3 -limit1

) ) )a u sa u ( qu 11Ž . Ž . Ž .K K 2

and

yb ) u sb ) ) u . 12Ž . Ž . Ž .K K

Ž . Ž . Ž . Ž .Eqs. 9 , 10 and Eqs. 11 , 12 guarantee that thenon diffractive imaginary part of As vanishes.3r2

w xThey used to be called ‘‘duality constraints’’ 5 .We neglect lower lying trajectories such as the

XŽ . Ž .r Is1,tsy1 and the f Is0,tsq1 in the0Ž .t-channel as well as their SU 3 partners in the

u-channel. Were we to include them they should alsobe taken as exchange degenerate.

It is customary to write the residue function of ther trajectory as

b tŽ .r

b t s . 13Ž . Ž .rG a tŽ .Ž .Ž Ž .. Ž .Since G a t sinpa t is a very smooth function of

t, no harm is done in using at small t the approxima-tions

'G a t sinpa t fG a 0 sinpa 0 s p ,Ž . Ž . Ž . Ž .Ž . Ž .14Ž .

b t fb 0 , 15Ž . Ž . Ž .r r

and in writing the r trajectory contribution to At as1

b 0Ž .r 0.5qtA s, small t s 1q iexp yip t s .Ž . Ž .Ž .r 'p

16Ž .

An exactly similar reasoning gives for the f trajec-2

tory contribution to At0

A s, small tŽ .f 2

b 0Ž .r3 0.5qts y1q iexp yip t s , 17( Ž . Ž .Ž .2 'p

while for the K ) and K ) ) trajectories contribu-tions to Au one writes1r2

A ) s, small uŽ .K

)b 0Ž .K 0.5qus 1q iexp yip u s , 18Ž . Ž .Ž .'p

A ) ) s, small uŽ .K

)b 0Ž .K 0.5qusy y1q iexp yip u s , 19Ž . Ž .Ž .'p

with3

)b 0 s b 0 20Ž . Ž . Ž .rK 4

Ž .in the SU 3 limit.Putting everything together and using the crossing

Ž .matrices given in Eq. 1 , our Regge model for Kp

scattering is now completely defined by the ampli-tudes

i b 0Ž .rs b t 0.5qtPA s, small t s b 0 e sq sŽ . Ž .1r2 P' '6 2 p

3ib 0Ž .r yip t 0.5qtq e s , 21aŽ .'2 p

b 0Ž .rs 0.5quA s, small u s s , 21bŽ . Ž .1r2 '2 p

and

i b 0Ž .rs b t 0.5qtPA s, small t s b 0 e sy s ,Ž . Ž .3r2 P' '6 p

22aŽ .

b 0Ž .rs 0.5quA s, small u sy s . 22bŽ . Ž .3r2 'p

3. S-wave rescattering phases

The remaining task is now to extract from Eqs.Ž . Ž . Ž .21 , 22 the lls0 partial wave amplitudes a s1r2

Ž .and a s . Neglecting p and K masses, we have,3r2

up to irrelevant real factors

0 sa s A dtA s,t . 23Ž . Ž . Ž .HI Iys

( )D. Delepine et al.rPhysics Letters B 429 1998 106–110´ 109

Ž .From the physical ideas underlying Eqs. 21 ,Ž .22 it is clear that outside the forward and backward

Ž .regions, the integral in Eq. 23 gives a negligiblyŽ .small contribution to a s . We thus writeI

0 0s sa s A dtA s, small t q duA s, small u .Ž . Ž . Ž .H HI I It uo o

24Ž .Ž .With the explicit expressions given in Eqs. 21 ,

Ž . Ž .22 , the integrals in Eq. 24 are trivial to perform.Furthermore, the integrated contributions at the to

Ž 2 .and u boundaries around 1 GeV are considerablyo

smaller than at the boundary 0 of both integrals inŽ .Eq. 24 . Neglecting these contributions, one thus

obtains

i b 0 b 0 1Ž . Ž .rP 1r2a s s sq sŽ .1r2 ' 'b ln s6 pP

3i ln sq ip1r2q b 0 s 25Ž . Ž .r 2 2'2 p ln s qpŽ .

and

i b 0 b 0 1Ž . Ž .rP 1r2a s s sy2 s 26Ž . Ž .3r2 ' 'b ln s6 pP

Ž .Im a sIŽ .from which the tan d s are straightfor-IŽ .Re a sI

Ž . Ž .ward to compute. Note that both tan d and tan d1 3

depend on one single phenomenologically deter-mined parameter namely

'p b 0Ž .Pxs . 27Ž .

b b 0Ž .rP

w xFrom fits 6 to p p, pp and Kp total crossŽ . Žsections in the energy range given in Eq. 4 again

.using factorization , we find

'p b 0Ž .Ps2.9"0.2. 28Ž .

b 0Ž .r

Ž .From Eq. 7 , we thus conclude that x is close toone

xs1.07"0.17. 29Ž .Similar results are obtained using the fits given in

w xRef. 7 for a larger energy range.With these values for x, the range for the FSI

angle at the D mass is calculated to be

d m2 'd m2 yd m2 s 85"6 8 30Ž . Ž .Ž . Ž . Ž .D 3 D 1 D

in spectacular agreement with the recent analysis ofw xCLEO data 2

d m2 s 96"13 8. 31Ž . Ž .Ž .D

We stress that both the analysis of CLEO data andour calculation are based on the quasi-elastic approx-imation.

At the B mass, we predict a sizeable angle closeto 20 degrees, namely

d m2 s 17"3 8. 32Ž . Ž .Ž .B

Ž 2 .Before commenting on our prediction for d m ,B

it may be worthwhile to point out a few facts aboutŽ .our calculation of d s

Ø it is a no-parameter calculation: x is determineddsw xfrom the data on total cross-sections 6 and ’sdt

w x4 ;Ž .Ø in performing our calculation of d s , we have

made several approximations a.o. we neglectedlower trajectories as well as the intermediate re-gion in the S-wave projection integral. Theseapproximations are certainly sound from a phe-nomenological point of view and they becomebetter and better as s increases. At the D masswe do not believe that our end result should betrusted to better than 10–20% but in any case,agreement with the data remains excellent;

Ø the calculations presented here for Kp scatteringcan of course be repeated for pp or KK scatter-ing. A detailed account and discussion of these

w xcalculations will be presented elsewhere 8 . Herewe simply point out that the results of bothcalculations are once again in excellent agreement

w x ppwith the data available 2 at the D mass : d isK Kfound to be around pr3 and d around pr6.

These results considerably strengthen our confi-dence in a simple Regge parametrization ofhadronic scattering amplitudes.

4. Conclusions

The main results of this letter are given by Eqs.Ž . Ž .30 – 32 and can be summarized as follows: aRegge model for Kp scattering explains the largeS-wave rescattering phase difference d observed at

p2Ž .the D meson mass namely d m f , and predictsD 2

Ž 2 .d m f208.B

( )D. Delepine et al.rPhysics Letters B 429 1998 106–110´110

Such a sizeable FSI angle at the B meson massleads to important implications for B™Kp decaysw x w x9 : it invalidates the Fleischer-Mannel bound 10 onthe Cabibbo-Kobayashi-Maskawa angle g and im-

Ž "plies a potentially large CP asymmetry, a, in B ™".Kp decays:

af4 sing %. 33Ž . Ž .Strong interaction hadronic phases can be

Ž .parametrized a la Regge for any quasi two bodyŽdecay mode of the B meson pp , KK as already

) ) .mentioned, but also pr, K p , K r, etc. .The fact that our quasi-elastic treatment of the

scattering amplitudes for Kp ,pp and KK agrees sowell with the data at the D meson mass is a strongargument for neglecting inelastic effects on hadronicphases.

In view of the previous comments, a generalparametrization for all two-body decay modes of theB mesons naturally suggests itself. The decay ampli-tude can be written as a sum of reduced matrix

²² Ž . ::elements BNH N M M , I of the effectiveW 1 2

weak hamiltonian, multiplied by the appropriatehadronic FSI phases edI. These reduced matrix ele-ments are in general complex numbers which can besystematically calculated in terms of tree-level, coloursuppressed, penguin, exchange or annihilation quarkdiagrams. Of course, no isospin violating ‘‘scatteringphases’’ are allowed between these diagrams and

w xfurthermore, as already shown elsewhere 9 , classesof diagrams which would naıvely be excluded can¨

Ž id I .reappear due to factors of the type 1ye . On theother hand, penguin diagrams can provide an absorp-

Ž .tive i.e. imaginary component to the reduced ma-w xtrix elements 11 . But these imaginary parts are very

model-dependent and probably quite small. There-

w xfore we suggest 12 , as a first approximation tosimply ignore these ‘‘quark phases’’ whenever thehadronic phases are sizeable. This was assumed in

w xRef. 9 . This happens to be the case for B™Kp

decays.

Acknowledgements

We are grateful to Christopher Smith and FrankWurthwein for useful discussions and comments.¨

References

w x1 R. Godang et al., CLEO 97-27, CLNS 97r1522, hep-exr9711010.

w x2 M. Bishai et al., CLEO Collaboration, Phys. Rev. Lett. 78Ž .1997 3261.

w x3 For earlier attempts, see e.g. H. Zheng, Phys. Lett. B 356Ž .1995 107; J.F. Donoghue, E. Golowich, A.A. Petrov, J.M.

Ž .Soares, Phys. Rev. Lett. 77 1996 2178; B. Blok, I. Halperin,Ž .Phys. Lett. B 385 1996 324; A.N. Kamal, C.W. Luo, Univ.

of Alberta preprint Thy-08-97, 1997, hep-phr9705396; A.F.Falk, A.L. Kagan, Y. Nir, A.A. Petrov, Johns Hopkins Univ.preprint TIPAC-97018, 1997, hep-phr9712225.

w x Ž .4 R. Serber, Phys. Rev. Lett. 13 1964 32.w x5 See for example, J. Mandula, J. Weyers, G. Zweig, Ann.

Ž .Rev. Nucl. Sc. 20 1970 289.w x Ž .6 V. Barger, R.J.N. Phillips, Nucl. Phys. B 32 1971 93.w x Ž .7 A. Donnachie, P.V. Landshoff, Phys. Lett. B 296 1992 227.w x8 J.-M. Gerard, J. Pestieau, J. Weyers, in preparation.´w x9 J.-M. Gerard, J. Weyers, UCL preprint IPT-97-18, 1997,´

hep-phr9711469.w x10 R. Fleischer, T. Mannel, Univ. of Karlsruhe preprint TTP-97-

17, 1997, hep-phr9704423.w x11 M. Bander, D. Silverman, A. Soni, Phys. Rev. Lett. 43

Ž .1979 242.w x12 For another point of view, see M. Neubert, CERN preprint

THr97-342, 1997, hep-phr9712224.

11 June 1998


Chiral corrections to vector meson decay constants

J. Bijnens a, P. Gosdzinsky b, P. Talavera c

a Dept. of Theor. Phys., UniÕ. Lund, SolÕegatan 14A, S-22362 Lund, Sweden¨b NORDITA, BlegdamsÕej 17, DK-2100 Copenhagen Ou, Denmark

c Dept. de Fısica i Enginyeria Nuclear, UPC, E-08034 Barcelona, Spain´

Received 3 February 1998; revised 10 March 1998Editor: R. Gatto

Abstract

Ž . 3r2We calculate the leading quark mass corrections of order m log m , m and m to the vector meson decay constantsq q q q

within heavy vector meson chiral perturbation theory. We discuss the issue of electromagnetic gauge invariance and theheavy mass expansion. Reasonably good fits to the observed decay constants are obtained. q 1998 Elsevier Science B.V. Allrights reserved.

PACS: 11.30Rd; 12.39.Fe; 13.40.Hq; 14.40.Cs; 13.35.DxKeywords: Chiral symmetry; Chiral perturbation theory; Heavy vector meson theory

1. Introduction

Ž .In this letter our aim is to determine within the heavy meson effective theory HMET the decay constants forw xthe vector nonet. HMET was introduced 1 as the non-relativistic limit of an interacting theory between vector

Ž . Ž . w xmesons heavy mesons and a pseudoscalar meson background light mesons –see also 2,3 . The theory isformulated in terms of operators involving the hadronic fields. The main reason to introduce such anon-relativistic formalism is to recover a well defined power counting in small masses and momenta. In therelativistic theory this is complicated by the presence of the heavy mass, an additional mass parameter that is notsmall and thus spoils naive dimensional counting. This is similar to the heavy baryon chiral perturbation theoryw x4 .

Being a decoupled effective theory, the HMET lagrangian is expanded in operators with increasing order inmomenta and masses with coefficients not restricted by symmetry. In principle these coefficients or low-

Ž .energy-constants LEC’s should be determined directly from QCD. Alternatively we could, obtain them as inthe light meson sector, directly from experiment. However, as in the baryon sector, limited experimentalknowledge together with the rapid increase in the number of LECs when we introduce higher orders in theeffective lagrangian constitute the main drawback of this method for the heavy meson sector. To overcome thisproblem we fit their values to experiment, using Zweig’s rule to limit the number of relevant constants, just as

w xwas done for the vector meson masses in 5 . Our expectation is that the picture emerging after this fit is


( )J. Bijnens et al.rPhysics Letters B 429 1998 111–120112

consistent with perturbation theory, i.e. we will obtain a converging series for the observables under study witha reasonable set of parameters. Here we fit 5 new data with 3 new parameters.

The vector decay constants are experimentally determined through the branching ratios of r 0,v,f™

eqey,mqmy, via an electromagnetic current in the matrix element, or the branching ratios of ty™n ry,n K )y,t t

via the vector part of the weak current.3 ŽFor their study we will classify the terms needed up to order p order 3 in the expansion of small meson

. 0 2masses and momenta . The lowest order is p , and the leading corrections are m log m and m at order p .q q q3r2 Ž 3.Loop diagrams also contribute at this order. We also include the order m or p . Two-loop contributionsq

2 w xstart appearing at order m . We also need some terms classified in 5 and their coefficients as estimated there.q

We first discuss our notation and a few definitions. In Section 3 we discuss how gauge invariance can beused to connect terms at different orders in the heavy mass expansion. Using this we proceed in Section 4 to listall the terms in the Lagrangian that are needed. Here we also describe the slight extension needed for the weakcurrents. Then we give our main results, the vector decay constants up to order p3 in the HMET expansion, firstthe general expressions and then approximate expressions for the vector isospin states. We finally present thenumerical results and compare with the experimental values.

2. Definitions and notation

We define here our notation and the basis of chiral transformations. For a general introduction to chiralw xperturbation theory see 6 .

Ž . Ž .Under SU 3 =SU 3 global chiral rotations the goldstone fields can be collected in an unitary matrixL RŽ .field, U f transforming as:

U f ™g U f g† , g =g gSU 3 =SU 3 1Ž . Ž . Ž . Ž . Ž .L RR L L R

Ž . Ž . Ž .For the chiral coset space SU 3 =SU 3 rSU 3 our choice of coordinates allows us to write:L R V

0p hq qq p K' '2 6

a a 0l f l f p ha a2 y 0U f sexpi su f , s , 2Ž . Ž . Ž .p y q K'F ' '2 2 6

2h0yK K y '6

Ž .where F;F s92.4 MeV and u f transforms asp

y1 y1u f ™g u f h f , g sh f , g u f g , 3Ž . Ž . Ž . Ž . Ž . Ž .R L

Ž . Ž .with h f, g the so-called compensator field, an element of the conserved subgroup SU 3 . In the nonet caseV

the annihilation modes for the effective vector meson fields are collected in a 3=3 matrix given by

0v qrm m q )q, r , Km m'20a v yrW sl w s , 4Ž .m mm a m y ) 0r , , Km m'2

) 0)yK , K , fmm m

( )J. Bijnens et al.rPhysics Letters B 429 1998 111–120 113

while its hermitian conjugate, W †, parametrizes the creation modes. In what follows we are only concerned withm

the transverse components of the W fields, i.e. ÕPWsÕPW † s0, Õ is the chosen reference velocity for them

heavy vector meson. The ‘longitudinal’ component of W can easily be expressed in terms of the transversem

components. This is due to the fact that a massive vector meson has 4 components, but only 3 degrees offreedom. This should be kept in mind in the remainder.

Under chiral symmetry the effective vector fields transform as

W ™h f , g W h† f , g , W † ™h f , g W †h† f , g . 5Ž . Ž . Ž . Ž . Ž .m m m m

As has been already mentioned, our purpose is to compute the vector decay constants in the effective theory.They are defined through the matrix elements:

² :0Nq g q NW sF ´ , 6Ž .i m j W m

for a vector meson state normalized to 1 and with momentum m Õ . m is the mass of the relevant meson andW m W

´ its polarization vector.m

Ž .In order to introduce photons we extend the formalism to weak currents in Section 1 we use the externalw xfield formalism 6 , coupling the pseudoscalar fields to external hermitian matrix fields, a , Õ defined as:m m

r sÕ qa seQAe x t q . . . , l sÕ ya seQAe x t q . . . , 7Ž .m m m m m m m m

Ž . Ž .where Q is the diagonal quark charge matrix, Qs 1r3 diag 2,y1,y1 .The inclusion of the fields a , Õ promotes the global chiral symmetry to a local one, allowing thus to definem m

a covariant derivative and a connection:

1† †D UsE Uy ir Uq iUl , G s u E y ir uqu E y il u . 8Ž .Ž .m m m m m m m m m2

Instead of using the fields r and l , we will use the combinationm m

Q se u†Qu"uQu† , 9Ž .Ž ."

Ž . Ž .†where Q transforms as Q ™h f, g Q h f, g under chiral transformations." " "

Ž .Notice that the insertion of an external field a photon or a weak current does not modify the powerŽ .counting, i.e OO e s1.

3. Gauge and reparametrization invariance

w xIn this section we discuss briefly the constraints stemming from reparametrization invariance 7 and theadditional constraints from gauge invariance on the HMET lagrangian. We will discuss it simply in terms of asingle neutral vector meson and the photon. It is convenient to start from the relation between the relativistic and

w x w xthe effective fields 8 , see also the discussion of 5 :

1m yim ÕP x m im ÕP x †m mV VV s e W qe W qW , 10Ž .Ž .Õ Õ I2m( V

where the subindex Õ denotes the velocity of the heavy meson particle referred to an inertial observer and WÕ

has momenta small compared to m Õ and satisfies ÕPW s0. The longitudinal component is suppressed byV Õ

w x1rm , see Section 4 in 5 . Choosing a second reference frame related to the first one by a LorentzV


transformation we should get the same description of the physics. This fact relates the expression of the vectorfield in both frames:

Õ™wsÕqq , W n seyi mV qP x W n qwnqPW , 11Ž .Ž .Õ w w

q is infinitesimal and satisfies ÕPqs0 since Õ2 sw2 s1. This determines some 1rm coefficients in theVw xchiral expansion, that can not be modified by non-perturbative corrections 7 .

For the photon we also need to consider gauge invariance. The lagrangian has to be invariant under

A x ™A x qE e x , 12Ž . Ž . Ž . Ž .m m m

Ž .where e x is the gauge field. This transformation is defined in a particular frame, for fixed Õ. Relevantmomenta fall around "m Õ and 0, we therefore perform a Fourier decomposition into low and high momentaV

in all the fields entering in the gauge transformation:

yi mV ÕP x ˆ imV ÕP x † Ã x se A x qe A x qA xŽ . Ž . Ž . Ž .m m m m

e x seyi mV ÕP xe x qeimV ÕP xe † x yqe x , 13Ž . Ž . Ž . Ž . Ž .ˆ ˆ ˜

ˆ ˜Ž . Ž . Ž . Ž . w xwhere A x , A x , e x and e x are low momentum fields. As was mentioned in 9 a similar˜ ˆm m

decomposition allows to take properly into account the low momentum component for the vector meson field.Ž . Ž . Ž . Ž .Since the low momentum electromagnetic U 1 is a subgroup of the low momentum SU 3 , the effect ofV

˜Ž .e x and A will be ignored in what follows, it is treated by the covariant derivatives defined above. In˜ m

Ž .contrast, the high momentum of the electromagnetic field, U 1 , is not included in that group. In order to obtaina gauge invariant lagrangian under this high momentum subgroup the electromagnetic field should transform as

ˆ Â x ™A x y im Õ e x qE e x . 14Ž . Ž . Ž . Ž . Ž .ˆ ˆm m V m m

Ž . Ž . Ž .The transformation in Eq. 14 is the equivalent of Eq. 11 because of gauge invariance. As Eq. 11 , itdetermines higher order coefficients in the 1rm expansion.V

For instance, if we take the following toy lagrangian:

† ˆm m ˆ ˆm † n ˆm m n n ˆmLLsW A qa E ÕPA qa ÕPE A qh.c.sW Õ Õ A qa E A qa E A qh.c. , 15Ž . Ž .Ž .Ž . ž /m 1 2 m n 1 2

Ž .a high momentum gauge invariance transformation Eq. 14 implies

i ia sy , a s , 16Ž .1 2m mV V

those coefficients cannot by modified by non-perturbative corrections. Reparametrization invariance requires inŽ .addition the separately gauge invariant term

1n †m n m m nˆ Ê W E A yE A qh.c. 17Ž .Ž .2mV

with the coefficient fixed. Notice that this is precisely the combination of terms that the relativistic termE n V m E nA m yE mAn produces, taking into account ÕPW s0. Therefore, terms that are not gauge invariant atŽ . Õ

first sight can be made gauge invariant by adding terms of higher order in the heavy mass expansion. We willwork in a fixed gauge, ÕPAs0 to avoid this complication.


4. The effective lagrangian

To construct the relevant terms in the lagrangian we use both

ˆÕPW s0 and ÕPAs0 , 18Ž .Õ

Ž .corresponding to the temporal gauge. Together with Eq. 9 , which involves vertices with photons andpseudoscalars, we also use the following building blocks:

x su†x u† "ux †u , u s iu†E Uu†, 19Ž ." m m

Ž .where x contains the quark mass matrix, xs2 B diag m ,m ,m .0 u d s

We now construct the most general structure involving pseudoscalar, vector meson and photon fields whichshould be invariant under Lorentz transformations, chiral transformations, charge conjugation, parity and timereversal.

Ž .Making use of the lowest order equation of motion, ÕPD W s0, which eliminates terms removable bym

field redefinitions, and also using all previous mentioned constraints, we have the following non anomalous² : Ž .lagrangian to leading order in 1rN , with N the number of colors and B s tr B :c c

† ˆm † ˆm² : ² :� 4LL sl W Q A ql W x ,Q A qh.c. 20Ž .1 1 m q 2 m q q

To next order in 1rN :c

† ˆm † ˆm² :² : ² :² :LL sl W Q x A ql W x Q A qh.c. 21Ž .2 3 m q q 4 m q q

Notice that in the three flavour case, terms involving only a trace over Q vanish, so they never appear. Terms"

Ž . Ž 4.at next-to-leading order in N are Zweig rule suppressed so that we can treat the terms in Eq. 21 as OO p .c

At lowest order, the odd intrinsic parity sector of the lagrangian is given by:

† mna b † ˆ mna b² : ² :� 4LL s ig W ,W u Õ e q il W Q ,u A Õ e qh.c. . 22Ž .� 4 ž /3 m n a b 5 m q a n b

Ž . Ž . Ž . 1In Eq. 20 , Eq. 21 and Eq. 22 , all the coupling constants are real numbers .In fact, reparametrization invariance requires the presence of higher order terms proportional to g and l .5

However they only contribute to the vector decay constants at order p4. The connection between a relativisticw xformulation and the present one can be done as in 5,9 but the external fields need to be split up as was done for

the photon field in Section 3.We now present the main features required to incorporate charged weak current effects into our formalism.

The neutral weak current is not phenomenologically relevant at present. To do so one needs to extend the leftŽ .current defined in Eq. 7 to:

e†l seQA q WW T qWW T 23Ž .Ž .m m m q m y'2 sinuW

Ž . qwhere sin u is Weinberg’s angle, WW parametrizes the spin-1 gauge boson fields – it creates an W gaugeW m

1 We need to use C, P and T separately here to prove this. The HMET is not a relativistic field theory so the CPT theorem is not valid. Cconnects W and W T. It is only using both the requirement of a hermitian lagrangian and T , which also connects W with W †, that we canm m m m

conclude that the l are real.i


boson field and destroys an Wy one - and we have introduced the T matrices in terms of the relevant Cabibbo–Kobayashi–Maskawa factors

0 0 00 V Vud u s† V 0 0T s , T sT s . 24ud Ž .0 0 0q y q� 0 � 0V 0 00 0 0 u s

As in the QED sector, the requirement of not breaking chiral symmetry forces us to split the WW field inm

different components in the momenta space according to

yi mV ÕP x ˆ imV ÕP x ˆ † † yimV ÕP x ˆ imV ÕP x ˆ †WW se WW qe WW , WW se WW qe WW . 25Ž .m qm ym m ym qm

The inclusion of the charged currents is now achieved by replacinge

† †ˆ ˆ ˆ ˆeQ A by eQ A q u T WW qT WW u 26Ž .ž /q m q m q ym y qm'2 sinuW

Ž . Ž . Ž .in lagrangians Eq. 20 , Eq. 21 and Eq. 22 . Where now P violation is allowed.

5. Calculation of the vector decay constants

The vector meson leptonic widths are given by2 28pa Fem Wq y 2 2 2 2(G W™ l l s m q2m m y4m 27Ž . Ž .Ž .W l W l53 mW

q y † mˆ< <for a q q hadronic system which decays via W™g™ l l with a eF A W effective coupling. Otherwise,j W miŽ .if a weak decay current is involved i.e. a t lepton decay , the width is determined by

22 2 2 2 2 2 2< <V G F m ym m q2mŽ . Ž .C K M F W t W t WG t™Wn s , 28Ž . Ž .38p m mW t

2 2 2' Ž . wwhere G s 2 e r 8sin u M is the Fermi constant and an effective coupling F erF W WW W† m q q )qˆ'Ž . x2 2 sinu V W WW has been used. V is equal to V or V for t decay to r n or K nW C K M m q C K M ud u s t t

respectively, m is the mass of the vector meson, WW is the weak Wq boson and M s80.3 GeV its mass.W WW

Ž .The only difference with the definition 6 is that we use the electromagnetic current for the neutral vectorbosons.

Ž . Ž .To determine within the effective theory the constants F of Eqs. 27 and 28 one has to compute theW

diagrams depicted in Fig. 1 and in addition, the contribution coming from the vector meson wave functionŽ .renormalization w.f.r. .

Ž .We first define the physical vector meson basis T , to be used in what follows for the external vectore x tŽ Ž 2 2 ..propagators. This is the basis that fully diagonalizes the vector–vector two point Green function G p ,m2 ph y s

Fig. 1. The three effective diagrams contributing to the width. A double line is a vector meson, a dashed line a pseudoscalar meson and thecircled cross a vertex from LL .1,2,3


Fig. 2. One of the relativistic diagrams contributing to the decay constant.

w x Ž 4.at the 1-loop level. This is done using the results of Ref. 5 for G up to OO p . We then compute the w.f.r. as2

defined by 2

Ey1Z s G k ,m N , 29Ž .Ž .V 2 W ph y s onyshel lE kW

3 Ž . 2 2to order p . In Eq. 29 , k sp ym Õ and p sm , k is the momentum in the HMET. The externalWm Wm V m W W Wa Ž .basis, denoted by T a 3=3 flavour matrix is also the one needed for the external vector mesons.e x t

Ž . ŽWe also define an internal vector basis T , used in the inner propagator of the sunrise type diagram lasti n t. Ž 2 .diagram in Fig. 1 which is obtained by diagonalizing the vector–vector propagators to OO p . Similarly, we

Ž M . Ž .define a pseudoscalar meson matrix T where the lowest order mass term p–h has been diagonalized. Thelatter two diagonalizations are required in order to simplify the calculations.

Ž .The contributions from the diagrams in Fig. 1 are then found by direct calculation using Eq. 20 and Eq.Ž .22 . We define

0 0x s x and Q sQ 30Ž .q q q qus0 us0

to be the quantities Q and x with us0. The extension of Q needed for the charged case is defined in Eq.q qŽ . Ž . a26 . The contribution of the tree diagram first diagram of Fig. 1 to the decay constant corresponding to Te x t

is 3

² a 0 : ² a 0 0 :l T Q ql T x ,Q , 31Ž .� 41 e x t q 2 e x t q q

Ž .where as1, . . . ,9. The tadpole type diagram second diagram of Fig. 1 only has the vertex proportional to l1

due to power counting and is given by

l1 a M M † 0² :T T , T ,Q m . 32Ž .Ý e x t q M24FpMs1,8

Ž .Finally, the sunrise type diagram using two vertices of Eq. 22 is written:

4 gl5 c† a M c 0 M †² :² :T ,T T T Q ,T K p ,Dm ,m , 33Ž . Ž .� 4 � 4Ý Ý i n t e x t i nt q e x t c M2FpMs1,8 cs1,9

Ž . Ž . Ž . Ž .where the integrals functions m and K p ,Dm ,m are defined in Eq. 38 . The traces in Eqs. 31 , 32M e x t c MŽ . Ž . Ž . Ž . a a a aand 33 are from expanding the terms in Eqs. 20 , 21 and 22 via W sW T or W sW T the choicem m e x t m m i n t

depends on whether we have an internal or external vector line and on the expansion of Q and x in terms ofqthe pseudoscalar matrices, f M T M.

Ž . Ž . Ž .The full result is given by the sum of Eqs. 31 , 32 and 33 divided by the square root of the relevant Z .V

We have also kept some higher order terms required by reparametrization invariance. This corresponds to usingŽ . w x Ž .instead of the function K p ,Dm ,m the full combination G,V defined in 39 .e x t c M

We have checked that the non-analytical pieces of the relativistic diagram in Fig. 2 are fully recovered by thew xsecond diagram in Fig. 1, as was explicitly shown in 9 for the scalar form-factor case.

2 w xNote that in this way we include the electromagnetic corrections 10 to the masses and mixings.3 0'Ž . Ž .We need to extract a factor of e, respectively er 2 2 sinu V , compared to the definition of Q in 30 to obtain the decayW C K M q

constant F .W


6. Vector decay constants in the unmixed basis

In this section we show the formulae for the contribution coming from the effective diagrams of Fig. 1,where we take the approximation of non-diagonal fields inside T and T , i.e. we use the pure isospin 1 statee x t i nt

for r 0, pure isospin 0 for v and f and the f as the pure strange vector state, i.e. we neglect here r 0 yvyf

mixing.For the tree level contribution we find

'8 2'0F s 2 l q B l m q2mŽ .r 1 0 2 d u3

' '2 8 2F s l y B l m y2mŽ .v 1 0 2 d u3 3

2 16F sy l y B m lf 1 0 s 23 3

F qs2l q8 B m qm lŽ .r 1 0 u d 2

F )qs2l q8 B m qm l . 34Ž . Ž .K 1 0 u s 2

For the tadpole type diagrams, we find

l10 q qF s m q2mŽ .r K p2'2 Fp

l1qF s mv K2'2 Fp

l1qF sy mf K2Fp

l m q m 01 K K 2 2q q 0F s q qm qc m qs mr p p h2 ž /2 2Fp

0 q 0l m m m m2 21 K p h p' ')q qF s m q q q 3 cqs q cy 3 s . 35Ž .Ž . Ž .K K2 ž /2 2 4 4Fp

And finally the contribution coming from the sunrise diagram is

2 2cs s c cs 10 q )qw x w x0F sL q h ,v q q h ,r q K , Kr ½ ž / ž /' ' ' ' '3 6 2 3 2 3 6 3 2

2 2'2 c cs s cs0 ) 0 0 0 0w x w xq K , K q y p ,v q y p ,r 5ž / ž /' ' ' '3 2 3 6 3 2 3 6

2 2 'c cs cs s 1 20 q )q 0 ) 0w x w x w xF sL q h ,v q q h ,r q K , K y K , Kv ½ ž / ž /' ' ' ' ' 39 2 6 6 3 2 3 2

2 2'2 s cs c csq q 0 0 0w x w xq p ,r q y p ,v q y p ,r 5ž / ž /' ' ' '3 9 2 6 3 2 6


2 2L 4c 4 sq )q 0 ) 0 0w x w x w xF sy h ,f y K , K q2 K , K q p ,ff ½ 53 3 3

2 2c 1 1 sq q ) 0 0 )q q 0 qw x w x w x w xqF sL h ,r q K , K q K , K q p ,v q p ,rr ½ 53 2 2 3

2L 1 c 1 1)q q q q 0 q q ) 0w x w x w x w x)qF s ys h , K q K ,v q K ,f q K ,r q K ,r q p , KK 0ž /½ '2 2 2 23

21 s0 )qw xq cq p , K , 36Ž .ž / 5'2 3

with

16 gl5Ls , cosusc and sinuss. 37Ž .2Fp

Ž . Ž . Ž .In addition to Eq. 34 , Eq. 35 and Eq. 36 one has the w.f.r. terms.We have defined the following integrals

ddq q q 1m ni Kg qLÕ Õ sŽ . Hmn m n d 2 2ÕPqyvq ih q ym q ih2pŽ .

d 2 2d q 1 im mim s s ly log , 38Ž .Hm d 2 2 2 2ž /q ym 16p m2pŽ .

Ž .with ls1reygq log 4p q1 and ds4y2e , and

pPÕ m2 EGw xG ,V s 1q y K m ,m ym ypPÕ 39Ž . Ž .G V Vž /m 2m E mV V V

Here h and p stand for the physical pseudoscalar fields.0

7. Numerical results and conclusions

Ž .From the decay widths of Eqs. 27,28 we obtain the experimental results of the second column of Table 1.The Cabibbo-Kobayashi-Maskawa mixing angles, lifetimes, masses and branching ratios are taken from the

w xparticle data book 11 . The naive prediction, using all the pure isospin states, is

' '0 q )q2 F s3 2 F sy3F sF sF . 40Ž .r v f r K

This is satisfied to about 5% for rq,r 0,v, to about 13% for rq, K )q and about 32% for rq,f. The signs weŽ .have fixed to agree with 40 .

w xThe input parameters used are scenarios III,IV and VI from Ref. 5 . III and VI were fitted only using themasses for two different values of g, one high and one low, fit IV also included the ryv mixing in the inputbut otherwise it is similar to fit III. We find a reasonable fit to all the decay constants for reasonable values ofl . The higher order corrections are also reasonable, below 40%. The fit for scenario IV is worse for the1,2,5

w x 4following reason: using the mixings from 5 for the v, including p effects, the contributions at tree level fromthe l term essentially cancel, leaving the loop diagrams of Fig. 1 and l as the main contributions. This makes1 2

the predictions for the v somewhat unstable. We also have a rather large w.f.r. factor for the f. The total size


Table 1Ž . Ž . qThe experimental decay constants and various good fits. The column labeled LO lowest order is Eq. 40 with F as input. For anr

w xexplanation of the scenarios and the values of the other constants see Table 1 in 5 .

Scenario exp LO III IV VI3r2Ž . Ž . Ž . Ž .100l GeV – 6.42 6.97 36 7.55 39 7.74 381y1r2Ž . Ž . Ž . Ž .100l GeV – 0.00 y0.26 50 0.40 49 1.21 6921r2Ž . Ž . Ž . Ž .100l GeV – 0.00 3.56 75 5.12 71 7.24 1.435

3r2Ž . Ž .0F GeV 0.0946 23 0.0908 0.0951 0.0955 0.0937r3r2Ž . Ž .F GeV 0.0287 5 0.0303 0.0286 0.0278 0.0282v

3r2Ž . Ž .y F GeV 0.0565 10 0.0428 0.0563 0.0544 0.0534f3r2Ž . Ž .qF GeV 0.1284 10 input 0.1275 0.1234 0.1280r

3r2Ž . Ž .)qF GeV 0.1447 45 0.1284 0.1460 0.1571 0.1554K

Ž .of the higher order corrections can be judged by comparing the results from Eq. 40 , column LO, with those ofthe three scenarios.

< theo e x p <2

What we have minimized in order to get the values of l in Table 1 is F sÝ F rF y1 .Ž .1,2,5 f i t Ws1,5 W W

The error of the l corresponds to changes in F by about 0.01, i.e. at most 10% for an individual F ,i f i t W

minimizing the other two l simultaneously.i

In conclusion, we have calculated the corrections to the vector decay constants in heavy vector meson chiralperturbation theory and found acceptable fits to all the measured ones. We have determined 5 observables interms of 3 parameters.

Acknowledgements

PG acknowledges a grant form the Spanish Ministry for Education and Culture. PT received support fromCICYT research project AEN95-0815.

References

w x Ž .1 E. Jenkins, A. Manohar, M. Wise, Phys. Rev. Lett. 75 1995 2272.w x Ž .2 H. Davoudiasl, M. Wise, Phys. Rev. D 53 1996 2523.w x3 C-K. Chow, S-J. Rey, hep-phr9708432.w x Ž .4 E. Jenkins, A. Manohar, Phys. Lett. B 255 1991 558.w x Ž .5 J. Bijnens, P. Gosdzinsky, P. Talavera, Nucl. Phys. B 501 1997 495.w x Ž .6 G. Ecker, Prog. Part. Nucl. Phys. 35 1995 1; J Gasser, Aspects of Chiral Dynamics, hep-phr9711503.w x Ž .7 M. Luke, A.V. Manohar, Phys. Lett. B 286 1992 348.w x Ž .8 H. Georgi, Phys. Lett. B 240 1990 447.w x Ž .9 J. Bijnens, P. Gosdzinsky, P. Talavera, JH EP01 1998 014.

w x Ž .10 J. Bijnens, P. Gosdzinsky, Phys. Lett. B 388 1996 203.w x Ž .11 Review of Particle Physics, R.M. Barnett et al., Phys. Rev. D 54 1996 1

11 June 1998


A modified Balitskii-Fadin-Kuraev-Lipatov equationwith unitarity

Hsiang-nan LiDepartment of Physics, National Cheng-Kung UniÕersity, Tainan, Taiwan, ROC

Received 17 February 1998Editor: H. Georgi

Abstract

We propose a modified Balitskii-Fadin-Kuraev-Lipatov equation from the viewpoint of the resummation technique,which satisfies the unitarity bound. The idea is to relax the strong rapidity ordering and to consider the variation of the gluondistribution function with the momentum fraction, when evaluating the BFKL kernel. It is shown that the power-law rise ofthe gluon distribution function with the small Bjorken variable x turns into a logarithmic rise at x™0. q 1998 Published byElsevier Science B.V. All rights reserved.

PACS: 12.38.Cy; 11.10.Hi

Ž . w x Ž .It is known that the Balitskii-Fadin-Kuraev-Lipatov BFKL equation 1 sums leading logarithms ln 1rx , xbeing the Bjorken variable, produced from reggeon ladder diagrams with rung gluons obeying the strongrapidity ordering. The gluon distribution function, governed by this equation, is found to increase at small x,which has been confirmed by the recent HERA data of the proton structure function F involved in deep2

Ž . w xinelastic scattering DIS 2 . However, the increase is power-like, such that F and the DIS cross section s2 tot

rise as a power of x, i.e., as a power of s for xsQ2rs, Q being the momentum transfer and s the total energy.This behavior does not satisfy the Froissart bound s Fconst.= ln2s, and violates unitarity. Hence, the BFKLtot

equation can not be the final theory for small x physics. Though it has been expected that the inclusions ofŽ . w x Ž . w xnext-to-leading ln 1rx 3 and of higher-twist effects from the exchange of multiple ladders pomerons 4

w xmay soften the BFKL rise, the attempts have not yet led to a concrete conclusion. In 5 a renormalization groupŽ .RG improved double factorization formula was developed, and the obtained structure functions satisfy theunitarity bound. However, the logarithmic rise appears as the consequence of the RG, instead of BFKL,evolution.

In this letter we shall propose a modified evolution equation appropriate for small x from the viewpoint ofw xthe Collins-Soper-Sterman resummation technique 6 , which satisfies the unitarity bound. This technique was

developed originally for the organization of double logarithms ln2 Q, and applied recently to the all-orderw xsummations of various single logarithms contained in parton distribution functions 7 . The known evolution

Ž . w xequations, such as the Dokshitzer-Gribov-Lipatov-Altarelli-Parisi DGLAP equation 8 , which sums lnQ, thew x Ž . w xBFKL equation 1 , and the Ciafaloni-Catani-Fiorani-Marchesini CCFM equation 9 , which embodies the


( )H.-n. LirPhysics Letters B 429 1998 121–126122

w xabove two equations, can be derived in a simple and unified way 7 . It was observed that the BFKL rise isattributed to real gluon emissions. We point out that the assumption of the strong rapidity ordering employed inthe evaluation of the BFKL kernel, however, overestimates these contributions, and the predicted structurefunction grows as a power of x. By relaxing the strong rapidity ordering, the BFKL rise will be moderated intoa logarithmic rise.

w xWe review the derivation of the BFKL equation using the resummation technique 7,10 . The unintegratedŽ .gluon distribution function F x,k , describing the probability of a parton carrying longitudinal momentumT

kqsxpq and transverse momenta k , is defined byT

1 dyy d2 y q yT 1yi Ž x p y yk P y . q y mqT T ² < < :F x ,k s e p ,s F y , y F 0 p ,s , 1Ž . Ž . Ž .Ž .ÝH HT m T2qp 2p 4ps

< : m q mqin the axial gauge nPAs0. The ket p,s denotes the incoming proton with light-like momentum p sp d

and spin s . An average over color is understood. Fq is the field tensor. To implement the resummationm

Ž 2 .technique, we allow n to vary arbitrarily n /0 . It will be shown that the BFKL kernel turns out to ben-independent, i.e., gauge invariant. After deriving the equation, n can be chosen in such a way that makes thedefinition of the gluon distribution function coincide with the conventional one. That is, the vector n appearsonly at the intermediate stage of the derivation and as an auxiliary tool. Note that we do not specify the

Ž .renormalization scheme in which Eq. 1 is given. The renormalization schemes are not essential here, since allof them are equivalent in the sense that the distribution functions defined in different schemes contain the samenonperturbative information and exhibit the same evolution features. Hence, we can work in any renormaliza-tion scheme.

The standard procedure of the resummation is to obtain the derivative pqdFrdpq first. For a fixed partonmomentum kq, F varies with pq implicitly through xskqrpq. That is, the derivative of F with respect to xis related to the derivative with respect to pq. Because of the scale invariance in n of the gluon propagator

mn Ž . 2yiN l rl with

n mln qnn l m l mln

mn mn 2N sg y qn , 2Ž .2nP l nP lŽ .q Ž .2 2 q qF must depend on p via the ratio pPn rn . Hence, there exists a chain rule relating p drdp to drdn

w x w x6 . Combining the above arguments, we have 7,10

d d n2 dqyx Fsp Fsy Õ F , 3Ž .bqdx dp ÕPn dnb

Ž .Õ sd being a vector along p. To derive Eq. 3 , we need only the fact that F depends on the vectors p andb bqn, instead of the rigorous definition of F, as stated above. The operator drdn applying to a gluon propagatorb

gives

d 1X X Xnn n bn n nbN sy l N q l N . 4Ž . Ž .

dn nP lb

n Ž nX.The loop momentum l l contracts with a vertex in F, which is then replaced by a special vertex

2 Ž . Ž . Ž .Õ sn Õ r ÕPnnP l . This special vertex can be read off the combination of Eqs. 3 and 4 .b b

The contraction of ln leads to the Ward identity,am lg ag agyiN kq l yiN k yiN k yiN kq lŽ . Ž . Ž . Ž .

nl G syi y , 5Ž .mnl2 2 2 2k kkq l kq lŽ . Ž .G being the triple-gluon vertex. Summing all the diagrams with different differentiated gluons, thosemnl

( )H.-n. LirPhysics Letters B 429 1998 121–126 123

embedding the special vertices cancel by pairs, leaving the one in which the special vertex moves to the outerw xend of the parton line 6 . We arrive at

dyx F x ,k s2 F x ,k , 6Ž . Ž . Ž .T Tdx

Ž .described by Fig. 1 a , where the new function F contains one special vertex represented by a square. Thecoefficient 2 comes from the equality of the new functions with the special vertex on either of the two partonlines.

We then factorize the subdiagram containing the special vertex out of F. It is known that factorization holdsonly in leading regions. The leading regions of the loop momentum l flowing through the special vertex are soft

2 w xand hard, since the factor 1rnP l with n /0 in Õ suppresses collinear enhancements 6 . For soft and hard l,b

Ž . Ž .the subdiagram is factorized according to Figs. 1 b and 1 c , respectively. A straightforward evaluation showsŽ . qthat the contribution from the first diagram of Fig. 1 c is less important in the considered region with small k .

Ž . Ž .Therefore, we drop Fig. 1 c , and concentrate on Fig. 1 b . The color factor is extracted from the relationŽ .f f syN d , where the indices a,b, . . . have been indicated in Fig. 1 b , and N s3 is the number ofabc b dc c ad c

colors. The factorization formula is written as4 lgd l yiN xpŽ .

2 nbF x ,k s iN g G Õ yiN lŽ . Ž .ˆHs T c mnl b4 y2 xpP l2pŽ .

=

2 2u k y lŽ .T T2 q q < <2p id l F xq l rp , k q l q F x ,k , 7Ž . Ž . Ž .Ž .T T T2l

Ž .where iN comes from the product of the overall coefficient yi in Eq. 5 and the color factor yN extractedc c

above. The triple-gluon vertex for vanishing l is given by

G syg xp yg xp q2 g xp . 8Ž .mnl mn l nl m lm n

The denominator y2 xpP l is the consequence of the eikonal approximation for the gluon propagator,Ž .2 Žxpy l fy2 xpP l. The first term in the brackets corresponds to the real gluon emission, where F xqq q < <. q ql rp , k q l implies that the parton coming out of the proton carries the momentum components xp q lT T

and k q l in order to radiate a real gluon of momentum l. The second term corresponds to the virtual gluonT T

emission, where the u function sets the upper bound of l to k to ensure a soft momentum flow.T T

It can be shown that the contraction of p with a vertex in the quark box diagram the partons attach, or with aŽ .2vertex in the distribution function, leads to a contribution down by a power 1rs, ss pqq , compared to the

Ž .contribution from the contraction with Õ . Eq. 7 then becomesb

4 2 2d l Õ Õ u k y lˆ Ž .b n T T2 nb 2 q q < <F x ,k s iN g N l 2p id l F xq l rp , k q l q F x ,k .Ž . Ž . Ž . Ž .Ž .Hs T c T T T4 2ÕP l l2pŽ .9Ž .

Ž q. Ž .The eikonal vertex Õ comes from the last term xp divided by xp in Eq. 8 , and the eikonal propagatorn n

Ž . Ž . Ž . Ž .Fig. 1. a The derivative yxdFrdx in axial gauge. b The soft structure and c the ultraviolet structure of the O a subdiagrams

containing the special vertex.


Ž . Ž . mg1rÕP l from 1r xpP l , which is represented by a double line in Fig. 1 b . The remaining metric tensor g hasbeen absorbed into F.

To derive the conventional BFKL equation, we simply assume the strong rapidity ordering, xq lqrpq4x,

Ž q q < <. Ž < <.for the real gluon emission, namely, approximate F xq l rp , k q l by its dominant value F x, k q l .T T T T

The integrations over ly and lq to infinity lead to2 q 2

`a d l 2 l ns T q 2 2< <F x ,k s dl F x , k q l yu k y l F x ,k , 10Ž . Ž . Ž .Ž . Ž .H Hs T T T T T T2y q2 q 22 p 0 2n l qn lŽ .T

2 2a d l yn qs T l s` 2 2q< < <s F x , k q l yu k y l F x ,k , 11Ž . Ž .Ž . Ž .H l s0 T T T T Ty y q2 q 22 p 2n 2n l qn lŽ .T

q yŽ .where a sN a rp , and ns n ,n ,0 has been assumed for convenience. The second expression of thes c sŽ .above formulas demonstrates how n disappears from the BFKL kernel. Substituting FfF into Eq. 6 , wes

obtain2dF x ,k d lŽ .T T 2 2< <sa F x , k q l yu k y l F x ,k , 12Ž . Ž .Ž . Ž .Hs T T T T T2dln 1rx p lŽ . T

which is the BFKL equation.Ž .As stated before, the BFKL Eq. 12 predicts a power-law rise for the gluon distribution function at x™0,

which violates unitarity. We point out that the assumption of the strong rapidity ordering may be the cause forq Ž q q < <. Ž < <.the unitarity violation. For most values of l , F xq l rp , k q l is much smaller than F x, k q l , andT T T T

replacing the former by the latter in the whole integration range of lq overestimates the contribution from realgluon emissions. Hence, we shall not adopt the assumption, and derive a new evolution equation that satisfies

Ž q q < <.the unitarity bound. Reexpressing F xq l rp , k q l asT T

q q< < < < < <F x , k q l q F xq l rp , k q l yF x , k q l , 13Ž .Ž . Ž . Ž .T T T T T T

Ž .Eq. 10 becomes2a d ls T 2 2< <F x ,k s F x , k q l yu k y l F x ,kŽ . Ž .Ž . Ž .Hs T T T T T T22 p lT

2 q 2`a d l 2 l ns T q q q < < < <q dl F xq l rp , k q l yF x , k q l . 14Ž .Ž . Ž .H H T T T T2y q2 q 22 p 0 2n l qn lŽ .T

To maintain the gauge invariance of the evolution equation, we drop the l dependence of the arguments of FT

in the last term, and perform the integration over l . Using the variable change jsxq lqrpq, the modifiedT

equation is given by2dF x ,k d l F j ,k yF x ,kŽ . Ž . Ž .1T T T T2 2< <sa F x , k q l yu k y l F x ,k q2a dj .Ž .Ž . Ž .H Hs T T T T T s2dln 1rx jyxp lŽ . xT

15Ž .Ž .where the upper bound of j is introduced by the vanishing of F at the momentum fraction js1. Eq. 15 ,

resembling the mixture of the BFKL and DGLAP equations, can be regarded as a new evolution equation.Obviously, the last term represents the correction to the conventional BFKL equation from relaxing the strongrapidity ordering.

Ž .We shall show that Eq. 15 moderates the BFKL power-law rise into a logarithmic rise at small x. An initialŽ . Ž0.Ž . Ž .condition F x ,k sF x ,k must be assumed when solving Eq. 15 , x being the initial momentum0 T 0 T 0

( )H.-n. LirPhysics Letters B 429 1998 121–126 125

fraction below which F evolves according to the BFKL equation. For instance, a ‘‘flat’’ gluon distributionw xfunction 11

Ng 5Ž0. 2 2F x ,k s3 1yx exp yk r 1 GeV , 16Ž . Ž . Ž . Ž .T T21 GeVŽ0.Ž .N being a normalization constant, has been adopted. Hence, the initial function F j ,k should beg T

Ž . Ž .substituted for F j ,k in Eq. 15 as j)x .T 0Ž 2 2 . w xTo simplify the analysis, the u function for the virtual gluon emission is replaced by u Q y l 10 , where0 T

the parameter Q can be determined from data fitting. This modification is acceptable, because the virtual gluon0

contribution only plays the role of a soft regulator for the real gluon emission, and setting the cutoff of l to QT 0Ž .serves the same purpose. Fourier transformation of Eq. 15 into the b space conjugate to k givesT

˜ ˜ ˜dF x ,b F j ,b yF x ,bŽ . Ž . Ž .1˜sS b F x ,b q2a 1rb dj , 17Ž . Ž . Ž . Ž .Hsdln 1rx jyxŽ . x

with

S b sy2a 1rb ln Q b qgy ln2 , 18Ž . Ž . Ž . Ž .s 0

K being the Bessel function, and g the Euler constant. The argument of a has been set to the natural scale0 s

1rb.˜ ˜ ylŽ .Before solving Eq. 17 , we extract the behavior of F analytically. Substituting a guess FAx into Eq.

Ž .17 , l being a parameter, we obtainyl

jrx y1Ž .1lsSq2a dj . 19Ž .Hs

jyxx

˜It can be numerically verified that a solution of l, 0-l-S, exists for x-x . That is, F increases as a power0

of x, consistent with the results from the conventional BFKL equation. However, the correction term diverges˜Ž .as ln 1rx at x™0, and no solution of l is allowed, implying that F can not maintain the power-law rise at

˜ < Ž .extremely small x. We then try another guess F A ln 1rx with a milder rise. It is found that thex ™ 0˜ ˜Ž . Žcorrection term is of the same order of the first term SFA ln 1rx the derivative term yxdFrdxA1 is

˜. Ž . <negligible , and Eq. 17 holds approximately. At last, we assume F A const. as a test. In this trial the firstx ™ 0

term becomes dominant, and the correction term and the derivative term vanish, i.e., no const./0 exists. These˜ Ž .observations indicate that F should rise as ln 1rx at most when x approaches zero. We conclude that the

˜modified evolution equation predicts a rapid power-like rise of F for x-x and a milder logarithmic rise at0

x™0.The behaviors of other quantities can be deduced. The gluon density, defined by

2`d kQ T2 ˜xg x ,Q s F x ,k s2Q dbJ Qb F x ,b , 20Ž . Ž . Ž . Ž .Ž . H HT 1

p0 0

possesses a similar dependence on x, where J is the Bessel function. The structure function F is written, in1 2

terms of the k -factorization theorem, asT

dj d2 k1 T2F x ,Q s H xrj ,k ,Q F j ,k , 21Ž . Ž . Ž .Ž . H H2 T Tj px

where the hard scattering subamplitude H denotes the contribution from quark box diagrams. Because ofŽ . Ž . 2Ž .F j ,k F ln 1rj at j™0, F rises as ln 1rx at most as indicated by the j integration, and thus satisfiesT 2

the unitarity bound.w xWe present the explicit numerical results of the gluon density xg in Fig. 2 with N f3 and Q f0.3 10 ,g 0

which, obviously, behave in the way stated above. The deviation of the curve from a straight line at x™0 is theconsequence of relaxing the strong rapidity ordering. The comparison of our predictions for F with the2


Fig. 2. The dependence of xg on x.

w x Ž .experimental data 2 and the connection of the modified evolution Eq. 15 to the other approaches in theliterature will be published elsewhere.

Acknowledgements

This work was supported by the National Science Council of R.O.C. under the Grant No. NSC87-2112-M-006-018.

References

w x Ž . Ž .1 E.A. Kuraev, L.N. Lipatov, V.S. Fadin, Sov. Phys. JETP 45 1977 199; Ya. Ya. Balitskii, L.N. Lipatov, Sov. J. Nucl. Phys. 28 1978Ž .822; L.N. Lipatov, Sov. Phys. JETP 63 1986 904.

w x Ž . Ž .2 ZEUS Collaboration, M. Derrick et al., Z. Phys. C 65 1995 379; H1 Collaboration, T. Ahmed et al., Nucl. Phys. B 439 1995 471.w x Ž . Ž .3 V.S. Fadin, L.N. Lipatov, Nucl. Phys. B 406 1993 259; A.R. White, Phys. Lett. B 334 1994 87; C. Coriano, A.R. White, Nucl.´

Ž .Phys. B 451 1995 231.w x Ž .4 A.H. Mueller, B. Patel, Nucl. Phys. B 425 1994 471.w x5 R.D. Ball, S. Forte, Report No. hep-phr9703417; hep-phr9706459.w x Ž . Ž .6 J.C. Collins, D.E. Soper, Nucl. Phys. B 193 1981 381; H-n. Li, Phys. Rev. D 55 1997 105.w x Ž .7 H-n. Li, Phys. Lett. B 405 1997 347.w x Ž . Ž .8 V.N. Gribov, L.N. Lipatov, Sov. J. Nucl. Phys. 15 1972 428; G. Altarelli, G. Parisi, Nucl. Phys. B 126 1977 298; Yu. L.

Ž .Dokshitzer, Sov. Phys. JETP 46 1977 641.w x Ž . Ž .9 M. Ciafaloni, Nucl. Phys. B 296 1988 49; S. Catani, F. Fiorani, G. Marchesini, Phys. Lett. B 234 1990 339; Nucl. Phys. B 336

Ž . Ž .1990 18; G. Marchesini, Nucl. Phys. B 445 1995 49.w x Ž .10 H-n. Li, Report No. NCKU-HEP-97-04, to appear in Phys. Lett. B.; Report No. hep-phr9703328 unpublished .w x Ž .11 J. Kwiecinski, A.D. Martin, P.J. Sutton, Phys. Rev. D 53 1996 6094.´

11 June 1998


BFKL pomeron in the next-to-leading approximation

V.S. Fadin a, L.N. Lipatov b

a Budker Nuclear Physics Institute and NoÕosibirsk State UniÕersity, NoÕosibirsk 630090, Russiab Petersburg Nuclear Physics Institute and St. Petersburg State UniÕersity, Gatchina 188350, Russia

Received 16 February 1998; revised 6 April 1998Editor: P.V. Landshoff

Abstract

We find one-loop correction to the integral kernel of the BFKL equation for the total cross section of the high energyscattering in QCD and calculate the next-to-leading contribution to anomalous dimensions of twist-2 operators near js1.q 1998 Published by Elsevier Science B.V. All rights reserved.

'The BFKL equation is very important for the theory of the Regge processes at high energies s in thew x w xperturbative QCD 1 . In particular, it can be used together with the DGLAP evolution equation 2 for the

description of structure functions for the deep inelastic ep scattering at small values of the Bjorken variable2 Ž .xsyq r 2 pq , where p and q are the momenta of the proton and the virtual photon correspondingly. But up

to recent years the integral kernel for the BFKL equation was known only in the leading logarithmicŽ .approximation LLA , which did not allow one to find its region of applicability, including the scale in

2Ž .transverse momenta fixing the argument of the QCD coupling constant a ck and the longitudinal scale s(H 0

for the minimal initial energy. In this paper we calculate the QCD radiative corrections to this kernel.In LLA the gluon is reggeized and the Pomeron is a compound state of two reggeized gluons. One can

neglect multi-gluon components of the Pomeron wave function also in the next-to-leading logarithmicŽ . Ž .approximation NLLA and express the total cross-section s s for the high energy scattering of colourless

Ž . Ž X.particles A, B in terms of their impact factors F q and the t-channel partial wave G q,q for the reggeizedi i v

gluon scattering at ts0:

X v2 2d q d q dv saqi`X X

s s s F q F q G q ,q . 1Ž . Ž . Ž . Ž . Ž .H H HA B vXX2 2 ž /2p i q q2p q 2p q ayi`

Here q and qX are transverse momenta of gluons with the virtualities yq2 'yq2 and yqX 2 'yqX 2

correspondingly, ss2 p p is the squared invariant mass of the colliding particles with momenta p and p .A B A B

Note, that the dependence of the Regge factor from q and qX is natural from the point of view of theWatson-Sommerfeld representation for high energy scattering amplitudes. The change of the energy scale in thisfactor leads generally to the corresponding modification of the impact factors and the BFKL equation for G butv

the physical results are not changed.


( )V.S. Fadin, L.N. LipatoÕrPhysics Letters B 429 1998 127–134128

Using the dimensional regularization in the MS-scheme to renormalize the QCD coupling constant and toremove infrared divergencies in the intermediate expressions, we write the generalized BFKL equation for

Ž X.G q,q in the following formv

v G q ,qX sd Dy 2 qyqX q d Dy 2q K q ,q G q ,qX . 2Ž . Ž . Ž . Ž . Ž .˜ ˜ ˜Hv v

Here

K q ,q s2 v q d ŽDy2. q yq qK q ,q , 3Ž . Ž . Ž . Ž . Ž .1 2 1 1 2 r 1 2

Ž .and the space-time dimension is Ds4q2´ for e™0. The gluon Regge trajectory v q and the integralŽ .kernel K q ,q , related with the real particle production, are expanded in the series over the QCD couplingr 1 2

constant

v q sv q qv Ž2. q q . . . , K q ,q sK B q ,q qK Ž1. q ,q q . . . , 4Ž . Ž . Ž . Ž . Ž . Ž . Ž .B r 1 2 r 1 2 r 1 2

where the Born expressions, corresponding to LLA, are

2 2 y2 ´2 q 4 g m 1m2 Bv q syg q2ln , K q ,q s . 5Ž . Ž . Ž .B m r 1 22 1q´ 2ž /´ m p G 1y´Ž . q yqŽ .1 2

Here

2g N G 1y´Ž .m c2g sm 2q´4pŽ .

Ž .for the colour group SU N and g is the QCD coupling constant fixed at the normalization point m in thec m

MS-scheme.The program of calculating next-to-leading corrections to the BFKL equation was formulated several years

w xago 3 . It was shown, that the corrections can be expressed through the Born production amplitudes in thequasi-multi-regge kinematics with the use of the unitarity conditions in the s and t channels. These amplitudeswere constructed in terms of various reggeon-particle vertices. The gauge-invariant action containing all such

w xvertices was formulated recently 4 .Ž2.Ž . w xThe two-loop correction v q to the gluon Regge trajectory is known 5 and for massless quarks can be

written as follows

2 2 211 2 n 1 q 67 p 10 n 1 q 404f fŽ2. 4 2v q syg y y ln q y y q2ln yŽ . m 2 2 2ž / ž / ž /ž /ž / ž /3 3 N 9 3 9 N ´ 27´ m mc c

56 nfq2z 3 q . 6Ž . Ž .

27 Nc

where n is the number of light quarks. The poles in ´s0 correspond to infrared divergencies cancelled in thef

total cross-section.Ž1.Ž .The one-loop correction to the integral kernel K q ,q is obtained as a sum of two contributions. The firstr 1 2

w xone is related with the one-loop virtual correction to the one-gluon production cross-section 6 and the second

( )V.S. Fadin, L.N. LipatoÕrPhysics Letters B 429 1998 127–134 129

w x w xone is determined by the Born cross-sections for production of two gluons 7 and quark-antiquark pair 8 . TheŽ1.Ž .final result for K q ,q can be written as followsr 1 2

° ´4 2y2 ´ 24g m 1 11 2n 1 q yq pŽ .m f 1 2Ž1. 2~K q ,q s y 1y 1y´Ž .r 1 2 1q´ 2 2 ž /ž / ž /3 3N ´ 6p G 1y´ mž /Ž . q yqŽ . c1 2¢´2 2q yq 67 p 10 n 404 56 nŽ .1 2 f f

q y y q´ y q14z 3 qŽ .2 ž /ž /ž / 9 3 9 N 27 27 Nm c c

222 2 2 2n 2 q q y3 q q 2 2 1 1 q 1 qŽ .f 1 2 1 2 1 1y 1q q q y ln y ln3 2 2 2 2 2 2 2 2 2ž / ž /ž / ž /N 16q q q q q q q qq yqŽ .c 1 2 2 1 2 1 2 21 2

42 2 2 2 2 2 22 q yq 1 q q q q yq q qŽ .Ž .1 2 1 1 2 1 2 1 2q ln ln qL y yL y2 2 2 4 2 22 2ž / ž / ž /2 q q qž /ž /q yq q qqŽ . Ž . 2 2 1q qqŽ .1 2 1 2 1 2

1qx2 dx ln2 2 2 2 4 4

`n q qq 2 q q y3q y3qŽ . 1yxf 1 2 1 2 1 2 2y 3q 1q 1y y q qŽ . H1 23 2 2 4 4 2 2 2ž / ž /N 8q q 16q q q qx qž / 0c 1 2 1 2 1 2

2zqŽ .1 ¶dz ln2 22 2`q yq qŽ .Ž . 11 2 2 •y 1y y , 7Ž .H H2 2 2ž /ž / 0 1q yq q qq q yzqŽ . Ž . Ž .1 2 1 2 2 1 ß

where`z dt

ynL z s ln 1y t , z n s k .Ž . Ž . Ž . ÝHt0 ks1

Ž1.Ž .Note, that the gluonic part of the above expression for K q ,q is different from the corresponding piecer 1 2w xof the so called ‘‘irreducible part’’ of the kernel, which was constructed by the authors of Ref. 9 and it leads to

different estimates of radiative corrections to the intercept of tw x.he BFKL Pomeron. This difference is due to the fact, that the result of Ref. 9 is incomplete. So-called

scale-dependent contributions were not calculated by these authors and in their opinion there is an ambiguity infixing these contributions. We stress again, that physical results do not depend on the energy scale in the Regge

Ž .factors in Eq. 1 because its change is compensated by the corresponding modification of the impact factors andthe kernel. Moreover, this modification does not have any influence on the correction to the Pomeron intercept.

In our approach the contribution to the total cross section from one gluon production in the central rapidityregion can be presented as follows

Ž X .Ž . 2 v qX 2 v q2 2d q d q db k s bk rq HHX Xgs s F q F q R q ,q , 8Ž . Ž . Ž . Ž .H H H2 ™ 3 A B XX2 2 X ž / ž /b q b q k2p q 2p q q k rs HH

X Ž . Ž .where k sqyq is the emitted gluon transverse momentum and b is its Sudakov variable, bs kp r p p .H B A B

Note, that acting more accurately one should introduce in the impact factors and in the limits of integration overb some intermediate cut-offs in such way, that the physical results do not depend on them. The quantity

g Ž X.R q,q is the total contribution to the BFKL kernel from one gluon production and it is expressed through the


Ž w x.product of the Reggeon-Reggeon-gluon vertices G see 6 with a subtraction of the terms proportional to them

product of the gluon Regge trajectories v and the vertices G B in the Born approximationB m

m4 m4X Xg B BR q ,q ;G G y v q ln qv q ln G G .Ž . Ž . Ž .m m B B m mX2 2 2 2ž /q k q kH H

Ž .The necessity of this subtraction is related with the renormalization of the Regge factors in Eq. 8 inw xcomparison with their definition in Ref. 6 .

The ultraviolet divergency in the integral over the relative gluon rapidity for the contribution to the BFKLw xkernel from the two gluon emission 7 is cancelled with the corresponding divergency, related with the infrared

cut-off in the relative rapidity for the produced gluons in the multi-Regge kinematics. The analogousŽ .cancellation is implied in the loop corrections to the impact factors F q .i

The solution of the inhomogeneous BFKL equation can be presented as a linear combination of a completeset of the solutions of a homogeneous equation proportional to spherical harmonics in the Dy2 dimensional

Ž . vPspace. The biggest eigen value v , leading to a rapid increase of total cross-sections s s ;s with energy,P

corresponds to a spherically symmetric eigen function. Therefore it is natural to average the BFKL kernel overthe angle between the momenta q and q :1 2

2 ´4 °y2 e 2 2 24g m 1 11 2n 1 q yq pm f 2 1Ž1. 2~K q ,q s y 1q´Ž .r 1 2 1qe 2 22 2 ž /ž / ž /3 3N e 3p G 1ye max q ,qŽ . q yq Ž .¢ Ž .�c 1 22 1

´42 2 2q yq 5pŽ .2 1 2y 1q´3 ž /2 2 2 6� 0 0m max q ,qŽ .Ž .1 2

´42 2 2q yq 67 p 10 n 404 56 n2 1 f f

q y y q´ y q14z 3 qŽ .3 ž /2 2 2 ž /9 3 9 N 27 27 N� 0 c cm max q ,qŽ .Ž .1 2

22 21 n 2 2 1 1 q 1 qf 1 1y 1q q q y ln y ln3 2 2 2 2 2 2 2 2ž / ž /ž / ž /< <32 N q q q q q q yq qc 2 1 2 1 2 1 2 2

22 2`n 3 q qq dx 1qxŽ .f 1 2

y 3q 1q y lnH3 2 2 2 2 2ž / ž /4 1yxN 32 q q q qx qž / 01 2 1 2

¶2 2 21 p q q1 2 •q y4L min , . 9Ž .2 2 2 2ž ž /ž /3q qq q q ß2 1 2 1

Instead of the dimensional regularization we can remove the infrared divergencies in the kernel byŽ . Ž . Ž .introducing a fictious gluon mass l. Using Eqs. 4 - 6 , it is possible to verify that the averaged kernel 9 at


´™0 is equivalent to the expression2 2 2 2 2 2a m N q a m N 11 2n q m pŽ . Ž .s c 1 s c f 1

K q ,q sy2 ln q y ln ln qŽ .1 2 2 2 2 2½ ž /ž /4p 3 3N 12p l l lc

2 2 2 2 2 2< <67 p 10 n q a m N u q yq ylŽ . Ž .f 1 s c 1 22 2q y y ln y3z 3 d q yq qŽ . Ž .1 22 2 2 25ž / < <9 3 9 N l p q yqc 1 2

22 2 2 2< <a m N 11 2n q yq 67 p 10 nŽ .s c f 1 2 f= 1y y ln y y y2 2 2ž /½ 5ž /ž /4p 3 3N 9 3 9 Nmax q ,q mŽ .c c1 2

22 2 2 2 2a m N 1 n 2 2 1 1 q 1 qŽ .s c f 1 1y 1q q q y ln q ln3 3 2 2 2 2 2 2 2 2½ ž / ž /ž / ž /< <324p N q q q q q q yq qc 2 1 2 1 2 1 2 2

22 2`n 3 q qq dx 1qxŽ .f 1 2

q 3q 1q y lnH3 2 2 2 2 2ž / ž /4 1yxN 32 q q q qx qž / 01 2 1 2

1 p 2 q2 q21 2

y y4L min , , 10Ž .2 2 2 2 5ž /ž /ž /3q qq q q2 1 2 1

defined in the two-dimensional transverse space, with l™0. Of course, the dependence from l disappearsŽ .when the kernel acts on a function. Moreover, the representation 10 permits to find such form of the kernel for

which this cancellation is evident:2 ° 2 2a m N 1 min q ,qŽ . Ž .s c 1 2Dy2 2 2 2 2~d q K q ,q f q s dq f q y2 f qŽ . Ž . Ž . Ž .H H2 1 2 2 2 2 12 2 2 2¢< < ž /p q yq q qqŽ .1 2 1 2

=

22 2 2 2< <a m N 11 2n q yq 67 p 10 nŽ .s c f 1 2 f1y y ln y y y2 2 2ž / ž /ž /ž /4p 3 3N 9 3 9 Nmax q ,q mŽ .c c1 2

22 2 2a m N 1 n 2 2 1 1 q 1 qŽ .s c f 1 12y f q 1q q q y ln q lnŽ .2 3 2 2 2 2 2 2 2 2ž / ž /ž / ž /< <4p 32 N q q q q q q yq qc 2 1 2 1 2 1 2 2

22 2`n 3 q qq dx 1qxŽ .f 1 2

q 3q 1q y lnH3 2 2 2 2 2ž / ž /4 1yxN 32 q q q qx qž / 01 2 1 2

2 2 2 ¶1 p q q1 2 •y y4L min ,2 2 2 2 ßž /ž /ž /3q qq q q2 1 2 1

a 2 m2 N 2 5p 2 11 2nŽ .s c f 2q 6z 3 y y f q . 11Ž . Ž .Ž .12 ž /ž /12 3 3N4p c

The m-dependence in the right hand side of this equality leads to the violation of the scale invariance and isrelated with running the QCD coupling constant.

Ž . 2Žgy1.The form 11 is very convenient for finding the action of the kernel on the eigenfunctions q of the2

Born kernel:gy12 2 2q a q N a q NŽ . Ž .2 s 1 c s 1 cDy2d q K q ,q s x g qd g , 12Ž . Ž . Ž . Ž .H 2 1 2 2 ž /ž / p 4 pq1


were within our accuracy we expressed the result in terms of the running coupling constant

a m2 a m2 N 11 2n q2Ž . Ž .s s c f2 2a q s ,a m 1y y ln .Ž . Ž .s s 22 2 ž / ž /ž /4p 3 3N ma m N 11 2n qŽ . cs c f1q y ln 2ž / ž /4p 3 3N mc

Ž .The introduced quantity x g is proportional to the eigenvalue of the Born kernel

x g s2c 1 yc g yc 1yg , c g sGX

g rG g , 13Ž . Ž . Ž . Ž . Ž . Ž . Ž . Ž .Ž .and the correction d g is given below

211 2n 1 67 p 10 nf fX X2d g sy y x g yc g qc 1yg y y y x g y6z 3Ž . Ž . Ž . Ž . Ž . Ž .Ž .ž / ž /3 3N 2 9 3 9 Nc c

p 2cos pg n 2q3g 1ygŽ . Ž .f XX XXq 3q 1q yc g yc 1ygŽ . Ž .2 3ž /ž /3y2g 1q2gsin pg 1y2g N Ž . Ž .Ž . Ž . c

3py q4f g . 14Ž . Ž .

sin pgŽ .

Ž .The function f g is

dx dt1 1gy1 ygf g sy x qx ln 1y tŽ . Ž . Ž .H H

1qx t0 x

` c nq1qg yc 1 c nq2yg yc 1Ž . Ž . Ž . Ž .ns y1 q . 15Ž . Ž .Ý 2 2nqg nq1ygŽ . Ž .ns0

Ž . w xNote, that the contribution in the last line of Eq. 14 is different from the result of Ref. 9 , which leads todifferent estimates of the next-to-leading corrections to the intercept of the BFKL pomeron, for the reasonsmentioned before.

Ž .Almost all terms in the right hand side of Eq. 12 except the contribution

a 2 m2 N 2 11 2n 1Ž .s c f X XD g s y c g yc 1ygŽ . Ž . Ž .Ž .2 ž /3 3N 24p c

Ž .are symmetric to the transformation gl1yg . Moreover, it is possible to cancel D g if one would redefine2Žgy1. Ž Ž 2 . Ž 2 ..y1r2the function q by including in it the logarithmic factor a q ra m .s s

Ž .Because the radiative correction to v is negative, it is convenient to introduce the relative correction c g byŽ . Ž . Ž .the definition d g syc g x g . In particular, for the symmetric point gs1r2, corresponding to the

rightmost singularity of the t-channel partial wave, we obtain

1 11 2n 67 p 2 10 nf fc s2 y ln2y q qž / ž /2 3 3N 9 3 9 Nc c

31 1 dt p 27 11 n1 f'q 16 arctan t ln q22z 3 q qŽ .Ž .H 3ž / ž /4ln2 1y t t 2 16 16 N0 c

n nf fs25,8388q0.1869 q10.6584 , 16Ž .3N Nc c


w x Ž 2 .which is almost two times larger, than its estimate in Ref. 9 . For example, if a q s0.15, where the BornsB Ž 2 .intercept is v s4N a q rp ln2s .39714, the relative correction for n s0 is very big:P c s f

v 1 a q2Ž .P ss1yc N s0.0747.cB ž /2 4pvP

Ž 2 .The maximal value of v ,0.1 is obtained for a q ,0.08. These numerical estimates show, that in theP s

kinematical region of HERA probably it is not enough to take into account only the next-to-leading correction.The value of this correction strongly depends on its representation. For example, if one takes into account thenext-to-leading correction by the corresponding increase of the argument of the running QCD coupling constant,the intercept of the pomeron turns out to be only two times smaller, than its Born value.

Due to the effect of running the coupling constant the eigenfunctions of the NLLA kernel, which can beeasily found, do not coincide with q2Žgy1.; moreover, the position and the nature of the Pomeron singularity are

w xstrongly affected by the nonperturbative effects 10 . The BFKL equation with the next-to-leading correctionsŽ .can be considered as the quantized version of the renormalization group equations. Expression 16 can be used

only for a rough estimate of the power of the energy dependence of the total cross section.The above results can be applied for the calculation of anomalous dimensions of the local operators in the

vicinity of the point vsJy1s0. To begin with, it is necessary to emphasize the difference between theg BŽ 2 .t-channel partial wave GG q for the reggeized gluon scattering off the colourless particle B at ts0v

d2qX

X Xg B 2GG q s G q ,q F q 17Ž . Ž . Ž .Ž . Hv v BX 22p qg BŽ 2 .and the deep-inelastic moments FF q defined as followsv

yv` ds s

g B 2 g B 2FF q s s q ,s , 18Ž .Ž . Ž .Hv 2ž /2 s qq

whereX v2d q dv saqi`

X Xg B 2s q ,s s F q G q ,q . 19Ž . Ž . Ž .Ž . H HB vXX 2 ž /2p i q q2p q ayi`

This difference was not essential in LLA but becomes important in NLLA. Whereas the t-channel partialg BŽ 2 . Ž . Žwave G q obeys the integral equation of the type 2 with the same kernel and the inhomogeneous termv

Ž . 2 . g BŽ 2 .equal to F q r2p q , the kernel of the corresponding equation for the deep inelastic moments FF q inB v

NLLA is

1 qX 2X X XDy 2 B BK q ,q sK q ,q y d q K q ,q ln K q ,q , 20Ž . Ž . Ž . Ž . Ž .H1 2 1 2 1 222 q1

BŽ X . 2Žgy1.where K q ,q is the LLA kernel. The action of the modified kernel on the Born eigen functions q can2 2

be calculated easily:gy12 2 2q a q N a q NŽ . Ž .2 s 1 c s 1 cDy2 ˜ ˜d q K q ,q s x g qd g , 21Ž . Ž . Ž . Ž .H 2 1 2 2 ž /ž / p 4 pq1

where

˜ Xd g sd g y2 x g x g . 22Ž . Ž . Ž . Ž . Ž .

The anomalous dimensions

gsg a rv qa g a rvŽ . Ž .0 s s 1 s


of the twist-2 operators near point vs0 are determined from the solution of the equation

a N a Ns c s c˜vs x g qd gŽ . Ž .ž /p 4 p

a N 1s c 2s qO gŽ .ž /p g

a 2N 2 11q2n rN 3 n 10q13rN 2 395 11 p 2 n 71 p 2Ž .s c f c f c fy q q y2z 3 y q yŽ .2 2 3 ž /ž 9g N 27 3 6 27 94 p 3g Nc c

qO g 23Ž . Ž ./3 ˜Ž . Ž .for g™0. The singularity in d g ,y2rg for g™0 is exactly cancelled in the transition to d g because

Ž . XŽ . 32 x g x g ,y2rg in the same limit. It gives a possibility to find the corrections to the anomalousdimensions. In particular for the low orders of the perturbation theory we reproduce the known results andpredict the higher loop correction for v™0:

2a N 1 11 n a n N 5 13s c f s f cgs y y qO v y q qO vŽ . Ž .3 2ž /ž / ž /p v 12 p 6 v 36N 6Nc c

3 2 2 41 a N 395 11 p n 71 p as c f sy y2z 3 y q y qO v qO . 24Ž . Ž . Ž .2 3 4ž / ž / ž /ž /p 27 3 6 27 94v N vc

Note, that our formulas for the anomalous dimensions are different from the expressions obtained by authorsw xof Ref. 9 . Their results are not complete because they did not calculate the contributions of so called

energy-scale dependent parts of the kernel.

Acknowledgements

We want to thank Universitat Hamburg and DESY for the hospitality during our stay in Germany. The¨fruitful discussions with J. Bartels, M. Ciafaloni and J. Blumlein were very helpful. The work was supported by¨

Ž .the INTAS and RFFI grants. One of us L.N.L. is thankful to Deutsche Forschunggemeinschaft for the grant,which gave him a possibility to work on this problem at the Hamburg University.

References

w x Ž . Ž .1 V.S. Fadin, E.A. Kuraev, L.N. Lipatov, Phys. Lett. B 60 1975 50; Ya. Ya. Balitsky, L.N. Lipatov, Sov. J. Nucl. Phys. 28 1978 822.w x Ž . Ž .2 V.N. Gribov, L.N. Lipatov, Sov. J. Nucl. Phys. 15 1972 438; L.N. Lipatov, Sov. J. Nucl. Phys. 20 1975 94; G. Altarelli, G. Parisi,

Ž . Ž .Nucl. Phys. B 26 1977 298; Yu.L. Dokshitzer, Sov. Phys. JETP 46 1977 641.w x Ž .3 L.N. Lipatov, V.S. Fadin, Sov. J. Nucl. Phys. 50 1989 712.w x Ž . Ž .4 L.N. Lipatov, Nucl. Phys. B 452 1995 369; Phys. Rep. 286 1997 132.w x Ž . Ž .5 V.S. Fadin, R. Fiore, M.I. Kotsky, Phys. Lett. B 359 1995 181; B 387 1996 593.w x Ž . Ž .6 V.S. Fadin, L.N. Lipatov, Nucl. Phys. B 406 1993 259; V.S. Fadin, R. Fiore, A. Quartarolo, Phys. Rev. D 50 1994 5893; V.S.

Ž .Fadin, R. Fiore, M.I. Kotsky, Phys. Lett. B 389 1996 737.w x Ž . Ž .7 V.S. Fadin, L.N. Lipatov, Nucl. Phys. B 477 1996 767; V.S. Fadin, M.I. Kotsky, L.N. Lipatov, Phys. Lett. B 415 1997 97.w x Ž . Ž .8 S. Catani, M. Ciafaloni, F. Hautman, Phys. Lett. B 242 1990 97; Nucl. Phys. B 366 1991 135; G. Camici, M. Ciafaloni, Phys. Lett.

Ž . Ž .B 386 1996 341; Nucl. Phys. B 496 1997 305; V.S. Fadin, R. Fiore, A. Flashi, M.I. Kotsky, BUDKERINP-97-86, hep-phr9711427.w x Ž .9 G. Camici, M. Ciafaloni, Phys. Lett. B 412 1997 396.

w x Ž . Ž .10 L.N. Lipatov, JETP 63 1986 904; G. Camici, M. Ciafaloni, Phys. Lett. B 395 1997 118.

11 June 1998


The collision energy dependence of dijet cross sectionsas a probe of BFKL physics

Lynne H. Orr a, W.J. Stirling b

a Department of Physics and Astronomy, UniÕersity of Rochester, Rochester, NY 14627-0171, USAb Departments of Physics and Mathematical Sciences, UniÕersity of Durham, Durham, DH1 3LE, UK

Received 18 January 1998; revised 9 April 1998Editor: H. Georgi

Abstract

Ž .The dependence of the subprocess cross section for dijet production at fixed transverse momentum on the large rapidityŽ .nseparation D y of the dijets can be used to test for ‘BFKL physics’, i.e. the presence of higher-order a log D ys

contributions. Unfortunately in practice these subtle effects are masked by the additional, stronger dependence arising fromthe parton distributions. We propose a simple ratio test using two different collider energies in which the parton distributiondependence largely cancels. Such a ratio does distinguish between the ‘asymptotic’ analytic BFKL and lowest-order QCDpredictions. However, when subasymptotic effects from overall energy-momentum conservation are included, the BFKLpredictions for present collider energies change qualitatively. In particular, the real gluon emission contributions to theBFKL cross section are suppressed. The proposed ratio therefore provides an interesting laboratory for studying the interplayof leading-logarithm and kinematic effects. q 1998 Published by Elsevier Science B.V. All rights reserved.

Quantum Chromodynamics has been very successful in describing jet production in high energy colliderexperiments. Perturbative QCD at fixed order in the strong coupling constant a provides sufficient predictives

power for a wide variety of high energy phenomena. However in some regions of phase space, large logarithmscan multiply the coupling, spoiling the good behavior of fixed-order expansions. In certain cases these large

Ž . w xlogarithms can be resummed, as in the Balitsky, Fadin, Kuraev and Lipatov BFKL equation 1 , and thepredictive power of the theory is then in principle restored. The latter statement should be subject toexperimental test, since the predictions of BFKL are distinct from those of fixed-order QCD.

Being asymptotic, the regions for which BFKL resummation is presumed to be necessary have until recentlybeen beyond the reach of experiments, and therefore the predictions of BFKL have been difficult to test. Thelast few years, however, have seen experiments at the HERA ep collider and the Fermilab Tevatron thatapproach the kinematic regimes relevant to BFKL. At HERA the relevant regime is small parton momentumfraction x, where BFKL predicts a sharp rise in the structure function F as x decreases. Further discussion can2

w xbe found in Refs. 2,3 and references therein; here we are concerned with BFKL physics at hadron colliders.


( )L.H. Orr, W.J. StirlingrPhysics Letters B 429 1998 135–144136

At the Tevatron pp collider, and at hadron colliders in general, BFKL resummation applies to dijetproduction when the rapidity separation D of the two jets is large, D41, while fixed-order QCD should be

Ž .adequate for DsOO 1 . BFKL predicts, among other things, a rise in the dijet subprocess cross section s as Dˆw xincreases 4 , in contrast to the s™ constant behavior expected at lowest order. In practice such behavior canˆ

Ž .be difficult to observe because s gets folded in with parton distribution functions pdfs , which decrease with Dˆmuch more rapidly than the subprocess cross section increases. In particular, the BFKL rise with D gets killedby kinematic constraints. The challenge is to find measurable quantities in dijet production that are insensitive tothe pdfs, but that retain the distinctive behavior characteristic of BFKL resummation. Some possibilities have

w xbeen discussed in Refs. 5–10 , including the azimuthal decorrelation of the two jets: the multiple emission ofsoft gluons between the leading jets predicted by BFKL leads to a stronger decorrelation than does fixed-orderQCD, and the prediction is relatively insensitive to the pdfs.

w xAnother possibility is to look for the increase in s with D by considering different collision energies 4 .ˆThe idea is to choose D’s that correspond to the same parton momentum fractions at different energies so that

œthe pdf dependence is the same for both, thereby allowing the D dependence to be extracted. The D0Collaboration has in fact recently reported the results of a preliminary study of dijet cross section ratios at fixed

w xparton momentum fraction 11 .In this paper we investigate the dependence of the dijet cross section on collision energies in leading-order

QCD and in the BFKL approach. We define a cross section ratio at two collider energies in which the pdfdependence largely cancels, and in which the predictions of leading-order QCD and ‘naive’ BFKL can be

Ž Ž .ndistinguished. By naive BFKL, we mean the predictions obtained by resumming leading-logarithm a Ds

contributions arising from the emission of soft real and virtual gluon contributions in the absence of overall.kinematic constraints. However, one difficulty with the BFKL approach is that even for the largest D values

which are accessible at present, subasymptotic effects, for example from energy-momentum conservation, arelikely to be important. This has led to ‘improved’ BFKL dijet cross section calculations, in which some of these

w xeffects are taken into account. In this study we will use the BFKL Monte Carlo approach developed in Ref. 10Ž w x.see also 9 . We find that at present collider energies the predictions of naive BFKL are indeed substantiallymodified when subasymptotic kinematic effects are included.

Our calculations will therefore give us a sequence of predictions – full leading-order QCD, asymptoticleading-order QCD, naive BFKL, improved BFKL – to confront experiment. In what follows we develop eachof these approximations in turn.

We begin with the lowest-order QCD inclusive two-jet production cross section as a function of the two jetrapidities y , y and their common transverse momentum p . For simplicity we will consider the symmetric1 2 T

1situation where the two rapidities are equal and opposite. Setting y syy s D gives for the differential cross1 2 2

section

ds 122 2 < <s xf x ,m xf x ,m MM ab™cd ,Ž .Ž . Ž .Ý Ýa b12 4 4

1d y d y d p 256p p cosh DŽ .1 2 T T 2y syy s D a,b ,c ,dsq , g1 2 2

1Ž .

2Ž .where Ý denotes the appropriate sums and averages over colors and spins and f x,m are the partona,b

densities. Here

2 pT 1xs cosh D , 2Ž .Ž .2's

( )L.H. Orr, W.J. StirlingrPhysics Letters B 429 1998 135–144 137

and the subprocess matrix elements are functions only of the rapidity difference D. In the limit of large D thegg, qg and qq subprocess matrix elements squared become equal, up to overall color factors of C 2, C C andA A F

2 Ž .C respectively. This defines the effectiÕe subprocess approximation to Eq. 1 :F

22ds x GG x ,mŽ . 2< <, MM gg™gg , 3Ž . Ž .Ý12 4 41d y d y d p 256p p cosh DŽ .1 2 T T 2y syy s D1 2 2

where

4GGsgq qqq . 4Ž . Ž .Ý9q

Ž .In an experiment and in the BFKL formalism to be discussed below , we are interested in events with jetsabove some transverse momentum threshold P . Integrating over the jet transverse momentum p )P thenT T T

gives

2 2< <MM gg™ggds 1 duy2Ž .Ý 2X 2, x GG x ,m , 5Ž .Ž .H14 2 4 xsX u1d y d y 256p cosh D P uŽ . 11 2 T2y syy s D1 2 2

1 'Ž . Ž . Ž .with Xsp min rp max s2 P cosh D r s . In the asymptotic limit where D is large, the D dependenceT T T 2

in the prefactor disappears so that

2< <MM gg™ggŽ .Ý 1 2 2™ p C a , 6Ž .A s214256p cosh DŽ .2

and hence

2 2 2ds p C a duy2 2A s X 2 2, x GG x ,m 's FF X ,m , 7Ž .Ž . Ž .ˆH 02 4 xsX u1d y d y 2 P u11 2 Ty syy s D1 2 2

where

p C 2 a 2A s

s s 8Ž .ˆ0 22 PT

Ž .and FF contains the integration over the parton distribution functions. Eq. 7 defines the asymptotic QCD LOcross section to which we will compare the BFKL predictions below.

Ž . Ž Ž .. Ž .Fig. 1 shows the QCD LO dijet cross sections calculated using i the 2™2 matrix elements Eq. 1 , iiŽ Ž .. Ž .the effective subprocess approximation Eq. 5 , and iii the asymptotic form of the effective subprocess

'Ž Ž ..approximation Eq. 7 , for two pp collider energies s s630, 1800 GeV. The jet p threshold is P sT Tw x20 GeVrc, the partons are CTEQ4L with msP 12 , and a is evaluated in leading order also at scale msPT s T

Ž .the numerical value is 0.170 . The effective subprocess approximation is seen to reproduce the exact result towithin a few per cent over the entire D range. The asymptotic form is approached at large D, as expected.

At higher orders in QCD perturbation theory the subprocess cross section receives large logarithmiccorrections from soft gluon emission in the rapidity interval between the two leading jets. For fixed coupling as

we have

nsss 1q a a D q . . . , 9Ž . Ž .ˆ ˆ Ý0 n s

nG1


Ž .Fig. 1. The dependence of the leading-order 2™2 dijet cross sections on the dijet rapidity separation. The partons are the CTEQ4L setw x Ž . Ž . Ž . Ž12 . The three curves at each collider energy use: i exact matrix elements solid lines , ii the effective subprocess approximation dashed

. Ž . Ž . Ž .lines , and iii the asymptotic D41 form of the latter dotted lines .

with s as defined above. The BFKL formalism to be discussed below resums the logarithms and in theˆ0

leading-logarithm, fixed-coupling ‘naive’ approximation gives a subprocess cross section which is not constantŽ .but grows asymptotically with D: s;exp lD with ls4C ln2a rpf0.5.ˆ A s'At fixed s and minimum transverse momentum P , both s and FF depend on D. In fact because of theˆT

shape of the parton distributions the latter quantity decreases rapidly with increasing D and vanishes at they1 'Ž .kinematic limit Ds2cosh s r2 P , as in Fig. 1. It is therefore difficult to observe the relatively slow riseT

Ž w x. w xwith D of s see for example Fig. 8 of Ref. 6 . The original idea of Ref. 4 was to increase the colliderˆ'energy s as D increases such that X and therefore FF remains fixed. Any observed rise in the cross section

could then only arise from higher-order contributions to s .ˆ' 'Ž .If two collider energies are available s and s say one can make use of this idea by comparing the dijet1 2Ž .cross sections at two rapidity separations D and D for which the asymptotic leading-order cross sections are1 2

equal. Specifically, if for a given D we define D such that1 2

1 'cosh D sŽ . 112s 10Ž .1 'cosh D sŽ . 222

' 'Ž . Ž . Ž .then the cross sections defined by Eq. 7 will be equal, i.e. ds s ,D sds s ,D . Using for s and(1 2 11 2

s , respectively, the two Tevatron collider energies 630 and 1800 GeV, we have the following values for D( 2 1

and D :2

Ž .D s s630 GeV 0.0 1.0 2.0 3.0 4.0 5.0 6.0 6.8(1 1

Ž .D s s1800 GeV 3.42 3.68 4.33 5.19 6.13 7.11 8.10 8.90(2 2

Ž . Ž .Note that for large D Eq. 10 reduces to D sD q ln s rs sD q2.1, as can be seen in the table.1 2 1 2 1 1


' 'Fig. 2. The ratio R of the dijet cross sections at the two collider energies s s630GeV and s s1800GeV, as defined in the text. The12 1 2Ž . Ž . Ž . w x Ž . w xcurves are: i the exact leading-order 2™2 predictions using CTEQ4L solid curve 12 and GRV94LO partons dashed curve 13 , both

Ž . Ž .with ms P s20GeV, and ii the ‘naive BFKL’ prediction dot-dashed curve . Note that the asymptotic leading-order prediction isT

R s1.12

The equality of the asymptotic cross sections at lowest order for the two D’s leads us to define a crosssection ratio

'ds s ,DŽ .1 1R s 11Ž .12 'ds s ,DŽ .2 2

Ž .as a function of D , with D given by Eq. 10 . This ratio has the advantage of being directly accessible1 2

experimentally, since it depends on the rapidity difference between the two ‘outside’ jets, a quantity more easilymeasured than for example the parton momentum fractions themselves. By construction, R s1 in asymptotic12

LO QCD, but notice that the ratio is not equal to unity for the leading-order cross section at subasymptoticrapidity separations calculated using exact matrix elements, or using the effective subprocess approximation.

Ž .This is because the prefactor multiplying the integral in for example Eq. 5 only becomes independent of D at'large D. This is illustrated in Fig. 2, which shows R as a function of D for s s630 GeV and12 1 1's s1800 GeV, using the parameters of Fig. 1. The effective subprocess approximation curve is very close to2

the exact curve and is not shown. The ratio approaches unity at large D , as expected. Not surprisingly, the1

‘exact matrix element’ ratio is insensitive to the choice of parton distributions. The dashed line shows the ratiow xobtained using GRV94LO distributions 13 . The curves are almost identical. Similarly, the scale choice

dependence is also very weak. For example, using the more natural ‘running’ choice msp instead of msPT T

in the parton distributions and in a has a negligible effect on the ratios.s1 Ž .nThe BFKL formalism resums the leading-logarithm a D higher-order corrections which arise froms

multiple emission of real and virtual gluons in the rapidity interval between the two leading jets. The theoreticalŽ w x.details of how this is achieved can be found elsewhere see for example Ref. 15 and will not be repeated here.

1 Ž n ny1 . w xIn fact, sub-leading logarithms a D etc. can also be resummed in principle 14 . In practice, however, only the leadings

contributions are known at present in a form which is phenomenologically useful.


In fact the naive BFKL prediction, in which for example subasymptotic effects from overall energymomentum conservation are ignored, has an analytic representation which leads to a simple expression for R12

in the large D limit. If the coupling a is assumed to be constant, all kinematic constraints are ignored, and1 s

only the leading logarithms are resummed, then the result is

a Cs Asss C D , 12Ž .ˆ ˆ0 0 ž /p

with

q`1 d z2 tx Ž z .0C t s e ,Ž . H0 122p z q ery` 4

1x z sRe c 1 yc q iz . 13Ž . Ž . Ž .Ž .0 2

Ž .The asymptotic large t, equivalently large D behavior is

14log 2 tC t ; e . 14Ž . Ž .0 1

p 7z 3 t( Ž .2

It follows immediately that the naive BFKL prediction for the ratio R at large rapidity separations is12

a Cs AC D0 1ž /p

R s . 15Ž .12 a Cs AC D0 2ž /p

The prediction is shown in Fig. 2. Note that now R is well below the QCD LO curves, and in fact at first sight12

this appears to be a primary test of the presence of higher-order ‘BFKL-like’ corrections to the dijet crossŽ .section. The formalism is of course only supposed to be valid for large D which implies large D also and so1 2

we only show the predictions for D )2, which is roughly where the leading-order prediction begins to1

approach its asymptotic limit R s1. The asymptotic large D behavior of the ratio is readily obtained from12 1Ž . Ž .Eqs. 14 and 15 :

ls a C1 s AR ™ , ls 4log2 . 16Ž .12 ž /s p2

One interesting aspect of this result is that one is tempted to use the experimentally measurable quantityŽ . Ž .KKs log R rlog s rs to determine the effective BFKL exponent l, i.e.12 1 2

a Cs AKK s0 , KK ™ 4log2s0.45 , 17Ž .LO asymp. naive BFKL

p

where the numerical value is obtained from using a evaluated at msP s20 GeV, as in Fig. 2. However it iss T

clear from the figure that the asymptotic behavior is attained only very slowly. For this choice of parameters weexpect R ™0.39 as D ™`, but we see that the ratio has only decreased to R s0.45 at the kinematic limit12 1 12

D s6.9. Furthermore, as we shall see below, an improved BFKL calculation does not lead to such a simple1

prediction.So far we have considered the predictions of naive BFKL, which can be written analytically but at the cost of

making several assumptions which do not hold in experimental situations. In particular, the analytic solutionsresult from integrating over arbitrarily large values of the transverse momenta of emitted gluons, and the


Fig. 3. The dependence of the BFKL and asymptotic QCD leading-order dijet cross sections on the dijet rapidity separation. The pdfs are thew x Ž . Ž . Ž . Ž .CTEQ4L set 12 . The three curves at each collider energy use: i ‘improved’ BFKL MC solid lines , ii ‘naive’ BFKL dashed lines , and

Ž . Ž . Ž .iii the asymptotic D41 form of QCD leading order dotted lines .

analytic phase space integrations at the subprocess level prevents a proper incorporation of parton distributions.In addition, a is assumed to be fixed. The BFKL approach can be improved and these assumptions avoided bys

w xrecasting the BFKL cross section as an event generator 9,10 using an iterated solution to the BFKL equation.Kinematic constraints and the running of a can then be included. The effect is to restrict the growth of s withˆs

increasing D compared to naive BFKL, since the emission of relatively large transverse momentum gluons,which give a positive contribution to the resummed cross section, is suppressed. Details of the calculation and

w xapplication to the azimuthal decorrelation can be found in 10 .w xHere we repeat the calculation of the cross section and ratios R using the Monte Carlo of Ref. 10 to get12

an ‘improved’ BFKL prediction. One feature of the MC calculation that will be relevant to our results is worthmentioning. The MC solution to the BFKL equation is obtained by separating the contributions from real gluonemission into ‘resolved’ and ‘unresolved’ contributions, the idea being that below a certain energy scale m ,0

emitted real gluons are not detectable in practice. The contribution from unresolved real gluons is combinedwith the virtual gluon contribution to give an overall suppressing form factor. The differential cross sectiontakes the form, for fixed a , 2

s

a C Drps A2 2 2ds a C mˆg g s A 0s RR p , p ,D y , 18Ž . Ž .T 1 T 22 2 2 2 2d p d p dD y p p pT 1 T 2 T 1 T 2 T 1

and RR can be written as a sum over resolved real gluon emissions:`

Žn.RR p , p ,D s RR p , p ,D , 19Ž . Ž . Ž .ÝT 1 T 2 T 1 T 2ns0

Ž0. Ž . Žn. w xwith RR sd p qp and RR for n)0 given in 10 . The point is that even for ns0 the form factorT 1 T 22 a C Drpm s A0 appears in the cross section and represents a suppression of the probability that no resolvable2pT 1

2 w xWe show the fixed a expression for simplicity; see 10 for the corresponding running a expression.s s


' 'Fig. 4. The ratio R of the dijet cross sections at the two collider energies s s630GeV and s s1800GeV, as defined in the text. The12 1 2Ž . w x Ž . Ž .curves are: i the ‘improved’ BFKL MC predictions using CTEQ4L 12 pdfs solid curve , with ms P s20GeV, ii the ‘naive’ BFKLTŽ . Ž . Ž .prediction dashed curve , and iii the asymptotic leading-order prediction dotted curve R s1.12

Ž .gluons will be emitted. The inclusion of resolvable gluon contributions nG1 counteracts this suppression andremoves the dependence on m .0

Fig. 3 shows the dijet cross section as a function of D for the naive BFKL and improved BFKL MC cases atthe two collision energies, with P s20 GeV. Asymptotic QCD LO is also shown for reference. There areT

Ž .several features worth noting here. First, the naive BFKL cross section dashed curve is always largest, becauseŽ .it includes the analytic subprocess cross section given in Eq. 12 , which allows emission of any number of

gluons with arbitrarily large energies. The curve falls off rather than increasing because s is multiplied byˆparton densities, but even those incorporate only lowest order kinematics in this case.

When exact kinematics for entire events are included in both the subprocess cross section and the partonŽ . 3densities, as in the BFKL MC solid curve , there is a dramatic suppression of the total cross section. In fact

the suppression is so strong that it drives the BFKL MC cross section below that for asymptotic QCD LO. Thereason is due to simple kinematics: the QCD LO cross section contains only two jets, but the BFKL MC crosssection also includes additional jets, each of which increases the subprocess center-of-mass energy and elicits acorresponding price in parton densities. In the naive BFKL calculation, the contribution to the subprocessenergy from additional jets is ignored and their net effect is to combine with the virtual gluons to increase the

Ž .subprocess cross section. Interestingly, the effect is compounded by the fact that the naive BFKL increase of s

Ž 2 . Ž 2 . Ž . Ž 1.only starts at OO a in perturbation theory, i.e. C s1qOO a in Eq. 13 . The OO a correction is zeros 0 s s

because the real and virtual gluon contributions exactly cancel. However the effect of the pdfs is to suppressonly the real contributions, giving an overall negative correction at this order.

The ratio R as calculated in the improved BFKL MC is shown in Fig. 4, again with the naive BFKL and12

asymptotic QCD LO predictions. Here the effects of kinematic suppression and the parton distributions are quitedramatic. The BFKL MC curve does not fall between the naive BFKL and QCD LO curves as one might

3 The running of a , which we include, also contributes to the suppression, but it has a much smaller effect than kinematics.s


expect. Instead, it is everywhere greater than or equal to the QCD LO curve, and in particular, R is12

everywhere greater than 1.This behavior can be understood as follows. We can represent R schematically as12

1qa D CC q . . .s 1 1R ; , 20Ž .12 1qa D CC q . . .s 2 2

where the CC are coefficients modified by the effects of the kinematic suppression. By the arguments giveni

above, we expect CC -0. In the formal 4 limit D ™0, the nonleading terms in the numerator vanish, thei 1

denominator is -1 and hence R )1. On the other hand, in the limit D ™D , there is no phase space12 1 max

available for any gluon emission, hence CC ™0 and R ™1. The observation of a ‘BFKL effect’ wouldi 12

require a region of phase space where D 41 but CC ,1. Increasing the collider energy at fixed transversei i

momentum threshold would eventually lead to this situation. In such a region we would expect to see R -1,12

as predicted by the naive BFKL calculation shown in Fig. 4. Finally, why does the BFKL MC curve in Fig. 4rise again at large D , instead of approaching 1 as simple kinematics would suggest? The origin of this behavior1

Ž Ž .. Ž .arises in the form factor see Eq. 18 which suppresses the probability of no real i.e. resolved gluon emissionin the rapidity interval between the dijets. As D increases towards its maximum allowed value there is an

Ž .effective upper kinematic limit on the transverse momentum of each emitted gluon. This in turn leads to asuppressing form factor with the ‘artificial’ parameter m replaced by a physical parameter determined by0

kinematics and the pdfs. In other words, the all-orders leading-log contribution from virtual gluon emission thatis built into the overall form factor is not completely compensated by corresponding real emission because thekinematics do not allow it. The kinematic suppression increases with D, hence D )D implies R )1.2 1 12

The arguments above illustrate the general behavior of the cross section ratio – starting out at a value greaterthan 1, falling and then increasing again at large D – as predicted by the ‘improved’ BFKL Monte Carlo

Žcalculation. The details, however – how far the ratio falls and in particular whether it falls below unity as.predicted in the naive calculation and for how long – depend on the specifics of the experimental

configuration, such as the collider energies and the transverse momentum threshold. In this study we haverestricted ourselves to parameter values which are currently accessible at the Tevatron. Clearly it would bedesirable to have more than the two Tevatron pp energies available for such a measurement. In principle, theLHC at 14 TeV and RHIC in pp mode at 500 GeV would expand the range of D. In practice, one is limited byD at the lower energy. Furthermore, comparing jet measurements at different machines with differentmax

detectors can be challenging, and ideally one would like to construct ratios of measurements made at a singlemachine in a single detector, so that some systematic uncertainties would cancel. For example, running the LHCat 10 and 14 TeV would make such a measurement possible, although ideally the two energies should be moredifferent. The essential point is that higher collider energies appear to be necessary in order to prevent the BFKLeffects being swamped by kinematical constraints.

In summary, we have shown that the effects of the increase in the dijet subprocess cross section predicted inthe naive BFKL approach can in principle be detected by measuring the ratio of cross sections at energies andrapidities chosen such that the asymptotic QCD LO ratio is equal to 1. Forming the ratio minimizes the effectsof the parton distribution functions. However an improved BFKL calculation which includes the subasymptoticeffects of the conservation of energy and momentum gives a result for the ratio that is qualitatively differentfrom that of naive BFKL. In particular, for dijet production at the two Tevatron energies, the improved BFKLcalculation gives either agreement with lowest-order QCD or deviations opposite to those predicted by naiveBFKL. If our arguments about the kinematic suppression of higher-order real gluon emission are correct, then it

4 We do not of course expect the BFKL formalism to apply for small rapidity separations; here our aim is simply to explain the behaviorof the predictions of the BFKL MC calculation in Fig. 4.


would appear that a fixed-order perturbation theory approach would be a more appropriate calculationalframework for the cross section ratio at present collider energies than all-orders resummation. Therefore, animportant next step is to perform a calculation of R in exact next-to-leading-order QCD, to determine whether12

it provides a satisfactory description of the data.Finally we emphasise that this study has concentrated on the cross section ratio only. It may well be that

other quantities are better suited to revealing underlying BFKL effects. Indeed, we note that the azimuthalw x Ž . w xdecorrelation of the dijet pair 5,6 , as calculated in the naive BFKL approach, is already non-zero at OO a 6 ,s

w xand that its effects are not completely swamped by kinematics in an improved BFKL calculation 10 .

Acknowledgements

œWe acknowledge many useful discussions with members of the D0 Collaboration, and in particular withAnna Goussiou, whose ideas motivated this study. L.H.O. is grateful to the UK PPARC for a VisitingFellowship. This work was supported in part by the US Department of Energy, under grant DE-FG02-91ER40685and by the US National Science Foundation, under grant PHY-9600155.

References

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w x Ž . Ž .2 J. Kwiecinski, Nucl. Phys. Proc. Suppl. 39BC 1995 58.´w x Ž .3 V. Del Duca, in: Proc. Intl. Workshop on Deep Inelastic Scattering and Related Phenomena DIS96 , Rome, Italy, April 1996, preprint

Edinburghr96r7, 1996, hep-phr9608426.w x Ž .4 A.H. Mueller, H. Navelet, Nucl. Phys. B 282 1987 727.w x Ž .5 V. Del Duca, C.R. Schmidt, Phys. Rev. D 49 1994 4510.w x Ž .6 W.J. Stirling, Nucl. Phys. B 423 1994 56.w x Ž .7 V. Del Duca, C.R. Schmidt, Phys. Rev. D 51 1995 215.w x Ž . Ž . Ž .8 V. Del Duca, C.R. Schmidt, Nucl. Phys. Proc. Suppl. 39BC 1995 137; preprint DESY 94-163 1994 , presented at the 6th

Rencontres de Blois, Blois, France, June 1994.w x Ž .9 C.R. Schmidt, Phys. Rev. Lett. 78 1997 4531.

w x Ž .10 L.H. Orr, W.J. Stirling, Phys. Rev. D 56 1977 5875.w x œ11 A. Goussiou for the D0Collaboration, presented at the International Europhysics Conference on High Energy Physics, Jerusalem,

August 19–26, 1997.w x Ž .12 CTEQ Collaboration: H.L. Lai et al., Phys. Rev. D 55 1997 1280.w x Ž .13 M. Gluck, E. Reya, A. Vogt, Z. Phys. C 67 1995 433.¨w x Ž . Ž .14 V.S. Fadin, L.N. Lipatov, Sov. J. Nucl. Phys. 50 1989 712; Nucl. Phys. B 406 1993 259; V.S. Fadin, R. Fiore, Phys. Lett. B 294

Ž . Ž .1992 286; V.S. Fadin, R. Fiore, Quartarolo, Phys. Rev. D 50 1994 2265, 5893; V.S. Fadin, R. Fiore, M.I. Kotskii, Phys. Lett. B 359Ž . Ž . Ž .1995 181; B 387 1996 593; V.S. Fadin, L.N. Lipatov, Nucl. Phys. 477 1996 767; V.S. Fadin, M.I. Kotskii, L.N. Lipatov, preprint

Ž . Ž .BUDKERINPr96r92, 1996, hep-phr9704267; V. Del Duca, Phys. Rev. D 52 1995 1527; D 54 1996 989, 4474; preprintŽ .Edinburghr96r4, 1996, hep-phr9605404; M. Ciafaloni, G. Camici, Phys. Lett. B 386 1996 341; preprint DFFr264r01r97, 1997,

hep-phr9701303.w x15 R.K. Ellis, W.J. Stirling, B.R. Webber, QCD and Collider Physics, Cambridge University Press, Cambridge, 1996.

11 June 1998


Further numerical results on non-factorizable correctionsto eqey™4 fermions

A. Denner a, S. Dittmaier b, M. Roth c

a Paul Scherrer Institut, Wurenlingen und Villigen, CH-5232 Villigen PSI, Switzerland¨b Theory DiÕision, CERN, CH-1211 GeneÕa 23, Switzerland

c Paul Scherrer Institut, Wurenlingen und Villigen, CH-5232 Villigen PSI, Switzerland¨and Institut fur Theoretische Physik, ETH-Honggerberg, CH-8093 Zurich, Switzerland¨ ¨ ¨


Abstract

Numerical results on non-factorizable corrections to eqey™WW™4 fermions with semi-leptonic and hadronic finalstates, as well as to eqey™ZZ™4 fermions, are presented. The corrections turn out to be small in comparison to theexperimental uncertainty of LEP2, but they might compete with the expected accuracy at future eqey-colliders. q 1998Elsevier Science B.V. All rights reserved.

In order to match the experimental accuracy ofroughly 1% at LEP2, the precision of the predictionsfor the cross sections of eqey™4 fermions shouldbe at, or rather exceed the per-cent level. This preci-sion requires us to go beyond the narrow-widthapproximation for W- and Z-boson-pair productionas well as to include electroweak radiative correc-tions. For general aspects and further details of thestrategy for such precision calculations we refer to

w xreview articles 1 .Ž . q yThe full electroweak OO a corrections to e e

™WW,ZZ™4 fermions are not known and will notbe available in the near future. A few decay widthsabove the threshold for gauge-boson-pair productionthese corrections should be sufficiently well de-scribed by the so-called double-pole approximation,which consists of taking into account only thosecontributions that are enhanced by two resonant W

Ž .or Z bosons. In this approximation the OO a correc-tions can be separated into factorizable and non-factorizable corrections. The former can be associ-ated either with the gauge-boson-pair production orthe gauge-boson decay subprocesses and are well-known. The latter include the corrections in whichthe two W- or Z-boson resonances are not indepen-dent; they are due to the exchange of soft photonsbetween the different subprocesses. The non-factorizable corrections have recently been discussed

w xin the literature 2–5 .In this letter we supplement our results on non-

w xfactorizable corrections presented in Ref. 5 , whichagree 1 analytically as well as numerically with those

1 For more details of the comparison with the results of Refs.w x w x3,4 we refer to Ref. 5 .


( )A. Denner et al.rPhysics Letters B 429 1998 145–150146

w xof Ref. 4 . We start by briefly recalling the salientfeatures of the non-factorizable corrections. Already

w xbefore their explicit calculation it was known 6 thatthey are non-vanishing only if not both invariantmasses of the W or Z bosons are integrated over, i.e.,for instance, that they do not influence total crosssections.

w xIn Ref. 5 we explained why the actual form ofthe non-factorizable corrections is non-universal inthe sense that it depends on the choice for theparametrization of phase space for the real photoniccorrections. We have adopted the usual choice andtaken the invariant masses of decay fermion pairs asindependent variables. Since all corrections that arerelated to the initial state drop out, the correctionsneither depend on the gauge-boson-pair productionangle nor on the initial state itself.

The non-factorizable corrections to the processes

eq p qey pŽ . Ž .q y

™V k qk qV k qkŽ . Ž .1 1 2 2 3 4

™ f k q f k q f k q f k 1Ž . Ž . Ž . Ž . Ž .1 1 2 2 3 3 4 4

Ž .V V sWW,ZZ are proportional to the lowest-order1 2

differential cross section:

ds sd ds , 2Ž .nf nf Born

and the relative correction factor can be written asw x5

aqbq1d k ,k ;k ,k s y1 Q QŽ . Ž .Ý Ýnf 1 2 3 4 a b

a s 1,2 b s 3,4

=a

Re D k qk ,k ;k qk ,k , 3� 4Ž . Ž .1 2 a 3 4 bp

Ž .where Q is1,2,3,4 denotes the relative charge ofi

fermion f . The function D is explicitly given in Ref.iw x5 and has the symmetry

D k qk ,k ;k qk ,kŽ .1 2 a 3 4 b

sD k qk ,k ;k qk ,k . 4Ž . Ž .3 4 b 1 2 a

In all numerical evaluations up to now, only thepurely leptonic process eqey™WW™n llq ll Xy

n Xll ll

has been considered. In this case the non-factorizablecorrections are of the order of 1% in the LEP2energy range, but rapidly tend to zero for higher

energies. The corrections to the single-invariant-massdistributions dsrd M are identical and shift the"

peaks of the distributions by an amount of 1–2 MeVfor typical LEP2 energies, which is in fact negligibleat LEP.

The invariant-mass distributions for the hadronicdecay channels are of particular importance for thereconstruction of the W-boson mass from the W-bo-son decay products. The non-factorizable correctionsto the invariant-mass distributions are different fordifferent final states and in general also for theintermediate Wq and Wy bosons. 2 The invariant-mass distributions to the intermediate W " bosonscoincide only if the complete process is CP-symmet-ric. In this context, CP symmetry does not distin-guish between the different fermion generations,since we work in double-pole approximation andneglect fermion masses; in other words, the argu-

q yment also applies to final states like n e m n ande m

udsc, which are not CP-symmetric in the strict sense.Thus, we end up with equal distributions for the W "

bosons in the purely leptonic and purely hadronicchannels, respectively, but not in the semi-leptoniccase.

Fig. 1 shows the non-factorizable corrections tothe single-invariant-mass distributions for leptonic,hadronic, and semi-leptonic final states at variouscentre-of-mass energies 3. We observe the samequalitative features for all final states; the correctionsare positive below resonance and negative above.Quantitatively the differences between the correc-tions to the different final states are small; we notethat the slopes of the corrections on resonance, which

2 w x w xIn Ref. 4 and in the preprint version of Ref. 5 it has beenargued that the relative non-factorizable corrections to pure invari-ant-mass distributions are identical for all final states in eq ey™

WW™4 fermions and vanish for Z-pair-mediated four-fermionŽproduction. This was deduced from the assumption that up to

.charge factors the non-factorizable corrections become symmetricunder the separate interchanges k l k and k l k after inte-1 2 3 4

gration over all decay angles. Although the function D for therelative correction has this property, this assumption is not correct,because the differential lowest-order cross section is not symmet-ric under these interchanges.

3 For details concerning the implementation of the correctionw xfactor d in EXCALIBUR 7 and for the input parameters wenf

w xrefer to Ref. 5 .

( )A. Denner et al.rPhysics Letters B 429 1998 145–150 147

Fig. 1. Relative non-factorizable corrections to the single-invariant-mass distributions dsrd M for eqey™WW™4 fermions with"

different final states for various centre-of-mass energies.

are responsible for the shift in the maximum of thedistribution, are maximal for the leptonic final state.Therefore, we conclude that the W-boson mass de-termination by invariant-mass reconstruction at LEP2is not significantly influenced by non-factorizablecorrections.

w x w xThe authors of Ref. 4 have also calculated 8the non-factorizable corrections to the single-in-

'variant-mass distributions shown in Fig. 1 for s s172 GeV and 184 GeV. They find good agreementwith our results for positive invariant masses. How-ever, their corrections are antisymmetric and there-fore differ from our results for negative invariantmasses. The differences are of the order of non-dou-bly-resonant corrections and due to differentparametrizations of the corrections. For a discussion

w xof these differences we refer to Ref. 5 .

In our discussion of the non-factorizable correc-q y w xtions to e e ™WW™4 leptons in Ref. 5 we also

investigated their influence on various angular andenergy distributions with fixed invariant masses forthe final-state fermion pairs. We have repeated thisanalysis for hadronic and semi-leptonic final statesand found corrections of the same order of magni-tude, viz. of typically 1% at LEP2 energies.

For the production channels via a resonant Z-bo-son pair, eqey™ZZ™4 fermions, we have f s f1 2

and f s f . Owing to Bose symmetry the lowest-3 4

order cross section ds is invariant under the setBornŽ . Ž .of interchanges k ,k l k ,k . This symmetry,1 2 3 4

which is respected by the non-factorizable correc-w Ž .xtions see 4 , implies that the single-invariant-mass

distributions to each of the final-state fermion pairsof the two Z-boson decays are equal. CP invariance


leads to the additional symmetry with respect toŽ . Ž .p ,k ,k l p ,k ,k ; after integration over theq 1 2 y 4 3

Z-pair production angle this substitution reduces toŽ . Ž .k ,k l k ,k . In view of non-factorizable cor-1 2 4 3

rections it is also interesting to inspect the behaviourof ds under the replacements k lk and k lBorn 1 2 3

k separately, since terms in ds that are sym-4 Born

metric in at least one of these substitutions do notcontribute to ds if all decay angles are integratednf

over. This is a direct consequence of the antisymme-try of d in each of the substitutions k lk andnf 1 2

Ž .k lk , which follows from 3 and Q sQ , Q s3 4 1 2 3

Q .4

In order to study the behaviour of ds underBorn

the replacements k lk and k lk , it is conve-1 2 3 4

nient to consider the helicity amplitudes for the twosignal diagrams for eqey™ZZ™4 fermions, whichcontain two resonant Z-boson propagators. Theseamplitudes are proportional to the right- and left-handed couplings g "sÕ .a of each fermion f si i i i

f , f to the Z boson. As can be seen from the1 3

explicit form of the amplitudes, the substitution k1

lk transforms the helicity amplitudes to those2

with reversed helicities of the fermions f and f s f1 2 1

apart from changing the couplings g " into g ..1 1

Therefore, the differential lowest-order cross section,i.e. the squared helicity amplitudes summed over allfinal-state polarizations, can be split into two parts:one is symmetric in k lk and proportional to1 2wŽ q. 2 Ž y. 2 x 2 2g q g r2sÕ qa , the other is anti-sym-1 1 1 1

wŽ y. 2 Ž q. 2 xmetric and proportional to g y g r2s1 1

2Õ a . The analogous reasoning applies to the substi-1 1

tution k lk . After performing the angular integra-3 4

tions, we finally find that the lowest-order crossŽ 2 2 .Ž 2 2 .section is proportional to Õ qa Õ qa , and the1 1 3 3

non-factorizable correction proportional to4Q Õ a Q Õ a , where the charge factors Q stem1 1 1 3 3 3 i

from the correction factor d . Comparing pure in-nf

variant-mass distributions for different final states,the ratios of the non-factorizable corrections shouldbe of the same order of magnitude as the ratios ofthe corresponding coupling factors,

4Q Õ a Q Õ a1 1 1 3 3 3Fs . 5Ž .2 2 2 2Õ qa Õ qaŽ . Ž .1 1 3 3

The factor F takes the values shown in Table 1. Thereason for the smallness of the factor F is different

Table 1Values of the factor F

f f ll ll ll u ll d uu ud dd1 3

F 0.04 0.09 0.06 0.21 0.14 0.10

ll , u, d generically refer to leptons, up-type quarks anddown-type quarks, respectively.

for leptons and quarks: for leptons the suppression isdue to the small coupling Õ to the vector current, fori

quarks the factor F is reduced by the relative chargesQ .i

Fig. 2 shows the non-factorizable corrections tothe single-invariant-mass distributions dsrd M ,1,2

where M denote the invariant masses of the first1,2

and second fermion–anti-fermion pairs, respectively.The ratios of the different curves are indeed of theorder of magnitude of the ratios of the factor F givenin Table 1. For equal signs of Q and Q the shape1 3

of the corrections is similar to the shape of thecorrections to eqey™WW™4 fermions, for oppo-site signs of Q and Q the shape is reversed. The1 3

corrections by themselves are very small and phe-nomenologically unimportant. The smallness of thesecorrections can be qualitatively understood by com-

Ž .paring the factor F of 5 for the ratios of thecouplings with the corresponding one for the W-pair-mediated processes. For eqey™ WW ™

4 leptons we simply have Fs1, because in theLEP2 energy range the purely left-handed t-channeldiagram dominates the cross section, and no system-atic compensations are induced by symmetries.

Fig. 2. Relative non-factorizable corrections to the single-in-variant-mass distributions ds rd M for eq ey ™ ZZ ™1,2'4 fermions with different final states for s s192GeV.

( )A. Denner et al.rPhysics Letters B 429 1998 145–150 149

Fig. 3. Relative non-factorizable corrections to the angular distributions dsrd M d M dcosf and dsrd M d M dcosu q y in eqey™ZZ1 2 1 2 m ty q y q '™m m t t for fixed values of the invariant masses M and s s192 GeV.1,2

Therefore, the factors in Table 1 should directly givean estimate for the suppression of d for eqey™nf

ZZ™4, fermions with respect to four-lepton produc-tion via a W-boson pair. Comparing the correctionsfor energies with the same distance from the respec-tive on-shell pair-production thresholds, i.e. the curve

' Ž .for s s184 GeV in the W-boson case Fig. 1 with'the curves for s s192 GeV in the Z-boson case

Ž .Fig. 2 , we find reasonable agreement with ourw xexpectation. The authors of Ref. 4 have reproduced

the corrections shown in Fig. 2 with good agreementw x8 .

Finally, we inspect the impact of non-factorizablecorrections to some angular distributions in Z-pair-mediated four-fermion production for fixed values ofthe invariant masses M . Since the presence of the1,2

suppression factor F relies on the assumption thatthe phase-space integration is symmetric under k l1

k and k lk , this suppression in general does not2 3 4

apply to angular distributions. However, partial sup-pressions occur, e.g., if the integration is still sym-metric under one of these substitutions and, in partic-ular, for quarks in the final state because of theirsmaller charges. Two examples for angular distribu-tions without any suppression are illustrated in Fig. 3for the purely leptonic final state mymqtytq . Theangle f is defined by the two planes spanned by themomenta of the two fermion pairs in which the Zbosons decay,

k =k k =kŽ . Ž .1 2 3 4cosfs , 6Ž .

< < < <k =k k =k1 2 3 4

and u q y denotes the angle between the momentam t

of the mq and the ty, respectively. The shapes ofthe curves in Fig. 3, specifically the curves for thedistribution in cosf, nicely reflect the approximateanti-symmetric behaviour in the angular dependence,which leads to the suppression in the invariant-massdistributions. The size of the corrections turns out tobe at the level of a few per cent, i.e. they are notnecessarily negligible in precision predictions. Note,however, that the cross section for Z-pair productionis only one tenth of the W-pair production crosssection.

In conclusion, we find that the non-factorizablecorrections to W-pair production are not identical butsimilar for all final states and negligible compared tothe LEP2 accuracy. In the case of Z-pair production,the corrections are smaller for the invariant-massdistribution, but larger for some angular distribu-tions. In view of the smallness of the cross sectionfor Z-pair production, even these relatively largecorrections are not relevant for LEP2. The size of thenon-factorizable corrections might, however, com-pete with the expected experimental accuracy offuture eqey-colliders with higher luminosity.

Acknowledgements

We thank W. Beenakker, F.A. Berends, and A.P.Chapovsky for discussions and for performing acomparison with our numerical results on the invari-ant-mass distributions.


References

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11 June 1998


hg Z anomaly from the h™gmqmy decay

J. Bernabeu, D. Gomez Dumm, J. Vidal´ ´( )Departament de Fısica Teorica, IFIC, CSIC – UniÕersitat de Valencia, Dr. Moliner 50, E-46100 Burjassot Valencia , Spain´ ` ` `

Received 2 April 1998Editor: R. Gatto

Abstract

We show that the hg Z anomaly can be measured by analysing parity-violating effects in the h™gmqmy decay. In thissense, we find that the longitudinal polarization of the outgoing mq is an appropriate observable to be considered in futurehigh-statistics h factories. The effect is expected to lie in the range 10y5–10y6 in the Standard Model. q 1998 Published byElsevier Science B.V. All rights reserved.

The production of h mesons with high statistics in future experiments will improve the present knowledge ofrare h decay processes, enabling new possible tests of the Standard Model predictions. In particular, asufficiently large number of events would allow the experimental observation of weak interaction effects, whichare in general hardly suppressed by the large mass of the W " and Z gauge bosons. Parity-violating observablesare obvious candidates to get relevant information in this sense.

Among the different h decay channels, let us focus our attention on those which involve the effectivecoupling hgg , with g either a real or virtual photon. These processes deserve a significant theoretical interest,since the hgg vertex is governed by the axial anomaly, i.e., it provides a direct evidence of the presence of

Ž . w xquantum corrections breaking the U 1 symmetry of the QCD Lagrangian 1 . A consistent description forAŽ .these decays can be obtained within the framework of Chiral Perturbation Theory ChPT , where the anomaly is

Ž . w xintroduced through the Wess-Zumino-Witten WZW functional 2 . As it is well known, this leads also toŽsuccessful predictions for processes involving a pgg coupling, such as the decays p™gg where the anomaly

. q y w xwas actually discovered or p™g e e 3 .Now, if weak interactions are taken into account, the presence of an anomalous effective vertex hg Z is

expected as well. The latter should be correctly described in an analogous way as the hgg one, the quarkelectromagnetic couplings being replaced by the vector part of the corresponding weak neutral currents. Wepoint out that this ‘‘Z anomaly’’ has never been measured experimentally. Clearly, in order to get an observableeffect, it would be necessary to search for an asymmetry that could disentangle the Z contribution. In this letter,we show that the h™gmqmy channel is an appropriate one for this purpose. Indeed, owing to theparity-violating nature of the weak interactions, this process offers the possibility of constructing the requiredasymmetry by looking at the polarization of one of the final muons. We perform here an explicit calculation of

q Ž ythe longitudinal and transverse polarizations of the m the easiest to be measured, in view of the m capture.produced in the polarimeters , and give a numerical estimate of the effects that can be expected.


( )J. Bernabeu et al.rPhysics Letters B 429 1998 151–157´152

Fig. 1. Diagram for the h™gmqmy decay. The circle stands for the anomalous vertex.

Our analysis will be carried out within the framework of ChPT. First of all, let us remark that the treatmentof the hgg anomaly is not completely equivalent to the pgg one: there is an additional difficulty arising fromthe so-called hyh

X mixing. As it is well known, in the isospin limit the h mass state is given in general by aŽ .mixing between the Is0 states h and h , octet and singlet respectively under chiral SU 3 . The problem is8 0

Ž .that, due to the presence of the axial anomaly, the singlet state h cannot in principle be treated as an0

approximate Goldstone boson of the theory, and consequently its interactions are not described by ChPT. Still,however, it is possible to take into account the approximation of large number of colours. It can be shown that

Ž .in the large N limit, the U 1 symmetry of the Lagrangian is restored at the quantum level, and the h field isc A 0w xindeed incorporated as a ninth Goldstone boson 4 . In this way, it is possible to get definite predictions for the

interactions of the h by performing a double expansion in momenta and Ny1. This is the procedure we will0 c

follow in this work. In fact, when looking at the hg Z anomaly, it is found that the contribution of the h is8

significantly suppressed, so that the h part turns out to be the dominant one. This means that the measurement0

of the observables proposed here would represent an important test not only for the Z anomaly itself, but alsofor the viability of the large N approximation.c

Let us concentrate on the h™gmqmy decay channel. We begin by writing down the squared amplitude forthe process, which is represented by the diagram in Fig. 1. One has in general

C C gg g Zrna b Žl.) 2 y q y l l qMMs ie q k ´ f q e u p g Õ p q u p g g qg g Õ p ,Ž . Ž . Ž . Ž .Ž . Ž .a b r n n V A 52 2 2cosuq M WZ

1Ž .

where e Žl. is the photon polarization four-vector, g l and g l are the lepton weak neutral couplings, and C andr V A g

C stand for the anomalous vertices hgg ) and hg Z ) , respectively. Notice that we have included a form factorg ZŽ 2 .f q for the off-shell photon; we will make use of a single-pole approximation, taking

y12q2f q s 1y . 2Ž .Ž . 2ž /L

y2 w xAn experimental fit of the slope parameter L has been done some time ago 5 resulting in Ls720"90MeV, in good agreement with the hypothesis of r meson dominance 1.

In order to evaluate the anomalous vertices, let us first separate the contributions of the h and h states. As0 8

mentioned above, the mass eigenstate h is given in general by a mixing

< : < : < :h scosu h ysinu h , 3Ž .P 8 P 0

where the angle u is a parameter that can be estimated either through the diagonalization of the h-hX massP

1 y2 Ž 2 . ) w xIn fact, the averaged slope L turns out to be slightly smaller ;1rm when data from gg ™h processes are also included 6 .r

Ž 2 .However, these measurements have been performed at relatively large y q values and require an extrapolation. We keep here the resultw x q yof Ref. 5 , which was taken directly from the h™gm m decay.

( )J. Bernabeu et al.rPhysics Letters B 429 1998 151–157´ 153

matrix, or by analysing the phenomenology of h and hX decays. Both procedures are consistent in ChPT at

w x Ž .one-loop order, leading to a value of u of about y208 7 . In this way, the couplings C and C in 1 can beP g g Z

conveniently written as

C sC Ž8. cosu yC Ž0. sinu . 4Ž .g ,g Z g ,g Z P g ,g Z P

The values of C Ž8. and C Ž8., i.e., those which correspond to the octet state, can be easily obtained from theg g Z

WZW effective Lagrangian. One has

N a 1 acŽ8. 2C s Tr Q l s ,g 8 'p f p f3h h8 8

N eg 1 e gcŽ8. 2w xC s Tr Qg l s 1y4sin u , 5Ž .Ž .g Z V 8 W2 2'8p f cosu 16p f cosu3h W h W8 8

Ž u d s. Ž u d s .where Q and g are defined as diag Q ,Q ,Q and diag g , g , g respectively, and the parameter f isV V V V h8

equal to the pion decay constant f in the chiral limit. Notice that C Ž8. is found to be suppressed by a factorp g ZŽ 2 . w x1y4sin u 8 , as it is demanded by the lack of anomalies in the SM: quark and lepton contributions have toW

amount to the same magnitude and opposite sign.The evaluation of C Ž0. is more subtle. As stated above, since h is not a Goldstone boson in the chiral limit,g ,g Z 0

its couplings are in principle not described by ChPT. However, we can take into account the large N limit incŽ . y1order to get analogous expressions to those in 5 . At leading order in N , the chiral symmetry is enlarged toc

Ž .U 3 , and the WZW Lagrangian can be extended to incorporate the h field. One gets in this way0

' '2 N a 2 2 acŽ0. 2C s Tr Q s ,g ' 'p f p f3 3h h0 0

' '2 N eg 2 e gcŽ0. 2w xC s Tr Qg s 1y2sin u , 6Ž .Ž .g Z V W2 2' '8p f cosu 4p f cosu3 3h W h W0 0

where once again the relation f s f is expected to hold at the lowest order in the chiral expansion. In fact, ifh p0

f is identified with the axial current decay constant corresponding to the h state, one finds at next to leadingh 88

Ž . w xorder NLO in ChPT 9

f ,1.3 f , 7Ž .h p8

and then, from the experimental value of the hX™gg decay,

f ,1.1 f . 8Ž .h p0

w xIt can be seen 10 that this value shows a very good agreement with the NLO prediction given by ChPT in theŽ .U 3 symmetric limit, thus giving important support to the large N approximation.c

Ž . Ž 2 . Ž0.It is worth to notice from 6 that the 1y4sin u suppression factor is not present in the case of C . InW g Z

fact, for a mixing angle u ,y208, we see that the h state contribution to C is enhanced by about a factorP 0 g Z

10 with respect to that of the h , and largely dominates the hg Z anomalous coupling.8

We proceed now to identify an observable that could be sensitive to the Z anomaly. As stated, in order todisentangle the Z ) contribution to the h™gmqmy amplitude, one is led to search for a parity-violatingasymmetry.


We will consider two possible candidates, namely the longitudinal and transverse polarizations of the finalq Ž .m resulting from the decay. Let us recall the amplitude in 1 , and perform the sum over spins and helicities

for the my and photon final states respectively. We find

2B 2V 2 22 2 2 q y 2< <MM s f q 2 q qPk y2 p Pk p Pk q4 m qPkŽ . Ž . Ž . Ž .Ž .Ž . Ž .Ý m4qyŽ .s m ,l

) 2Re B B qŽ .V A 2 y y 2 yy f q 8 m qPk sPp p Pk q qPk sPk yq sPk p Pk ,Ž . Ž . Ž . Ž . Ž . Ž . Ž .Ž . m2 ½ 52q

9Ž .a Ž 0 . qwhere s s s ,s stands for the m polarization four-vector, and B and B correspond respectively to theV A

Ž . Ž < < 2 < < 2 .vector and axial vector muon couplings in 1 we have assumed B < B . Considering the mixing in Eq.A VŽ .4 , we have

C Ž0. g g l q2 C Ž0.g V g ZŽ8. Ž8.B se C cosu y sinu y C cosu y sinuV g P P g Z P PŽ8. 2 Ž8.ž / ž /2cosuC M CWg Z g Z

g g l C Ž8. C Ž0.A g Z g Z

B sy cosu y sinu , 10Ž .A P P2 Ž8.ž /2cosu M CW Z g Z

Ž . Ž .and then, from relations 5 and 6 ,

23 f4a h2 8'< <B , cosu y2 2 sinuV P P2 ž /f3p f hh 08

2 fG a hF 8) 'Re B B , cosu y2 2 sinuŽ .V A P P2 2 ž /' f6p f2 hh 08

=fh82 2'1y4sin u cosu y4 2 1y2sin u sinu , 11Ž .Ž . Ž .W P W Pž /fh0

where we have kept only leading terms in powers of the weak effective coupling G .F

In order to deal with the phase space, we will define our observables in the h rest frame 2. Let us choose thez axis along the mq three-momentum, and take as independent variables the mq energy E and the angles u andw determining the direction of the outgoing photon. For each differential phase space volume dF'dE dV

Ž . qwith dVsdcosu dw , it is possible to define the polarization of the m along a given direction s byˆ

dG Žq.rdFydG Žy.rdFP E,u ,w ;s ' , 12Ž . Ž .Žq. Žy.dG rdFqdG rdF

Ž" . Ž . qwhere G 'G "s are the widths to final states with opposite m polarization vectors. This observable is

2 Notice that the longitudinal and transverse polarizations of the outgoing particles are in general not invariant under Lorentztransformations. In fact, some small dilution of the effect can be expected when the h mesons are produced in flight.


clearly parity-violating, hence it will be dominated by the g ) yZ ) interference term in the squared amplitude.Ž . Ž .From Eq. 9 , the numerator in 12 explicitly reads

Žq. Žy. ) < < 2dG ydG Re B B m y2 E P f qŽ . Ž .Ž .V A hs =FF E,cosu ;s , 13Ž . Ž .4 20dE dV 128p < <m 2k ym m yEq P cosuŽ .Ž .h h h

Ž . 0where the function FF corresponds to the expression in curly brackets in 9 , and k and P stand for the photonenergy and the mq three-momentum respectively; in terms of E and u ,

m m r2yEŽ .h h2 0 0 2 2< <q sm m y2k , k s , P s E ym . 14Ž .(Ž .h h m< <m yEq P cosuh

Ž < <.The form of FF depends on the chosen direction of s. In the longitudinal case ssEPrm P , one hasm

3 28 m m m mh h h m0 02 0 02 0FF s E yk k q m y2 E yEyk y k m yEyk 15Ž .Ž . Ž .L h hž / ž /ž /< <P 2 2 mh

whereas for a transverse s, we find

2mh2 0 0 0FF s8 m m k y m yk Eqk sinu 16Ž . Ž .Ž .T m h h2

Žhere, it is understood that s is oriented within the decay plane, hence its direction is determined by the angle. Ž .w . On the other hand, notice that the normal polarization this means, normal to the decay plane is expected to

be very small in this scenario, since it is related to CP- or T-odd effects.Ž . q yThe denominator in 12 is nothing but the differential width for the h™gm m process. In this case the

Ž .contribution of the virtual Z can be safely neglected, and Eq. 9 leads to

22Žq. Žy. 2< < < <dG qdG B m y2 E P f qŽ .Ž . Ž .V h, =FF E,cosu , 17Ž . Ž .04 20dE dV 128p < <m m y2k m yEq P cosuŽ .Ž .h h h

with

m mh h2 0 02 0 2 02FF s4 m m yk k q m y2 E yEyk qm k . 18Ž .Ž .0 h h h mž / ž /ž /2 2

Ž .By integrating the expression in 17 over the whole phase space, one obtains a prediction for the total width< < 2 Ž .that can be compared with the experimental results. Using the value of B in 11 , a mixing angleV

Ž . Ž .u ,y208, and taking f and f as in 7 and 8 respectively, we findP h h8 0

G h™gmqmy ,3.6=10y7 MeV , 19Ž . Ž .Ž . y7 w xin good agreement with the value of 3.7"0.6 =10 MeV from the Particle Data Group 11 .

Ž . Ž . Ž .Now, from 13 and 17 , the asymmetry defined in Eq. 12 is given by

q2 Re B B) FF E,cosu ;sŽ .Ž .V AP E,u ,w ;s sy . 20Ž . Ž .22 FF E,cosu< < Ž .f q BŽ . 0V

As expected, one finds here a strong suppression factor, arising from the ratio between the Z ) and g )

contributions to the decay amplitude. A rough estimate of the order of magnitude for the effect yields< < 2 < < y5 y6P ;m B rB ;10 –10 .h A V


Let us finally perform a more detailed numerical analysis, considering the expected mq polarization for aŽ .finite region DF of the phase space. In analogy with 12 , it is possible to define the asymmetry

dG Žq.y dG Žy.H HP DF ;s ' , 21Ž . Ž .

Žq. Žy.dG q dGH Hwhere the integrals extend to the volume DF . We will concentrate on the longitudinal mq polarization, looking

Ž . Žat the dependence of both numerator and denominator in 21 with the variables E and u introduced above the.integration in w is trivial . In general, we expect the value of P to be optimized by choosing a convenient

Ž .region of the phase space. By analysing the expression in 17 , it can be seen that the differential decay width isŽ q.sharply peaked backwards i.e., when the photons are produced with opposite direction to that of the m , with

Ž .more than 70% of the events in the y1FcosuFy0.5 region. Unfortunately, the numerator in Eq. 21 shows< <a similar behaviour, and we cannot get a significant enhancement in P by introducing a cut in cosu . On the

q Ž .other hand, the m energy spectrum is shown in Fig. 2. The plotted curves result from the functions in Eqs. 17Ž . Ž . Ž .solid and 13 dashed , after integrating over all possible directions of the final photon; as anticipated, thedifference between the rates to opposite mq polarizations is about six orders of magnitude lower than the total

Ž .decay width notice the different scales at both sides in Fig. 2 . By looking at the figure, it is seen that the value< < qof P can be increased by performing a lower cut in the m energy range. Indeed, a convenient region is that

given by 140 MeV QEFm r2, in which we find P,y2.4=10y6 , with 80% of the total number of events.h

Though it is still possible to obtain higher values of P by moving the cut towards the upper limit of E, theŽgrowth is found to be slow in comparison with the reduction of statistics e.g. for EG220 MeV, we get

y6 .P,y3.8=10 , while only 20% of the events remain .The transverse mq polarization is less favoured from the experimental point of view. It can be seen from Eq.

Ž .16 that the asymmetry presents in this case an additional m rm suppression, which is indeed expected fromm h

Ž .chirality arguments no transverse polarization is obtained in the limit of vanishing muon mass . Moreover, foreach event, the observable requires the identification of the decay plane, which defines the direction of the

< < y6 y7polarization vector. The values of P obtained in this case fall typically in the range 10 –10 .Summarizing, we have analysed here the Z contribution to the decay h™gmqmy. We have shown that this

channel can be an appropriate one to find an observable effect of the anomalous coupling hg Z, which has neverbeen measured experimentally up to now. Our analysis has been performed using ChPT, together with large-Nc

Fig. 2. Differential decay rates for the process h™gmqmy, in terms of the energy of the final mq. The solid line stands for the total width,while the dashed one corresponds to the difference between rates to opposite longitudinal mq polarizations. Notice the different ordinatescales at both sides of the figure.


considerations. This framework allows to deal with the interactions involving not only the h but also the h8 0

component of the h mass eigenstate. In fact, it turns out that the h part is that which dominates the hg Z0

anomalous vertex. In order to disentangle the contribution of the Z boson to the decay amplitude, we haveconsidered the polarization of the final muons, which give rise to parity-violating effects. In particular, thelongitudinal polarization of the mq is shown to be an adequate candidate for the measurement of the Z anomalyin future h-factory experiments. The value of this observable in the h rest frame is found to lie in the range10y5 –10y6 in the Standard Model.

Acknowledgements

D.G.D. has been supported by a grant from the Commission of the European Communities, under the TMRŽ . Ž .programme Contract N8 ERBFMBICT961548 . This work has been funded by CICYT Spain under grant No.

AEN-96-1718.

References

w x Ž . Ž .1 S.L. Adler, Phys. Rev. 177 1969 2426; J.S. Bell, R. Jackiw, Nuovo Cim. A 60 1969 47.w x Ž . Ž .2 J. Wess, B. Zumino, Phys. Lett. B 37 1971 95; E. Witten, Nucl. Phys. B 223 1983 422.w x Ž .3 For a review of anomalous processes in ChPT, see J. Bijnens, Int. J. Mod. Phys. A 8 1993 3045.w x Ž .4 E. Witten, Nucl. Phys. B 156 1979 269.w x Ž .5 R.I. Dzhelyadin et al., Phys. Lett. B 94 1980 548.w x Ž . Ž .6 H. Aihara et al., Phys. Rev. Lett. 64 1990 172; H.-J. Behrend et al., Z. Phys. C 49 1991 401.w x Ž . Ž .7 J.F. Donoghue, B.R. Holstein, Y.-C.R. Lin, Phys. Rev. Lett. 55 1985 2766; F.J. Gilman, R. Kauffman, Phys. Rev. D 36 1987 2761;

recent phenomenological analyses have been done by E.P. Venugopal, B.R. Holstein, hep-phr9710382; A. Bramon, R. Escribano,´M.D. Scadron, Preprint UAB-FT 97r431, ULB-TH 97r20, November 1997, hep-phr9711229.

w x Ž .8 J. Bernabeu, G.A. Gonzalez-Sprinberg, J. Vidal, Z. Phys. C 69 1996 431.´ ´w x Ž .9 J. Gasser, H. Leutwyler, Nucl. Phys. B 250 1985 465.

w x Ž .10 J. Bijnens, A. Bramon, F. Cornet, Phys. Rev. Lett. 61 1988 1453.´w x Ž .11 Review of Particle Properties, Phys. Rev. D 54 1996 1.

11 June 1998


Leptonic contribution to the effective electromagneticcoupling constant up to three loops

M. SteinhauserMax-Planck-Institut fur Physik, Werner-Heisenberg-Institut, D-80805 Munich, Germany¨

Received 11 March 1998; revised 8 April 1998Editor: P.V. Landshoff

Abstract

In this note the leptonic contribution to the running of the electromagnetic coupling constant is discussed up to thethree-loop level. Special emphasis is put on the evaluation of the double-bubble diagrams. The new term obtained is twoorders of magnitude smaller than the hadronic uncertainty. q 1998 Elsevier Science B.V. All rights reserved.

The dominant correction to electroweak observables is provided by the running of the electromagneticcoupling constant from its value at vanishing momentum transfer to high energies. The main part is delivered

Ž 2 .from the leptonic contribution, Da , where QED corrections to the photon polarization function, P q , havelep

to be considered. The large logarithms of the ratio between the lepton masses, M , and the mass of the Z boson,l

M , is responsible for the size of the corrections. On the other hand the small ratio ensures that it is enough toZŽ 2 . 2 2consider an expansion of P q for q 4M . Below in addition to the leading term we will include the firstl

order mass corrections for the one- and two-loop case and demonstrate that the numerical impact is small.w x Ž 2 . y4In 1 it is shown that the two-loop term for Da M amounts to approximately 0.78=10 whichlep Z

though is less than 0.3% of the one-loop value is of the same order of magnitude as the error of the hadronicŽ5. w xcontribution, Da 1 . Hence, it is interesting to have a look at the three-loop contributions also because therehad

for the first time quadratic logarithms of M 2rM 2 appear.l ZŽ 2 .In this note the three-loop diagrams contributing to P q , respectively, Da , are discussed. Thereby onlylep

the electron, muon and tau lepton are considered. Virtual quark loops which are present at this order are nottaken into account. However, the formulae given below could in principle also be applied to this case.

At one- and two-loop level and for the diagrams relevant for the so-called quenched QED only one massscale – in the following called M – is involved. Concerning the double-bubble diagrams, however, in general1

Ž Ž ..two mass scales, M and M , appear, where M is the mass of the second fermion loop see Fig. 1 a . Having1 2 2Ž . Ž .in mind the electron, muon and tau lepton three different cases are of practical interest: i M 4M , ii1 2

Ž .M sM and iii M <M .1 2 1 2Ž . Ž 2 . Ž 2 .The quantity which enters the relation between a 0 and a M is the polarization function, P q ,Z

2 2 Ž .evaluated for q sM and normalized in such a way that P 0 s0. This means that one has to compute theZ

current correlator both in the limit q24M 2, M 2, where the masses may be neglected, and for qs0. It is1 2


( )M. SteinhauserrPhysics Letters B 429 1998 158–161 159

Ž . Ž .Fig. 1. Double-bubble diagram a and quark self-energy diagram which is needed for the computation of the mass counterterm b .

convenient to perform the calculation in a first step in a renormalization scheme where the charge and massesare defined in the MS scheme. The transformation to the on-shell quantities is done afterwards.

The polarization function can be cast into the form2a a a

2 Ž0. Ž1. Ž2. Ž2. Ž2. Ž2.P q s P q P q P qP qP qP , 1Ž .Ž . Ž .A l F hž /4p p p

Ž . Ž .where the double-bubble contributions are given by the last three terms corresponding to the cases i , ii andŽ . Ž2. Ž2.iii , respectively. P and P depend on both masses whereas in all other expressions only M appears.l h 1

Ž0. Ž1. w x w xP and P where already computed in 2 . The results for large external momentum can be found in 3–5 .Ž . Ž . Ž .Note that in this limit the results for the three cases i , ii and iii are identical as mass corrections are

neglected. Thus the differences arise from the evaluation of the polarization function for qs0. The correspond-w x Ž .ing results for P , P and P can be extracted from 6,4,5 . To the knowledge of the author the case iii wasA l F

Ž .never discussed. Therefore we want to present the result in more detail. For the mass constellation of case iii itis not possible to set M s0 from the very beginning. Rather one has to apply the so-called hard mass1

w x 2 2procedure 7 in the limit M <M . Furthermore one encounters a two-loop mass counterterm which has to be1 2Ž Ž ..taken into account see Fig. 1 b , which, however, is not yet available in the literature.

Before presenting the results let us in a first step introduce some notation. Throughout the paper on-shellmasses are denoted with capital letters; MS ones with small letters. Bare masses are accompanied with an index

Ž . Ž .‘‘0’’. The double-bubble diagrams from case i are multiplied with n , the ones from ii with n and the onesl FŽ .from iii with n .h

w xThe relation between the bare and the MS mass reads 8 :22 2a m 3 a m 9 3 1 5Ž . Ž .

0m sm 1q y q y q n qn qn y q . 2Ž . Ž .1 1 l F h2 2ž / ž /ž /½ 5p 4´ p 64´ 48´32´ 8´

Note that the pole parts for the double-bubble diagrams are identical, i.e., they are independent of the massconfiguration – a special feature of MS-like schemes. The transformation to the pole mass is performed with thehelp of:

22 2 2 2a m m a m mŽ . Ž .3 7 15 3 21m sM 1q y1y ln q q y q3ln2 z 2 y z 3 q lnŽ . Ž .Ž .1 1 4 128 8 4 322 2ž /½ ž /p pM M1 1

m2 m2 m2 m2 m29 71 1 13 1 143 13 12 2 2q ln qn q z 2 q ln q ln qn yz 2 q ln q lnŽ . Ž .l F32 96 2 24 8 96 24 82 2 2 2 2ž / ž /M M M M M1 1 1 1 1

2 2 2 2m m m m89 13 1 12qn y q ln y ln q ln ln . 3Ž .h 288 24 8 42 2 2 2 5ž /M M M M2 2 1 2

( )M. SteinhauserrPhysics Letters B 429 1998 158–161160

w xThe terms in the first three lines can be found in 9 . The computation is reduced to the evaluation of two-loopw xon-shell integrals and can be be performed, e.g., with the help of the program package SHELL2 10 . The terms

Ž .proportional to n arise from the diagram pictured in Fig. 1 b which has to be evaluated in the limith2 2 2 w xM 4M sp where p is the external momentum. The bare diagram has already been computed in 11 .2 1

Using the relation2 2 2a 1 m m m

2a m sa 1q n ln qn ln qn ln , 4Ž .Ž . l F h2 2 2ž /p 3 M M M2 1 2

Ž .also the coupling can be transformed to the on-shell scheme. Note that the overall factor a in Eq. 1 is notŽ 2 .affected by this change of parameters. The results for the separate contributions to P q then read:

M 2120 4Ž0.P s y L q8 q . . . , 5Ž .qM9 3 21 q

M 215Ž1.P s y4z 3 yL y12 L q . . . , 6Ž . Ž .qM qM6 21 1q

121 99 1Ž2.P sy q y5q8ln2 z 2 y z 3 q10z 5 q L q . . . , 7Ž . Ž . Ž . Ž . Ž .A qM48 16 8 1

116 4 38 14 5 4 1 1Ž2. 2P sy q z 2 q z 3 q L q y z 3 L q L y L L q . . . , 8Ž . Ž . Ž . Ž .Ž .l qM qM qM qM qM27 3 9 9 18 3 6 31 2 1 1 2

307 8 545 11 4 1Ž2. 2P sy y z 2 q z 3 q y z 3 L y L q . . . , 9Ž . Ž . Ž . Ž .Ž .F qM qM216 3 144 6 3 61 1

37 38 11 4 1Ž2. 2P sy q z 3 q y z 3 L y L q . . . , 10Ž . Ž . Ž .Ž .h qM qM6 9 6 3 62 2

Ž 2 2 . Ž 2 2 . 2 2with L s ln yq rM and L s ln yq rM . The dots represent subleading terms in M rM . TheqM 1 qM 2 l Z1 2

logarithmic dependence on q of the double-bubble diagrams coincides, the constant terms are, however,different. It is also worth to mention that P Ž2. gets independent of M , however, only after the mass ish 1

transformed to the on-shell scheme. Note also that P Ž2. becomes dependent on M after the result is expressedl 2

in terms of the on-shell coupling.Ž 2 . Ž Ž 2 2 ..For the numerical evaluation of Da M the quantity yRe P q sM has to be considered taking intolep Z Z

account the contributions from the electron, muon and tau lepton. Special care has to be taken for thosecontributions where two masses are involved: In the case of the electron P Ž2. is not present and P Ž2. has tol h

be evaluated with M sM and M sM . For the muon P Ž2. is used with M sM and P Ž2. with M sM .2 m 2 t l 2 e h 2 t

For the contribution from the tau lepton P Ž2. has to be evaluated with M sM and M sM .l 2 e 2 m

Ž 2 .In Table 1 the numerical results for Da M separated into the contributions from the different leptonlep Z

species and the number of loops is listed. It can be seen that at three-loop order for each lepton the sum of thedouble-bubble diagrams is larger by a factor of 30 to 40 as compared to the quenched part. The total

Table 1Ž 2 . 4Contributions to Da M =10 .lep Z

4Da =10 1-loop 2-loop 3-loop sumlep

quenched inner lepton

A e m t sum

e 174.34653 0.37983 y0.00014 0.00287 0.00084 0.00025 0.00382 174.73018m 91.78419 0.23600 y0.00009 0.00266 0.00084 0.00025 0.00366 92.02385t 48.05934 0.16034 y0.00007 0.00214 0.00082 0.00025 0.00314 48.22282eqmqt 314.19007 0.77617 y0.00030 – – – 0.01063 314.97686

( )M. SteinhauserrPhysics Letters B 429 1998 158–161 161

Ž 3. Ž 2 .contribution of OO a amounts to f1.4% of the OO a result and is thus roughly an order of magnitude largerthan expected using the naive estimation arpf2=10y3. This can be traced back to the squared logarithmswhich are actually only present in P Ž2., P Ž2. and P Ž2.. It is also remarkable that the contributions to thel F h

double-bubble diagrams are essentially dominated by the mass of the inner lepton which can be understood by aŽ . Ž .closer look to Eqs. 8 – 10 .

A comment concerning the mass corrections M 2rM 2 is in order. Obviously the most important contributionl Z

comes from the t lepton. It amounts at one-loop level to approximately y2=10y6 which is of the same orderof magnitude as the one from the three-loop diagrams. The quadratic mass corrections of order a 2 are alreadytwo orders of magnitude smaller. Therefore we neglect the mass corrections at the three-loop level. In principlealso terms of the form M 2rM 2 are present in the correlator P Ž2.. However, they are also small and will be1 2 h

neglected.To summarize, the leptonic contribution to the effective electromagnetic coupling constant is computed up toŽ 3.OO a . At three-loop level it turns out that the double-bubble diagrams are most important as quadratic

2 2 Ž 2 . y6logarithms of the ratio M rM appear. The total contribution to Da M amounts to 10 which is roughlyl Z lep Z

two orders of magnitude smaller than the hadronic uncertainty.

I would like to thank J.H. Kuhn for inspiring and valuable discussions.¨

References

w x1 J.H. Kuhn, M. Steinhauser, Report Nos. TTP98-05, MPI-PhTr98-12 and hep-phr9802241.¨w x Ž .2 G. Kallen, A. Sabry, K. Dan. Vidensk. Selsk. Mat.-Fys. Medd. 29 1955 No. 17.¨ ´w x Ž . Ž .3 S.G. Gorishny, A.L. Kataev, S.A. Larin, Phys. Lett. B 259 1991 144; L.R. Surguladze, M.A. Samuel, Phys. Rev. Lett. 66 1991 560,

Ž . Ž .2416 E ; K.G. Chetyrkin, Phys. Lett. B 391 1997 402.w x4 P.A. Baikov, D.J. Broadhurst, Presented at 4th International Workshop on Software Engineering and Artificial Intelligence for High

Ž .Energy and Nuclear Physics AIHENP95 , Pisa, Italy, 3–8 April 1995, Published in Pisa AIHENP, 1995, p. 167.w x Ž .5 K.G. Chetyrkin, J.H. Kuhn, M. Steinhauser, Nucl. Phys. B 482 1996 213.¨w x Ž .6 D.J. Broadhurst, A.L. Kataev, O.V. Tarasov, Phys. Lett. B 298 1993 445.w x Ž .7 For a review see e.g.: V.A. Smirnov, Mod. Phys. Lett. A 10 1995 1485.w x Ž .8 R. Tarrach, Nucl. Phys. B 183 1981 384.w x Ž .9 N. Gray, D.J. Broadhurst, W. Grafe, K. Schilcher, Z. Phys. C 48 1990 673.

w x Ž .10 J. Fleischer, O.V. Tarasov, Comput. Phys. Commun. 71 1992 193.w x Ž .11 L.V. Avdeev, M.Yu. Kalmykov, Nucl. Phys. B 502 1997 419.

11 June 1998


Luminosity spectrum measurement in future eqey linear collidersusing large-angle Bhabha events

N. Toomi a,1, J. Fujimoto a,2, S. Kawabata a,3, Y. Kurihara a,4, T. Watanabe b,5

a High Energy Accelerator Research Organization Tsukuba, Ibaraki 305, Japanb Department of physics, Kogakuin UniÕersity Shinjuku, Tokyo 160, Japan

Received 16 January 1998; revised 1 April 1998Editor: L. Montanet

Abstract

In future eqey linear colliders, the colliding energy cannot be monochromatic due to the large effect of beam-beaminteractions before eqey collisions. The luminosity must have a continuous spectrum which distributes around the nominalenergy and has a long tail toward the low-energy region. An accurate measurement of the luminosity spectrum isindispensable for a precise experiment at high-luminosity eqey linear colliders.

It has been found that the luminosity spectrum can be determined by measuring the acollinearity angle of Bhabha eventsby a tracking device with good angular resolution. The beam parameters, i.e. the beam-energy spread, the topological beamsize and the number of particles in a beam bunch, can be also extracted from the measured luminosity spectrum by alikelihood fitting. q 1998 Elsevier Science B.V. All rights reserved.

PACS: 29.17.qw; 29.90.q rKeywords: Luminosity measurement; Linear collider

1. Introduction

The eqey linear colliders at the center of massenergy from 300 to 500 GeV are studied as thefuture projects of high energy physics. In such a highenergy region, the cross sections for the most ofeqey processes are around or less than 1pb. The

1 E-mail: [email protected] E-mail: [email protected] E-mail: [email protected] E-mail: [email protected] E-mail: [email protected].

luminosity of more than 1033 cmy2 secy1 is requiredto obtain sufficient event rates. Several design stud-ies have been made to achieve the high luminosityand reached a common design-principle that thebeam size should be of the order of nano-meter andthe number of particles be more than 1010 per bunch.As a result of such a high density beam, electronsand positrons may lose their energies due to thebeam-beam effect before they collide. Consequently,

'the colliding energy, s , is no more monochromatic,but shows a continuous spectrum.

In the current eqey colliding experiments, theluminosity is measured by counting the number of

0370-2693r98r$19.00 q 1998 Elsevier Science B.V. All rights reserved.Ž .PII: S0370-2693 98 00464-X

( )N. Toomi et al.rPhysics Letters B 429 1998 162–168 163

Bhabha events under the assumption that the collid-ing energy is monochromatic. That number of eventscan be converted to the luminosity dividing it by theeffective cross section, which is calculated taking thedetection efficiency into account. At the linear col-liders, however, since the colliding energy has a

'continuous spectrum, Bhabha events with various svalues are detected. The current method is not appli-cable to the precise determination of the luminosity.

ŽIn addition, the beam-energy spread of around 1% ;. w xa few GeV! 1 makes the luminosity measurement

more difficult. A precise determination of the lumi-Žnosity spectrum the distribution of the center-of-

.mass energy is indispensable for the experimentalstudies, especially at the particle threshold, such as

w xin t-quark physics 2 .Miller et al. proposed a method to measure the

luminosity spectrum using the acollinearity angle ofw xlarge-angle Bhabha events 3 , caused by the asym-

metric collision of electrons and positrons. The po-tentiality of this idea is studied by means of our

w xluminosity spectrum generator, LL uminous 4 , and aBhabha-event generator produced by the GRACE

w xsystem 5 .In next section, the method of the event genera-

tion and basic assumptions in this work are de-scribed. Section 3 is devoted to the direct measure-ment of the luminosity spectrum. The effect of thedetector resolution to this method is also discussed.Using the analysis introduced in Section 3, thebeam-parameter fitting by a likelihood method isperformed in Section 4 to see how the luminosityspectrum is reproduced. The integrated luminosity isestimated from the result of beam parameter fittingin Section 5.

2. Event generation

Yokoya et al. have developed simulation codes,w x w xABLE 6 and CAIN 7 , to calculate a detailed

beam-beam interaction, and have shown a precisedistribution of the colliding energy of the eqey

colliders as well as those of the eg and gg colliders.They also proposed an empirical function expressingthe energy distribution of colliding electrons orpositrons based on that detailed simulation in a

w xrelatively simple form 8 . In the experimentally

Table 1Accelerator parameters for JLC

E rGeV 250 s rnm 260.0beam x10N r10 0.7 s rnm 3.04particles y

y6e r10 mrad 3.3 s rmm 90x zy6e r10 mrad 0.048 f 150y rep

)b rmm 10.0 n 85x bunch)b rmm 0.1y

'interesting energy region of more than s s450 GeV, the function can reproduce CAIN’s resultwell.

The event rate of Bhabha scattering can be writtenby

' 'd LL s ds sŽ . Ž . 'Ns dV d s , 1Ž .HH ' dVd s

where the first factor is the luminosity spectrum andthe second is the differential cross section of Bhabhascattering.

To obtain the luminosity spectrum, we developeda program package LL uminous, which generates theenergy of colliding electrons or positrons efficiently

Ž .Fig. 1. A Luminosity spectrum weighted by the Bhabha crossŽ .section without with the beam-energy spread, shown by the solid

Ž . Ž .dashed histogram. B The same distribution calculated based onŽ .the acollinearity angle using Bhabha events without with any

Ž .beam-energy spread, shown by the solid dashed histogram.

( )N. Toomi et al.rPhysics Letters B 429 1998 162–168164

using the empirical function as the probability den-sity function. Through this work, it is assumed thatthe nominal beam energy is known, and both elec-tron and positron beams have completely identicalbeam parameters for simplicity, as listed in Table 1.

Besides the beam-beam interactions, the beam-en-ergy spread also distorts the luminosity spectrum.Since the exact shape of the beam-energy spread isnot known well, a flat distribution around the nomi-

Ž .nal beam energy "1% is assumed in the analysis.Ž .This effect is visualized in Fig. 1 A .

To calculate the matrix element of Bhabha scatter-ing, an automatic Feynman amplitude calculationsystem, GRACE, is used. In this study only thelowest order diagrams of Bhabha scattering are taken

Ž .into account. The integration Eq. 1 is carried outnumerically by BASES and the unweighted Bhabha

w xevents are generated by SPRING 9 . When we studythe effect of the angular resolution of the detector,the production angle of each charged particle issmeared with the Gaussian distribution indepen-dently.

3. Luminosity spectrum measurement

Miller et al. proposed to use the acollinearityangle for calculating the effective CM energy of theBhabha event, since the energies of the final electronand positron could not be measured accuratelyenough. The momentum difference, DP, betweenthe initial electron and positron can be approximatelydescribed by the acollinearity angle, u , of theA

Bhabha event as

u PA bDP, , 2Ž .

sinu

where P stands for the nominal beam momentum,b

and u is the average polar angle of the final electronand positron. The CM energy, s , can be approx-( meas

imately obtained by

s s s yDP , 3Ž .( (meas 0

where s is the nominal CM energy obtained from( 0

an independent energy measurement. Since theacollinearity angle is positive definite, s is al-( meas

ways less than or equal to s . Although the beam-( 0

energy spread may produce those events with theeffective CM energy above the nominal value as

Ž .shown in Fig. 1 A , the measured CM energy dis-Ž .tributes only below that value, as shown in Fig. 1 B .

This is one of the limitations of this method. Whenthe beam has a finite energy-spread, a direct mea-surement of the luminosity spectrum requires otherinformation in addition to the acollinearity angle andthe nominal beam energy.

In order to see the effect of the detector resolutionclearly, the beam-energy spread is assumed to bezero hereafter in this section. The spectrum measuredby such an ideal detector which determines the pro-duction angle of a charged particle without error iscompared with that by a detector with the angularresolutions of 3mrad and 5mrad, as shown in Fig. 2.The result with the resolution of 1mrad is hardlydistinguished from that by the ideal detector. Themeasured spectra with a worse resolution than 3mradshow visible deviations from that of the ideal case.

The CM energy distribution of Bhabha eventsaround nominal energy is essentially important be-cause almost all the events observed in the experi-ment come from this energy region as shown in Fig.

Ž1 and Fig. 2. Consider a quantity call the effective.luminosity hereafter defined by

'` d LL sŽ . ' 'Is s s d s , 4Ž .Ž .H 'd s495

' 'Ž .where d LL s rd s is either the measured lumi-

Fig. 2. Luminosity spectrum calculated from the measuredacollinearity angle of Bhabha events by a tracker with an angular

Ž . Ž .resolution of perfect solid histogram , 3mrad dashed histogramŽ .and 5mrad dotted histogram .


'Ž .nosity spectrum or the generated one and s s isthe total cross section of Bhabha scattering. Fig. 3shows how the ratio of the measured effective lumi-nosity, I , to the generated one, I , depends onmeas gen

the angular resolution of the detector.Even with the ideal detector the measured CM

energy distribution can not perfectly reproduce thegenerated one. When, for example, the electron andpositron lose the same amount of energy by thebeam-beam effect, the shape of this event is collinearin spite that its CM energy is not equal to thenominal one. This is another limitation of thismethod. A deviation of 2.5% for the ideal detectorcase shown in Fig. 3 is due to this effect.

For the small-angle region, the resolution of DPŽ .in Eq. 2 is deteriorated by the projection factor on

Ž .the beam axis 1rsinu . When only those eventsdetected around the central part of the detector areused, they give a good DP resolution. On the otherhand, the statistics of the Bhabha events increasesvery rapidly along with decreasing the minimum

Žpolar angle, u , of the detector acceptance Aminy3 .sin u . If u of the Bhabha events is set equalmin min

to 208, instead of 458, the statistics increases byabout one order of magnitude. Even if the angularresolution can be kept constant down to 208, and if itis larger than 1mrad, the measured spectrum is dis-torted according to the decrease of u as shown inmin

Fig. 3. If the resolution were kept at 1mrad, theacceptance could be set lower to 208.

Fig. 3. Measured cumulative luminosity above a CM energy of495.0GeV I normalized by the generated one I v.s. themeas gen

angular resolution. Results with two different minimum accep-tance-angles are shown.

4. Beam-parameter fitting

The beam-energy distribution just before colli-sions is described by the beam parameters, such as:the horizontal, vertical and longitudinal beam sizesŽ .s ,s ,s , and the number of particles in a beamx y z

bunch, N. In the empirical function, there are twoŽ .independent parameters, Nr s qs and s . In-x y z

cluding the beam-energy spread, DE, we have threeindependent parameters under our assumption thatthe electron and positron beams have completelyidentical beam parameters. Among three combina-tions of them, the simultaneous determinations of

Ž . Ž .Nr s qs and s , and that of Nr s qs andx y z x y

DE are studied, while another remaining parameteris assumed to be known for each case.

For this study we prepare an ‘‘experimental data’’of Bhabha events generated with the following con-ditions:Ø The nominal values of beam parameters are taken

from Table 1.Ø The number of events is corresponding to the

y1 Žintegrated luminosity of 10 fb . about one-.year’s data accumulation

Ø The events are observed with the detector ofwhich polar-angle acceptance is between 458 and1358 and the angular resolution is 1mrad.

'The measured CM energy, s , of each eventmeas

is calculated from the acollinearity angle by the Eqs.Ž . Ž .2 and 3 , and its histogram for all events is made,where the CM energy region between 375 and500GeV is divided into 50 bins. Only for this energyregion our likelihood fitting is applied, since it cov-ers more than 85% of the whole events and theempirical function for the beam-energy spectrumgives a good approximation in this region.

The estimates of two unknown beam parametersŽ .say j and j are searched for by a grid search1 2

method, where each parameter varies between "3%Ž .of its nominal value. At each point j , j in the1 2

j -j parameter space a simulation data of Bhabha1 2Ž .events is produced with the corresponding j , j1 2

values. The statistics of each simulation data is100 fby1, i.e. 10 times that of the experimental data.A histogram of the effective CM energy for allevents of the simulation data is made with the identi-cal binning to that for the experimental data and isfurther normalized to unity. This resultant histogram


'Ž .is used as the probability density function, PP s ,and can be written as

'PP s sPP j ,j 'N rN , 5Ž . Ž .Ž . i 1 2 i

'where i indicates the bin number where the s valueof an event falls in, N is the number of simulationi

events in the i-th bin, and N is the total number of'events with s G 375 GeV in the simulation data.

Ž .For each point j ,j in the j -j parameter1 2 1 2

space the following log-likelihood is calculated:n

jln LL j ,j ' ln PP , 6Ž . Ž .Ý1 2js1

where n is the total number of experimental eventsand PP j is the value of the probability density func-tion of the j-th event. If the effective CM energy,'s , of the j-th event falls in the k-th bin of the

Ž .probability density histogram defined by the Eq. 5 ,the value of the probability density function, PP j, isgiven by

PP j sPP j ,j . 7Ž . Ž .k 1 2

The estimates of two parameters are given bythose values where the log-likelihood has its maxi-mum. The 68.3% joint likelihood region in two-parameter space, spanned by j and j , is inside the1 2

contour defined by the equation

ln LL j ,j s ln LL max y1.15, 8Ž . Ž . Ž .1 2

Ž .where ln LL max is the maximum value of log-like-lihood. This region is corresponding to the 1s inter-val in the one-parameter case.

Ž .Fig. 4 A shows the log-likelihood distribution inŽ .the two-parameter space of Nr s qs and s ,x y z

where the innermost contour represents the boundaryof the 68.3% joint likelihood region and for eachstep of Dln LLs1.15 a contour is drawn. The pa-

Ž .rameter Nr s qs is quite sensitive to the shapex y

of the luminosity spectrum, but s is not, and noz

correlation between them is found. Since there aretwo maxima in the distribution due to small statisticsof the experimental data, it is not an appropriatesituation to say anything about the estimate anderror. For the experimental data of 100 fby1 we cansee only one maximum in the distribution, as shown

Ž .in Fig. 4 B .The determination of s by this method requiresz

at least a few years’ data accumulation, and is not

ŽFig. 4. Log-likelihood distributions for the parameters of Nr sx.qs and s . Each parameter is varied between "4% of they z

nominal value. The distribution with 10-times more statistics ofŽ .the experimental data than usual B . The innermost contour in

these distributions shows a 1s deviation.

practicable. This is due to the weak sensitivity of sz

to the shape of the luminosity spectrum. In otherwords, an inaccurate knowledge about s does notz

affect to the determination of the other parameters.Furthermore, since the bunch length, s , is a parame-z

ter of the bunch compressor, and is known witharound 10 % accuracy, we can take this value as aknown parameter for s .z

The log-likelihood distribution as a function ofŽ . Ž .Nr s qs and DE is shown in Fig. 5 A , wherex y

the innermost contour shows the boundary of the

ŽFig. 5. Log-likelihood distributions for the parameters of Nr sx.qs and the beam-energy spread for an experimental data. Eachy

parameter is varied between "3% of the nominal value. Theinnermost contour in these distributions shows the 1s deviationŽ .A . The distributions show 100 times more statistics of the

Ž .experimental data than usual B .


Fig. 6. Fitting accuracy of the beam-energy spread as a function ofthe angular resolution of the detector. The mean value is calcu-lated by fitting the likelihood distribution to a Gaussian form

Ž .when the parameter Nr s qs is nominal.x y

68.3% joint likelihood region. The estimates ofŽ .Nr s qs and DE are slightly different from thex y

nominal ones. This uncertainty of the central value isdue to the small statistics of the experimental data of10 fby1. In order to confirm the validity of thismethod, the same fitting procedure is performed forthe experimental data of 1000 fby1. In this case, theestimated beam parameters are much closer to the

Ž .given values, as shown in Fig. 5 B .The fitting accuracy of DE as a function of the

angular resolution of the detector is shown in Fig. 6,where the central value is estimated by fitting thelikelihood distribution with a Gaussian form when

Ž .the parameter Nr s qs is assumed to be itsx y

nominal value.

5. Luminosity measurement

For the measurements of the cross sections ofvarious elementary processes in the experiment weneed to know the precise value of the integrated

'luminosity as a function of s . For this purpose, wecalculate the integrated luminosity using the empiri-cal function. Fig. 7 shows the luminosity spectrumcalculated by using the beam parameters determinedby the likelihood fitting in the previous section. Theintegrated luminosity is obtained by the formula

LLsN PP Bmeas ,DEmeas rs , 9Ž . Ž .Ýevents i iis1

Fig. 7. Luminosity spectrum calculated by the empirical functionŽ .using the given value solid histogram and the parameters deter-

Ž .mined by a likelihood fitting closed circles .

where N is the observed number of events andevents

PP is a probability density function defined by theiŽ . meas Ž .Eq. 5 . B is the estimate of Nr s qs , andx y

DEmeas is that of the beam-energy spread. s is thei

cross section of Bhabha scattering for i-th bin in-cluding the detector effect. The luminosity is calcu-

'lated as a function of s , and the integrated luminos-'ity is estimated by integrating the luminosity over s .

The integrated luminosity of the experimental datacorresponding to 10 fby1 is calculated to be 9.990"

0.070 fby1 when the beam-parameter values ob-tained by the likelihood fitting are used. The error isestimated by

< meas meas < 2s s PP B yPP B "dŽ . Ž .ÝLL i i B½is1

1r22meas meas< <q PP DE yPP DE "d ,Ž . Ž .i i D E 5

10Ž .Ž .where d is the fitting error of Nr s qs and dB x y D E

is that of DE. The estimate of the integrated lumi-nosity agrees well with the given value.

6. Conclusion

The potentiality of measuring the luminosity spec-trum using the acollinearity angle of the large-angleBhabha events was studied by means of the luminos-ity spectrum generator, LL uminous, and a Bhabha-


event generator produced by the GRACE systemwhich contains only the lowest order diagrams. Inthis study, for the energy distribution of the electronor positron beam before collisions the empiricalfunction proposed by Yokoya et al. was used, which

Ž .has two independent parameters, Nr s qs andx y

s , besides the nominal beam energy. The nominalz

beam energy is assumed to be known by a differentmeasurement. The electron and positron beams areassumed to have completely identical parameters.For the beam-energy spread,DE, a flat distributionwith the half width of 1% of the nominal beamenergy is assumed in the calculations.

An advantage of our method described in thispaper is that since the empirical function is ex-pressed by the basic beam parameters in a relativelysimple form, the fitting result of the parameters doesnot only allow us to estimate the luminosity, but alsoto compare the estimated parameters with the actualbeam parameters. Among three combinations of threebeam parameters, the simultaneous determination of

Ž . Ž .Nr s qs and s , and that of Nr s qs andx y z x y

the beam-energy spread, DE, were studied. As aresult of the former combination, the parameter

Ž .Nr s qs is quite sensitive to the shape of thex y

luminosity spectrum, but s is not, and no correla-z

tion between them has been found. This is due to aweak sensitivity of s to the shape of the luminosityz

spectrum. Since an inaccurate knowledge about theparameter, s , does not affect to the determination ofz

other parameters, we can take the s value expectedz

from the bunch compressor as being a known param-Žeter. In the fitting of the latter combination, Nr sx

.qs and DE can be determined with the accura-y

cies of 0.5 and 1.0%, respectively, for the experi-mental data of the integrated luminosity of 10 fby1

with the minimum acceptance angle, u , of 458.min

When u is lowered to 208 with keeping the angu-min

lar resolution of 1mrad, the equivalent accuracies ofthe beam parameters can be achieved even with theexperimental data of 1 fby1.

Acknowledgements

The authors wish to appreciate Prof. D.J. Miller,Prof. K. Yokoya and Minami-Tateya Collaborationfor their fruitful discussions and suggestions. Thiswork is supported in part by Ministry of Education,Science, and Culture, Japan under Grant-in-Aid for

ŽInternational Scientific Research Program No..09044359 .

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11 June 1998


A measurement of the inclusive b™sg branching ratio

ALEPH Collaboration

R. Barate a, D. Buskulic a, D. Decamp a, P. Ghez a, C. Goy a,J.-P. Lees a, A. Lucotte a, E. Merle a, M.-N. Minard a, J.-Y. Nief a, B. Pietrzyk a,

R. Alemany b, G. Boix b, M.P. Casado b, M. Chmeissani b, J.M. Crespo b,M. Delfino b, E. Fernandez b, M. Fernandez-Bosman b, Ll. Garrido b,1,

E. Grauges b, A. Juste b, M. Martinez b, G. Merino b, R. Miquel b, Ll.M. Mir b,Ì.C. Park b, A. Pascual b, J.A. Perlas b, I. Riu b, F. Sanchez b, A. Colaleo c,

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P. Tempesta c, A. Tricomi c,2, G. Zito c, X. Huang d, J. Lin d, Q. Ouyang d,T. Wang d, Y. Xie d, R. Xu d, S. Xue d, J. Zhang d, L. Zhang d,

W. Zhao d, D. Abbaneo e, U. Becker e, P. Bright-Thomas e, D. Casper e,M. Cattaneo e, V. Ciulli e, G. Dissertori e, H. Drevermann e, R.W. Forty e,

M. Frank e, R. Hagelberg e, J.B. Hansen e, J. Harvey e, P. Janot e, B. Jost e,I. Lehraus e, P. Mato e, A. Minten e, L. Moneta e,3, A. Pacheco e, J.-F. Pusztaszeri e,4,

F. Ranjard e, L. Rolandi e, D. Rousseau e, D. Schlatter e, M. Schmitt e,5,O. Schneider e, W. Tejessy e, F. Teubert e, I.R. Tomalin e,

H. Wachsmuth e, Z. Ajaltouni f, F. Badaud f, G. Chazelle f, O. Deschamps f,A. Falvard f, C. Ferdi f, P. Gay f, C. Guicheney f, P. Henrard f, J. Jousset f,

B. Michel f, S. Monteil f, J-C. Montret f, D. Pallin f, P. Perret f, F. Podlyski f,J. Proriol f, P. Rosnet f, J.D. Hansen g, J.R. Hansen g, P.H. Hansen g, B.S. Nilsson g,

B. Rensch g, A. Waananen g, G. Daskalakis h, A. Kyriakis h, C. Markou h,¨¨ Ë. Simopoulou h, I. Siotis h, A. Vayaki h, A. Blondel i, G. Bonneaud i, J.-C. Brient i,

P. Bourdon i, A. Rouge i, M. Rumpf i, A. Valassi i,6, M. Verderi i, H. Videau i,É. Focardi j, G. Parrini j, K. Zachariadou j, M. Corden k, C. Georgiopoulos k,

D.E. Jaffe k, A. Antonelli l, G. Bencivenni l, G. Bologna l,7, F. Bossi l, P. Campana l,G. Capon l, F. Cerutti l, V. Chiarella l, G. Felici l, P. Laurelli l, G. Mannocchi l,8,

F. Murtas l, G.P. Murtas l, L. Passalacqua l, M. Pepe-Altarelli l, L. Curtis m,A.W. Halley m, J.G. Lynch m, P. Negus m, V. O’Shea m, C. Raine m, J.M. Scarr m,


( )R. Barate et al.rPhysics Letters B 429 1998 169–187170

K. Smith m, P. Teixeira-Dias m, A.S. Thompson m, E. Thomson m,O. Buchmuller n, S. Dhamotharan n, C. Geweniger n, G. Graefe n, P. Hanke n,¨G. Hansper n, V. Hepp n, E.E. Kluge n, A. Putzer n, J. Sommer n, K. Tittel n,

S. Werner n, M. Wunsch n, R. Beuselinck o, D.M. Binnie o,W. Cameron o, P.J. Dornan o,9, M. Girone o, S. Goodsir o, E.B. Martin o,N. Marinelli o, A. Moutoussi o, J. Nash o, J.K. Sedgbeer o, P. Spagnolo o,M.D. Williams o, V.M. Ghete p, P. Girtler p, E. Kneringer p, D. Kuhn p,

G. Rudolph p, A.P. Betteridge q, C.K. Bowdery q, P.G. Buck q, P. Colrain q,G. Crawford q, A.J. Finch q, F. Foster q, G. Hughes q, R.W.L. Jones q,N.A. Robertson q, T. Sloan q, M.I. Williams q, I. Giehl r, A.M. Greene r,

C. Hoffmann r, K. Jakobs r, K. Kleinknecht r, G. Quast r, B. Renk r,E. Rohne r, H.-G. Sander r, P. van Gemmeren r, C. Zeitnitz r,

J.J. Aubert s, C. Benchouk s, A. Bonissent s, G. Bujosa s, J. Carr s,9,P. Coyle s, F. Etienne s, O. Leroy s, F. Motsch s, P. Payre s, M. Talby s,A. Sadouki s, M. Thulasidas s, K. Trabelsi s, M. Aleppo t, M. Antonelli t,

F. Ragusa t, R. Berlich u, W. Blum u, V. Buscher u, H. Dietl u, G. Ganis u,¨H. Kroha u, G. Lutjens u, C. Mannert u, W. Manner u, H.-G. Moser u, S. Schael u,¨ ¨

R. Settles u, H. Seywerd u, H. Stenzel u, W. Wiedenmann u, G. Wolf u, J. Boucrot v,O. Callot v, S. Chen v, A. Cordier v, M. Davier v, L. Duflot v, J.-F. Grivaz v,

Ph. Heusse v, A. Hocker v, A. Jacholkowska v, D.W. Kim v,10, F. Le Diberder v,¨J. Lefrancois v, A.-M. Lutz v, M.-H. Schune v, E. Tournefier v, J.-J. Veillet v,

I. Videau v, D. Zerwas v, P. Azzurri w, G. Bagliesi w,9, G. Batignani w, S. Bettarini w,T. Boccali w, C. Bozzi w, G. Calderini w, M. Carpinelli w, M.A. Ciocci w,

R. Dell’Orso w, R. Fantechi w, I. Ferrante w, L. Foa w,11, F. Forti w, A. Giassi w,`M.A. Giorgi w, A. Gregorio w, F. Ligabue w, A. Lusiani w, P.S. Marrocchesi w,

A. Messineo w, F. Palla w, G. Rizzo w, G. Sanguinetti w, A. Sciaba w, G. Sguazzoni w,`R. Tenchini w, G. Tonelli w,12, C. Vannini w, A. Venturi w, P.G. Verdini w,

G.A. Blair x, L.M. Bryant x, J.T. Chambers x, M.G. Green x, T. Medcalf x,P. Perrodo x, J.A. Strong x, J.H. von Wimmersperg-Toeller x, D.R. Botterill y,

R.W. Clifft y, T.R. Edgecock y, S. Haywood y, P.R. Norton y, J.C. Thompson y,A.E. Wright y, B. Bloch-Devaux z, P. Colas z, S. Emery z, W. Kozanecki z,

E. Lancon z,9, M.-C. Lemaire z, E. Locci z, P. Perez z, J. Rander z, J.-F. Renardy z,A. Roussarie z, J.-P. Schuller z, J. Schwindling z, A. Trabelsi z, B. Vallage z,

S.N. Black aa, J.H. Dann aa, R.P. Johnson aa, H.Y. Kim aa, N. Konstantinidis aa,A.M. Litke aa, M.A. McNeil aa, G. Taylor aa, C.N. Booth ab, C.A.J. Brew ab,

S. Cartwright ab, F. Combley ab, M.S. Kelly ab, M. Lehto ab, J. Reeve ab,L.F. Thompson ab, K. Affholderbach ac, A. Bohrer ac, S. Brandt ac,¨

G. Cowan ac, C. Grupen ac, P. Saraiva ac, L. Smolik ac, F. Stephan ac, G. Giannini ad,

( )R. Barate et al.rPhysics Letters B 429 1998 169–187 171

B. Gobbo ad, G. Musolino ad, J. Rothberg ae, S. Wasserbaech ae, S.R. Armstrong af,E. Charles af, P. Elmer af, D.P.S. Ferguson af, Y. Gao af, S. Gonzalez af,´

T.C. Greening af, O.J. Hayes af, H. Hu af, S. Jin af, P.A. McNamara III af,J.M. Nachtman af,13, J. Nielsen af, W. Orejudos af, Y.B. Pan af, Y. Saadi af,

I.J. Scott af, J. Walsh af, Sau Lan Wu af, X. Wu af, G. Zobernig af

a ( )Laboratoire de Physique des Particules LAPP , IN2P3-CNRS, F-74019 Annecy-le-Vieux Cedex, Franceb ´ 14( )Institut de Fisica d’Altes Energies, UniÕersitat Autonoma de Barcelona, E-08193 Bellaterra Barcelona , Spain`

c Dipartimento di Fisica, INFN Sezione di Bari, I-70126 Bari, Italyd Institute of High-Energy Physics, Academia Sinica, Beijing, People’s Republic of China 15

e ( )European Laboratory for Particle Physics CERN , CH-1211 GeneÕa 23, Switzerlandf Laboratoire de Physique Corpusculaire, UniÕersite Blaise Pascal, IN2P3-CNRS, Clermont-Ferrand, F-63177 Aubiere, France´ `

g Niels Bohr Institute, DK-2100 Copenhagen, Denmark 16

h ( )Nuclear Research Center Demokritos NRCD , GR-15310 Attiki, Greecei Laboratoire de Physique Nucleaire et des Hautes Energies, Ecole Polytechnique, IN2P3-CNRS, F-91128 Palaiseau Cedex, France´

j Dipartimento di Fisica, UniÕersita di Firenze, INFN Sezione di Firenze, I-50125 Firenze, Italy`k Supercomputer Computations Research Institute, Florida State UniÕersity, Tallahassee, FL 32306-4052, USA 17,18

l ( )Laboratori Nazionali dell’INFN LNF-INFN , I-00044 Frascati, Italym Department of Physics and Astronomy, UniÕersity of Glasgow, Glasgow G12 8QQ, United Kingdom 19

n Institut fur Hochenergiephysik, UniÕersitat Heidelberg, D-69120 Heidelberg, Germany 2 0¨ ö Department of Physics, Imperial College, London SW7 2BZ, United Kingdom 19

p Institut fur Experimentalphysik, UniÕersitat Innsbruck, A-6020 Innsbruck, Austria 21¨ ¨q Department of Physics, UniÕersity of Lancaster, Lancaster LA1 4YB, United Kingdom 19

r Institut fur Physik, UniÕersitat Mainz, D-55099 Mainz, Germany 2 0¨ ¨s Centre de Physique des Particules, Faculte des Sciences de Luminy, IN2P3-CNRS, F-13288 Marseille, France´

t Dipartimento di Fisica, UniÕersita di Milano e INFN Sezione di Milano, I-20133 Milano, Italyù Max-Planck-Institut fur Physik, Werner-Heisenberg-Institut, D-80805 Munchen, Germany 2 0¨ ¨

v Laboratoire de l’Accelerateur Lineaire, UniÕersite de Paris-Sud, IN2P3-CNRS, F-91405 Orsay Cedex, France´ ´ ´ ´w Dipartimento di Fisica dell’UniÕersita, INFN Sezione di Pisa, e Scuola Normale Superiore, I-56010 Pisa, Italy`

x Department of Physics, Royal Holloway & Bedford New College, UniÕersity of London, Surrey TW20 OEX, United Kingdom 19

y Particle Physics Dept., Rutherford Appleton Laboratory, Chilton, Didcot, Oxon OX11 OQX, United Kingdom 19

z CEA, DAPNIArSerÕice de Physique des Particules, CE-Saclay, F-91191 Gif-sur-YÕette Cedex, France 22

aa Institute for Particle Physics, UniÕersity of California at Santa Cruz, Santa Cruz, CA 95064, USA 23

ab Department of Physics, UniÕersity of Sheffield, Sheffield S3 7RH, United Kingdom 19

ac Fachbereich Physik, UniÕersitat Siegen, D-57068 Siegen, Germany 2 0äd Dipartimento di Fisica, UniÕersita di Trieste e INFN Sezione di Trieste, I-34127 Trieste, Italyàe Experimental Elementary Particle Physics, UniÕersity of Washington, Seattle, WA 98195, USA

af Department of Physics, UniÕersity of Wisconsin, Madison, WI 53706, USA 24

Received 30 March 1998Editor: L. Montanet

Abstract

The flavour changing neutral current decay b™sg has been detected in hadronic Z decays collected by ALEPH at LEP.The signal is isolated in lifetime-tagged bb events by the presence of a hard photon associated with a system of highmomentum and high rapidity hadrons. The background processes are normalised from the data themselves. The inclusivebranching ratio is measured to be

3.11"0.80 "0.72 =10y4 ,Ž .stat syst

consistent with the Standard Model expectation via penguin processes. q 1998 Published by Elsevier Science B.V. All rightsreserved.


1. Introduction

The flavour changing neutral current decay b™sg may proceed via an electromagnetic penguin diagram inwhich the photon is radiated from either the W or one of the quark lines. The next-to-leading order calculations

w xare now available for this transition 1 and the Standard Model predicts the inclusive b™sg branching ratio toŽ . y4 Ž . Ž . y4 w xbe 3.51"0.32 =10 for measurements at the F 4S resonance and 3.76"0.30 =10 at the Z peak 2 .

Virtual particles in the loop may be replaced by non-Standard Model particles, such as charged Higgs bosons orw xsupersymmetric particles. These additional contributions could either enhance or suppress the decay rate 3,4

making it sensitive to physics beyond the Standard Model.Ž . )The b™sg process is expected to be dominated by two-body resonant decays such as B™K g , and final

Ž . w xstates with soft gluon emission non-resonant which kinematically resemble two-body decays 5,6 . In this case,the photon energy in the b-hadron rest frame, denoted Ew, has a spectrum which is peaked at approximatelyg

half the b-hadron mass. In contrast, charged current radiative b decays such as b™cW )g and b™uW )g

Ž ) .where W represents a virtual W boson produce photons with a spectrum resembling that from bremsstrahlungw xand hence of much lower energy 7 . The decay b™dg can also be mediated by penguin diagrams but is

y5 w xCabibbo suppressed, with a predicted branching ratio F 2.8 = 10 8 . In the ALEPH data sample of 4.1million hadronic Z decays, collected between 1991 and 1995, the Standard Model predicts the production of; 660 b™sg decays. It is thus important, if a b™sg signal is to be observed, that both the signal acceptanceand background rejection be high and that Ew be accurately reconstructed. The latter requires preciseg

determination of the parent b-hadron’s momentum and direction.In the analyis described in this paper, bb events are tagged by lifetime in one hemisphere. The signal for the

b™sg decay in the other hemisphere is characterised by the presence of a hard photon associated with a systemof high momentum and high rapidity hadrons, originating from a displaced secondary vertex. The extraction of

w xthe signal is based on a Monte Carlo simulation constructed using Heavy Quark Effective Theory 5 , and frommeasurements and assumptions about exclusive decay rates.

1 Permanent address: Universitat de Barcelona, 08208 Barcelona, Spain.2 Also at Dipartimento di Fisica, INFN, Sezione di Catania, Catania, Italy.3 Now at University of Geneva, 1211 Geneva 4, Switzerland.4 Now at School of Operations Research and Industrial Engineering, Cornell University, Ithaca, NY 14853-3801, USA.5 Now at Harvard University, Cambridge, MA 02138, USA.6 Supported by the Commission of the European Communities, contract ERBCHBICT941234.7 Also Istituto di Fisica Generale, Universita di Torino, Torino, Italy.`8 Also Istituto di Cosmo-Geofisica del C.N.R., Torino, Italy.9 Also at CERN, 1211 Geneva 23, Switzerland.

10 Permanent address: Kangnung National University, Kangnung, Korea.11 Now at CERN, 1211 Geneva 23, Switzerland.12 Also at Istituto di Matematica e Fisica, Universita di Sassari, Sassari, Italy.`13 Ž .Now at University of California at Los Angeles UCLA , Los Angeles, CA 90024, USA.14 Supported by CICYT, Spain.15 Supported by the National Science Foundation of China.16 Supported by the Danish Natural Science Research Council.17 Supported by the US Department of Energy, contract DE-FG05-92ER40742.18 Supported by the US Department of Energy, contract DE-FC05-85ER250000.19 Supported by the UK Particle Physics and Astronomy Research Council.20 Supported by the Bundesministerium fur Bildung, Wissenschaft, Forschung und Technologie, Germany.¨21 Supported by Fonds zur Forderung der wissenschaftlichen Forschung, Austria.¨22 Supported by the Direction des Sciences de la Matiere, C.E.A.`23 Supported by the US Department of Energy, grant DE-FG03-92ER40689.24 Supported by the US Department of Energy, grant DE-FG0295-ER40896.


2. The ALEPH detector

w xA detailed description of the ALEPH detector and its performance is given in 9,10 . What follows is a briefdescription of those parts of the detector relevant to this analysis. Charged particles are detected in the central

Ž .part of the detector, consisting of a high precision vertex detector VDET , a cylindrical multi-wire driftŽ . Ž .chamber ITC and a large time projection chamber TPC . The VDET consists of two concentric layers of

double sided silicon detectors surrounding the beam pipe, positioned at average radii of 6.5 and 10.8 cmcovering 85% and 69% of the solid angle, respectively. The intrinsic spatial resolution of the VDET is 12 mmfor the ryf coordinate and between 11 and 22 mm for the z coordinate, depending on the polar angle of thecharged particle. The ITC, at radii from 16 to 26 cm, provides up to eight coordinates per track in the ryf

view while the TPC measures up to 21 three-dimensional points per track at radii between 40 and 171 cm. TheTPC also serves to separate charged particle species with up to 338 measurements of their specific ionization.The three detectors are immersed in an axial magnetic field of 1.5 T and together provide a transverse

Ž . y3 Ž .y1momentum resolution of s 1rp s0.6=10 GeVrc for high momentum tracks.TŽ .Electrons and photons are identified and measured in the electromagnetic calorimeter ECAL which is

formed by a barrel surrounding the TPC, closed at each end by end-cap modules. It consists of 45 layers of leadinterleaved with proportional wire chambers. The position and energies of electromagnetic showers aremeasured using cathode pads each subtending a solid angle of 0.98 by 0.98 in u and f and connected internallyto form projective towers. Each tower is read out in three segments, known as storeys, with depths of 4, 9 and 9radiation lengths. The inactive zones of this detector represent 2% of the solid angle in the barrel and 6% in theend-caps. The iron return yoke of the magnet is instrumented with streamer tubes to form a hadron calorimeterŽ .HCAL , with a thickness of over 7 interaction lengths. It is surrounded by two additional double layers ofstreamer tubes used for muon identification.

w xAn energy flow algorithm, which is described in 10 , is used to improve the energy resolution of events. Thealgorithm links charged tracks to calorimeter clusters and uses the resultant redundancy in energy measurementsto assign neutral particle energy. Particle identification methods are used in this algorithm to distinguishbetween particle species and the resulting objects are labelled energy flow objects. The algorithm to identify

w xphotons in ECAL is also described in detail in 10 . The clusters found by the algorithm are retained ascandidate photons if their energy is greater than 0.25 GeV and there is no charged track impact at a distance ofless than 2 cm from the cluster barycentre. The photons are detected with angular resolution of s su ,fŽ . Ž .( (2.5r E GeV q0.25 mrad and energy resolution of s rEs 0.25r E GeV q0.009 . The detector’sŽ . Ž .E

ability to resolve a p 0 into two g clusters decreases for energies greater than 10 GeV above which the two g

clusters overlap to form one larger cluster which tends to be more elliptical than single g clusters. A momentsanalysis of the energy sharing between neighbouring detector elements within the cluster enables the length ofthe major axis of the shower ellipse, s , to be measured. This quantity is used in the analysis which follows tol

0 Ž . Ž .separate high energy p mesons s ) 2.3 cm from single photons s - 2.3 cm .l l

3. Monte Carlo simulation

w xMonte Carlo events were generated using JETSET 11 with the ALEPH standard parameter set. Theseevents were processed through a detailed simulation of the ALEPH detector and the ALEPH reconstructionprogram. A sample of 9.9 million hadronic Z decays — simulated without the b™sg decay and henceforthcalled the standard Monte Carlo — is used to model background processes in this analysis. The rate of

Ž .production of photons from final state radiation FSR in the standard MC is reweighted by a factor 1.21 asw xrequired by the ALEPH measurements of this process over the full kinematic range 12 .

Ž .The composition of the signal b™sg Monte Carlo event sample is based primarily on predictions fromŽ ) ) . Ž . ) )

) )Heavy Quark Effective Theory for the exclusive ratios R sG B™K g rG B™X g , where K isK s


Table 1The composition of the inclusive b™sg Monte Carlo sample

Ž .Exclusive decay channel Exclusive decay fraction, R %) Ž .B ™K 892 g 13.6 " 5.1uŽd .Ž .B ™K 1270 g 5.5 " 2.1uŽd . 1Ž .B ™K 1400 g 7.2 " 2.6uŽd . 1

) Ž .B ™K 1410 g 6.7 " 2.6uŽd .) Ž .B ™K 1430 g 20.6 " 7.5uŽd . 2Ž .B ™K np g 22.0 " 19.9uŽd .

´ 75.6Ž .B ™f 1020 g 2.0 " 0.8sŽ .B ™h 1380 g 0.8 " 0.3s 1Ž .B ™ f 1420 g 1.1 " 0.4s 1Ž .B ™f 1680 g 1.0 " 0.4s

X Ž .B ™ f 1525 g 3.0 " 1.1s 2Ž .B ™KK np g 3.3 " 3.0s

´ 11.2Ž .L ™ L 1116 g 13.2 " 4.1b

w x)any particular K-resonance and the B meson is either a B or B 5 . R is taken as the ratio of the CLEOu d K Ž892.

) Ž . w xmeasurements of the exclusive B™K 892 g branching ratio 13 and the inclusive b™sg branching ratiow x

)14 , giving R s18.1"6.8 %. The f-resonances, produced in B penguin decays, are assumed to be inK Ž892. s

the same relative proportions as the K-resonances. The exclusive ratios for resonant decays are summed and theremaining inclusive branching ratio is completed with non-resonant penguin decays, where X is a multi-bodys

state involving a kaon and n pions, hadronised using JETSET. The photon energy spectrum in the non-resonantdecays is obtained from a fully inclusive spectator model including gluon bremsstrahlung and higher order

w x 2radiative effects 6 . The model parameters used were m s 150 MeVrc , where m is the spectator quarkq q

mass, and p s 265 MeVrc, where p is the Fermi momentum of the b quark in the hadron, taken fromF Fw xCLEO fits to the inclusive lepton spectrum in B™Xln decays 15 . Baryonic penguins are modelled by one

channel, L ™Lg , where the L is produced directly from hadronisation or from the decay of a heavier bb b

baryon. The resulting signal Monte Carlo composition is shown in Table 1, where the b-hadron productionw xfractions are taken to be: B 75.6% ; B 11.2% ; and L 13.2% 16 .uŽd . s b

4. Inclusive reconstruction of b™sg decays

The event hemisphere opposite to a b™sg decay contains a typical b-hadron decay and is used to ‘b-tag’events in a largely unbiased way. The b™sg hemisphere, in contrast, has low multiplicity, a single displaceddecay vertex, and contains a high energy photon. A dedicated inclusive b™sg reconstruction algorithm is usedto assemble the hadronic system accompanying the photon by distinguishing between objects from the b-hadrondecay from those produced by hadronisation at the primary vertex.

In each hemisphere containing a candidate photon, p 0 and K 0 mesons are searched for. The p 0 mesons areS

reconstructed from two candidate photons when the gg invariant mass is compatible with the p 0 mass. K 0S

mesons are reconstructed similarly from any two oppositely signed charged tracks which form a vertex and haveŽ . 0an invariant mass assuming both charged tracks are pions consistent with the K mass. Each hemisphere thenS

consists of reconstructed p 0 and K 0 mesons and the remaining charged tracks and neutral electromagnetic andS

hadronic clusters. Any neutral hadronic cluster is assumed to be a K 0 and given the kaon mass. All chargedL

tracks are given the pion mass and, depending on the length of the major axis of its shower ellipse, s , a neutrallŽ . Ž 0.electromagnetic cluster is given either zero mass photon or the pion mass unresolved p .


Ž . Ž .Fig. 1. The momentum-rapidity a and impact parameter significance b distributions for signal charged tracks and the correspondingŽ . Ž . Ž .hadronisation charged track distributions c and d . Plot e shows the momentum probability distribution in the rapidity bin, 2.5 -

Ž . Ž . Ž .rapidity - 2.75, and f shows the total charged track impact parameter significance probability distribution. The curves in e and f arethe fitted parameterisations.

The b™sg probability for each object is calculated as a function of its momentum, its rapidity with respectto the b-hadron direction, and, if the object is a charged track with VDET hits, its three-dimensional impact

Ž .parameter significance. The probability functions were produced using signal Monte Carlo events. Plots a andŽ .b of Fig. 1 show the momentum-rapidity and impact parameter significance distributions for charged tracks

Ž . Ž .from the b™sg decay. Plots c and d of Fig. 1 show the corresponding hadronisation charged trackdistributions. The b™sg decay objects have a much harder momentum spectrum than the hadronisation objectsand possess larger rapidities. Similarly, due to the long lifetime of the b-hadron, the signal objects have moresignificant impact parameters than the hadronisation objects, where the spectrum is dominated by the impactparameter resolution. In a given momentum bin the b™sg probability, P , is defined as:m om

No. of b™sg decay objectsP sm om No. of b™sg decay objectsqNo. of hadronisation objects

If the object is a charged track with VDET hits, the independent impact parameter significance probability is


calculated in the same way as P and the two probabilities, P and P , are combined to give a finalm om m om i p

probability, P:

P PPm om i pPs

P PP q 1yP P 1yP PrŽ . Ž .m om i p m om i p

where r is the average ratio of the total number of signal decay charged tracks with VDET hits to the totalŽ . Ž .number of hadronisation charged tracks with VDET hits in a b™sg hemisphere rs0.298 . Fig. 1 e shows

Ž .the momentum probability distribution in the rapidity bin, 2.5 -Rapidity- 2.75, and Fig. 1 f shows thecharged track impact parameter significance probability distribution.

The momentum probability functions are constructed for ten different rapidity bins which are used tointerpolate to the probability corresponding to any value of rapidity and momentum. The probabilities arecalculated separately for identified K 0 and K 0 mesons and all other objects, since kaons from the b™sgL S

decay have a harder momentum spectrum than other X decay products and can therefore be separated froms

hadronisation kaons more easily.These probability functions are used in the jet reconstruction algorithm now described. The reconstruction of

Ž 0 0an assumed b™sg decay begins with the candidate signal photon onto which objects K and p mesons,S.charged tracks and neutral calorimeter objects in the same hemisphere as the photon are added in order of

decreasing b™sg probability. The addition of objects stops if, by adding the next highest probability object,the mass of the reconstructed jet is further from the mean B meson mass of 5.28 GeVrc2 than it would be ifthe object had not been added. This reconstruction is performed in two stages, as follows:1. An initial estimate of the b-hadron flight direction is obtained by using only those objects with rapidity with

respect to the thrust axis greater than 1.0 and then calculating the probabilities as a function of momentumand impact parameter significance only.

2. The 4-momenta of the candidate photon and its accompanying high rapidity objects are then summed andused as a new estimate of the b-hadron flight direction with which a better estimate of the object rapiditiescan be made. The above process is then repeated but with the rapidity cut removed and the momentumprobability function replaced by a 2-dimensional momentum-rapidity probability function.The resulting jets are accepted as possible b™sg decays if the jet mass is within 0.7 GeVrc2 of the mean

B meson mass of 5.28 GeVrc2; the mass of the hadronic system, X , is less than 4.0 GeVrc2; and the Xs sŽ 0 0 .object multiplicity K and p mesons, charged tracks and neutral calorimeter objects is greater than one andS

less than eight. After this procedure, studies of the signal Monte Carlo show that b™sg decays are wellw xreconstructed with a resolution in momentum of 1.5 GeVrc and angle of 0.38 17 .

5. Event selection and data analysis

A sample of 4.06 million hadronic Z decays is preselected according to the standard ALEPH hadronic eventw xselection 10 . To ensure the event is well contained within the detector volume, the cosine of the polar angle of

the thrust axis is required to be less than 0.9, where the thrust axis is determined using all energy flow objects.w xTo allow the calculation of variables used to tag b events, jets are reconstructed using the JADE algorithm 18

with a y of 0.02, where each event must have at least one track with VDET hits and a minimum of two jetscutw xwith momentum greater than 10 GeVrc and polar angle greater than 5.78 19 .

A set of cuts is used to produce a data sample enriched in b™sg events. To reduce the background from p 0

decays, events are selected with at least one electromagnetic cluster containing a single candidate photon ofenergy more than 10 GeV that cannot be combined with another candidate photon to make an invariant mass ofless than 0.2 GeVrc2. The hemisphere opposite to the candidate photon is required to be b-like with P o p p lesshem

than 0.1, where P o p p is the probability that all charged tracks in the hemisphere are consistent with havinghem


Table 2Predicted efficiencies for each of the exclusive b™sg decay channels after the background rejection cuts. The uncertainties are the MonteCarlo statistical uncertainties. The b™sg efficiency is the average of the exclusive channels weighted by their production rates, as given inTable 1. The final two columns give the means and standard deviations of the exclusive E) spectra, after all the cuts described in section 5g

)Ž . Ž . Ž .Decay channel Efficiency % -E ) GeV s GeVg

) Ž .B ™K 892 g 14.5 " 0.5 2.49 0.19uŽd .Ž .B ™K 1270 g 15.0 " 0.6 2.42 0.17uŽd . 1Ž .B ™K 1400 g 13.5 " 0.5 2.39 0.18uŽd . 1

) Ž .B ™K 1410 g 13.0 " 0.5 2.38 0.18uŽd .) Ž .B ™K 1430 g 13.8 " 0.5 2.39 0.18uŽd . 2Ž .B ™K np g 12.4 " 0.4 2.37 0.21uŽd .

Ž .B ™f 1020 g 17.7 " 1.7 2.52 0.15sŽ .B ™h 1380 g 14.7 " 1.5 2.40 0.19s 1Ž .B ™ f 1420 g 14.0 " 1.5 2.38 0.19s 1Ž .B ™f 1680 g 13.0 " 1.7 2.36 0.21s

X Ž .B ™ f 1525 g 12.1 " 1.4 2.41 0.15s 2Ž .B ™K K np g 13.5 " 1.1 2.34 0.21s

Ž .L ™ L 1116 g 7.4 " 0.4 2.45 0.23b

b™sg 12.8 " 0.3 2.41 0.20

w xoriginated from the reconstructed primary vertex 19 . In the Monte Carlo the hemisphere probabilities arew xcorrected using the impact parameter smearing algorithm described in 20 which improves the agreement

between data and Monte Carlo. In the remaining events, jets are reconstructed and selected as described inSection 4, and the decay objects — including the photon — are transformed into the rest frame of their parentjet. The angle of the photon, u w, in this frame relative to the jet direction, is required to have cosu w less thang g

0.55 since cosu w peaks at unity for background processes whereas photons are emitted isotropically in b™sgg

w xdecays. Furthermore, the boosted sphericity, S , in the jet’s rest frame 21 is required to be less than 0.16,b

because this quantity is peaked at zero for two-body b™sg decays but has a broad distribution for backgroundprocesses.

The expected efficiencies for the various exclusive b™sg channels after all these cuts are shown in Table 2along with the means and standard deviations of their reconstructed E) spectra. From the original data sample,g

Ž1560 hadronic Z decays remain, each containing only one candidate photon two events with more than one.candidate photon were rejected . The expectation from the background Monte Carlo is 1443 events with a single

candidate photon of which 34% are p 0 mesons from b™c decays; 30% are p 0 mesons from non-b decays;Ž . Ž . Ž .13% are prompt photons from q™qg final state radiation FSR of which 23% are from light uds quarks,

Ž .53% are from c quarks and 24% are from b quarks; 17% are photons from other non-FSR sources, half ofwhich are from h decays; 4% are p 0 mesons from b™u decays; and the remaining 2% are of undetermined

Žorigin because there is no one-to-one correspondence between the reconstructed objects and ‘‘true’’ simulated.particles .

The final sample of 1560 hadronic events is split into eight sub-samples, defined in Table 3. The b™sgdecay populates mainly sub-sample 4. Dividing the data up in this way allows the relative normalisations of theremaining background processes to be measured outside the signal region.

The first four sub-samples have s - 2.3 cm and hence have candidate photons resembling prompt photonsl0 Ž .while the second set of sub-samples have s ) 2.3 cm and are p -like, see Fig. 2 a . The data are furtherl

divided into sub-samples of relatively high or low b-purity, containing jets of relatively high or low energy, bybinning in ylog P o p p and E respectively. This is done to separate signal photons from FSR photons and tohem jet

0 Ž . 0 Ž .distinguish between p mesons produced in b hadron decays and other non-b p mesons, see Fig. 2 b,c .


Table 3Defining the sub-samples, where f in column 5 is the fraction of b™sg decays populating each sub-sample and E is the energy ofb ™ sg jet

the reconstructed b™sg candidatesoppŽ . Ž . Ž .Sub-sample E GeV ylogP s cm f %jet hem l b ™ sg

1 - 32 1.2 – 2.2 - 2.3 8.4 g-like low b-purity2 32 – 50 1.2 – 2.2 - 2.3 19.2 ’’ ’’3 - 32 ) 2.2 - 2.3 17.1 ’’ high b-purity4 32 – 50 ) 2.2 - 2.3 47.0 ’’ ’’

05 - 32 1.2 – 2.2 ) 2.3 0.7 p -like low b-purity6 32 – 50 1.2 – 2.2 ) 2.3 2.8 ’’ ’’7 - 32 ) 2.2 ) 2.3 1.1 ’’ high b-purity8 32 – 50 ) 2.2 ) 2.3 3.7 ’’ ’’

The Ew distributions for the eight subsamples are shown in Fig. 3 together with the background Monte Carlog

Ž .dashed curve which is absolutely normalised.The background Monte Carlo gives a reasonable representation of the data, but there is an excess where the

Ž ) .b™sg signal is expected sub-sample 4 in the range 2.2 GeV -E - 2.8 GeV . Due to the b™sgg

reconstruction algorithm, background events in which the jet contains a FSR photon tend to produce peaks inthe Ew spectrum in the vicinity of the b™sg signal region. From Monte Carlo studies it is found that theg

residual FSR Ew peaks are most prominent in sub-samples 1 and 3, and are almost absent in sub-samples 2 andg

4 where the signal is greatest. Thus, uncertainty in the FSR rate does not cause significant uncertainty in thenumber of background events in the b™sg signal region. To measure the uncertainties in the backgroundlevels, and improve the agreement of the Monte Carlo and the data, a multivariate fit is performed where thenormalisation of the four major backgrounds and the b™sg signal rate are allowed to be free parameters.

A binned log-likelihood fit of the Ew data distributions is performed for the eight sub-samples using theg

w xcorresponding distributions for the signal and background simulations, and the HMCMML fitting package 22which correctly incorporates uncertainties due to the finite Monte Carlo statistics in each bin. The fiveparameters in the fit are N , N , N 0 , N 0 and N , which are, respectively, the totalb™ sg FSR Žb™ c.p Žnonyb.p o ther

Ž . 0 Ž . 0number of signal, FSR, b™c p , nonyb p , and ‘other’ background events which make up the remainingdata. The shapes of the Ew distributions are taken from the standard Monte Carlo. The sensitivity of the fit tog

b™sg enters mainly through sub-sample 4 where there is little FSR expected and, according to Monte Carlothe b purity is 96.6%.

6. Results

6.1. Extraction of the b™sg signal

The results of the multivariate fit are shown in Table 4. A significant amount of b™sg is required to fit thedata while the adjustments of the background normalisations lead to an improved agreement between data andthe background Monte Carlo. These adjustments are within the accuracy of the fit for N , N ando ther FSR

w x0N as would be expected from previous ALEPH measurements 12,23 . However, significant modifica-Žnonyb.pŽ . 0tion is required to the b™c p normalisation. This is not unexpected since there is a 50% uncertainty in the

w xbranching ratio for such decays 16 . The correlation matrix for the fit is shown in Table 5.The Ew plots for the sub-samples are shown in Fig. 3 for data and Monte Carlo before and after the fit.g

Before the fit, the x 2 is 87.1 for 68 degrees of freedom and after the fit the x 2 is 66.6 for 63 degrees offreedom — where, in both cases, only bins with at least four entries are included in the x 2 calculation. Fig. 4


Fig. 2. Monte Carlo distributions for the quantities used to define the sub-samples. The vertical lines at s s 2.3 cm, E s 32 GeV andl je t

ylog P o p p s 2.2 define the eight event sub-samples, as in Table 3.h em

shows the improved agreement between data and Monte Carlo after the fit in each of the main variables used inthe analysis, both in the event selection and the fit itself. These distributions are dominated by backgroundprocesses and this agreement indicates that the shapes of the backgrounds are well modelled in the Monte Carlo,in the selected range.

Ž . )Fig. 5 a shows the E distribution for sub-sample 4 in data and Monte Carlo after the normalisations of theg

background processes have been corrected for. The remaining excess after subtraction of the background is seenŽ .in Fig. 5 b with the fitted signal distribution superimposed. The shape of this distribution is consistent with the

observed excess within statistical uncertainties indicating a signal for b™sg . If the fit is repeated with Nb™ sgŽ .constrained to zero, the excess distribution shown in Fig. 5 b remains almost the same. The reason for this is

Ž )that the background is mainly constrained by the data outside the signal region 2.2 GeV -E - 2.8 GeV ing

. Ž .sub-sample 4 and is hardly influenced by the data in the signal region. The excess in Fig. 5 b is thusdemonstrated to be insensitive to the details of the fit.

6.2. Consistency checks of the b™sg hypothesis

Ž . Ž w .For the purest sample of b™sg events sub-sample 4 and in the signal region 2.2 GeV -E - 2.8 GeVg

there is evidence in the remaining data events, after subtraction of the corrected background, of lifetime in theŽ .same hemisphere as the photon. This is consistent with that expected from b™sg , see Fig. 6 a,b .

In the signal region of sub-sample 4 there is seen to be an excess of high momentum kaons, that isŽ .strangeness, in the data which is again consistent with b™sg , see Fig. 6 c,d . Charged kaons are required to

have ionisation in the TPC within one standard deviation of the kaon hypothesis and greater than two standard


Table 4Results of the five parameter fit to the data

Ž . Ž .Fit parameter Standard normalised MC value Fitted value Change in value %

N 336.8 " 11.7 390.7 " 66.0 q16 " 20other

N 223.0 " 10.5 239.6 " 36.5 q7 " 16FSR

0N 431.8 " 13.3 507.7 " 78.6 q18 " 18Žnonyb.p0N 482.0 " 14.0 352.7 " 59.1 –27 " 12Žb ™ c.p

N 0 69.4 " 19.7 –b ™ sg

Table 5Correlation matrix for the five parameter fit

0 0N N N N Nother FSR Žnonyb.p Žb ™ c.p b ™ sg

N 1 y0.20 y0.45 y0.23 y0.03other

N – 1 y0.27 0.27 y0.37FSR

0N – – 1 y0.54 0.12Žnonyb.p0N – – – 1 y0.18Žb ™ c.p

N – – – – 1b ™ sg

deviations away from either the electron or pion hypothesis while the identification of K 0 and K 0 mesons wasL SŽ 0 0described in Section 4. The data distribution contains 38 kaon candidates 20 K mesons, 11 K mesons and 7L S

. Ž 0 0charged kaons . After the fit the Monte Carlo predicts a background of 20 kaons 9.9 K mesons, 3.4 KL S. Ž 0 0mesons and 6.8 charged kaons and a signal of 16.2 kaons 5.4 K mesons, 5.0 K mesons, and 5.8 chargedL S

.kaons .w Ž Ž ..Finally, on combining sub-samples 4 and 8 for the E signal region, the excess s distribution Fig. 6 eg l

Ž .peaks at about 2 cm which is characteristic of single photons which have -s )s1.991 " 0.007 cm ratherl0 Ž .than p mesons which have -s )s2.251 " 0.034 cm . This is well modelled by the fitted signal Montel

Ž . Ž .Carlo, as shown in Fig. 6 f . The mean s for data 2.062 " 0.033 cm tends to be photon-like, and is 3.0lŽ .standard deviations less than the mean s for the background Monte Carlo 2.187 " 0.026 cm which consistsl

mainly of p 0 decays. Hence the excess is not consistent with a fluctuation of the p 0 background.

6.3. The inclusiÕe b™sg branching ratio.

These observations show that the excess in Fig. 5 is photon-like and tends to have a lifetime distribution anda strangeness content consistent with the b™sg process. This evidence supports the hypothesis that the excessis due to b™sg decays. The inclusive b™sg branching ratio is then evaluated as follows:

N 1b™ sgBr b™sg s PŽ .

´ 2 N Rb™ sg had b

where ´ is the efficiency with which inclusive b™sg decays pass the event selection and is given inb™ sg

Table 2; N is the number of hadronic Z decays in the data after the standard ALEPH hadronic eventhad

w Ž . Ž .Fig. 3. The E distributions in data error bars and the Monte Carlo histograms before and after the fit for the eight sub-samples. Theg

Ž . Ž .dashed histogram is the standard Monte Carlo no b™ sg before the fit. The solid histogram is the Monte Carlo including b™ sg afterthe fit. The shaded histogram is the b™ sg contribution from the fitted Monte Carlo.


Ž . Ž .Fig. 4. Comparing data and Monte Carlo events before standard MC and after corrected MC the fit in each cut variable and fit variable.

selection; and R is the Z ™bb hadronic branching fraction which is set to its Standard Model value of 0.2158bw x25 . The resulting branching ratio and its total statistical uncertainty is:

Br b™sg s 3.11"0.88 =10y4 .Ž . Ž .

7. Statistical and systematic uncertainties

Table 6 gives a breakdown of the statistical and systematic uncertainties on the branching ratio measurement.The methods used to evaluate these uncertainties are described below, where each uncertainty is referred to byits label in the table.


Fig. 5. The energy of the photon in the rest frame of the reconstructed jet. The top figure shows the data and Monte Carlo backgroundŽ .events in sub-sample 4 see section 5 , after the correction of the background processes’ relative normalisations. The bottom figure shows

the excess in data after subtraction of the corrected Monte Carlo background and the signal distribution resulting from the multivariate fit.Also shown is the excess remaining in the data when a fit is performed without b™sg .

Ž . Ž .The total Br b™sg fit uncertainty D has a small contribution, D , from the b™sg Monte Carlof i t 1

statistics. The covariance matrix produced in the fit is used to extract the uncertainties in the branching ratio forw xb™sg due to uncertainties in the four other parameters in the fit by adopting the method described in 24 ,

Ž .giving a combined systematic uncertainty of D . The background normalisation uncertainty D is added in2 2Ž .quadrature to the signal Monte Carlo statistical uncertainty D , and the remaining uncertainty in D is1 f i t

Ž . Ž .assumed to be the statistical uncertainty on Br b™sg due to the finite data statistics D . As the fit resultsstat

give a background Monte Carlo composition which is different from the standard ALEPH Monte Carlo, aŽ .conservative systematic uncertainty D is evaluated as follows. The b™sg branching ratio is recalculated3

using the standard Monte Carlo prediction for the background normalisations and allowing only N to varyb™ sg

in the fit; then D is taken to be the difference between this branching ratio and the value derived in Section 6.3

The x 2 for this one parameter fit is 72.4 for 67 degrees of freedom.The accuracy of the fit depends on how well the Monte Carlo represents the shape of the data in Ew and ong

the relative proportion of each background source in each sub-sample. The systematic uncertainty due to apossible imprecision of the Monte Carlo in this regard is assessed by observing the change in the measuredbranching ratio as the boundaries between the eight sub-samples are varied. In this way it is possible to vary thebackground in the signal region leaving the signal itself approximately unchanged. For example, as the Ejet

boundary is decreased it is possible to increase the FSR background by up to a factor 3 and the otherbackgrounds by a factor 2 leaving the signal approximately constant in sub-samples 2 and 4. Similarlyincreasing the s boundary allows the other backgrounds to increase by up to a factor of 1.7 leaving the FSRl


Ž .Fig. 6. Supporting evidence for the b™sg signal: a displays the impact parameter hemisphere probability in the same hemisphere as thecandidate photon for data and the corrected background Monte Carlo, while the excess of data above the background, along with the fitted

Ž . Ž . Ž .signal distribution, are shown in b ; c and d show the leading kaon momentum distributions for data, background Monte Carlo andŽ . Ž .signal Monte Carlo; e and f show the shower major axis length distributions for data, background Monte Carlo and signal Monte Carlo.

The events are selected as described in the text.

and the signal in regions 2 and 4 almost unchanged. As these changes are made the branching ratio changes byD and this is taken as a systematic uncertainty, although such changes are compatible with statistical4

fluctuations. The branching ratio measurement is found to be insensitive to variation of the cut values in thew o p p w xother variables, cosu , S , E and ylog P 17 .g b g hemŽ .The total jet energy E for candidate b hadrons is obtained by adding the energies contributing to the jetje t

Ž .of the charged tracks ;30% of the total energy on average ; all neutral hadronic clusters which are assumed to0 Ž . Ž . Ž .be K ; 4% on average ; and electromagnetic clusters ;66% on average . For those jets 1% of the totalL

which lie between cosu of 0.9 and 0.95 relative to the beam axis, where there are uncertain energy losses due toimperfections in the detector coverage at these small angles, the calibration uncertainty is taken to be 10%. For


Table 6Ž .Uncertainties on the Br b™sg measurement. Note that D is the total statistical uncertainty which comes from the multivariate fitfit

y4Ž .Source of uncertainty Uncertainty =10

D Signal Monte Carlo statistics 0.0651

D Background Monte Carlo statistics 0.3762

D Background Monte Carlo composition 0.1853

D Background shapes 0.4624

D Energy calibration uncertainties 0.18250D K detection efficiency 0.0316 L

D s scale 0.0977 l

0.200D B meson production fractions 0.01211

D b quark Fermi momentum 0.02812

D Assumption for B ™X g exclusive branching fractions 0.01413 s ss

2 2 2D Data statistics –- D s D yÝ D 0.80(stat stat f i t is1 i

13 2D Total systematic uncertainty s Ý D 0.72(sys is1 i

2 2D Total uncertainty s D q D 1.08(tot st at s y s

the remaining jets, the calibration uncertainty of the energy scale is taken to be 1.0% and is obtained byŽ . Ž . Ž .weighting the calibration uncertainty of the ECAL 1.5% , the HCAL 4% and charged tracks 0.2% by the

average fraction of the energy in each, as given above. Varying E within these uncertainties, for theje t

appropriate events, results in a shift in the fitted branching ratio of D . The uncertainty in the HCAL detection5

efficiency for K 0 mesons in HCAL is taken to be 4% and since this affects a quarter of b™sg decays itLŽ .corresponds to a 1% systematic uncertainty on the branching ratio D .6

A fit to the s data distribution using Monte Carlo background p 0 mesons and photons, for which the meanlŽs values are allowed to vary in the fit, results in a negligible shift in the mean s for photons but a q1.6 "l l

. 00.8 % shift for p mesons. The change in the b™sg branching ratio measurement when the s scale isl

increased by 1.6% for Monte Carlo p 0 mesons is D .7

The remaining systematic uncertainties are evaluated by reweighting signal and background Monte Carloevents to take into account modelling uncertainties, repeating the fit, and measuring the resultant changes in the

Ž .branching ratio measurement. The b™u decay rate used in the fit 1.5% is varied over the range of itsŽ .uncertainty " 50% , giving a shift in the measurement of D . The systematic uncertainty due to theoretical8

Ž .uncertainty in the relative fractions of exclusiÕe penguin decays in the inclusiÕe b™sg model D is9

calculated by repeating the fit with two extreme signal Monte Carlo compositions. Firstly the fractions ofresonant penguins are all increased to their upper limits — as shown in Table 1 — while the numbers ofnon-resonant penguins are reduced to compensate; then the fractions of resonant penguins are all decreased totheir lower limits while the numbers of non-resonant penguins are increased to compensate. The fit is alsorepeated with the fraction of baryonic b™sg decays set to zero, giving a shift of D . The production fractions10

w xof B mesons are varied within their experimental limits 16 to give a combined uncertainty of D . The central11Ž .Fermi momentum value used in the simulation of B™K np , 265 MeV, is changed by its uncertainty, "25

w xMeV 15 , which gives a maximum shift of D . Finally, a systematic uncertainty of D is evaluated to account12 13

for the assumption that the exclusive B penguin decays are produced in the same relative fractions as thes

exclusive B penguin decays by measuring the effect of weighting the B penguin decay branching fractionsuŽd . sw xaccording to their spin multiplicity 26 instead of those given in Table 1.


8. Conclusions

The inclusive b™sg branching ratio is measured to be:

3.11"0.80 "0.72 =10y4 .Ž .stat syst

This could include a small contribution from b™dg . The result is consistent with both the Standard ModelŽ . y4 w x Ž . y4prediction of 3.76"0.30 =10 2 and the only previous measurement of 2.32"0.57"0.35 =10 by

w x w x w xCLEO 14 . The L3 27 and DELPHI 28 collaborations at LEP have previously performed searches forradiative charmless b decays and placed 90% confidence level upper limits of 1.2=10y3 and 5.4=10y4 ,respectively, on the branching ratio.

Acknowledgements

We wish to congratulate our colleagues in the CERN accelerator divisions for the excellent performance ofthe LEP machine. We are grateful to the engineers and technicians at all the collaborating institutions for theircontribution to the success of ALEPH. Those of us from non-member states are grateful to CERN for itshospitality. We would like to thank A. Ali, T. Ohl and M. Neubert for helpful discussions, C. Greub forproviding us with a b™sg spectator model program, and R. Poling for explaining the CLEO method fordetermining the b quark Fermi momentum.

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( )J.Z. Bai et al.rPhysics Letters B 429 1998 188–194 189

a Institute of High Energy Physics, Beijing 100039, People’s Republic of Chinab Boston UniÕersity, Boston, MA 02215, USA

c California Institute of Technology, Pasadena, CA 91125, USAd China’s UniÕersity of Science and Technology, Hefei 230026, People’s Republic of China

e Colorado State UniÕersity, Fort Collins, CO 80523, USAf Massachusetts Institute of Technology, Cambridge, MA 02139, USA

g Shandong UniÕersity, Jinan 250100, People’s Republic of Chinah Stanford Linear Accelerator Center, Stanford, CA 94309, USA

i UniÕersity of Hawai’i, Honolulu, HI 96822, USAj UniÕersity of California at IrÕine, IrÕine, CA 92717, USA

k UniÕersity of Texas at Dallas, Richardson, TX 75083-0688, USAl UniÕersity of Washington, Seattle, WA 98195, USA

Received 28 November 1997Editor: K. Winter

Abstract

Purely leptonic decays of the charged D meson have been studied using the reaction eqey™D)qDy at a center ofmass energy of 4.03 GeV. A search was performed for D™mn recoiling against a D0 or Dq which had beenm

reconstructed from its hadronic decay products. A single event candidate was found in the reaction eqey™D)qDy, where)q q 0 0 y q y y yD ™p D with the D ™K p , and the recoiling D decaying via D ™m n . This yields a branching fractionm

Ž . q0.16q0.05 q180q80value B D™mn s0.08 %, and a corresponding value of the pseudoscalar decay constant f s300m y0.05y0.02 D y150y40

MeV. q 1998 Elsevier Science B.V. All rights reserved.

1. Introduction

Purely leptonic decays of the Dq 1 meson pro-ceed via annihilation of the charm and anti-downquarks into a virtual W boson. The decay rate of thisCabibbo-suppressed process is determined by thewavefunction overlap of the two quarks at the origin,and is parametrized by the D meson decay constant,f . The leptonic decay width of the D can be writtenD

w xas 1222 2< <G V mF cd ll2 2G D™ lln s f m m 1y ,Ž .ll D D ll 2ž /8p mD

1Ž .

where G is the Fermi constant, V is the c™dF cdw x qCKM matrix element 2 , m is the mass of the DD

meson, and m is the lepton mass.ll

Theoretical models predict values of f and fD Ds

Ž .the D meson decay constant which vary from 90s

1 Throughout the paper, reference to a particular charge config-uration implies reference to the charge conjugate configuration aswell.

w xto 350 MeV 1,3–8 . The measurements of f andD

f have special relevance to theoretical calculationsDs

of f , whose value is of considerable importance toB0 0 w xpredictions of B B mixing 9 . However, the deter-

mination of f is very difficult, since the branchingB

fraction for B™mn is expected to be very small.m

Hence information on leptonic decays of charmedmesons is very useful. To date, there are experimen-

w xtal measurements of f from the WA75 10 , CLEODs

w x w x11 and BES 12 groups. For D™mn , only am

Žbranching fraction upper limit of 0.07% correspond-.ing to f F290 MeV at 90% C.L. has been set byD

w xthe Mark III Collaboration 13 .In this paper the results of a search for the

Cabibbo-suppressed decay D™mn are reported.m

The data were collected using the Beijing Spectrom-eter at the Beijing eqey Collider. A total integratedluminosity of 22.3 pby1 was taken at c.m. energy

) ) )4.03 GeV. At this energy D D , D D, DD, and) 0 ) 0D D events are produced. The final states D DS S

) 0 0 qand D D yield no D mesons, since the decayD) 0 ™Dqpy is kinematically forbidden. Also, thecross section values for DqDy and D)qD)y pro-duction are much smaller than that for D)qDy

( )J.Z. Bai et al.rPhysics Letters B 429 1998 188–194190

production. In addition, the D)qD)y final stateyields two undetected low momentum final stateparticles in addition to the two D’s, and this forcesthe missing mass squared variable used to defineleptonic decay candidates into the region of semi-

Ž .leptonic background see below . For these reasons,the search for Dq leptonic decay is restricted to theD)qDy final state, which is characterized by Dq

mesons in the momentum range 370–650 MeVrcŽ )q qtaking into account the decay D ™D g , which

w x. 0has a branching fraction ;1% 2 , or by D mesonsin the range 465–550 MeVrc. These regions are

) )inaccessible to D mesons from the D D and DDfinal states. Candidate D)qDy events are definedby requiring that a D0 or Dq, reconstructed from itshadronic decay products, have momentum in theappropriate range; this D meson is referred to as thetagging D. The system recoiling against the taggingD is then searched for the presence of a D™mnm

candidate. For such events, only the charged tracksfrom D decay and the recoil muon are fully recon-structed. A pq from D)q decay has momentumless than 80 MeVrc, and is absorbed by the beampipeand inner wall of the central drift chamber. Theexistence of the m is inferred from the missing massn

recoiling against the muon and the tagging D. Thisis small due to the neutrino mass, and the fact thatthe undetected pion or photon from D)q decay haslow momentum.

2. BES detector

The Beijing Spectrometer is a solenoidal magneticw xdetector 14 . A four-layer central drift chamber

Ž .CDC located just outside the beampipe is used inthe event trigger. Each charged track is recon-structed, and its energy loss measured, in a 40-layer

Ž .main drift chamber MDC which covers 85% of thetotal solid angle. The momentum resolution is

2 Ž .(1.7% 1qp p in GeVrc , and the dErdx resolu-tion is 11% for hadron tracks. An array of 48 barrel

Ž .scintillation counters provides time-of-flight TOFmeasurement for charged tracks, with a resolution of450 ps for hadrons. A 12-radiation-length, lead-gas

Ž .barrel shower counter BSC , operating in self-quenching streamer mode, measures the energies of

electrons and photons over 80% of the total solid' Ž .angle with an energy resolution of 22 %r E GeV .

The solenoidal magnet provides a 0.4 T magneticfield in the central tracking region of the detector.Three double-layer muon counters instrument themagnet flux return, and serve to identify muons ofmomentum greater than 0.5 GeVrc. Endcap time-of-flight and shower counters extend coverage to theforward and backward regions.

The event trigger requires at least one barrel TOFhit within a time window of 40 ns, one hit in theouter two layers of the CDC and one charged trackreconstructed by the on-line trigger logic using thehit pattern in the MDC, and a total energy in theBSC above 200 MeV.

3. Analysis method

The analysis begins with the selection of thetagging D decays. Two Dq decay modesŽ y q q 0 q. 0K p p , K p and three D decay modesSŽ y q y q q y 0 q y.K p , K p p p , K p p have been consid-S

ered, where K 0 ™pqpy. Each charged track notS

from a K 0 candidate was required to come fromS

within 1 cm of the run-dependent interaction point inthe transverse plane, and from within 15 cm alongthe beam direction. For each charged track, the polar

Ž . < <angle u had to satisfy cosu F0.85 in order thatthere be reliable tracking and barrel TOF informa-tion. The corresponding dErdx and TOF measure-ments were required to be consistent with the masshypothesis assigned to the track, and the kaon as-signment further required x 2 -x 2, where the x 2 isK p

the joint chi-squared of the available dErdx andTOF information for the track in question. For theK 0, the pqpy invariant mass was limited to 498"S

30 MeV. The momentum of the tagging D wasrestricted to the range 440–620 MeVrc; for theD)qDy final state, this is the interval correspondingto D) ™Dp decay, extended at each end by twicethe momentum resolution. This choice, together withparticle identification requirements, serves almost toeliminate contamination from the DqDy final stateS S

at the small cost of reduced acceptance for D)q™

Dqg decays. With this momentum requirement, theinvariant mass distributions for the five tagging D


decay modes are as shown in Fig. 1; in each case Dproduction is evident with a signal superimposedover a background created by random Kaon and Pioncombinations.

The number of D)qDy events produced wasextracted from the signal due to Dq™Kypqpq in

Ž .Fig. 1 a . A fit to this distribution yielded an esti-mate, N obs, of 1409"66 Dq decays; the curveshown results from the fit. The number of D)qDy

events produced, N prod, was then obtained from

N obs sN prod =´=B Dq™KypqpqŽ .= 1qB D)q™p 0DqŽ .ŽqB D)q™g Dq , 2Ž . Ž ..

where ´ , the efficiency for reconstructing Dq™

Kypqpq, was found to be 22.6"0.4% from MonteCarlo simulation. This gave B prod ;52000, and a

Ž q y )q y.cross section value s e e ™D D s2.33"

0.23 nb.The events of Fig. 1 containing a tagging D

candidate were defined by requiring that the effec-tive mass lie within three standard deviations of the

w xrelevant D mass 2 . The recoil system in each ofthese events was then checked for consistency withD™mn decay. It was required that there be am

single charged track with momentum between 700and 1250 MeVrc having dErdx, TOF and BSCinformation consistent with the muon hypothesis.

�Ž . Ž .4Fig. 1. The invariant mass distribution for the charged a , b�Ž . Ž . Ž .4and neutral c , d , e D meson candidates selected as described

in the text.

This track was then extrapolated through the muonsystem, and was identified as a muon only if it hadassociated hits in at least two layers. For an eventwith a D0 tagging mode, no isolated photons 2 wereallowed to be present. However, an event with a Dq

tagging mode could have a low momentum photonor p 0 in addition to a recoiling Dy. Such an eventwas rejected if it had more than two isolated pho-tons, or if it contained a photon having energygreater than 400 MeV. Only six muonic decay candi-dates survived these selection procedures.

The scatter-plot of muon momentum versus miss-ing mass squared recoiling against the muon andtagging D is shown in Fig. 2 for the six candidateevents. The contours at lower missing mass squaredrepresent the region of the plot corresponding to theD)qDy final state with one of the resulting D’sdecaying via the tagging mode, the other via mn .m

These contours were defined by means of MonteCarlo simulation 3, and thus take into account resolu-tion effects. The contour lines are similar to lines ofequal altitude on a topography map except instead ofaltitude, events per unit area are used on the twodimensional plot in Fig. 2. The contour lines separateregions that have roughly equal events per unit area.Also the events per unit area in a region proportion-ally increases as one steps up to next inner region.

To identify sources of background in the muonicdecay data sample, 5=106 D)qDy and 106 DqDy

S S

events were generated by Monte Carlo simulation,and subjected to the selection criteria applied to thedata. The D)q, Dq, D0, and D were allowed toS

decay according to their known branching fractionsw x2 . For all D tagging modes, the main backgroundresulted from D)qDy events in which one D diddecay via the tagging mode, while the other decayedvia p K 0, mn K 0, or mn p 0, with the K 0 or p 0

L m L m L

undetected. The contours at higher missing masssquared in Fig. 2 were obtained, as for the lower

2 An isolated photon is defined as an e.m. shower of energy)60 MeV and separated by at least 18 degrees from the directionof the nearest charged track.

3 In the simulation, 20000 D)q Dy events were generated foreach tagging D mode, with the other D decaying to mn . Them

contours at lower missing mass squared in Fig. 2 represent thesuperposition of the reconstructed events for each tagging modewhich survive the muonic decay selection criteria.


Fig. 2. The scatter-plot of muon momentum versus missing masssquared recoiling against the muon and tagging D for the surviv-

Ž .ing D muonic decay candidates dots ; the contours are describedin the text.

missing mass squared contours, by simulatingD)qDyevents in which one D decayed via a tag-ging mode, and one other via a mode which is asource of background.

The contours of Fig. 2 indicate that, although thesignal and background regions overlap substantiallyin muon momentum, they are quite well-separated inmissing mass squared. Consequently, the signal re-gion corresponding to D™mn is defined simplym

by requiring that the missing mass squared be lessthan 0.7 GeV 2. Only one event satisfies this furthercriterion. Its properties are listed in Table 1, and theevent display is shown in Fig. 3.

The event, which is tagged by the D0 ™Kypq

mode, is very clean, with two hits in the muonsystem for the muon track, no extra photons, andKy and pq tracks which are well-identified bydErdx and the barrel TOF system. The pq fromD)q™D q

0 is calculated to have a momentum ;55p

MeVrc; it should generate no hits in the CDC, andno hits are observed. The calculated momentum of

y ythe neutrino from D ™m n is ;890 MeVrc,m

with polar angle ;69 degrees, and azimuthal angle;164 degrees i.e. it passes through the BSC andmuon system well within the fiducial volume; noBSC or muon counter activity should be generated in

Ž .this region, and none is observed cf. Fig. 3 . It

Table 1The properties of the D muonic decay candidate

0Tagging D D Mass Muon momentum Missing mass2Ž . Ž . Ž .decay mode MeV MeVrc squared GeV

y qK p 1850 1048 0.533

Fig. 3. The event display for the D™mn candidate; track 2 ism

the my, and tracks 1 and 3 are the Ky and pq from D0 decay,respectively.

follows that the event kinematics and the detectorresponse are quite consistent with the interpretationof this event as being due to eqey™D)qDy, with

y yD ™m n .m

For each tagging D mode, the expected back-ground was estimated by Monte Carlo generation of2 = 105 events for each of the contributingp K 0, mn K 0, and mn p 0 modes. The number ofL m L m

events satisfying the selection criteria and havingmissing mass squared less than 0.7 GeV 2 was thenrenormalized to correspond to a luminosity of 22.3pby1. The resulting background levels, which arelisted in the fourth column of Table 2, are all verysmall, and are consistent with the observation of onlyone event in the five tagging modes. The predictedbackground for the D0 ™Kypq tagging decaymode is 0.03 events and thus the Poisson probabilitythat such a background could have produced theobserved candidate event is 3%.

Table 2A summary of the data concerning the tagging D decay modes forthe D)q Dy final state

Ž .Tagging D Number of Efficiency ´ EstimatediŽ .decay mode non tagging including m % background

q Ž . Ž .D C bgi i

y qK p 1418 18.8"0.3 0.03y q q yK p p p 2865 9.5"0.3 0.03

0 q y-2d K p p 1875 3.2"0.2 0.01y q qK p p 3016 15.6"0.3 0.120 qK p 908 5.5"0.2 0.01

Total 10082 0.20


The corresponding Monte Carlo signal for miss-ing mass squared greater than 0.7 GeV 2 is 5.2events. In this region, an additional contribution is

Ž .expected from D™tn . Eq. 1 , together with thet

appropriate efficiency factor, would imply that forone D™mn event there should be ;0.3 eventm

resulting from D™tn with t™mn n . Such ant m t

event has three missing neutrinos, and so would fallin the background region of missing mass squared inFig. 2. It follows that the total number of backgroundevents expected is ;5.5, in good agreement withthe 5 events observed.

4. Results

In order to extract a value for the D muonicbranching fraction, B, a likelihood function wasconstructed as the product of the Poisson probabilityfunctions for the individual tagging modes. For aDq tagging mode, i, the expected number of signalevents is

N exp s2 N prodi

= B D)q™p 0Dq qB D)q™g DqŽ . Ž .Ž .=B Dq™ i =´ =BŽ . i

sC =´ =B , 3Ž .i i

prod Ž .where N is from Eq. 2 ; ´ is the over alli

efficiency taking account of the muon, and the factor2 occurs since either charged D can decay muoni-cally. Similarly, for a D0 tagging mode, i, theexpected number of signal events is

N exp sN prod =B D)q™pqD 0Ž .i

=B D0 ™ i =´ =BŽ . i

sC =´ =B . 4Ž .i i

Ž . Ž .In Eqs. 3 and 4 , C is the number of charged Di

mesons produced in association with tagging mode iŽ q.i.e. the number of non-tagging D , and is calcu-

obs Ž qlated using N the observed number of D de-y q q.cays to K p p , ´ , and the appropriate function

)q Ž Ž ..of D and D branching fractions see Eq. 2 .The values of the C and ´ are listed in the secondi i

and third columns of Table 2. The expected numberŽof observed events in tagging mode i is then C =´i i

.=Bqbg , where bg is the expected number ofi i

Ž .Fig. 4. The dependence of the likelihood function on a the valueŽ .of the D™mn branching fraction, and b the value of f ; them D

unshaded areas correspond to the one standard deviation errorsdescribed in the text.

background events, and the likelihood function isgiven by

Lsexp yB C =´Ž .Ý i iž /is1,5

= C =´ =Bqbg , 5Ž . Ž .1 1 1

where tagging mode 1 corresponds to D0 ™Kypq;Ž .a factor exp yÝ bg has been removed, since itis1,5 i

does not depend on B. It should be noted that onlythe background estimate for tagging mode D0 ™

Kypq appears in this function. The dependence ofŽ .L on B is shown in Fig. 4 a , and the maximum

value occurs for

1 bg1Bs y . 6Ž .

C =´C =´Ž .Ý 1 1i iis1,5

This gives Bs0.08q0 .16%, where the errors resulty0.05

from the values of B corresponding to 68.3% of thearea under the curve above and below the maximum

Ž Ž ..position i.e. the unshaded area of Fig. 4 a .The systematic errors on B are estimated from the

uncertainties in the D and D)q branching fractionsw x q2 , from the error on the number of D decays toKypqpq, by varying bg by 50%, and, predomi-1

nantly, by varying the event selection criteria, andthereby the efficiencies. The final result is

B D™mn s0.08q0 .16q0.05 % , 7Ž . Ž .y0 .05y0.02


where the second errors are systematic. From Eq.Ž . q w x1 , with D life-time 1.057 ps 2 ,

f 2 s1.136=B , 8Ž .D

with f in GeV and B in %. Substituting B in termsDŽ .of f into Eq. 5 , the dependence of the likelihoodD

Ž .function on f is as shown in Fig. 4 b . The proce-D

dure followed for B yields

f s300q180q80 MeV , 9Ž .D y150y40

where the systematic errors have been obtained fromŽ .those on B by using Eq. 8 .

ŽAlthough the uncertainties in the values of B D.™mn and f obtained in this experiment are large,D

it should be emphasized that the analysis procedureis independent of measured luminosity and theDq) Dy cross section value, and does not require

Žmodel-dependent assumptions. The result for B D.™mn is consistent with the upper limit set by the

ŽMark III experiment which obtained no candidate.events , while that for f is comparable to the valuesD

obtained for f in recent experiments, as expectedDS

theoretically.

5. Conclusions

We have searched for the leptonic decay D™mn .m

One event candidate was observed and a branchingfraction was estimated. The estimate of the branch-ing fraction based on one event is equal to the upperlimit set by the Mark III experiment, which did notdetect any events. From theoretical estimations, fD

is expect to be comparable to f , and in this respectDS

the present result is consistent with other recentmeasurements.

Acknowledgements

This work is supported in part by the NationalNatural Science Foundation of China under ContractNo. 19290400 and the Chinese Academy of Sciences

Ž .under contract No. KJ85 IHEP ; by the Departmentof Energy under Contract Nos. DE-FG02-91ER40676Ž .Boston University , DE-FG03-92ER40701Ž . ŽCaltech , DE-FG03-93ER40788 Colorado State

. Ž .University , DE-AC02-76ER03069 MIT , DE-

Ž .AC03-76SF00515 SLAC , DE-FG03-91ER40679Ž . Ž .UC Irvine , DE-FG03-94ER40833 U Hawai’i ,

Ž .DE-FG05-92ER40736 UT Dallas , DE-AC35-Ž .89ER40486 SSC Lab ; by the US National Science

ŽFoundation, Grant No. PHY9203212 University of.Washington ; and by the Texas National Research

Laboratory Commission under Contract Nos.Ž .RGFY91B5, RGFY92B5 Colorado State , and

Ž .RCFY93-316H UT Dallas . We would like to thankthe staffs of the BEPC accelerator and the Comput-ing Center at the Institute of High Energy PhysicsŽ .Beijing .

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11 June 1998


Proton proton bremsstrahlung at 797 MeVrc

COSY TOF Collaboration

R. Bilger d, A. Bohm b, H. Brand a, S. Brand a, K.-Th. Brinkmann b, H. Clement d,¨P. Cloth e, M. Dahmen e, S. Dshemuchadse f, W. Eyrich c, D. Filges e,

H. Freiesleben b, M. Fritsch c, R. Geyer e, A. Hassan g, J. Hauffe c, P. Herrmann a,B. Hubner b, P. Jahn e, K. Kilian e, H. Koch a, J. Kress d, J. Krug a, E. Kuhlmann b,¨

J.S. Lange b, A. Metzger c, P. Michel f, K. Moller f, H.P. Morsch e, Ch. Nake e,¨H. Nann h, B. Naumann f, L. Naumann f, P. Ringe a, E. Roderburg e, M. Rogge e,

A. Schamlott f, P. Schonmeier b, A. Schulke f, M. Steinke a, F. Stinzing c,¨ ¨P. Turek e, G.J. Wagner d, S. Wirth c, U. Zielinski a

a Institut fur Experimentalphysik, Ruhr-UniÕersitat Bochum, D-44780 Bochum, Germany¨ ¨b Institut fur Kern- und Teilchenphysik, Technische UniÕersitat Dresden, D-01062 Dresden, Germany¨ ¨

c Physikalisches Institut, UniÕersitat Erlangen, D-91058 Erlangen, Germany¨d Physikalisches Institut, UniÕersitat Tubingen, D-72076 Tubingen, Germany¨ ¨ ¨

e Institut fur Kernphysik, Forschungszentrum Julich, D-52425 Julich, Germany¨ ¨ ¨f Institut fur Kern- und Hadronenphysik, Forschungszentrum Rossendorf, D-01314 Dresden, Germany¨

g Atomic Energy Authority NRC, Kairo, Egypth IUCF Bloomington, IN 47408, USA

Received 1 April 1998Editor: L. Montanet

Abstract

At COSY pp-bremsstrahlung was measured at a beam momentum of 797 MeVrc using an external proton beam. Datawere taken with a wide angle spectrometer covering a solid angle of approximately 1 sr. The complete data set is presentedin a series of c.m. angular distributions as well as a single Dalitz plot. The absence of final state interaction effects isunderstood as being due to a general insensitivity of the ppg reaction to the spin-singlet component of the NN interaction.

Ž .Coplanar angular distributions in the laboratory system are well reproduced by recent model calculations; also goodw Ž . xagreement is found with the original TRIUMF data K. Michaelian et al., Phys. Rev. D 41 1990 286 when omitting the

rescaling factor of 2r3. q 1998 Published by Elsevier Science B.V. All rights reserved.

PACS: 13.75.Cs; 25.10.qs


( )R. Bilger et al.rPhysics Letters B 429 1998 195–200196

In recent years NN bremsstrahlung has gainedrenewed interest both experimentally and theoreti-cally. Roughly 30 years ago when first data weretaken with detectors covering only a small fractionof phase space the main goal was to check on the

w xvarious potential models then in use 1,2 . It wasespecially hoped for that the ppg reaction wouldallow an insight into the off-shell behaviour of thestrong interaction and hence to differentiate betweenthe various models. It turned out, however, that allthe different model predictions more or less agreewith each other and that they also reproduce most ofthe data; in those cases, where they failed, they didso in a very similar way indicating that some impor-tant contributions were still missing. This can exem-plarily be shown by the data from the coplanar

w xTRIUMF experiment 3 which was performed witha polarized proton beam and where all three outgo-ing particles were detected. The data were comparedto theoretical predictions and reasonable agreementwas found. This, however, could only be achievedby application of a somewhat arbitrary rescalingfactor of 2r3. Since then much theoretical effort hasbeen raised to explain the apparent mismatch, andsizeable progress has been made. The importance ofthe previously underestimated meson exchange cur-rents had to be acknowledged and non-negligiblecontributions from NN creation and annihilation as

w xwell as Dydegrees of freedom were found 4,5 . Itwas also regarded to be highly desirable to check onthe published TRIUMF data. A first attempt has beenmade in an internal target experiment with the Indi-

w xana cooler ring 6 , albeit with no definite conclu-sions since only very forward angles were covered.

In the present paper we report on first results onŽ .pp bremsstrahlung at 797 MeVrc T s293 MeV .kin

The two outcoming protons were detected in theCOSY TOF spectrometer set up in its most basicversion on an external beam line of the COSYaccelerator complex at the Forschungszentrum Julich.¨As shown schematically in Fig. 1, it consists of ascintillator hodoscope for charged particles, hascylindrical symmetry and allows measurements forpolar angles as small as 38. The liquid hydrogentarget is contained within a cylindrical 4 mm = 6

w xmm B cell 7 . Entrance and exit window foils weremade from 0.9 mm thick mylar foils. Right behindthe target the outcoming reaction products traverse a

w xcircular and segmented start detector 8 made from2P16 trapezoidally shaped, 0.5 mm thin scintillator

Ž . w xplates BC418 . The stop detector 9 placed 87 cmbehind the target consists of a 3-layer scintillator

Ž .hodoscope BC404 with 48 wedge-shaped elementsin the first layer and 24 left- and 24 right-woundelements in the following ones with each layer being5 mm thick. The maximum angle obtained with thissetup was 338. To avoid secondary reactions of theejectiles, the whole system is housed within a vac-uum tank. Its main vault is separated from the targetarea by a thin foil and can be pumped down to about10y3 hPa. The luminosity was measured throughobservation of pp elastic scattering. For this purposeadditional scintillator slabs were mounted within thetank at q-values ranging from 578 to 648.

For each detected particle the primary observablesŽ .are q ,f as well as the time of flight and hence b ,

which together with a mass hypothesis allows one toinfer its four-momentum components. By means of

Ž . Ž .Fig. 1. Sketch of the experimental setup not to scale showing the principal method for measuring the point of impact q ,f as well as thetime of flight between start and stop detector. The distance between start and stop detector is 87 cm.

( )R. Bilger et al.rPhysics Letters B 429 1998 195–200 197

reconstructing the missing mass distributions a checkon each hypothesis is made and simultaneously onecan identify the looked for bremsstrahlung events. At797 MeVrc only the reaction channels ppg , ppp 0

and dpq are accessible. Elastic scattering eventscannot be recorded since the summed scattering an-gles are always larger than the opening angle of thestop detector. In order to discriminate against back-ground reactions with nuclei of the target foils andfrozen gas molecules, empty target runs were takenat repeated intervals.

The present data stem from about 36 h of beamtime. Full target runs were taken for an integratedluminosity of 2.28 nby1. Fig. 2 shows the result of amissing mass analysis, where the squared mass m2

3

of the unobserved particle is plotted. Full and emptytarget spectra are shown in the top, the differencespectrum in the bottom part. Two peaks, representing

2 Ž 2 .2the reaction channels ppg around m s0 GeVrc30 Ž 2 .2and ppp near 0.018 GeVrc , are visible. The

experimental data, given by points with statisticalerror bars, are compared to the result of a Monte

Ž .Carlo MC simulation which assumed phase-space

distributed 3-particle events and included particlestraggling and energy loss within the various scintil-lators as well as inefficiencies due to holes or gaps inor between individual detector elements. In generalgood agreement is found.

With the present set-up already a large fraction ofthe available phase space could be covered. Byfocussing onto the high energy end of thebremsstrahlung spectrum this percentage is of the

Ž . Ž .order of 80% )95% for the upper third fifth inE , respectively, allowing a deeper view into theg

nature of the hard photon component. Shown in Fig.Ž3 are six photon angular distributions in the initial

.proton proton center-of-mass frame where for eachdistribution the photon angle u is measured relativeg

to the direction of the low-energy proton. The secondŽproton has its momentum vector pointing within the

.same plane but on the opposite side from the photonin an angular range as indicated within square brack-ets in each of the six individual plots. Normalizedempty target runs have been subtracted. The dashedcurves show MC simulated phase space distributions.It was found that one common normalisation factor

Ž . Ž .Fig. 2. Top: Missing mass spectra as taken with a full solid and empty dashed line target, respectively; bottom: difference spectrumŽ .where experimental data are given by the dots and statistical error bars, the solid line shows the result of a Monte Carlo simulation.


Ž .Fig. 3. Photon angular distributions in the initial proton proton center-of-mass frame solid histograms compared to Monte Carlo simulatedŽ .phase space predictions dashed curves . The latter were normalized to the experimental data by using one common normalisation factor

Ž .see text . The numbers within brackets denote the angular range where the second proton has been observed.

could be used when distributing the simulated eventsŽ . 2according to w u s1r3qcos u with u denotingp p p

the opening angle of the two protons in the c.m.frame. We regard the necessity to use such an addi-tional distribution function as a simple means toaccount for the fact that in the ppg reaction, whichcontrary to the meson producing reactions mentionedabove is not connected to any threshold, surelyhigher partial waves with lG1 in the exit channelare present at a beam momentum close to 0.8 GeVrc.In a plain phase space simulation, however, one

Ž .assumes an isotropic i.e., s wave distribution. Thesize of the constant term was fixed on the groundsthat it had to be much smaller than the coefficient ofthe angle dependent term. It should be mentioned

Ž .that the inclusion of w u led to a remarkablep

agreement between simulated and experimental dataalso for other observables as, e.g., the photon energyspectrum.

Ž .A Dalitz plot of kinematically fittedŽ .bremsstrahlung events is shown in Fig. 4 top ; in the

bottom part the projection onto the x-axis is shown,

where the dashed curve represents MC simulateddata including the 1r3qcos2u distribution functionp

just mentioned. As a surprising result we note theŽ .absence of sizeable pp final state interaction FSI

effects which should show up at lowest M 2 valuespp

as an enhancement above the phase space simula-tions as observed in the corresponding Dalitz plots of

0 w x q w xthe ppp 10 and pnp reactions 11 at compara-ble bombarding energies. Since from phase space arather low yield for two-proton emission at near-zerorelative momentum is expected, it is not clear whetheronly statistical uncertainties fake the noted non-ex-istence. Yet there might be a deeper reason originat-ing from the structure of the ppg transition matrixelement for the magnetisation current operator whichis believed to govern the processes relevant in thehigh energy part of the photon spectrum. As has

w xbeen shown in 12 this matrix element is suppressedwhenever a spin-singlet state is involved thus leadingto a predominance of triplet ™ triplet transitions.Sizeable FSI effects, however, are only expected ifthe two protons are in the 1S state as in the p 0

0

( )R. Bilger et al.rPhysics Letters B 429 1998 195–200 199

Ž .Fig. 4. Top: Dalitz plot of kinematically fitted bremsstrahlungevents; bottom: projection onto the x-axis where the dashed curverepresents the Monte Carlo simulated distribution.

producing reaction which close to threshold proceedsvia the single channel 3P ™1S s .0 0 0

ŽSubsamples of coplanar ppg events in the labora-.tory system were extracted for several intervals of

Ž .the two proton laboratory angles q Fig. 5 . Allow-pŽing for a proton-noncoplanarity of F-58 for defini-

w x.tion see 13 , g angular distributions have beendeduced for three symmetric as well as three asym-metric q settings. The three basic intervals, denotedp

by the average angles q s158,238 and 29.58, arepw x w x w x108 y 208 , 208 y 268 and 268 y 338 , respec-tively. In general rather small differential cross sec-

Ž 2 .tions of only a few mbr sr rad were encountered.

For statistical reasons the two generally differingdistributions where the photon is emitted either onthe first or the second proton’s side have been added

w xtogether, as was also done for the calculated data 4Ž . Ž .Fig. 5, solid lines . In view of the large statisticaluncertainties structures predicted by theory are hardlyrecognized in the data. The comparison with themodel calculations, however, allows to demonstrateagreement in size; here an additional systematic errorof order 20% is assumed which results from uncer-tainties in background correction, detector accep-tance, luminosity determination and the pp elasticscattering cross section. It should be remarked that inkinematical situations as in the present experimentwhere rather high photon energies are encounteredonly calculations that additionally include two-bodycurrent contributions such as vector-meson decay

Ž . w xand NDg p ,r exchange 4,5 can give a consistentand reliable description of the data. When comparing

w xthese distributions with the TRIUMF data 3 andagain applying the summation just mentioned, goodagreement is observed if the rescaling factor 2r3 isomitted.

Integrating the 29.58,29.58 distribution of Fig. 5over q , a fourfold differential cross sectiong

Fig. 5. Differential ppg cross sections taken in coplanar geometrywith the protons detected at average q angles as indicated in thetop left corners. The solid lines are the result of recent potentialmodel calculations obtained by using computer codes developed

w xin 4 .


Fig. 6. Compilation of fourfold differential cross sections of theppg reaction at various beam energies taken in the same coplanar

w xgeometry. The data points given by triangles are taken from 1 ,the asterisk denotes the TRIUMF data point without the rescaling

w xfactor of 2r3 3 , and the solid dot gives the present result.

4 Ž .d sr dV dV is found which is shown in Fig. 6 in1 2

comparison with previous data taken in the samegeometry. The open triangles covering the region

w xbelow Ts250 MeV are from 1 , the TRIUMFw xmeasurement 3 without the rescaling factor of 2r3

is plotted as an asterisk. The present result is givenby the solid dot, where the shown error of order 15%

Ž .contains a statistical error 10% and an error due tothe background subtraction estimated to be of similarsize. A smooth and monotonic increase with bom-barding energy T is observed as was also predicted

w xfrom previous model calculations 2 . On the basis ofassuming the ppg data to be distributed purely ac-cording to phase space an estimate of the total crosssection can be deduced as follows. By folding thetotal number of observed events with the overallacceptance correction factor as given by the MCsimulation and putting in the deduced luminosity we

Ž .find a total cross section of ss 3.5"0.3 mb withan additional systematic error of about 0.7 mb. Inview of the fact that any 1rk dependence expectedfor the low energy part of the photon spectrum hasbeen neglected this value should be regarded as alower limit.

In summary we have measured the differentialcross section for pp ™ ppg at 797 MeVrc. The

two outgoing protons were detected in a wide angle,cylindrically symmetric time-of-flight spectrometercovering angles between 38 and 338. The whole dataset which is presented in a series of c.m. angulardistributions could be reproduced through MC simu-lations by just using one common normalisationfactor when distributing the data according to 1r3qcos2u . The absence of sizeable FSI effects is inter-p

preted as being due to the form of the electromag-netic transition operator which in turn induces asuppression of the 1S state. Coplanar photon angular0

Ž .distributions in the laboratory system were found tobe in good agreement with existing data as well asrecent model calculations. The absolute normaliza-tion of the data is based on a comparison with ppelastic scattering, where the overall systematic erroris estimated to be of the order of 20%.

Acknowledgements

We like to thank the COSY staff for their endur-ing help in preparing an extracted proton beam witha high duty cycle. Thanks are due to K. Nakayama,J. Eden and C. Wilkin for enlightening discussions.This work is based in parts on the Doctoral Thesis ofP. Herrmann. Financial support from the BMBF andFZ Julich is gratefully acknowledged.¨

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11 June 1998


Single- and multi-photon production in eqey collisionsat a centre-of-mass energy of 183 GeV

ALEPH Collaboration

R. Barate a, D. Buskulic a, D. Decamp a, P. Ghez a, C. Goy a, S. Jezequel a,J.-P. Lees a, A. Lucotte a, F. Martin a, E. Merle a, M.-N. Minard a, J.-Y. Nief a,

B. Pietrzyk a, R. Alemany b, G. Boix b, M.P. Casado b, M. Chmeissani b,J.M. Crespo b, M. Delfino, b, E. Fernandez b, M. Fernandez-Bosman b, Ll. Garrido b,1,

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A. Colaleo c, D. Creanza c, M. de Palma c, G. Gelao c, G. Iaselli c, G. Maggi c,M. Maggi c, S. Nuzzo c, A. Ranieri c, G. Raso c, F. Ruggieri c, G. Selvaggi c,L. Silvestris c, P. Tempesta c, A. Tricomi c,2, G. Zito c, X. Huang d, J. Lin d,Q. Ouyang d, T. Wang d, Y. Xie d, R. Xu d, S. Xue d, J. Zhang d, L. Zhang d,

W. Zhao d, D. Abbaneo e, U. Becker e, P. Bright-Thomas e, D. Casper e,M. Cattaneo e, V. Ciulli e, G. Dissertori e, H. Drevermann e, R.W. Forty e,

M. Frank e, F. Gianotti e, R. Hagelberg e, J.B. Hansen e, J. Harvey e, P. Janot e,B. Jost e, I. Lehraus e, P. Maley e, P. Mato e, A. Minten e, L. Moneta e,3, N. Qi e,

A. Pacheco e, F. Ranjard e, L. Rolandi e, D. Rousseau e, D. Schlatter e,M. Schmitt e,4, O. Schneider e, W. Tejessy e, F. Teubert e, I.R. Tomalin e,

M. Vreeswijk e, H. Wachsmuth e, Z. Ajaltouni f, F. Badaud f, G. Chazelle f,O. Deschamps f, A. Falvard f, C. Ferdi f, P. Gay f, C. Guicheney f, P. Henrard f,

J. Jousset f, B. Michel f, S. Monteil f, J-C. Montret f, D. Pallin f, P. Perret f,F. Podlyski f, J. Proriol f, P. Rosnet f, J.D. Hansen g, J.R. Hansen g, P.H. Hansen g,

B.S. Nilsson g, B. Rensch g, A. Waananen g, G. Daskalakis h, A. Kyriakis h,¨¨ ¨C. Markou h, E. Simopoulou h, A. Vayaki h, A. Blondel i, J.-C. Brient i,

F. Machefert i, A. Rouge i, M. Rumpf i, R. Tanaka i, A. Valassi i,5, H. Videau i,´E. Focardi j, G. Parrini j, K. Zachariadou j, R. Cavanaugh k, M. Corden k,

C. Georgiopoulos k, T. Huehn k, D.E. Jaffe k, A. Antonelli l, G. Bencivenni l,G. Bologna l,6, F. Bossi l, P. Campana l, G. Capon l, F. Cerutti l, V. Chiarella l,

G. Felici l, P. Laurelli l, G. Mannocchi l,7, F. Murtas l, G.P. Murtas l,

0370-2693r98r$19.00 q 1998 Published by Elsevier Science B.V. All rights reserved.Ž .PII S0370-2693 98 00468-7


L. Passalacqua l, M. Pepe-Altarelli l, M. Chalmers m, L. Curtis m, A.W. Halley m,J.G. Lynch m, P. Negus m, V. O’Shea m, C. Raine m, J.M. Scarr m, K. Smith m,

P. Teixeira-Dias m, A.S. Thompson m, E. Thomson m, J.J. Ward m, O. Buchmuller n,¨S. Dhamotharan n, C. Geweniger n, G. Graefe n, P. Hanke n, G. Hansper n,V. Hepp n, E.E. Kluge n, A. Putzer n, J. Sommer n, K. Tittel n, S. Werner n,

M. Wunsch n, R. Beuselinck o, D.M. Binnie o, W. Cameron o, P.J. Dornan o,8,M. Girone o, S. Goodsir o, E.B. Martin o, N. Marinelli o, A. Moutoussi o, J. Nash o,

J.K. Sedgbeer o, P. Spagnolo o, M.D. Williams o, V.M. Ghete p, P. Girtler p,E. Kneringer p, D. Kuhn p, G. Rudolph p, A.P. Betteridge q, C.K. Bowdery q,

P.G. Buck q, P. Colrain q, G. Crawford q, A.J. Finch q, F. Foster q, G. Hughes q,R.W.L. Jones q, A.N. Robertson q, M.I. Williams q, I. Giehl r, C. Hoffmann r,

K. Jakobs r, K. Kleinknecht r, G. Quast r, B. Renk r, E. Rohne r, H.-G. Sander r,P. van Gemmeren r, C. Zeitnitz r, J.J. Aubert s, C. Benchouk s, A. Bonissent s,

G. Bujosa s, J. Carr s,8, P. Coyle s, A. Ealet s, D. Fouchez s, O. Leroy s,F. Motsch s, P. Payre s, M. Talby s, A. Sadouki s, M. Thulasidas s,

A. Tilquin s, K. Trabelsi s, M. Aleppo, t, M. Antonelli t, F. Ragusa t,R. Berlich u, W. Blum u, V. Buscher u, H. Dietl u, G. Ganis u, H. Kroha u,¨

G. Lutjens u, C. Mannert u, W. Manner u, H.-G. Moser u, S. Schael u,¨ ¨R. Settles u, H. Seywerd u, H. Stenzel u, W. Wiedenmann u, G. Wolf u,

J. Boucrot v, O. Callot v, S. Chen v, M. Davier v, L. Duflot v, J.-F. Grivaz v,Ph. Heusse v, A. Hocker v, A. Jacholkowska v, M. Kado v, D.W. Kim v,9,¨

F. Le Diberder v, J. Lefrancois v, L. Serin v, E. Tournefier v, J.-J. Veillet v,I. Videau v, D. Zerwas v, P. Azzurri w, G. Bagliesi w,8, S. Bettarini w, T. Boccali w,

C. Bozzi w, G. Calderini w, R. Dell’Orso w, R. Fantechi w, I. Ferrante w, A. Giassi w,A. Gregorio w, F. Ligabue w, A. Lusiani w, P.S. Marrocchesi w, A. Messineo w,

F. Palla w, G. Rizzo w, G. Sanguinetti w, A. Sciaba w, G. Sguazzoni w, R. Tenchini w,`C. Vannini w, A. Venturi w, P.G. Verdini w, G.A. Blair x, L.M. Bryant x,J.T. Chambers x, J. Coles x, M.G. Green x, T. Medcalf x, P. Perrodo x,

J.A. Strong x, J.H. von Wimmersperg-Toeller x, D.R. Botterill y, R.W. Clifft y,T.R. Edgecock y, S. Haywood y, P.R. Norton y, J.C. Thompson y, A.E. Wright y,

B. Bloch-Devaux z, P. Colas z, B. Fabbro z, G. Faıf z, E. Lancon z,8,¨M.-C. Lemaire z, E. Locci z, P. Perez z, H. Przysiezniak z, J. Rander z,

J.-F. Renardy z, A. Rosowsky z, A. Roussarie z, A. Trabelsi z, B. Vallage z,S.N. Black aa, J.H. Dann aa, H.Y. Kim aa, N. Konstantinidis aa, A.M. Litke aa,

M.A. McNeil aa, G. Taylor aa, C.N. Booth ab, C.A.J. Brew ab, S. Cartwright ab,F. Combley ab, M.S. Kelly ab, M. Lehto ab, J. Reeve ab, L.F. Thompson ab,

K. Affholderbach ac, A. Bohrer ac, S. Brandt ac, G. Cowan ac, J. Foss ac, C. Grupen ac,¨L. Smolik ac, F. Stephan ac, G. Giannini ad, B. Gobbo ad, G. Musolino ad, J. Putz ae,

J. Rothberg ae, S. Wasserbaech ae, R.W. Williams ae, S.R. Armstrong af,


E. Charles af, P. Elmer af, D.P.S. Ferguson af, Y. Gao af, S. Gonzalez af,´T.C. Greening af, O.J. Hayes af, H. Hu af, S. Jin af, P.A. McNamara III af,J.M. Nachtman af,10, J. Nielsen af, W. Orejudos af, Y.B. Pan af, Y. Saadi af,

I.J. Scott af, J. Walsh af, Sau Lan Wu af, X. Wu af, G. Zobernig af

a ( )Laboratoire de Physique des Particules LAPP , IN2P3-CNRS, F-74019 Annecy-le-Vieux Cedex, Franceb ´ 11( )Institut de Fisica d’Altes Energies, UniÕersitat Autonoma de Barcelona, E-08193 Bellaterra Barcelona , Spain`

c Dipartimento di Fisica, INFN Sezione di Bari, I-70126 Bari, Italyd Institute of High-Energy Physics, Academia Sinica, Beijing, People’s Republic of China 12

e ( )European Laboratory for Particle Physics CERN , CH-1211 GeneÕa 23, Switzerlandf Laboratoire de Physique Corpusculaire, UniÕersite Blaise Pascal, IN2P3-CNRS, Clermont-Ferrand, F-63177 Aubiere, France´ `

g Niels Bohr Institute, DK-2100 Copenhagen, Denmark 13

h ( )Nuclear Research Center Demokritos NRCD , GR-15310 Attiki, Greecei Laboratoire de Physique Nucleaire et des Hautes Energies, Ecole Polytechnique, IN2P3-CNRS, F-91128 Palaiseau Cedex, France´

j Dipartimento di Fisica, UniÕersita di Firenze, INFN Sezione di Firenze, I-50125 Firenze, Italy`k Supercomputer Computations Research Institute, Florida State UniÕersity, Tallahassee, FL 32306-4052, USA 14 ,15

l ( )Laboratori Nazionali dell’INFN LNF-INFN , I-00044 Frascati, Italym Department of Physics and Astronomy, UniÕersity of Glasgow, Glasgow G12 8QQ, United Kingdom 16

n Institut fur Hochenergiephysik, UniÕersitat Heidelberg, D-69120 Heidelberg, Germany 17¨ ö Department of Physics, Imperial College, London SW7 2BZ, United Kingdom 16

p Institut fur Experimentalphysik, UniÕersitat Innsbruck, A-6020 Innsbruck, Austria 18¨ ¨q Department of Physics, UniÕersity of Lancaster, Lancaster LA1 4YB, United Kingdom 16

r Institut fur Physik, UniÕersitat Mainz, D-55099 Mainz, Germany 17¨ ¨s Centre de Physique des Particules, Faculte des Sciences de Luminy, IN2P3-CNRS, F-13288 Marseille, France´

t Dipartimento di Fisica, UniÕersita di Milano e INFN Sezione di Milano, I-20133 Milano, Italyù Max-Planck-Institut fur Physik, Werner-Heisenberg-Institut, D-80805 Munchen, Germany 17¨ ¨

v Laboratoire de l’Accelerateur Lineaire, UniÕersite de Paris-Sud, IN2P3-CNRS, F-91405 Orsay Cedex, France´ ´ ´ ´w Dipartimento di Fisica dell’UniÕersita, INFN Sezione di Pisa, e Scuola Normale Superiore, I-56010 Pisa, Italy`

x Department of Physics, Royal Holloway & Bedford New College, UniÕersity of London, Surrey TW20 OEX, United Kingdom 16

y Particle Physics Dept., Rutherford Appleton Laboratory, Chilton, Didcot, Oxon OX11 OQX, United Kingdom 16

z CEA, DAPNIArSerÕice de Physique des Particules, CE-Saclay, F-91191 Gif-sur-YÕette Cedex, France 19

aa Institute for Particle Physics, UniÕersity of California at Santa Cruz, Santa Cruz, CA 95064, USA 2 0

ab Department of Physics, UniÕersity of Sheffield, Sheffield S3 7RH, United Kingdom 16

ac Fachbereich Physik, UniÕersitat Siegen, D-57068 Siegen, Germany 17äd Dipartimento di Fisica, UniÕersita di Trieste e INFN Sezione di Trieste, I-34127 Trieste, Italyàe Experimental Elementary Particle Physics, UniÕersity of Washington, WA 98195 Seattle, USA

af Department of Physics, UniÕersity of Wisconsin, Madison, WI 53706, USA 21

Received 9 April 1998Editor: L. Montanet

Abstract

The production of final states involving one or more energetic photons from eqey collisions is studied in a sample ofy1 q y Ž .58.5 pb of data recorded at a centre-of-mass energy of 183 GeV by the ALEPH detector at LEP. The e e ™ nng g

q y Ž .and e e ™ gg g cross sections are measured. The data are in good agreement with predictions based on the StandardModel and are used to set upper limits on the cross sections for anomalous photon production in the context of twosupersymmetric models and for various extensions to QED. In particular, in the context of a super-light gravitino model a

q y ˜ ˜cross section upper limit of 0.38 pb is placed on the process e e ™ GGg , allowing a lower limit to be set on the mass ofthe gravitino. Limits are also set on the mass of the lightest neutralino in Gauge Mediated Supersymmetry Breaking models.In the case of equal ee)g and eeg couplings a 95% C.L. lower limit on M ) of 250 GeVrc2 is obtained. q 1998 Publishede

by Elsevier Science B.V. All rights reserved.


1. Introduction

In the framework of the Standard Model, eventsin which the only observable final state particles arephotons may be produced via two distinct reactions:

q y q yŽ . Ž .e e ™ nng g and e e ™ gg g .q y Ž .The reaction e e ™ nng g can proceed via

two processes which are theoretically well under-Ž q ystood: radiative returns to the Z resonance e e ™

.gZ with Z ™ nn , and t-channel W exchange withŽ .photon s radiated from the beam electrons or the W.

This reaction produces final states where one ormore photons are accompanied by significant miss-ing energy. These final states have been studiedextensively in eqey annihilations at lower centre-

w xof-mass energies 1,2 . Such final states are alsosensitive to new physics via the reactions eqey ™

XX and eqey ™ XY where Y is purely weaklyŽ .interacting and X decays radiatively to Y X ™ Yg .

In the Minimal Supersymmetric Standard ModelŽ .MSSM Y and X could be the lightest and next-to-

w xlightest neutralinos 3–5 , respectively. In GaugeŽ .Mediated Supersymmetry Breaking GMSB theories

w x6 Y and X could be the essentially massless grav-w xitino and the lightest neutralino 7,8 , respectively. In

w xthe super-light-gravitino scenario 9 the processq y ˜ ˜e e ™ GGg can have an appreciable cross sec-

tion.

The CDF collaboration has observed an unusualevent with two high energy electrons, two highenergy photons, and a large amount of missing trans-

w xverse energy 10 . The Standard Model explanationfor this event has a low probability, but it can beaccommodated by the SUSY models mentionedabove. The D0 collaboration has also searched for

w xthis process 11 and has found no significant excessof events. In the neutralino LSP scenario, the CDFevent could be explained by the Drell-Yan process

0 0 0 0 w xqq ™ ee ™ eex x ™ eex x gg 5 where the two˜˜ 2 2 1 1

x0’s escape detection, resulting in missing transverse1

energy. If this is the explanation for the CDF event,the best possibility for discovery at LEP2 is eqey

™ x0x

0 ™ x0x

0gg . In principle eqey ™ x0x

02 2 1 1 2 1

™ x0x

0g could be considered, however the pre-1 1

dicted cross section is uninterestingly small. In grav-itino LSP models, the CDF event could be explained

0 0 ˜ ˜ w xby qq ™ ee ™ eex x ™ eeGGgg 8 . In this sce-˜˜ 1 1

nario the best channel for discovery at LEP2 isq y 0 0 ˜ ˜e e ™ x x ™ GGgg . Limits derived from the1 1

ALEPH data are compared to the regions favouredby the CDF event within these models. In particular,in the case of GMSB theories, the data are comparedto the predictions of the Minimal Gauge-Mediated

w xMGM model of Ref. 8 which assumes that thelightest neutralino is pure bino, that the right-selectron mass is 1.1 times the neutralino mass and

1 Permanent address: Universitat de Barcelona, 08208 Barcelona,Spain.2 Also at Dipartimento di Fisica, INFN Sezione di Catania, Catania, Italy.3 Now at University of Geneva, 1211 Geneva 4, Switzerland.4 Now at Harvard University, Cambridge, MA 02138, USA.5 Supported by the Commission of the European Communities, contract ERBCHBICT941234.6 Also Istituto di Fisica Generale, Universita di Torino, Torino, Italy.`7 Also Istituto di Cosmo-Geofisica del C.N.R., Torino, Italy.8 Also at CERN, 1211 Geneva 23, Switzerland.9 Permanent address: Kangnung National University, Kangnung, Korea.

10 Ž .Now at University of California at Los Angeles UCLA , Los Angeles, CA 90024, USA.11 Supported by CICYT, Spain.12 Supported by the National Science Foundation of China.13 Supported by the Danish Natural Science Research Council.14 Supported by the US Department of Energy, contract DE-FG05-92ER40742.15 Supported by the US Department of Energy, contract DE-FC05-85ER250000.16 Supported by the UK Particle Physics and Astronomy Research Council.17 Supported by the Bundesministerium fur Bildung, Wissenschaft, Forschung und Technologie, Germany.¨18 Supported by Fonds zur Forderung der wissenschaftlichen Forschung, Austria.¨19 Supported by the Direction des Sciences de la Matiere, C.E.A.`20 Supported by the US Department of Energy, grant DE-FG03-92ER40689.21 Supported by the US Department of Energy, grant DE-FG0295-ER40896.


that the left-selectron mass is 2.5 times the neutralinomass.

q y Ž .The reaction e e ™ gg g proceeds via t-channel electron exchange and has been studied at

w xlower centre-of-mass energies 12 . Deviations fromthe expected QED differential cross section for theproduction of two photons could be evidence fornew physics due to, for example, eqeygg contactinteractions or excited electrons.

This letter is based on an analysis of 58.5 pby1 ofdata collected at a luminosity-weighted centre-of-mass energy of 182.7 GeV. Previously published re-

w x y1sults from ALEPH 1 based on 11.1 pb and10.6 pby1 of data taken at 161 GeV and 172 GeV,respectively, are taken into account when settingcross section limits on new physics processes.

2. The ALEPH detector and photon identification

The ALEPH detector and its performance arew xdescribed in detail elsewhere 13,14 . The analysis

presented here depends largely on the performanceŽ .of the electromagnetic calorimeter ECAL . The lu-

Ž .minosity calorimeters LCAL and SICAL , togetherŽ .with the hadron calorimeter HCAL , are used mainly

to veto events in which photons are accompanied byother energetic particles. The HCAL is instrumentedwith streamer tubes and, together with the muonchambers, is used to identify muons. The SICALprovides coverage between 34 and 63 mrad from thebeam axis while the LCAL provides coverage be-tween 45 and 160 mrad. Each LCAL endcap consistsof two halves which fit together around the beamaxis; the area where the two halves join is a region

Ž .of reduced sensitivity ‘‘the LCAL crack’’ . Thisvertical crack, which accounts for only 0.05% of thetotal solid angle coverage of the ALEPH detector,was instrumented with a veto counter for the 183 GeVrun. This counter consists of 2 radiation lengths oflead followed by scintillation counters. Energetic

Ž .electrons photons passing through the lead have aŽ .greater than 90% 70% chance of giving a veto

signal in the scintillation counters. The tracking sys-tem, composed of a silicon vertex detector, wire drift

Ž .chamber, and time projection chamber TPC , is

Ž .used to provide efficient )99.9% tracking of iso-< <lated charged particles in the angular range cosu -

0.96.The ECAL is a leadrwire-plane sampling

calorimeter consisting of 36 modules, twelve in thebarrel and twelve in each endcap, which provide

< <coverage in the angular range cosu -0.98. Inter-module cracks reduce this solid angle coverage by2% in the barrel and 6% in the endcaps. However,the ECAL and HCAL cracks are not aligned so thereis complete coverage in ALEPH down to 34 mrad.At normal incidence the ECAL comprises a totalthickness of 22 radiation lengths and is situated at185 cm from the interaction point. Anode wire sig-nals, sampled every 512 ns during their rise time,provide a measurement by the ECAL of the interac-tion time t of the particles relative to the beam0

crossing with a resolution better than 15 ns for show-ers with energy greater than 1 GeV. Cathode padsassociated with each layer of the wire chambers areconnected to form projective ‘‘towers’’, each sub-tending approximately 0.98=0.98. Each tower isread out in three segments in depth ‘‘storeys’’ offour, nine and nine radiation lengths. The high gran-ularity of the calorimeter provides excellent identifi-cation of photons and electrons. The energy calibra-tion of the ECAL is obtained from Bhabha andtwo-photon events. The energy resolution is mea-

' Ž .sured to be DErEs0.18r E q0.009 E in GeVw x14 .

Photon candidates are identified using an algo-w xrithm 14 which performs a topological search for

localised energy depositions within groups of neigh-bouring ECAL towers. In order to optimise theenergy reconstruction, photons that are not well-con-

Ž .tained in the ECAL near or in a crack have theirenergy measured from the sum of the localised en-ergy depositions and all energy deposits in the HCALwithin a cone of cosa)0.98. Photon candidatesmay also be identified in the tracking system if theyconvert in the material before the TPC, 6% of aradiation length at normal incidence, producing an

w xelectron-positron pair 14 .The trigger most relevant for photonic events is

the neutral energy trigger. This trigger is based onthe total energy measured on the wires of each of theECAL modules. For the 183 GeV run, this triggeraccepts events if the total wire energy is at least


1 GeV in any barrel module or at least 2.3 GeV inany endcap module. The efficiency of this trigger forthe selections presented below is estimated to be atleast 99.8%.

3. The Monte Carlo samples

q y Ž .The efficiency for the e e ™ nng g crosssection measurement and the background for theanomalous photon plus missing energy searches areestimated using the KORALZ Monte Carlo programw x w x15 . This generator uses the YFS 16 approach toexplicitly generate an arbitrary number of initial statephotons. It does not however include the small con-

Ž .tribution of order 0.2% where photons are directlyradiated from the W. This Monte Carlo is checked

w xby comparing to NUNUGG 17 at centre-of-massenergies below the W threshold and to CompHEPw x18 at higher energies.

The efficiency estimates for the reaction eqey

Ž .™ gg g are obtained using the GGG generatorw x 319 which contains contributions to order a withboth soft and hard photon emission. Events with fourhard photons observed in the detector are simulated

4 w xusing an order a generator 20 . The efficienciesfor the processes eqey ™ XX and eqey ™ XY

w xwith X ™ Yg are estimated using SUSYGEN 21assuming isotropic production and decay of X andtaking into account the effects of initial state radia-tion.

Background from Bhabha scattering, where initialor final state particles radiate a photon is studied

w xusing the UNIBAB 22 Monte Carlo program.

4. One photon and missing energy

4.1. EÕent selection

The selection of events with one photon andmissing energy follows that of the previous ALEPH

w xanalysis 1 and only a brief summary is given here.ŽEvents are selected with no charged tracks not

.coming from a conversion and exactly one photon< <inside the acceptance cuts of cosu -0.95 with

' Žp )0.0375 s where p is defined as the mea-H Hsured transverse momentum relative to the beam

.axis . Cosmic ray events that traverse the detectorare eliminated by the charged track requirement or ifthere are hits in the outer part of the HCAL. Residualcosmic ray events and events with detector noise inthe ECAL are removed by selection criteria based onthe ECAL information. The ‘‘impact parameter ofthe photon’’, calculated using the barycentre of thephoton shower in each of the three ECAL storeys, isrequired to be less than 25 cm. The compactness ofthe shower in the ECAL is calculated by taking anenergy-weighted average of the angle subtended atthe interaction point between the cluster barycentreand the barycentre of each of the ECAL storeyscontributing to the cluster. The compactness is re-quired to be less than 0.858. The interaction time ofthe event is required to be consistent with a beamcrossing.

To suppress background from Bhabha scattering,events are required to have no energy depositedwithin 148 of the beam axis and to have less than1 GeV of non-photonic energy. The selection is mod-ified to take advantage of the LCAL veto countersinstalled prior to the 183 GeV run. The requirementthat events with missing momentum around the

'LCAL crack region have a p )0.145 s is re-Hplaced by the requirement that there is no veto signaldetected in the LCAL veto counters.

q y ( )4.2. Measurement of the e e ™ nng g cross sec-tion

The efficiency of the above selection for theq y Ž .process e e ™ nng g is estimated from the

Monte Carlo to be 77%. This efficiency includes a2% loss, due to uncorrelated noise or beam-relatedbackground in the detector, estimated using eventstriggered at random beam crossings.

When this selection is applied to the data, 195one-photon events are found. The KORALZ MonteCarlo predicts that 187 events would be expectedfrom Standard Model processes. The cross section to

< <have at least one photon inside the acceptance cosu

'-0.95 and p )0.0375 s is measured to beH

q ys e e ™ nng g s4.32"0.31"0.13 pb.Ž .Ž .


Ž .Fig. 1. a The invariant mass distribution of the system recoilingŽagainst the photon candidate is shown for the data points with

. Ž . Ž . < <error bars and Monte Carlo histogram . b The cosu distribu-Ž .tion is shown for the data points with error bars and Monte

Ž . Ž .Carlo histogram . c The invariant mass of the system recoiling< <against the photon candidate versus cosu is shown for the data.

The missing mass and polar angle distributions ofthe selected data events are in good agreement withthe Monte Carlo expectations as shown in Fig. 1 22.

The estimate of the systematic uncertainty in theabove cross section includes contributions from thesources listed in Table 1. The simulation of theenergetic photon shower is checked with a sample ofBhabha events selected requiring two collinearbeam-momentum tracks and using muon chamber

22 Colour versions of the figures in this paper are available inencapsulated postscript form at http:rrrrrrrrrralephwww.cern.chrrrrrALPUBrrrrrpaperrrrrrpaper.html

Table 1Systematic uncertainties for the one-photon channel

Ž .Source Uncertainty %

Photon selection 0.6Converted photon selection 0.3Background -0.2

Integrated luminosity 0.5

Monte Carlo theoretical 3.0Monte Carlo statistical 0.4

Ž .Total in quadrature 3.1

information to veto mqmy events. The trackinginformation is masked from these events and thephoton reconstruction is redone. The efficiency toreconstruct a photon in the data is found to beconsistent at the 0.6% level with that predicted bythe simulation. The uncertainty in the number ofsimulated pair conversions is estimated to give a0.3% change in the overall efficiency. The 1% en-ergy calibration uncertainty is found to have a negli-gible effect. The level of cosmic ray and detectornoise background is measured by looking for eventsslightly out-of-time with respect to the beam cross-ing. No out-of-time events are observed in a timewindow five times larger than that used in the selec-tion. This leads to an estimate of less than 0.2 eventsexpected in the selected sample. The residual back-ground from Bhabha scattering is estimated fromMonte Carlo studies and is found to be negligible.From a comparison of different event generators thetheoretical uncertainty on the selection efficiency isestimated to be less than 3%. The total systematicuncertainty is obtained by adding in quadrature theindividual contributions.

4.3. Search for the process eqe y ™ XY ™ YYg

In order to search for the signal eqey ™ XY ™

YYg , a two-dimensional binned maximum likeli-hood fit is performed on the observed missing massversus cosu spectrum under the hypothesis that thereis a mixture of signal and background in the data.

w xDetails of the fitting procedure are given in Ref. 1 .w xData recorded at 161 GeV and 172 GeV 1 are in-


Fig. 2. The 95% C.L. upper limit on the production cross sectionin pb for the process eq ey ™ XY ™ YYg . The limits are valid

'for s s183 GeV assuming a br s threshold dependence,Ž .isotropic decays, short X lifetime t -0.1 ns and 100% branch-X

ing ratio for X ™ Yg .

cluded in the fit with a brs cross section depen-dence. The fit is performed for all possible X,Y masscombinations in steps of 1 GeVrc2 and the resultingupper limits on the cross section at 95% C.L. areshown in Fig. 2.

q y ˜ ˜4.4. Search for the process e e ™ GGg

˜If the gravitino G is very light the cross sectionq y ˜ ˜for the process e e ™ GGg can become apprecia-

ble. In order to search for this process a binnedmaximum likelihood fit is performed as above. Inthis case the missing mass and cosu distributions ofthe signal together with the cross section dependenceon the centre-of-mass energy are calculated from the

w xdifferential cross section given in Ref. 9 . From the'fit a cross section limit of 0.38 pb at s s183 GeV

is obtained at 95% C.L. This results in a 95% C.L.lower limit of 8.3=10y6 eVrc2 for the mass of the

w xgravitino 9 . In the same paper a more generalapproach gives a mass limit dependent on two freeparameters. In the worst case this would lead to alimit on the gravitino mass lower by factor two. Thesystematic uncertainty of 3.1% is taken into account

w xby means of the method of Ref. 23 and is found tohave a negligible effect on the above mass limit.

5. Two photons and missing energy

5.1. EÕent preselection

As described in the introduction, there are twoSUSY scenarios which can give acoplanar photons:the gravitino LSP and neutralino LSP scenarios. Thesignals differ in that the invisible particle is essen-tially massless in the first scenario and can havesubstantial mass in the second one. This leads to twoslightly different search criteria, as described in thesubsections below. However the similarity betweenthe two scenarios allows a common preselection ofevents with two photons and missing energy. Events

Žare selected with no charged tracks not coming from.a conversion and at least two photons, with energy

< <above 1 GeV, inside the acceptance of cosu -0.95.Since at least two photons are required, backgroundfrom cosmic rays and detector noise is less severe, sothe impact parameter and compactness requirementsare not imposed. Events with more than two photons

'are required to have at least 0.4 s of missing en-q y Ž .ergy. Background from the process e e ™ gg g

is effectively eliminated by requiring that the acopla-narity of the two most energetic photons be less than1778 and that there be less than 1 GeV of additionalvisible energy in the event. The total p is requiredHto be greater than 3.75% of the missing energy,reducing background from radiative events with finalstate particles escaping down the beam axis to anegligible level.

When this preselection is applied to the 183 GeVdata, 9 events are selected while 10.8 are predicted

q y Ž .from the process e e ™ nng g . This predictionis only known with an accuracy of around "10%.The missing mass and the energy of the second mostenergetic photon of these selected data events, and 3

w xevents selected at lower centre-of-mass energies 1 ,are shown together with Monte Carlo expectations inFig. 3.

5.2. Search for the process eqe y ™ XX ™ YYgg :Y massless

For this topology one additional cut is placed onthe energy of the less energetic photon E to sub-2

stantially reduce the remaining Standard Modelbackground. The energy distribution of the second


Ž .Fig. 3. a The invariant mass distribution of the system recoilingŽagainst the photon candidates is shown for the data points with

. Ž . Ž .error bars and Monte Carlo histogram . b The distribution ofthe energy of the second most energetic photon is shown for the

Ž . Ž .data points with error bars and Monte Carlo histogram . Bothplots contain data taken from centre-of-mass energies in the range161GeV to 183GeV.

most energetic photon is peaked near zero for thebackground, whereas for the signal both photonshave a flat distribution in an interval depending onthe neutralino mass and the centre-of-mass energy.

ŽThis cut is placed at E )25 GeV this is the opti-2w x .mised value in the MGM 8 model . After this final

cut is applied one event is found in the 183 GeV datawhile 1.43 events are expected from backgroundprocesses. Applying this increased E cut to the2

previously analysed data taken at 161 GeV to172 GeV no events are observed in the data while0.35 are expected from background processes. Theupper limit on the production cross section at183 GeV, obtained without performing backgroundsubtraction, is in the range of 0.10–0.12 pb for a100% X ™ Yg branching ratio and X masses in therange 45 GeVrc2 to 90 GeVrc2. The data recordedat lower energies are also used in the evaluation ofthis limit. The integrated luminosities are scaledaccording to the cross section predictions of the

w xMGM 8 model. The mass limit obtained for thismodel is

M 0 G84 GeVrc2x1

at 95% C.L. for a neutralino with lifetime -3 ns.The systematic uncertainty for this analysis is esti-mated to be 2%, dominated by the photon recon-struction efficiency. The effect of this uncertainty onthe cross section upper limit is less then 1% whentaken into account by means of the method of Ref.w x23 . The effect on the mass limit is negligible.

In the GMSB model the neutralino can have anon-negligible lifetime which depends directly on

'the SUSY breaking scale F . The lifetime of thew xneutralino is given by 7

5 42 '100 GeVrc Fct,130 mm.ž /ž /0M 100 TeVx1

The efficiency due to lifetime e to reconstruct at

photon resulting from a neutralino decay of a givenlifetime is found to be well parameterised by e s1t

Ž .yexp ylrgbct , where the average distance l forreconstruction is 2.5 m. The 95% C.L. exclusion

' 0limit obtained in the F , M plane using thisx1

parameterisation is shown in Fig. 4. For a neutralino

w xFig. 4. The excluded region of the MGM model 8 in the'neutralino mass, F plane.


of mass 84 GeVrc2 and lifetime 3 ns, the SUSYbreaking scale is at least 730 TeV at 95% C.L.

At LEP2 the production of bino neutralinos wouldproceed via t-channel selectron exchange. Right-selectron exchange dominates over left-selectron ex-change. Thus, the cross section for eqey ™ x

0x

01 1

depends strongly on the right-selectron mass. Thetheoretical cross section for eqey ™ x

0x

0 is calcu-1 1

lated at each M , M 0 mass point for right-selectrone x˜R 1

masses ranging from 70 GeVrc2 to 200 GeVrc2 andneutralino masses ranging from 30 GeVrc2 to86 GeVrc2 and compared to the experimental limitto obtain the exclusion region. The neutralino masslimits are also evaluated for various left-selectronmasses. The result is found to be robust at the"1 GeVrc2 level for left-selectron masses rangingfrom M s M to M 4M .e e e e˜ ˜ ˜ ˜L R L R

The experimentally excluded region in the neu-tralino, selectron mass plane is shown in Fig. 5.Overlayed is the ‘‘CDF region’’, the area in theneutralino, selectron mass plane where the propertiesof the CDF event are compatible with the process

0 0 ˜ ˜ Žqq ™ e e ™ eex x ™ eeGGgg taken from˜ ˜R R 1 1

Fig. 5. The excluded region in the neutralino, selectron mass planeŽ .at 95% C.L. for a pure bino neutralino light shaded area .

Overlayed is the CDF region determined from the properties of0 0the CDF event assuming the reaction qq ™ e e ™ eex x ™˜ ˜R R 1 1

˜ ˜ Ž w x.eeGGgg taken from the Ref. 24 . The dark shaded regioncorresponds to a topology not covered by this analysis.

w x.Ref. 24 . Most of the CDF region is excluded at95% C.L. by this analysis.

5.3. Search for the process eqe y ™ XX ™ YYgg :Y massiÕe

For massive Y a simple energy cut is not optimalsince the photons from the X ™ Yg decay can have

q y Ž .low energy. Here the fact that the e e ™ nng g

background peaks at small polar angles and has amissing mass near the Z mass is utilised. Events thathave missing mass between 82 GeVrc2 and100 GeVrc2 and the energy of the second mostenergetic photon less than 10 GeV are rejected. Thecosu cut is set using the N procedure, leading to a95

< <requirement of cosu -0.8. When this selection isapplied to the 183 GeV data 3 events are selectedwhile 2.8 events are expected from the eqey ™

Ž .nng g process. The upper limits obtained on thecross section as a function of the masses of X and Yare shown in Fig. 6. These upper limits are derivedwithout performing background subtraction but theobserved candidates are taken into account onlywhere they are kinematically consistent with a givenX,Y mass pairing. They are derived taking into

w xaccount lower energy data 1 with a brs thresholddependence and assuming a branching ratio for X™ Yg of 100%. The systematic uncertainties forthis analysis are the same as for the massless Y

Fig. 6. The 95% C.L. upper limit on the production cross sectionq y Žin pb for the process e e ™ XX ™ YYgg multiplied by BB X

'.™ Yg squared. The limit is valid for s s183GeV assumingbr s threshold behaviour and isotropic decays.


Fig. 7. The excluded region in the neutralino, selectron mass planeat 95% C.L. For this plot it is assumed that the x

0 is pure photino20 Ž 0 0 .and that the x is pure higgsino which implies BB x ™ x g s1 2 1

1. The lightly shaded region is for M s M . The darker shadede e˜ ˜L R

region refers to M 4 M . The mass limit is independent of thee e˜ ˜L R

x0 mass as long as the x

0 yx0 mass differences is greater than1 2 1

25GeVrc2. Overlayed is the CDF region labelled by the mass ofx

0 in GeVrc2. This is the area determined from the properties of10 0the CDF event assuming the reaction qq ™ ee ™ eex x ™˜˜ 2 2

0 0 Ž w x.eex x gg taken from Ref. 5 .1 1

scenario and the effect on the upper limits is againless than 1%.

The x0 LSP interpretation of the CDF event1

Žalong with the non-observation of other SUSY sig-.natures at Fermilab suggests a high branching ratio

for x0 ™ x

0g . A 100% branching ratio is achieved2 1

when the x0 is pure photino and the x

0 is pure2 1

higgsino. In this scenario, the lower mass limit of x02

as a function of the selectron mass is calculated andcompared to the region compatible with the CDFevent. In Fig. 7 two scenarios M s M ande e˜ ˜L R

M 4M are shown. With the assumption that thee e˜ ˜L R

x0 is pure photino and the x

0 is pure higgsino, these2 1

results exclude a significant portion of the region

compatible with the kinematics of the CDF eventgiven by the neutralino LSP interpretation.

6. Hard collinear photons

q y ( )6.1. EÕent selection and the process e e ™ gg g

An acceptance for events from the process eqey

Ž .™ gg g is defined to include events with at least< <two photons with polar angles such that cosu -0.95

'and energies above 0.25 s where the angle betweenthe two most energetic photons is at least 1608. Thebackground from Bhabha scattering is greatly re-duced by allowing at most one converted photon perevent and requiring that there be no tracks in theevent not associated with that photon. Cosmic rayevents which traverse the detector are eliminated ifthey leave hits in the outer part of the HCAL or iftheir measured interaction time is inconsistent with abeam crossing. The efficiency of this selection forevents within the acceptance is 84%.

The above selection is applied to the three datasamples collected at centre-of-mass energies of161 GeV, 172 GeV and 183 GeV. The number ofevents observed and expected at each of the energiesis given in Table 2. Summed over the three centre-of-mass energies a total of 713 events are selected ingood agreement with the total Monte Carlo predic-tion of 729 events, six of which are expected to

w xcome from the residual Bhabha background 22 .Also given in Table 2 are the number of observedand expected events that have one or more additionalphotons with energy above 1 GeV inside the angular

< <range cosu -0.95. A total of 33 such events areobserved, consistent with the expectation of 38events. Two events are observed in the data with

w xfour photons in agreement with a Monte Carlo 20

Table 2Ž .The selected number of observed and expected events which have two or more three or more photons inside the acceptance, the number of

expected background events and the measured and theoretical cross sections at the three different centre-of-mass energies

' Ž . Ž .s Observed Expected Exp. Bkg. Cross section pb Theor. cross section pb

Ž . Ž .161 114 7 124 6.4 1 12.0"1.1"0.2 13.20"0.14"0.13Ž . Ž .172 99 1 103 5.3 1 11.0"1.1"0.2 11.59"0.13"0.12Ž . Ž .183 500 25 496 26.3 4 10.1"0.5"0.2 10.11"0.11"0.10


expectation of 1.4 events. No events are observedwith more than four photons.

The lowest order differential cross section forelectron-positron annihilation into two photons isgiven by

ds a 2 1qcos2us .2ž / ž /dV s 1ycos uBorn

The observed cross section is modified by two ef-fects: higher order processes, in particular initialstate radiation, and detector effects. Due to initialstate radiation, the centre-of-mass frame of the twodetected photons is not necessarily at rest in thelaboratory. The events are therefore transformed intothe two-photon rest frame to define the productionangle u ) appropriately. The distribution of this pro-duction angle is in good agreement between data and

Ž 2Monte Carlo expectations x s17 for 19 degrees.of freedom as shown in Fig. 8. The background-sub-

tracted cross section for events inside the acceptanceis given in Table 2.

The systematic uncertainty in the above crosssection estimates includes contributions from the var-ious sources listed in Table 3. The uncertainties

Fig. 8. Predicted and observed lowest-order differential crosssection as a function of cosu ) for the reaction eq ey ™ gg . Thepredicted distribution includes a small contribution from theBhabha background. The errors shown here are purely statistical.

Table 3Systematic uncertainties in the eq ey ™ gg analysis

Ž .Source Uncertainty %

Photon selection 1.2Converted photon selection 0.6

Background 0.8

Integrated luminosity 0.5

Monte Carlo statistical 1.1Monte Carlo theoretical -1.0

Ž .Total in quadrature 2.2

coming from the photon selection efficiency aremeasured as in the single photon analysis. The uncer-tainty in the level of the Bhabha background isconservatively estimated to be equal to 100% of themeasured background of 0.8%. The effect of missinghigher orders in the Monte Carlo is estimated to beless than 1.0%. This estimate is obtained by compar-ing the number of observed and selected events in ahigh statistics data sample recorded at the Z peak.Added in quadrature, the total systematic uncertaintyis 2.2%. It is treated as an uncertainty in the overallnormalisation of the data.

6.2. QED cutoff parameters

Possible deviations from QED are usually charac-terised by cutoff parameters L and L corre-q ysponding to a modified differential cross section

2ds ds s2 )s 1" 1ycos u .Ž .4ž /dV dV 2 LQED "

In order to extract limits on the parameters L andqL a binned maximum likelihood fit is performedyon the background-subtracted cosu ) distribution un-der the assumption that it contains contributionsfrom both QED and the cutoff interaction. Since thecosu ) distribution of the cutoff interaction is onlyknown to lowest order, a bin-by-bin correction ismade by comparing the third order QED distributionto the corresponding lowest order distribution. Thisassumes that the effect of higher order corrections isthe same for both QED and the new physics. Afurther bin-by-bin correction is made to take into


Fig. 9. The ratio of the observed to predicted cross sections, forq y Ž . )the process e e ™ gg g , as a function of cosu . Also shown

are the 95% C.L. level limits on the QED cutoff model.

account the detector efficiency. The limit on LqŽ .L is obtained by integrating the likelihood distri-ybution over the physically allowed region L )0qŽ .L )0 . The 95% C.L. lower limits obtained foryL and L are 270 GeV and 230 GeV, respectively.q yThe systematic uncertainties are taken into account

w xusing the method of Ref. 23 and are found to havea negligible effect on the limits. Fig. 9 shows theratio of the observed cross section to that predictedby QED, as a function of cosu ). Also indicated, asdotted lines, are the modified cross sections corre-sponding to the 95% C.L. lower limits on L andqL .y

6.3. Contact interactions

An alternative description of extensions to QED isprovided by effective Lagrangians, which containnon-standard couplings of the form g eqey andgg eqey. The lowest order effective Lagrangians,describing these interactions, contain operators of

order 6, 7 and 8. These lead to modified differentialw xcross sections of the form 25

2ds ds s2 )s 1q 1ycos u ,Ž .4ž / ž /dV dV aLQEDq6 QED 6

ds ds s2

s q ,6ž / ž /dV dV 32pLQEDq7 QED 7

ds ds s2 m2e

s q .8ž / ž /dV dV 32pLQEDq8 QED 8

Fits are performed to extract limits on these parame-ters using the procedure outlined above. The 95%C.L. lower limits obtained for L , L and L are6 7 8

1100 GeV, 624 GeV and 18.8 GeV, respectively. Thesystematic uncertainties are again found to have anegligible effect on the limits.

6.4. Limits on M )e

The reaction eqey ™ gg can also proceed viathe exchange of an excited electron. In this case thecross section depends on two parameters: the massM ) of the excited electron and the ee)g coupling.e

w xThe simplest gauge-invariant form 26 of the inter-Ž .action the Low Lagrangian leads to the differential

w xcross section given in 27 . A fit is performed asabove and a 95% C.L. lower limit on M ) ofe

250 GeVrc2 is obtained in the case of equal ee)g

and eeg couplings.

7. Conclusions

Single- and multi-photon production is studied inthe ALEPH data collected at centre-of-mass energiesup to 183 GeV. The cross sections for the processes

q y q yŽ . Ž .e e ™ nng g and e e ™ gg g are measuredand are found to be compatible with the expectationsof the Standard Model.

Ž .The data from the photon s and missing energyanalyses are used to derive cross section upper limitsfor the processes eqey ™ XY ™ YYg , eqey ™˜ ˜ q yGGg and e e ™ XX ™ YYgg . A cross sectionupper limit of 0.38 pb is obtained of the eqey ™˜ ˜GGg process. From this cross section upper limits a95% C.L. lower limit of 8.3=10y6 eVrc2 at 95%


w xC.L. is set on the mass of the gravitino 9 . In thew xcontext of the MGM model 8 a 95% C.L. lower

limit on the x0 mass is found to be 84 GeVrc21

Ž . 0 Ž 0 .0t -3 ns . The lower limit on the x x mass asx 1 21

a function of selectron mass is determined and com-pared to the region compatible with the CDF event

Ž .for the gravitino neutralino LSP scenario.The data from the hard collinear photon analysis

are used to place limits on the parameters of anumber of extensions to the Standard model, notablythe presence of eqeygg contact interactions and theexchange of a massive excited electron in the t-chan-nel. The 95% C.L. lower limits on the QED cutoffparameters L and L are found to be 270 GeVq yand 230 GeV, respectively. The effect of excitedelectron exchange depends on both the mass andcoupling constant. In the simplest case, an assump-tion that the ee)g coupling is equal to the eeg

coupling yields a 95% C.L. lower limit on M ) ofe

250 GeVrc2.

Acknowledgements

We wish to congratulate our colleagues in theCERN accelerator divisions for their very successfuloperation of the LEP storage ring. We are grateful tothe engineers and technicians in all our institutionsfor their contribution towards the excellent perfor-mance of ALEPH. Those of us from non-membercountries thank CERN for its hospitality.

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Ph.D. thesis, Harvard Univ., 1970.

18 June 1998


Isovector pairing in odd-A proton-rich nuclei

J. Engel a, K. Langanke b, P. Vogel c

a Department of Physics and Astronomy, CB3255, UniÕersity of North Carolina, Chapel Hill, NC 27599, USAb Institute for Physics and Astronomy and Center for Theoretical Astrophysics, UniÕersity of Aarhus, DK-8000 Aarhus C, Denmark

c Physics Department, California Institute of Technology, Pasadena, CA 91125, USA

Received 11 March 1998Editor: W. Haxton

Abstract

Ž .A simple model based on the group SO 5 suggests that both the like-particle and neutron-proton components ofisovector pairing correlations in odd-A nuclei are Pauli blocked. The same effect emerges from Monte Carlo Shell-modelcalculations of proton rich nuclei in the full fp shell. There are small differences between the two models in theirrepresentation of the effects of an odd nucleon on the competition between like-particle and neutron-proton pairing, but they

Ž .can be understood and reduced by using a two-level version of the SO 5 model. On the other hand, in odd-odd nuclei withŽ .N/Z, SO 5 disagrees more severely with the shell model because it incorrectly predicts ground-state isospins. The shell

model calculations for any fp-shell nuclei can be extended to finite temperature, where they show a decrease in blocking.q 1998 Elsevier Science B.V. All rights reserved.

PACS: 21.10-k; 21.60.Fw; 21.60.Ka

Proton-rich nuclei play an important role in explosive nucleosynthesis and are increasingly accessible toexperiment. For this reason the subject of neutron-proton pairing has experienced a revival. Even-A nuclei havereceived most of the attention so far, but odd-A nuclei, in which pairing is affected by the odd nucleon, are

w x Ž .equally important. In a previous paper 1 we analyzed the competition between neutron-proton np andŽ .like-particle pp and nn pairing in even-even fp-shell nuclei and in odd-odd NsZ nuclei, arguing that a

Ž .simple model based on the group SO 5 captured the essentials of full shell-model calculations, despitedeformation, spin-orbit splitting, and other physics that the simple model omits. Here we turn to odd-A nuclei,discussing the same competition when an extra nucleon is present and pairing correlations are blocked. We also

w xtouch on odd-odd nuclei, of which only those with NsZ were treated in Ref. 1 . Though our focus is onground states, we briefly discuss the pairing competition at finite temperature as well.


( )J. Engel et al.rPhysics Letters B 429 1998 215–221216

w xWe begin with the predictions of the simple model, a full description of which is in Ref. 1 . Briefly, theisovector angular-momentum-zero nn, pp, and np pair creation operators, the corresponding annihilation

Ž .operators, the three isospin generators, and the number operator form the algebra SO 5 . The Hamiltonianconsists of three equally weighted pairing terms and has the ground-state expectation value

G ny6 ny6Esy n Vy yn Vy q t tq1 yT Tq1 , 1Ž . Ž . Ž .ž / ž /2 4 4

where G is a the pairing-force strength constant, n is the number of nucleons, V is half the number ofŽ .degenerate single-particle levels, T is the isospin, and n and t are the the seniority and ‘‘reduced isospin’’,which take the values ns0, ts0 in even-A ground states and ns1, ts1r2 in odd-A ground states. When

Ž .ns0 Eq. 1 becomes

G ny6Esy n Vy yT Tq1 . 2Ž . Ž .ž /2 4

When ns1 we have

G ny6 1Esy n Vy y Vq yT Tq1 . 3Ž . Ž .ž / ž /2 4 2

These equations imply a blocking in odd-A nuclei very similar to what would be present without np pairing;in fact in the ordinary like-particle seniority model the pairing energies along any even-Z isotope chain, in

Ž . Ž .which there are no odd-odd nuclei, differ only by a small constant from Eqs. 2 and 3 . This prediction isconfirmed by shell-model Monte Carlo calculations, which we describe shortly. Since the np pairing replacessome of the like-particle pairing when an isospin-symmetric interaction is used, it would seem that both pairingmodes are blocked in odd-A nuclei, though pp pairing is apparently almost unaffected by an odd neutron.

This can be seen explicitly by examining the competition among nn, pp, and np pairs, something wew x Ždiscussed in even-A nuclei in Ref. 1 . Defining pair ‘‘number operators’’ as in that paper, for a brief but

w x. w xexhaustive discussion of these operators see Ref. 2 , we use techniques described in Ref. 3 to obtain forodd-A ground states:

NNyT y3Ž .cNNyT 1yŽ .c ž /2V² :NN s T q1Ž .p p c 2T q3c

NNyT y2Ž .cNNyT 1yŽ .c ž /2V² :NN sn p 2T q3c

NNqT q1 NNŽ .cNNqT 1y q T qŽ .c cž /ž /2V V² :NN s T q1 4Ž . Ž .nn c 2T q3c

( )J. Engel et al.rPhysics Letters B 429 1998 215–221 217

Ž . Ž . Ž .if NN' ny1 r2 the total number of pairs and T 'Ty1r2 the core isospin are both even or both odd,c

and

NNyT y1Ž .cNNyT 1yŽ .c ž /2V² :NN sTp p c 2T q1c

NNyT y1Ž .cNNyT q1 1yŽ .c ž /2V² :NN sn p 2T q1c

NNqT y3 NNqTŽ . Ž .c cNNqT 1y q T y1 1yŽ . Ž .c cž / ž /2V V² :NN sT 5Ž .nn c 2T q1c

Žif NN and T 'Tq1r2 are both even or both odd the relation between T and T is different in the two sets ofc c. w xexpressions . Combining these results with those for even-A nuclei from Ref. 1 , we plot on the left-hand side

Ž .of Fig. 1 the numbers of each kind of pair scaled by 0.5; we discuss this factor shortly as neutron number

² : Ž . ² : Ž . ² : Ž .Fig. 1. The quantities V NN solid line , V NN dashed line , and V NN dotted line for the Cr isotopes. On the left are thep p n n n pŽ . Ž .SO 5 results described in the text with Vs10, half the number of single-particle levels in the fp shell scaled by a factor 0.5. On the right

are the results of the Shell Model Monte Carlo calculation with G s20rA MeV and the quadrupole coupling constant xs134 Ay1 1r3pair

MeVrfm4.


² :increases in the Cr isotopes. As the figure shows, NN oscillates sharply while the other pair-numbers staggernnŽ .less. The reason is that when N is even as well as Z, nn and pp pairing particularly the former for N)Z are

enhanced at the expense of np pairing. Adding a neutron to make N odd blocks the nn pairing, thereby² : ² :reducing NN , but also blocks np pairing to a degree so that NN cannot take advantage of the reducednn n p

² :coherence in the nn condensate. A little surprisingly, perhaps, NN is not able to take any advantage of thep p² :slight drop in np pairing; it doesn’t increase until N becomes even again and NN is reduced moren p

significantly by the increased strength of the nn condensate.w xThis picture augments that described in Ref. 1 and it is natural to ask how much it has to do with reality.

Since the strengths of the individual pairing modes and even the total isovector pairing strength are hard toŽ .extract from the limited data available, we again turn to a large-scale shell model Monte Carlo SMMC

calculation in the full fp shell to see if the physics plays out in the same way. Unfortunately in odd-A andŽ . w xodd-odd excepting NsZ nuclei the notorious sign-problem 4 inherent in SMMC studies with realistic

Ž . w xinteractions cannot be circumvented at low temperatures TF0.8 MeV by ‘‘g-extrapolation’’ 5 . We in largepart avoid the problem, however, by using ‘‘pairing plus multipole-multipole’’ interaction of the kind used in

w xRef. 6 . In even-even nuclei this interaction has been shown to do a good job with essential features of thespectrum, including isovector pairing correlations, which have been checked against the predictions of therealistic KB3 interaction. We fix the temperature at Ts0.4 MeV, which should be sufficient to cool a nucleus

Ž 49to near its ground state. Although a mild sign problem still affects the nuclei considered here the sign for Cr

² : ² :Fig. 2. Comparison of the two-level model and shell model. In the upper panel are the quantities V NN and V NN and in the lowerp p n p² :one V NN . The two level model is described in the text; the results are connected by full lines. The SMMC results, with the small errorn n

bars as indicated, are connected by the dashed lines.


.is 0.35"0.01 at Ts0.4 MeV SMMC calculations can be performed without g-extrapolation. The onlyconsequence of the residual sign problem is that statistical uncertainties for odd-A nuclei are slightly worse thanfor even-even nuclei, in which there is no sign-problem at all.

Ž .The right-hand sides of Fig. 1 contains the SMMC results for the same quantities discussed in SO 5 , and is² :clearly similar to the left-hand side. The minor differences between the two panels, most apparent in NN fornn

large NyZ, are most likely due to presence of several nondegenerate levels in the fp shell. We conclude, asŽ .before, that the presence of physics beyond SO 5 reduces the strength of each the three pairing modes by about

a factor of two, but does not drastically alter the balance of power among them.Where exactly does the factor of two come from? The shell-model contains many physical effects not

Ž .included in SO 5 , but we focus here on the role of spin-orbit splitting, which we can mock up in a two-levelŽ . Ž w x .version of the SO 5 model. For the formalism, applicable to seniority-zero states, see Ref. 7,8 . Although the

matrices one must diagonalize are no longer tiny, they are still small. To mimic the separation of the f level7r2

from the rest of the fp shell, we take our two levels to have V s4 and V s6 and split them by es10G, a1 2

number close to the real ratio of spin-orbit splitting to pairing-force strength. In Fig. 2 we plot the numbers ofŽthe 3 kinds of pairs together with the SMMC predictions, now just for even-A Cr isotopes i.e. for every other

.point in Fig. 1 . The splitting weakens the coherence of the pairs so that the numbers now agree quite well with² :the shell-model calculations without the factor of 2 scaling, except for NN at large NyZ. The disagreementnn

there clearly reflects the residual splitting between the other fp levels; it doesn’t appear until there are moreneutrons than can be accommodated in the f level. All this leads us to attribute the differences in scale7r2

Ž .between simple SO 5 and full shell-model calculations to spin-orbit splitting, or more generally to thenondegeneracy of the the single particle states, which can also reflect deformation.

² : Ž . ² : Ž . ² : Ž . Ž . Ž .Fig. 3. The quantities V NN solid line , V NN dashed line , and V NN dotted line in the odd-A top and odd-odd bottomp p n n n pŽ .isotopes of Mn. The left side contains the SO 5 results, the right the SMMC results.


² : Ž . ² : Ž . ² : Ž . 49Fig. 4. The quantities V NN circles , V NN squares , and V NN diamonds in the odd-A nucleus Cr as a function ofp p n n n p

temperature calculated in the SMMC.

We turn now to the Mn isotopes, which have Zs25 and are odd-odd as well as odd-even. Fig. 3 showsŽ .SO 5 and SMMC results for the pair numbers. The figure is split into 4 panels to separate the odd-A isotopes

Ž . Ž . Ž .from the even-A odd-odd isotopes. The agreement between scaled SO 5 and the SMMC is good in the upperpanels , but noticeably less so in lower panels. The problem is that the model predicts the wrong isospin for theground states in odd-odd isotopes with NyZ/0. The ground states in all these nuclei have TsT , while thez

model doesn’t even contain such states; they appear only when a pair is broken and, if you believe the model,should lie much higher in energy.

In the real world, however, the breaking of a pair is offset by the shell closure at Ns28 and by theŽ .quadrupole-quadrupole force, for the following reason: When NG28 AG53 here the lowest-lying states in

the pure single-particle model have TsT because the filled neutron shell prohibits the operator t fromz qgiving anything but zero. Higher T states correspond to particle-hole excitations, which have too muchsingle-particle excitation energy to be pulled all the way down when the pairing interaction is added. A similarargument in the Nilsson scheme implies that TsT in the deformed nucleus 52 Mn as well. Our version ofz

Ž .SO 5 includes only fully paired states even in the two-level model, and so overlooks the effects ofsingle-particle splitting and deformation. As a result, its description of odd-odd nuclei with N/Z is lacking. 1

1 Interestingly, the SMMC with our chosen strength of the pairing plus quadrupole interaction also fails to predict the correct ground-stateisospin in 52 Mn. The problem is clearly in the quadrupole-quadrupole force, which is a little too weak. Strengthening it by about 10% pullsthe T sT s1 state below the T s2 state, as it is in reality.z


ŽWe turn finally to the temperature dependence of pairing correlations in proton-rich nuclei related work onw x. 49nuclei with NsZ nuclei appears in Refs. 6,9 . As an example we have chosen Cr, an odd-A nucleus with

NsZq1. We again use the SMMC with the Hamiltonian and model space described above to calculatepair-numbers, but now for temperatures spanning the range Ts0.4y2 MeV. Fig. 4 shows the results. Theblocking effect is clearly visible in the pair correlations: at low temperatures the neutron-neutron correlations aresmaller than both the proton-neutron and proton-proton correlations, despite the presence of five neutrons toonly four protons. As the temperature increases both the pairing strengths and the blocking effect decrease, sothat for T)1.2 MeV the neutron-neutron pairing correlations are the largest and the neutron-proton theweakest, just as if there were no blocking. It appears that although both pairing correlations and Pauli blockingdecrease with temperature, the latter disappear first.

Ž .One can imagine using functional methods to study temperature-dependence in SO 5 as well, provided somehigher seniority states are included. We defer that exercise, however, and instead repeat in closing that although

Ž .SO 5 predicts the wrong isospin for odd-odd nuclei with N/Z, it makes the competition among the threekinds of isovector pairs in all other isotopes very easy to understand. Most of the effects that escape the modelare recovered by a simple modification: two sets of degenerate levels instead of one. These statements are trueeven when Pauli blocking, which affects all three modes and the competition among them, is at play.

We were supported in part by the U.S. Department of Energy under grants DE-FG05-94ER40827 andDE-FG03-88ER-40397, by the U.S. National Science Foundation under grants PHY94-12818 and PHY94-20470,and by the Danish Research Council.

References

w x Ž .1 J. Engel, K. Langanke, P. Vogel, Phys. Lett. B 389 1996 211.w x Ž .2 J. Dobes, Phys. Lett. B 413 1997 239.w x3 K.T. Hecht, The Vector Coherent State Method and Its Application to Problems of Higher Symmetries, Springer-Verlag, Berlin,

Heidelberg, 1987.w x Ž .4 S.E. Koonin, D.J. Dean, K. Langanke, Physics Reports 278 1997 1.w x Ž .5 Y. Alhassid, D.J. Dean, S.E. Koonin, G. Lang, W.E. Ormand, Phys. Rev. Lett. 72 1994 613.w x Ž .6 K. Langanke, P. Vogel, D.-C. Zheng, Nucl. Phys. A 626 1997 735.w x Ž .7 G.G. Dussel, E. Maqueda, R.P.J. Perazzo, J.A. Evans, Nucl. Phys. A 450 1987 164.w x Ž .8 O. Civitarese, M. Reboiro, P. Vogel, Phys. Rev. C 56 1997 1840.w x Ž .9 K. Langanke, D.J. Dean, P.B. Radha, S.E. Koonin, Nucl. Phys. A 613 1997 253.

18 June 1998


Manifestation of mixed symmetry components› X™3 ž /in the He e,e p pn reaction

S. Nagorny a, W. Turchinetz b

a NIKHEF, P.O. Box 41882, 1009 DB Amsterdam, The Netherlandsb MIT-Bates, P.O. Box 846, Middleton, MA 01949, USA

Received 24 March 1997; revised 19 January 1998Editor: J.-P. Blaizot

Abstract

› X™3 Ž .The A -asymmetries of the He e,e p pn reaction have been calculated using different models of the NN-interactionx, z

which provide analytic representation of the solutions of the Faddeev equations. Strong sensitivity to the mixed symmetrycomponents was discovered at P ;20y60 MeVrc. In the quasi-elastic region at P ;0 a large asymmetry is found to ber r

model-independent and arises from the FSI of the spectator nucleons. q 1998 Elsevier Science B.V. All rights reserved.

PACS: 25.30.Fj; 25.10.qs; 21.30.-x; 21.45.qvKeywords: Polarized electrons; Polarized 3He; Asymmetry; Mixed symmetry components; Faddeev equation; Final state interaction;Nucleon potential

Recently the possibility of examining the 3He› X™3Ž . Ž .wave function WF in He e,e p pn reaction was

w xshown 1 . However, the sensitivity of A -asymme-x , zŽ X.tries to the mixed symmetry component S and

their dependence upon the models of strong interac-tion was not investigated. Also, it is not evident why

w xsuch a large asymmetry was obtained in 1 at lowŽ . Ž . Ž w x.and zero recoil momenta P see Fig. 3 in 1 ,r

Ž .without final state interactions FSI . Since the pro-Ž .tons in the main full symmetric configuration S are

‘‘not polarized’’, the PWIA-asymmetry should beŽ .very close to zero at low P see below .r

In this letter we will show that the large asymme-try at low P arises due to FSI inside spectatorr

pn-pair. Moreover, this asymmetry at P ;0 isr

model-independent and may be used for the calibra-tion of the measurements. Also we will investigate

the sensitivity of asymmetries to the mixed symme-try components and different nuclear models, forwhich solutions of Faddeev equations with differentNN-potentials will be used.

The S-state part of the three-body wave functionŽ . w xWF may be represented 2 as:

C3He syC sj a qC

Xj

XX yCXXj

X . 1Ž .Ž . Syw aÕ e

Here C s is the fully symmetric space S-wave com-ponent, accounting for ;90% of WF 2. C

X, CXX are

the space SX-components with mixed symmetry,which indicate the deviation from the full symmetrystate due to spin-momentum correlations and accountfor ;2% of the WF 2. The spin-isospin pieces of theWF are the fully antisymmetric j a and the mixedsymmetry j

X, jXX configurations. The SX-components

.are intriguing objects: i their probability is strongly


( )S. Nagorny, W. TurchinetzrPhysics Letters B 429 1998 222–226 223

w x y2.1 .Xcorrelated with binding energy 3 as P ;E ; iiS B. 3they do not exist for the deuteron; iii for He they

w x 4Xhave 2 P ;1y2%, while for He we can expectS

Ž .Xtheir strong suppression P -0.1% due to theS

higher binding energies.In addition to S- and SX-components, the 3He WF

contains P- and D-waves. The P-state probabilitiesw x Ž .are extremely small 2 ;0.1% and we will not

discuss them here. Various D-wave components withw xa total probability estimated 2 at P ;8% arise dueD

to the tensor part of the NN-forces and becomeŽ w x.important only at high P see, for instance, 1,4,5 .r

Ž X .For the e,e p channel in quasi-elastic kinemat-Ž 2 2 .1r2 Žics: n;e q m qq ym m is nucleon mass,3

Ž .qs n ,q is 4-momentum of the photon and e is3.the binding energy proton-pole diagrams with either

singlet or triplet spectator pn-pairs will dominate atlow P . Their amplitudes are determined by twor

vertices G of the 3He break up with pn-pairs int, sŽ3 � 4 . Ž3the triplet He™pq pn and singlet He™pqt

� 4 . w xpn spin states 4 :s

XX XXs s' 'G s C yC r 2 ; G sy C qC r 2 .Ž . Ž .t s

2Ž .Thus, it is only owing to the SX-component that the

Ž .vertices 2 are not the same.For the main S-configuration the amplitude for

production of a singlet pn-spectator pair correspondsto absorption of the photon by a proton whose spin isoriented opposite to the nuclear spin direction, whilefor the amplitude with a triplet spectator pn-pairanother proton with its spin along the nuclear onewill absorb the virtual photons. In PWIA the squares

Ž 3 .of these amplitudes for polarized He will have thesame magnitude, but opposite signs. As a result,

› X™3 Ž .PWIA asymmetries of the He e,e p pn reactioncalculated on the basis of only full symmetric con-figuration will be equal to zero. This reflects the factthat the protons in the S-configuration of 3He WFare ‘‘not polarized’’. So, at low P the asymmetriesr

in PWIA may arise only due to SX-component andwe can expect their strong sensitivity to the mixedsymmetry configuration. However, the magnitudesof A should be very small, since the admixture ofx , z

SX is not more than 2%.3 � 4Using the explicit form of the He™pq pn s, t

w x 3vertices 4 with polarized He, nuclear electromag-Ž .netic EM currents of the proton-pole diagrams with

Ž Ž0.. Ž Ž1..singlet J and triplet J spectator pn-pairsm m

Ž .may be represented without re-scattering :

X XŽ1.J sG U p F p qm g qp r2m g U TŽ . Ž . Ž . Ž .ˆm t m a a 5 S

T= U p g yk r2m CU n , 3Ž . Ž . Ž . Ž .1 a a

XŽ0.J sG U p F p qm U TŽ . Ž . Ž .ˆm s m S

T= U p g CU n . 4Ž . Ž . Ž .1 5

XŽ .Here p p is the momentum of the observed protonŽ .before after photoabsorption; T and S are the target

momentum and spin vectors while p and n are the1

momenta of the un-observed proton and neutron:qqTspqp qn. The center of mass and relative1

momenta of the spectator pn-pair have the form:Ž . ŽP sp qn, ks p yn r2 in the lab. frame P sr 1 1 r

X. Xyp . Furthermore, F is the g p p-vertex, g arem m ,5Ž .4=4 Dirac matrices, psg p ; U p is a bi-spinorˆ m m

and C is the matrix of charge conjugation.The EM tensor of ultra-relativistic polarized elec-

trons has the form:

l s l Ž0.qll ŽS . ;mn mn mn

l ŽS .s2 i´ q k ,mn mngd g 1d

l Ž0.s2 k k qk k qq2 gŽ .mn 1m 2n 2 m 1n mn

Ž .where k is the momentum of initial final elec-1Ž2.Ž .tron, qsk yk , and ls1 y1 corresponds to the1 2

Ž .initial electron polarization along opposite to its3-momentum.

Ž . Ž . Ž . Ž .Taking the squares of 3 , 4 , using SU T U TS Sˆ ˆŽ .Ž . Žs TqM 1yg S and considering low P inT 5 r

quasi-elastic kinematics the relative momenta k will32 2 .be small too: k ; P yP q , and neglecting ther r4

Ž .3 Ž .3terms of order P rm , krm , we get a factorizedr

equation for the asymmetry of the exclusive› X™3 Ž .He e,e p pn reaction:

™™e p 2Ask P =A S,q ,n . 5Ž . Ž .Ž .p r

™™e p 2Ž .Here A S,q ,n has the meaning of the ‘‘quasi-freeproton asymmetry’’:

™™e p 2A S,q ,nŽ .XŽS . ˆl Sp p qm Sg F pqm FŽ . Ž .ˆ ˆ½ 5mn 5 n m

s , 6Ž .XŽ0.l Sp p qm F pqm FŽ . Ž .ˆ ˆ½ 5mn n m

( )S. Nagorny, W. TurchinetzrPhysics Letters B 429 1998 222–226224

Ž .while k P may be called the ‘‘effective protonp r ›3polarization’’ in He:

k P s G2 yG2 r G2 q3G2 . 7Ž . Ž .� 4 � 4p r s t s t

Ž . Ž .Substituting 2 into 7 , we see the quasi-elastic› X™3 Ž .PWIA asymmetry of the He e,e p pn reaction are

proportional to the admixture of the mixed symmetryconfiguration:

k P ;CXXrC s

<1. 8Ž . Ž .p r PWIA

The exact ‘‘covariant PWIA’’ calculations of theasymmetry are given in Fig. 1 by the dashed-dotted

Ž .curves 1 and 2 for the Reid Soft Core RSC and theŽ .Yamaguchi-Tabakin Y-T potential, respectively.

Without SX-components all PWIA calculations areequal to zero. So, the PWIA asymmetry is very

X Žsensitive to S -components, but its small value 1y. Ž .3% reflects insignificant ‘‘proton polarization’’ 8 .Now let us examine what transpires when FSI is

taken into account. At low P and high q the majorr

FSI will be between the spectator nucleons, sincetheir relative momenta k will be small due to en-ergy-momentum conservation, while the relative mo-

Ž < < < <.mentum of the struck proton p ; q with respect

to the spectator pn-pair will be large enough, so thatŽ w xtheir interaction may be neglected at least 1 at

.P -0.2 GeVrc . Thus, accounting for the FSI in ther3 � 4spectator pn-pairs of the He™pq pn verticess, t

in this particular case simply corresponds to replac-Ž .ing the functions G in Eq. 7 by the re-normalizeds, t

w x Žq.4 ones G , including additionally the loops withs, tŽ X .half-off-shell amplitudes f k ,k, E of the elastics, t k

pn-scattering in the 1S - and 3S -states:0 1

GŽq. P ,kŽ .s , t

1 f kX ,k , EŽ .s , t kX X3s d k d kyk qŽ .H 2 X 2½ 522p k yk y ie

=G P ,kX , 9Ž . Ž .s , t

Ž < < .First let us consider P ;0 which means k ;0 .rw xIn this case the Migdal-Watson approximation 6 for

the FSI of spectator nucleons will be available andthe ‘‘re-normalized’’ vertex GŽq. has the factorizeds, t

Žq .Ž . Ž . Ž 2 .form: G P , k s G P , k = D k withs, t s, t s, tŽ 2 . Ž 2 s, t.y1D k ; k rm q e for the ‘‘scatterings, t 0

Ž sŽ t . Ž .length’’ approximation e is the virtual real0Ž . .level in the singlet triplet pn-pair . As a result of

› X™3 Ž . Ž . ŽFig. 1. The A -asymmetry of the He e,e p pn reaction for the Raid Soft Core potential curves 1 and the Yamaguchi-Tabakin one curvesx. < < Ž .2 in the collinear kinematics at E s0.88 GeVrc and q s0.4 GeVrc as a function of P . Solid dashed curves reflect total calculationse r

X Ž .with SqS S configurations.

( )S. Nagorny, W. TurchinetzrPhysics Letters B 429 1998 222–226 225

the strong differences of the singlet and the tripletpn-interaction at low energies, the main contributions

Ž .in the numerator and denominator of Eq. 7 will befrom C s, which will be cancelled in the ratio. So,due to the FSI between spectator nucleons the effec-tive ‘‘proton polarization’’ will be very close to 1Ž s t .e re <1 at P ;0 and it will not depend upon0 0 r

the 3He structure:

k P ™0 ;1y4e sre t . 10Ž . Ž .p r 0 0PWIAqFSI

Ž . Ž . Ž .According to Eqs. 5 , 6 , 10 , quasi-elastic asym-Žmetries at P s0 will be rather large very close tor

.the asymmetries on the free polarized proton , andpractically model-independent. Thus, the polarized3 Ž .He-target is a simple polarized proton target 10 if

› X™3 Ž . w xP ™0, the same as for the He e,e p d reaction 5 ,r

except that the sign of A in the three-body chan-x , z

nel will be opposite. So, the asymmetry near zerorecoil momentum may be used for the calibration ofmeasurements. With increasing P the relative mo-r

< < Ž .mentum k will quickly increase and exact Eq. 9Ž .should be used in 7 .

Ž .The asymmetry calculated according to Eqs. 3 ,Ž . X4 on the basis of SqS components with exact

Ž .accounting of FSI inside spectator system via Eq. 9Ž .is shown in Fig. 1 for the RSC Y-T potential by the

Ž .solid curve 1 2 , while the same calculation, but on

the basis of only the S-component, are given by theŽ .dashed curve 1 2 . The same NN-potentials were

used to calculate the half-off-shell pn-amplitudes inŽ .9 , as were used in the Faddeev equations for thebound-state vertices. We see that at P s0 there isr

no sensitivity of A to the mixed symmetry compo-x

nents nor to the choice of NN-potentials. At P s20r

y60 MeVrc the differences of 1S y3S FSI and0 1

the contribution of SX-components become compara-ble and their interference decreases the asymmetryconsiderably. It is interesting that at P -60 MeVrcr

asymmetries calculated without SX for different po-Ž .tentials RSC and Y-T practically coincide. This

means that the Migdal-Watson factorization is validfor GŽq. up to P ;60 MeVrc. Then the nuclears, t r

Ž .structure will be cancelled in the ratio for k Pp r

when neglecting the SX-components, and the asym-metry is determined only by the low-energy be-haviour of the singletrtriplet pn-interaction, whichis the same for any realistic potentials. The presenceof mixed symmetry components prevents the nuclearstructure cancellation and changes results for variouspotentials in different amounts. The A -asymmetryz

has the same shape and sensitivity to SX but a factorof two smaller magnitude and we will not show ithere.

In Fig. 2 the same calculations of the asymmetry

< <Fig. 2. The same like in Fig. 1, but as a function of q at P s0.04 GeVrc.r

( )S. Nagorny, W. TurchinetzrPhysics Letters B 429 1998 222–226226

at P s40 MeVrc are plotted against the 3-momen-r< <tum transfer q . There is a strong sensitivity of Ax

X < <to the S -component, which increases with q , whilethe asymmetry does not depend upon the model ofthe full symmetric configuration, when neglectingthe SX-component.

The comparison of the exact calculations with thecorresponding results obtained on the basis of the

Ž .factorized form 5 shows that they deffer by aboutof 1% at P -100 MeVrc and can not be distin-r

guished in Fig. 1,2. So, simple analytical, factorizedŽ .representation 5 of the asymmetry in terms of the

Ž .‘‘quasi-free proton asymmetry’’ 6 and ‘‘effectiveŽ .proton polarization’’ 7 , together with vertex re-nor-

Ž .malization 9 , is a good approximation in the quasi-< <elastic region at high q and small P .r

To summarize, the A -asymmetries in collinearx , zŽ X .kinematics of the three-body e,e p break up of the

polarized 3He-target by polarized electrons appear tobe very sensitive to the mixed symmetry componentat P ;20y60 MeVrc. In addition, the dependencer

of the asymmetry on the SX-component admixturestrongly increases with increasing momentum trans-fer, while the calculations without SX are modelindependent at P -60 MeVrc. A large asymmetryr› X™3 Ž .of the He e,e p pn reaction at P ;0 arises onlyr

Ž .due to the FSI of spectator nucleons: i its value isdetermined by only the difference of the low energy

Ž .singletrtriplet pn-interaction, and ii it does notdepend on the nuclear structure. The asymmetry ismodel-independent near zero recoil momentum andits value may be used for the calibration of measure-ments. Finally, a factorized form of the asymmetry

› X™3 Ž .for He e,e p pn reaction in the quasi-elastic regionwas obtained.

Acknowledgements

This work was supported partially by FOMŽ .Fundamenteel Onderzoek der Materie of theNetherlands and partially by the USA Department of

Ž .Energy DOE under grant DE-FG05-90ER0570. Weare very grateful to J.-M. Laget who initiated thiswork, as well as W. Donnelly and P. Sauer for usefuldiscussions, H.P. Blok and L. Lapikas for the criticalreading this manuscript.

References

w x Ž .1 J.M. Laget, Phys. Lett. B 276 1992 398.w x Ž .2 Ch. Hajduk et al., Nucl. Phys. A 337 1980 13.w x Ž .3 J.L. Friar et al., Phys. Lett. B 161 1985 241.w x Ž .4 S.I. Nagorny et al., Phys. of Atomic Nuclei 57 1994 940.w x Ž .5 S.I. Nagorny, W. Turchinetz, Phys. Lett. B 389 1996 429.w x Ž .6 K.M. Watson, Phys. Rev. 88 1952 1163; A.B. Migdal, JETP

Ž .1 1955 2.

18 June 1998


EFT and NN scattering

J. Gegelia 1

School of Physical Sciences, Flinders UniÕersity of South Australia, Bedford Park, SA 5042, Australia

Received 21 January 1998; revised 16 March 1998Editor: W. Haxton

Abstract

Ž .It is shown that chiral perturbation theory in its original form by Weinberg can describe NN scattering with positive aswell as negative effective range. Some issues connected with unnaturally large NN 1S scattering length are discussed.0

q 1998 Elsevier Science B.V. All rights reserved.

The chiral perturbation theory approach to thew xlow-energy purely pionic processes 1 has been

generalised for processes involving an arbitrary num-w xber of nucleons 2,3 . Weinberg pointed out that for

the n-nucleon problem the power counting should beused for the ‘‘effective potentials’’ and not for thefull amplitudes. The effective potential is defined asa sum of time-ordered perturbation theory diagramsfor the T-matrix excluding those with purely nucle-onic intermediate states.

The full S-matrix can be obtained by solving aŽLippmann-Schwinger equation or Schrodinger equa-¨

.tion with this effective potential in place of theinteraction Hamiltonian, and with only n-nucleon

w xintermediate states 2 .Using renormalization points also characterised

by momenta of order of external momenta p or less,the ultraviolet divergences that arise in calculationsusing effective theory are absorbed into the parame-ters of the Lagrangian. After renormalization, the


w xeffective cut-off is of order p 3 . The Lagrangian isrich enough to contain all possible terms which areallowed by assumed symmetries, so all necessarycounter-terms are present in the Lagrangian.

There has been much recent interest in the chiralperturbation theory approach to nucleon-nucleon

w xscattering problems 4–10 . While some papers con-cern different constructive calculational and concep-

w xtual problems the authors of 6–10 came to theconclusion that Weinberg’s approach to EFT hassevere problems. These arise in the description ofinteractions with positive effective range. Moreoverthey conclude ‘‘there is no effective field theory ofNN scattering with nucleons alone’’ and the inclu-sion of pionic degrees of freedom does not solve the

w xproblem 10 . Below it is shown that the Marylandgroup encountered problems because some featuresof EFT approach were not taken into account consis-tently. Performing correct renormalization one seesthat the above discouraging conclusions are mislead-

Žing and nothing is wrong with the EFT chiral.perturbation theory approach to the nucleon-nucleon

scattering problem.


( )J. GegeliarPhysics Letters B 429 1998 227–231228

After realising that EFT does not suffer fromfundamental problems one is still left with the chal-

w xlenging problem 5 that unnaturally large scatteringlength of 1S wave nucleon-nucleon scattering re-0

stricts the validity of EFT to very small values ofŽ .energy external momenta . The use of the correct

renormalization procedure leads one to the naturalsolution of this problem via exploiting the freedomof the choice of normalisation condition.

The effective non-relativistic Lagrangian for verylow energy EFT, when the pions are integrated out is

w xgiven by 5

D 1 2† † †LLsN iE NyN Ny C N NŽ .t S2 M 2

1 12† † †y C N s N y C N DN N NŽ . Ž . Ž .T 22 2

qh.c.q... 1Ž .

where the nucleonic field N is a two-spinor in spinspace and a two-spinor in isotopic spin space and sare the Pauli matrices acting on spin indices. M isthe mass of nucleon and the ellipses refer to addi-tional 4-nucleon operators involving two or morederivatives, as well as relativistic corrections to thepropagator. C and C are couplings introduced byT S

w x Ž .y2Weinberg 2,3 , they are of dimension mass andŽ .y4C is of the order mass .2

The leading contribution to the 2-nucleon poten-tial is

V p , pX sC qC s ,s . 2Ž . Ž . Ž .0 S T 1 2

In the 1S wave it gives:0

V p , pX sC 3Ž . Ž .0

C Žwhere CsC y3C . If we define C ' where2S T 2 2 L

.L is a parameter of dimension of mass the next toleading order contribution to the 2-nucleon potentialin the 1S channel takes the form:0

p2 qpX 2X X 22V p , p sC p qp sC 4Ž . Ž .Ž .2 2 2ž /2 L

Formally iterating the potential V qV using the0 2

Lippmann-Schwinger equation one gets for on-shellŽ 2 . w xEsp rM s-wave T-matrix 9 :

21 C I y1Ž .2 3s y I p ,Ž .2 2T p CqC I qp C 2yC IŽ . Ž .2 5 2 2 3

5Ž .

d3kny3I syM k ;Hn 32pŽ .

d3k 1 iMpI p sM s I y , 6Ž . Ž .H 13 2 2 4pp yk q ih2pŽ .where p is the on-shell momentum and I , I and I1 3 5

are divergent integrals.w xThe authors of the paper 10 carried the renor-

malization a certain distance without specifying aregularization scheme, choosing as renormalised pa-rameters the experimental values of the scatteringlength a and the effective range r . C and C weree 2

fixed by demanding that

1 M 1 12 4sy y q r p qO p y ipŽ .ež /T p 4p a 2Ž .

7Ž .

Ž . 2Expanding 5 in powers of p and comparing withŽ .7 we see that imaginary parts agree and equatingcoefficients of terms of order 1 and p2 we get:

2M C I y1Ž .2 3s y I 8Ž .124p a CqC I2 5

and

2Mr M 1 1es q I y 9Ž .1 2ž /8p 4p a IC I y1 IŽ . 32 3 3

Rewritten in terms of a and r the scattering ampli-e

tude has the form:

1 Mr 4p a yp2I AŽ . 1Re s 10Ž .2ž /T p 1qp AŽ .with

y1Mr MeA' q I 11Ž .1ž /8p 4p a

( )J. GegeliarPhysics Letters B 429 1998 227–231 229

If one uses sharp cut-off to regularize the divergentintegrals then after removal of cut-off one gets the

MreŽ .finite result I ™`, A™0 and I A™ :1 1 8p

1 M 1 12sy y q r p y ip 12Ž .eT p 4p a 2Ž .

But one can remove regularization only if r F0.eŽ .The problem is that one cannot solve C from 9 if2

Žr is positive and cut-off l™` one gets a quadratice.equation for C which has no real solutions . The2

fact that one cannot take the cut-off to infinity andstill obtain positive effective range is not surprising.Wigner’s theorem states that if the potential vanishes

w xbeyond range R then for phase shifts we have 11 :

dd p 1Ž .GyRq sin 2d p q2 pR . 13Ž . Ž .Ž .

dp 2 p

w xFrom this one can derive 7 :

2 3R Rr F2 Ry q 14Ž .e 2a 3a

ŽSo, Wigner’s theorem derived from physical princi-.ples of causality and unitarity states that the zero

range potential cannot describe positive effectiverange. Consequently as far as the potential V qV0 2Ž .with removed cutoff is zero-range it fails to de-scribe actual nucleon-nucleon scattering with posi-

Žtive effective range. Does it mean that EFT with.removed cut-off fails to describe NN scattering with

positive effective range? The answer is no.Ž .One should remember that although 5 was ob-

tained from the quantum mechanics, it was written asan approximate expression of the QFT scattering

w xamplitude. Although in 9 it was shown that one canŽ .make 5 finite non-perturbatively, renormalizing

only two parameters, one should remember that toŽ .renormalize 5 consistently one should act in accor-

dance with rules of EFT. In particular, one shouldremove all divergences by subtracting all divergentintegrals or taking into account contributions of

Ž .counter-terms. If one takes the inverse of 5 andexpand in C and C , one finds that this expansion2

contains increasing powers of p2 with divergentcoefficients. To make the amplitude finite, one has to

include contributions of infinite number of counter-terms with number of derivatives growing up toinfinity, or more simply subtract all divergent inte-grals at some value of external momenta. One couldthink that this fact makes the theory completelyun-predictive, but it does not. Let us remind thatWeinberg’s power counting applies to the renor-malised quantities, i.e. after inclusion of contribu-tions of counter-terms. The theory possesses thepredictive power because we expect that renormal-ized higher dimensional couplings are heavily sup-pressed.

The non-perturbative finiteness of the above po-tential model is not a feature of the original fieldtheory. In addition one hardly can expect that thenext approximations to the potential lead to non-per-turbatively finite results. So the above given renor-malization scheme is not the one one should follow.Furthermore the power counting was performed inthe renormalized theory, so one can neglect thecontributions of higher order terms into physical

Ž .quantities like scattering length and effective rangeonly in this theory. If one is working in terms ofregularized integrals and ‘‘bare’’ coupling constants,then one can not neglect the contributions of higher

Žorder terms into effective range or into any other.physical quantity when the cutoff parameter is re-

moved: there are contributions of an infinite numberof terms with more and more severe divergences. So,

Ž . Ž .the Eqs. 8 - 9 are not reliable and hence beingŽ .unable to solve C in 9 in the removed cut-off limit2

does not mean that EFT is incapable of describingprocesses with a positive effective range. The only

Ž . Žcorrect way of renormalizing 5 consistent with.QFT is the approach by subtraction of integrals.

Otherwise one should refer to the original effectivefield theory: introduce all counter-terms and sum allrelevant renormalised diagrams up. Note that theabove non-perturbative expression for amplitude T isnothing else than the sum of infinite number ofperturbative diagrams. If one takes into account thecontributions of all relevant counter-terms, then onewill not encounter any problems like the impossibil-ity of removing the regularization for positive effec-tive range.

As far as renormalised amplitude contains contri-butions of an infinite number of counter-terms with

Ž .an increasing up to infinity number of derivatives

( )J. GegeliarPhysics Letters B 429 1998 227–231230

and consequently it does not correspond to anyŽ .zero-range potential the condition 14 can not keep

the effective range non-positive.It is easy to see that if we subtract all integrals in

Ž .the perturbative expansion of T p and after sumthese subtracted series we will get the non-perturba-

Ž Ž ..tive expression for the amplitude the inverse of 5with subtracted integrals.

Below we will proceed by performing subtrac-tions without specifying regularization. Subtracting

2 2 Ž .divergent integrals at p sym in 5 we get thefollowing expression:

1 1 M Ms q mq ipR 2 RT p 4p 4pC m q2 p C mŽ . Ž . Ž .2

°M 1~sy M4p¢ R 2 Ry C m q2 p C mŽ . Ž .Ž .24p

¶•ymy ip 15Ž .ß

RŽ . RŽ .where C m and C m are renormalised coupling2

constants.RŽ . RŽ . Ž .We fix C m and C m by fitting 15 to the2

Ž . R Reffective range expansion 7 . For C and C we2

have:

4p a pR R 2C s ; C s a r2 e2M 1yamŽ . M 1yamŽ .

16Ž .

Note that above we did not use any regularization atŽall, so if one uses the dimensional or cut-off or any

. Žother regularization and subtracts all integrals takesinto account the contributions of all counter-terms

.which are required by QFT approach one gets thesame results.

Ž .The expression 15 can describe positive as wellas negative effective range.

Ž . ŽThe expansion parameter in 15 if expanded in2 .powers of p is:

r p2e

ls 17Ž .2 1raymŽ .

If we take ms0 then

ar p2e

ls 18Ž .2

w x Ž .As considered in 5 , the result 18 is very discour-Ž .aging from the EFT point of view. From 18 we see4p 1Ž .that the expansion of pcotd p s ipq has theM T

2 Ž .radius of convergence p ; 1r ar . But a ;0Ž . 1y1r 8 MeV for the S channel and in general a0

Ž .blows up as a bound state or nearly bound stateapproaches threshold.

Ž .It is clear from 17 that the above problem has aquite natural and simple solution. One just needs toexploit the freedom of choice of normalisation point

r p2eŽ .m. For large a 17 becomes lf . If we takey2 m

m;p we get l;yr pr2 and the radius of con-e

vergence for 1S is ;2rr f146 MeV.0 eŽ . RIf we take ms140 MeV in 16 we get C s

Ž .2 2 Ž .2y1r 99 MeV and L f 147 MeV . So, we seethat L;m as was expected for the effective the-p

ory, where pions were integrated out. So, the effec-tive range expansion works quite well in this prob-

r2 elem. For a ™ ` we have L s y ;2 mŽ .Ž .2y 140 MeV if m;140 MeV. Note that although

this value of m is not of the order of externalmomenta it does not lead to any power countingproblems here.

An alternative solution of the problem of unnatu-rally large scattering length with similar power

w x w xcounting was given in 12 . The authors of 12 useddimensional regularization and an unusual subtrac-tion scheme.

Conclusions.One should be careful performingcalculations in the EFT approach. If one follows the

w xway described in 2,3 then one will not encounterw xthe fundamental problems described in 6–10 . One

can introduce regularization into the potential in theŽeffective theory of nucleons alone this potential is

.taken up to some order in EFT expansion and solveSchrodinger equation using this regulated potential.¨One can fit parameters of regularized theory usingthe technique of cut-off effective theory. Otherwise,if the regularization is supposed to be removed thenit is necessary to include contributions of an infinitenumber of counter-terms, or otherwise subtract alldivergences. The amplitude obtained without inclu-

( )J. GegeliarPhysics Letters B 429 1998 227–231 231

sion of counter-terms evidently satisfies the con-straint that the effective range has an upper boundwhich goes to zero in the removed cut-off limit. Onecan come to the incorrect conclusion that the theorycan not describe positive effective range. The analo-gous problems appear in the theory with included

Ž .pionic and other degrees of freedom and theyshould be treated in a similar way.

The problem of unnaturally large scattering lengthof 1S wave can be solved by appropriate choice of0

normalisation point within conventionally renor-malised theory. This solution does not depend onregularization.

I would like to thank B. Blankleider and A.Kvinikhidze for useful discussions.

This work was carried out whilst the author was arecipient of an Overseas Postgraduate ResearchScholarship and a Flinders University Research

Scholarship holder at Flinders University of SouthAustralia.

References

w x Ž .1 S. Weinberg, Physica A 96 1979 327.w x Ž .2 S. Weinberg, Phys. Lett. B 251 1990 288.w x Ž .3 S. Weinberg, Nucl. Phys. B 363 1991 3.w x Ž .4 C. Ordonez, L. Ray, U. van Kolck, Phys. Rev. C 53 1996

2086.w x5 D.B. Kaplan, M.J. Savage, M.B. Wise, Nucl. Phys. B 478

Ž .1996 629.w x Ž .6 T.D. Cohen, Phys. Rev. C 55 1997 67.w x Ž .7 D.R. Phillips, T.D. Cohen, Phys. Lett. B 390 1997 7,

nucl-thr9607048.w x8 K.A. Scaldeferri, D.R. Phillips, C.-W. Kao, T.D. Cohen,

Ž .Phys. Rev. C 56 1997 679, nucl-thr9610049.w x9 D.R. Phillips, S.R. Beane, T.D. Cohen, hep-thr9706070.

w x10 S.R. Beane, T.D. Cohen, D.R. Phillips, nucl-thr9709062.w x Ž .11 E.P. Wigner, Phys. Rev. 98 1955 145.w x12 D.B. Kaplan, M.J. Savage, M.B. Wise, nucl-thr9801034.

18 June 1998


Thermal phase transition in weakly interacting, large N QCDC

Joakim Hallin a,1, David Persson b,2

a Institute of Theoretical Physics, Chalmers UniÕersity of Technology and Goteborg UniÕersity, S-412 96 Goteborg, Sweden¨ ¨b Department of Physics and Astronomy, UniÕersity of British Columbia, VancouÕer, BC V6T 1Z1, Canada

Received 13 March 1998Editor: W. Haxton

Abstract

We consider thermal QCD in the large N limit, mainly in 1q1 dimensions. The gauge coupling is only taken intoC

account to minimal order, by projection onto colour singlets. An expression for the free energy, exact as N ™`, is thenC

obtained. A third order phase transition will occur. The critical temperature depends on the ratio N rL, where L is theCŽ .infinite spatial length. In the high temperature limit, the free energy will approach the same value as in the free theory,

² :whereas we have a mesonic like phase at low temperature. Expressions for the quark condensate, CC , are also obtained.q 1998 Published by Elsevier Science B.V. All rights reserved.

Considerable interest has recently been devoted to the assumed deconfinement phase transition in QCD athigh temperature andror density. Exact analytical results are incomprehensible sofar, and we have to rely onapproximate methods or computer simulations on the lattice. It has been well known for a long time that some

Ž .insight in the confinement mechanism may be gained by considering the large N limit, for an SU N gaugeC CŽ w x.theory. This limit is particularly fruitful in 1q1 dimensions see e.g. Refs. 1–5 . QCD in 1q1 dimensions

has also received attention recently due to the remarkable fact that different regularizations seems to lead tow xdifferent models, even within the same choice of gauge. In his pioneering work, ’t Hooft 1 employed a cutoff

w xaround the origin in momentum space, that later was shown to be equivalent to a principle value prescription 2 .Within this framework, no free quarks can propagate, but they are confined into mesons, consisting of quark

w xanti-quark pairs. Wu 3 , on the other hand, suggested another regularization, allowing for a Wick rotation toEuclidean space. The bound state equation is more complicated in this case, and no solution has yet been found.

w xBassetto and Nardelli with co-workers 6–9 have compared the two regularizations, and computed for examplethe expectation value of the Wilson loop. The expectation value of the Wilson loop with Wu’s regularization,

w xwas recently calculated by Staudacher and Krauth 10 , who showed that confinement is not enforced in thisw xcase, unlike using the ’t Hooft regularization. It has been suggested by Chibisov and Zhitnitsky 11 that the

models due to different regularizations could show up as different phases, that makes a study of the system at

1 Email address: [email protected] Email address: [email protected].


( )J. Hallin, D. PerssonrPhysics Letters B 429 1998 232–238 233

finite temperature intriguing. In 2 dimensional QCD, confinement is obvious as an infinite energy for colouredstates.

We shall in this letter take the interaction into account only by a projection onto colour singlets, i.e. weneglect the coupling unless it multiplies the infra red divergence. The quark contribution to the free energy in3q1 dimensions is then easily obtained from the free energy presented here in 1q1 dimensions through the

Ž . 3 Ž .3substitution, LHdpr 2p ™2VHd pr 2p . The projected free energy in 3q1 dimensions has earlier beenw xconsidered in the high temperature limit by Skagerstam 12 . The full interacting case in 2 dimensions, that is

troubled by infra red divergences, will be more extensively considered in another article. Including thew xinteraction in the strict large N limit LrN ™0, McLerran and Sen 13 , and employing a different methodC C

w xHansson and Zahed 14 showed that in the low temperature ’t Hooft meson phase, the partition functionŽ . w xlnZsOO L , due to confinement. Using a different resummation scheme for LrN ™`, McLerran and Sen 13C

Ž .also found another phase of almost free quarks, with lnZsOO LN . The phase transition itself was notC

obtainable, and could only take place as T™`, i.e. TAN . By considering finite ksN rL, we are here ableC C

to monitor the phase transition exactly, and for finite k , it will take place at finite T. However, due to thenegligence of the interaction we do not have confinement in the strict sense. Our low temperature phase consistsof quark–anti-quark pairs, showing up for example in Bose–Einstein distribution functions, but these are not the

Ž .confined mesons of ’t Hooft’s, since lnZsOO LN .Cˆyb HLet us now consider the unnormalized density matrix rse of free coloured fermions having theˆ

Hamiltonian

ˆ ˆ† ˆHs f p h p f p , 1Ž . Ž . Ž . Ž .Hp

where

h p sg 0 g pqm , 2Ž . Ž . Ž .1

dnpnand H sH , in n spatial dimensions. We shall here mainly discuss ns1. In the functional representationŽ .p 2p

w xthe expression for r is 15ˆ

r h )h ,h )h sN exp h ) qh ) V h qh qh )h yh )h , 3Ž . Ž .Ž . Ž .Ž .1 1 2 2 b 1 2 b 1 2 1 2 2 1

where

bN sexp 4N V ln cosh v , 4Ž .Hb C ž /ž /2p

b hV sytanh v 1 , 5Ž .b Cž /2 v

2 2(v p s p qm , 6Ž . Ž .Ž ) ) .and L denotes the spatial length. In the expression for r h h ,h h , summation over spinor, colour and1 1 2 2

momentum indices is understood. Moreover N denotes the number of colours and 1 is the unit matrix inC C

colour space. We now wish to calculate the partition function for colourless states. This is done by the operatorp ,ˆ

ps dup , 7Ž .ˆ ˆH u

ˆ ai a Qa Ž .that projects onto colour singlets. Here p se is a general global U N gauge transformation. Theˆu C

partition function Z is then

Zs tr pr . 8Ž .Ž .ˆ ˆ

( )J. Hallin, D. PerssonrPhysics Letters B 429 1998 232–238234

With the colour charge operator

t aa †ˆ ˆ ˆQ s f f , 9Ž .H

2p

the global gauge transformation p is in the functional representation:ˆu

) ) ) ) ) )p h h ,h h sN exp h qh V h qh qh h yh h . 10Ž . Ž .Ž . Ž .u 1 1 2 2 u 1 2 u 1 2 1 2 2 1

Here we have defined1 1 y1N sexp tr ln uqu q2 , 11Ž . Ž .u 2 4

and

uy1V s 1 . 12Ž .u suq1

Furthermore, tr sLH tr tr denotes the trace over all indices, momentum, spin and colour, while 1 is the unitp s C st a

i a a 2Ž .matrix in spinor space i.e. a 2=2-unit matrix 4=4 in ns3 . The group element is use . Multiplying r

and u one findsˆ

V qVb u) ) ) ) ) )² < < :h h ur h h sN N det 1qV V exp h qh h qh qh h yh h . 13Ž . Ž .Ž .ˆ ˆ Ž .1 1 2 2 b u b u 1 2 1 2 1 2 2 1½ 51qV Vb u

Finally we can then give the expression for the partition function,

Zs du tr p r s dudet 2 N N det 1qV V seyb E0 duexp L tr ln 1qj u 1qj uy1 ,Ž . Ž . Ž .ˆ ˆ Ž .Ž .H H H Hu b u b u C½ 5p

14Ž .

where E syLN H v is the vacuum energy. We have here defined0 C p

jseyb v , 15Ž .i a j Ž .and e denotes the N eigenvalues of the U N matrix u. On functions of eigenvalues the Haar measure duC C

takes the formNC a yai j2dus da sin . 16Ž .Ł ŁH H i 2is1 i-j

This gives

N NC Ca yai jyb E 2 i a yi a0 j jZse da sin exp L ln 1qj e 1qj e . 17Ž .Ž . Ž .Ł Ł ÝH Hi 2 pis1 i-j js1

Ž . Ž .In the large N -limit the integral over U N in 17 may be calculated by the steepest descent method. We takeC C

this limit by letting N and the spatial length L tend to infinity simultaneously keeping their quotientC

NCks , 18Ž .

L

w xconstant. We solve this by introducing the density of eigenstates, D , in the continuum limit 16,17 , i.e.NC a1 c

f a ™N dtf a t sN D a da f a . 19Ž . Ž . Ž . Ž . Ž .Ý H Hj C C0 ya cjs1


We find two distinct phases, depending on the inequality

2 11y G0. 20Ž .H bvk e y1p

Ž .As long as 20 is satisfied we are in the zero-gap phase, a sp , where D has support over the whole circle.cŽ .When 20 is not satisfied we are in the one-gap phase. In this case there is a gap in which D lacks support, i.e.

an interval where the eigenvalues give a vanishing contribution as N ™`.C

In the zero-gap phase lnZ is given by a sum over quark–antiquark pairs

N 2C

lnZsybE y ln 1yj p j q . 21Ž . Ž . Ž .HH0 2k p q

Ž .2Even though the states are mesonic-like, we do not have confinement since lnZA N rk sN Lrk , whereasC CŽ w x.in the confined phase of ’t Hooft mesons we have cf. 14 lnZAL. In the one-gap phase we find

2a 2 Nc C 12 2(lnZsybE qN ln sin q ln 1qjq j q2j cosa q1H ž /0 C c2ž /2 k p

N 2C

y ln 1yx p x q , 22Ž . Ž . Ž .HH2k p q

where we have defined

1 22(xs 1qjy j q2j cosa q1 . 23Ž .cac24j sin

2

Furthermore, the critical angle a is determined fromc

1qjy1 sk . 24Ž .H

2ž /p (j q2j cosa q1c

Ž 2 .In Fig. 1, we plot lnZr TN as a function of the temperature, for ksm. The critical temperature is then foundCŽ . Ž . 2as T f0.8m. The free energy FsyT lnZ in the 1-gap phase solid line grows as T for large T , and isc

Ž .approaching the free energy for free quarks, i.e. without the projection, dotted line , as T™`. Whereas theŽ . 3free energy in the 0-gap phase dashed line , continued above the phase transition would grow like T . The high

Ž . Ž . Ž .Fig. 1. The free energy for free quarks dotted line , in the 0-gap phase dashed line , and in the 1-gap phase solid line , for k s m. All inunits of N 2T 2rm.C


temperature behaviour in the 1-gap phase is obtained by an expansion for a <1, yielding a ,4pkrT. Asc c

T™`, a ™0, i.e. only us1 gives a contribution, corresponding to no projection.cŽ .In addition to the free energy, it is of interest to compare the entropy, and the renormalized energy density

in the two phases. They are given by

1 Ess T lnZ , 25Ž . Ž .LL E T

T 2 Ees lnZqbE , 26Ž . Ž .0 LL E T

respectively. Their behaviour is similar to the that of the free energy. In the interacting case it is well known thatw xa nontrivial chiral condensate appears, as shown for ms0 by Zhitnitsky 18,19 , and in the general case by

w xBurkardt 20 . However, when the interaction only is taken into account through the projection, the vacuum isŽ .trivial and no chiral condensate can appear in this case. Nevertheless, the renormalized expectation value of

cc , is an order parameter for the phase transition, and thus of interest. We find

E² :cc syT lnZqbE . 27Ž . Ž .0 LE m

Ž .In the 0-gap phase it is straightforward to evaluate this from 21 . However, in the 1-gap phase we need todefine

a Eccot T a 'ma, 28Ž .c2 E m

wherey1

j 1qj j 1yj 1Ž . Ž .as , 29Ž .H H3r2 3r2½ 5 ½ 52 2 vp pj q2j cosa q1 j q2j cosa q1Ž . Ž .c c

Ž . Ž . Ž .is obtained from 24 , using 15 , and 6 . We then find

T E x 1xs 1qj ay 1yj . 30Ž . Ž . Ž .

2m E m v(j q2j cosa q1c

This gives

°1 2 1

2 ~² :cc sN m y aq HC 22 k p (1qjq j q2j cosa q1¢ c

=

ac22j sinjqcosa jc 21q q z2 2v( (j q2j cosa q1 j q2j cosa q1� 0c c

1¶1qj ay 1yjŽ . Ž .2 x p x qŽ . Ž . v •y . 31Ž .HH2 21yx p x qk Ž . Ž .p q (j q2j cosa q1 ßc

² :We have been able to show that cc is continuous over the phase transition, whereas higher derivatives givecumbersome expressions.


However, in the limit of infinite massive quarks analytical results are obtained. As m™`, we must also letT ™`, such that their quotientc

zsmrT , 32Ž .c

is kept fixed, in order to keep a finite particle density. Furthermore, we define t according to TsT t . Thec

momentum integrals are now independent of p, so we write

™L. 33Ž .Hp

The critical temperature is then determined from

k 1 1s ™L . 34Ž .H v r T zc2 e y1 e y1p

In the 0-gap phase we then find2

L 12² :cc s2 N . 35Ž .c 2z rtž /k e y1

In the 1-gap phase the critical angle is now determined fromz rte q1

z rt 2z rt(1q2 e cosa qe s , 36Ž .c 1qkrL

that after some simplifications give

1 L k 12² :cc sN q 2q . 37Ž .c z rt½ 5ž /ž /2 k L e q1

Notice here the Bose–Einstein distribution in the low temperature 0-gap phase, as opposed to the Fermi–Diracdistribution in the 1-gap phase at high temperature. It is now a straightforward exercise to show that

2 2² :E rE t cc is discontinuous, so that it is a third order phase transition. This is in agreement with the study ofw xGattringer et al. 21 concerning static quarks on a line. Furthermore, in the corresponding lattice model Gross

w xand Witten 16 found a third order phase transition, so we conjecture that the phase transition most likely isthird order also for arbitrary mass.

Acknowledgements

We are grateful to Ariel Zhitnitsky for proposing the subject, as well as stimulating discussions; and to Hansw x ŽHansson for interesting remarks, and pointing out 14 . J.H.’s research was funded by NFR the Swedish Natural

. ŽScience Research Council , and D.P.’s by STINT the Swedish Foundation for Cooperation in Research and.Higher Education .

References

w x Ž .1 G. ’tHooft, Nucl. Phys. B 75 1974 461.w x Ž .2 C.G. Callan, N. Coote, D. Gross, Phys. Rev. D 13 1976 1649.w x Ž .3 T.T. Wu, Phys. Lett. B 71 1977 142.w x Ž .4 M.B. Einhorn, Phys. Rev. D 14 1976 3451.w x Ž .5 R. Brower, W.L. Spence, Phys. Rev. D 19 1979 3024.


w x Ž .6 A. Bassetto, L. Griguolo, Phys. Rev. D 53 1996 7385.w x Ž .7 A. Bassetto, G. Nardelli, Int. J. Mod. Phys. A 12 1997 1075.w x Ž .8 A. Bassetto, G. Nardelli, A. Shuvaev, Nucl. Phys. B 495 1997 451.w x Ž .9 A. Bassetto, D. Colferai, G. Nardelli, Nucl. Phys. B 501 1997 227.

w x10 M. Staudacher, W. Krauth, hep-thr9709101, 1997.w x Ž .11 B. Chibisov, A.R. Zhitnitsky, Phys. Lett. B 362 1995 105.w x Ž .12 B.S. Skagerstam, Z. Phys. C 24 1984 97.w x Ž .13 L.D. McLerran, A. Sen, Phys. Rev. D 32 1985 2794.w x Ž .14 T.H. Hansson, I. Zahed, Phys. Lett. 1993 .w x Ž .15 J. Hallin, P. Liljenberg, Phys. Rev. D 52 1995 1150.w x Ž .16 D.J. Gross, E. Witten, Phys. Rev. D 21 1980 446.w x Ž .17 G. Mandal, Mod. Phys. Lett. A 5 1990 1147.w x Ž .18 A.R. Zhitnitsky, Phys. Lett. B 165 1985 405.w x Ž .19 A.R. Zhitnitsky, Sov. J. Nucl. Phys. 44 1986 139.w x Ž .20 M. Burkardt, Phys. Rev. D 53 1996 933.w x Ž .21 C.R. Gattringer, L.D. Paniak, G.W. Semenoff, Annals Phys. 256 1997 74.

18 June 1998


In-medium kaon and antikaon propertiesin the quark-meson coupling model

K. Tsushima a,1, K. Saito b,2, A.W. Thomas a,3, S.V. Wright a,4

a Department of Physics and Mathematical Physics and Special Research Center for the Subatomic Structure of Matter,UniÕersity of Adelaide, SA 5005, Australia

b Physics DiÕision, Tohoku College of Pharmacy, Sendai 981, Japan

Received 16 December 1997; revised 12 February 1998Editor: J.-P. Blaizot

Abstract

Ž .The properties of the kaon, K , and antikaon, K , in nuclear medium are studied in the quark-meson coupling QMCŽ .model. Employing a constituent quark-antiquark MIT bag model picture, their excitation energies in a nuclear medium at

zero momentum are calculated within mean field approximation. The scalar, and the vector mesons are assumed to coupledirectly to the nonstrange quarks and antiquarks in the K and K mesons. It is demonstrated that the r meson inducesdifferent mean field potentials for each member of the isodoublets, K and K , when they are embedded in asymmetricnuclear matter. Furthermore, it is also shown that this r meson potential is repulsive for the Ky meson in matter with aneutron excess, and renders Ky condensation less likely to occur. q 1998 Elsevier Science B.V. All rights reserved.

PACS: 12.39.B; 14.40; 71.25J; 21.65; 13.75JKeywords: In-medium kaon and antikaon properties; The quark-meson coupling model; Effective mass; Kaon condensation; Neutron star

The study of the properties of the kaon, K , and antikaon, K , in a dense nuclear medium is one of the mostw xexciting new directions in nuclear physics. Stimulated by the pioneering work of Kaplan and Nelson 1 ,

intensive work has been performed about the possibility of the Ky meson condensation in a dense nuclearw xmedium, and its effect on the properties of neutron stars 2–10 . In addition, many investigations have been

w x w xmade 11,12 concerning KN and KN interactions 13–19 , and strangeness production in heavy ion collisionsw x12,20–24 , with a particular emphasis on the medium modification of the kaon and antikaon properties.

Although K and K mesons are Goldstone bosons in the chiral limit, they are also expected to reveal aŽ .quark-antiquark qq structure to some extent, because their mass is relatively heavy compared to that of the

1 E-mail:[email protected] E-mail: [email protected] E-mail: [email protected] E-mail: [email protected].


( )K. Tsushima et al.rPhysics Letters B 429 1998 239–246240

pion. Indeed, the naive constituent quark model has proven quite successful for studying the properties of K andw xK mesons in free space 25 .

However, not so many investigations have been performed on the properties of K and K mesons in nuclearw xmedium with explicit quark degrees of freedom 17 . One of the main reasons for this is that there has been no

Žappropriate model until recently, which can simultaneously describe the properties of the nuclear medium finite.nuclei , as well as hadron properties, based on quark degrees of freedom. It now seems possible to do this

Ž .because of the recent development of the quark-meson coupling QMC model, which was initiated by Guichonw x w x26 . This model has been successfully applied to investigate the properties of infinite nuclear matter 26–31

w xand finite nuclei 33–36 , with some extension to incorporate the self-consistent variation of the meson massesw x Ž .35 . Although the QMC model may be regarded as a general extension of Quantum Hadrodynamics QHDw x37 , the difference between the two models probably becomes distinctively clear when one investigates theproperties of mesons in nuclear medium, where their quark structure plays a vital role.

Let the mean values of the meson fields, the scalar, the time component of the vector isoscalar, and the timecomponent of the vector isovector in the third direction in isospin, be s , v and b, respectively, in a uniformlydistributed nuclear medium. Here, we assume that the K and K mesons are described by the static sphericalMIT bag, as are the nucleons. We also suppose that the s , v and r mesons only interact directly with thenonstrange quarks and antiquarks in the K and K mesons. The Dirac equations for the corresponding wavefunctions for up and down quarks are given by:

u10igPEy m yV .g V q V s0, 1Ž .Ž . Ž .q s v r2 ž /u

d10igPEy m yV .g V y V s0, 2Ž .Ž . Ž .q s v r2 ž /d

where V sg q s ,V sg q v and V sg qb with g q , g q and g q being, respectively, the corresponding quarks s v v r r s v r

Ž .and meson coupling constants. Here we assume SU 2 symmetry. Thus, the current masses for the quarks andantiquarks follow the relation, m 'm sm sm sm . The normalized, static solution for the ground stateq u d u d

for a nonstrange quark or antiquark in the kaon or antikaon may be written as:wyi e tr Ri Kc r sN e f r , for isu ,u ,d ,d , 3Ž . Ž . Ž .i i i

Ž .where N and f r are respectively the normalization factor and the corresponding spatial part of the wavei iw x wfunction 39 . The bag radius in medium, R , which depends on the hadron species in which the quarks belong,K

Ž . Ž . w xwill be determined self-consistently through Eqs. 6 and 7 similarly to those for the octet baryons 33,35,36 .w Ž . Ž .The quark eigenenergies in units of 1rR , e isu,u,d,d in Eq. 3 , are given by:K i

ee du 1 1w w w wsV "R V q V and sV "R V y V , 4Ž .Ž . Ž .K v r K v r2 2ež / ž /eu d

2w w 2 w w w q wwhere V s x q R m , with m sm yg s . The quark eigenfrequency in medium, x , is deter-( Ž .K q q q s

w xmined by the usual, linear boundary condition 32,39 . Then the excitation energies for the K and K mesonsq y 0 0Ž .with zero momenta, v isK , K , K , K , are given by:i

v 0qv K1 1K w wsm " V q V and sm " V y V , 5Ž .Ž . Ž .K v r K v r2 2yvž / ž /0vK K

w wwhere, the effective mass of the K and K mesons, m sm , is calculated using the MIT bag model:K K

V w qV yzs K 4w w 3m s q p R B , 6Ž .K K3wRK

( )K. Tsushima et al.rPhysics Letters B 429 1998 239–246 241

wE mKs0, 7Ž .

wE R RsR K

2w 2 w Ž .(with the strange-quark energy in units of 1rR , V s x q R m , and z in Eq. 6 parametrizes theŽ .K s s K s K

sum of the center-of-mass and gluon fluctuation effects.wAfter self-consistent calculation, the effective mass of the K and K mesons, m , can be parametrized in theK

w xapplied scalar field 32,33,35,36 :aK1w K N Nm 'm yg s s,m y g 1y g s s , 8Ž . Ž .Ž .K K s K s s3 2

N Ž . w x Kwhere g is the nucleon and s meson coupling constant in free space ss0 32,36 , and the relation, gs s1 N w x Ž Ž .s g , involves an error less than 0.5 % 33,35,36 . The quantity, a , in Eq. 8 is found to bes K3

y4 y1 .a s6.6=10 MeV . In this study, we chose the values, m sm s5 MeV and m s250 MeV, for theK u d s

current quark masses, and R s0.8 fm for the bag radius of the nucleon in free space. Other inputs andNŽparameters used, and some of the quantities calculated in the present study, are listed in Table 1. Note that r isq

Ž .the root-mean-square r.m.s. charge radius calculated using the MIT bag model wave functions obtained by.solving the set of equations including the strange quark. We stress that while the model has a number of

parameters, only three of them, g q , g q and g q, are adjusted to fit nuclear data – namely the saturation energys v r

and density of symmetric nuclear matter and the bulk symmetry energy. None of the results for nuclearproperties depend strongly on the choice of the other parameters – for example, the relatively weak dependenceof the final results on the chosen values of the current quark mass and bag radius is shown explicitly in Refs.w x32,33 .

Boosting the K and K bags in a uniformly distributed vector field, we can find the dispersion relation for thew xK and K mesons moving with momentum k as in Ref. 20 :

q 0v k v kŽ . Ž .K K1 1w 2 w 2( (s m qk " V q V and s m qk " V y V . 9Ž .Ž . Ž .K v r K v r2 2ž / ž /y 0v k v kŽ . Ž .K K

This is equivalent to the dispersion relation which is given by the gauge invariant effective Lagrangian densityw xat the hadronic level 9,10 :

†t t3 3K K m K m K m w 2LLs E q ig v q ig r K E q ig v q ig r K ym KKqLL , 10Ž .m v m r m v r K matterž / ž /2 2qK †where K s is the second quantized kaon field with KsK , and LL is the Lagrangian density of amatter0ž /K

w x wnuclear system 32,33,35 . In our approach, the effective mass, m , is calculated using the MIT bag model, Eq.KŽ . Ž .6 and the result is very well approximated by Eq. 8 .

Ž .In Fig. 1, we show the binding energy per nucleon, ErA ym , versus baryon density. Neutron matter andNŽ .nuclear matter in Fig. 1 denote matter with proton fractions 0 and 0.5 symmetric matter , respectively. This

notation will be used hereafter. The coupling constants, g N, g N and g N are determined so as to reproduce thes v r

Table 1Inputs, parameters and some of the quantities calculated in the present study. The quantities with star, w, are those quantities calculated at

y3 Ž .4normal nuclear density, r s0.15 fm . The values for the bag constant, and current quark masses are respectively, Bs 170.0MeV , and0

m s m s5 MeV and m s250 MeVu d s

w w wŽ . Ž . Ž . Ž . Ž . Ž .m MeV z R fm r fm m MeV R fm r fmN , K q q

Ž . Ž .N 939.0 input 3.295 0.800 input 0.582 754.6 0.786 0.594Ž .K 493.7 input 3.295 0.574 0.412 430.5 0.572 0.418


Fig. 1. Binding energy per nucleon for matter with different proton fractions. Neutron matter and nuclear matter denote matter with protonfractions 0 and 0.5, respectively. K s279.2 MeV is the value obtained for the nuclear incompressibility in the present model.

saturation properties of symmetric nuclear matter at normal nuclear density, r , namely, the binding energy per0

nucleon, y15.7 MeV, and bulk symmetry energy, 35.0 MeV. The values for the coupling constants determinedin this way are given in Table 2. Using these parameters, the nuclear incompressibility, K , is calculated to beKs279.2 MeV, which is well within the empirically required range.

In Fig. 2 we show the effective masses of the K and K mesons and the nucleon, as well as the scalar andvector mean field potentials. For the present we have omitted any effects of hyperons in the background nuclear

wmedium. We observe that neither the effective mass of the K and K mesons, m , nor that of the nucleonK

decreases linearly as the density increases. This feature may be ascribed to the quark structure in the presentw xapproach, and clearly differs from the linear decrease in QHD 37 . This behaviour can be also understood from

Ž . qthe parametrization, Eq. 8 . g v in Fig. 2 denotes the vector potential for the nonstrange quarks andK vqantiquarks in the K and K mesons, and we will explain this below, in connection with the K N potential.

It is known empirically that the KqN potential is slightly repulsive if one wants to be consistent with theq y3 w xK N scattering length, and the corresponding value at r s0.16 fm is estimated to be about 20 MeV 38 .B

On the other hand, the present model gives a very slightly attractive KqN potential. We believe that this minorshortcoming has its origin in the deficiencies of the bag model in dealing with the Goldstone nature of the K

qand K mesons. As a phenomenological means of compensation for this we rescale the coupling constant, g , tov

reproduce the KqN potential, q20 MeV, at r s0.16 fmy3. That is, we use, g q s1.4=g q , for the vB K v v

Ž .meson coupling constant to the nonstrange quark and antiquark in the K and K mesons. Note that thecoupling constant, g q , is the only parameter adjusted in the present study. None of our qualitative conclusionsK v

Table 2Values of the coupling constants determined required to reproduce the saturation properties of symmetric nuclear matter at normal nucleardensity, r s0.15 fmy3. For the relation between the coupling constants, g q and g N, or, the origin of the constant factor between them,0 s s

w xsee Refs. 32,33q N N 2 N 2 q 2 N 2 q 2Ž . Ž . Ž . Ž . Ž . Ž .g s g r 3=0.483 g r4p g r4p s 3g r4p g r4p s g r4ps s s v v r r

5.69 5.39 5.30 6.93


qŽ .Fig. 2. Effective masses of the nucleon, K and K meson and the mean field potentials for the nonstrange quarks and antiquarks. g v isK v

the v meson mean field potential for the nonstrange quarks and antiquarks in the K and K mesons.

would be altered if we did not make this adjustment. The v mean field vector potential calculated using thisrescaled coupling constant is denoted by g q v in Fig. 2.K v

In Fig. 3, we show the calculated kaon excitation energies at zero momentum versus the baryon density. It isinteresting to notice that although the excitation energies for the isodoublet members, Kq and K 0, aredegenerate in symmetric nuclear matter, this is no longer true in asymmetric nuclear matter. This is a

Ž . Ž .consequence of the r meson which couples to the nonstrange quarks antiquarks in the kaon antikaon .

Fig. 3. Kaon excitation energies at zero momentum.


Next, we show the antikaon excitation energies in Fig. 4, as well as the difference of the calculated chemicalpotentials for the neutron and proton, m ym , which is calculated by:n p

E Z, N E Z, N E E Z, NŽ . Ž . Ž .m ym , y sy , 11Ž .n p ž /E N EZ E x AZ N

Ž . Ž .where Z, N AsZqN and E Z, N rA are, respectively, the proton number, neutron number and the totalenergy per nucleon with proton fraction, x'ZrA. Because we have not included any effects which areexpected to lower the values of m ym , or raise the critical density for the onset of Ky meson condensationn pŽ w x w x ysuch as hyperons or muons 5,9,40 , the d meson 10 , non-zero momentum for the K mesons due to the

.thermal fluctuations or short-range correlations , the critical density found for each case may be regarded as ay 0lower limit. Again the excitation energies for the isodoublet members, K and K , are no longer degenerate in

asymmetric nuclear matter. In particular, the excitation energy for the Ky meson at a fixed density increases asthe neutron fraction increases. Thus, the r meson plays a role in making Ky meson condensation lessfavorable to occur in a matter with a larger neutron excess. This effect of the r field on the Ky meson,certainly should be taken into account when one studies Ky-condensation and its effect on the properties ofneutron stars.

Ž .If we use the parametrization of Eq. 8 , together with the explicit expressions for the vector mean fieldsŽ . Ž .using the isoscalar r sr qr and isovector r 'r yr baryon densities, the excitation energy for theB p n 3 p n

Ky meson at zero momentum, v y, can be expressed as:K

K N K Na g g g gK v v r r1w K K Nyv 'm yV y V ,m yg 1y g s sy r y r , 12Ž .Ž .K K v r K s s B 32 2 22 m 4mv r

1 1K N K q N K q N ywhere g s g , g s1.4=g s1.4= g and g sg sg . For a rough estimate of the K excitations s v v v r r r3 3N Ž .energy up to r ;r , one can use the approximate value for the scalar field g s,200r rr MeV .B 0 s B 0

In summary, we have studied the properties of kaon and antikaon in nuclear matter, using the QMC modelfor the first time. Although the model should eventually incorporate chiral symmetry in order to treat the kaonand antikaon as pseudo-Goldstone bosons, our present emphasis was on the role of the r meson in an

Fig. 4. Antikaon excitation energies at zero momentum, and the difference of the calculated chemical potentials for the neutron and proton,m ym .n p


asymmetric nuclear medium. In particular, in matter with a neutron excess, or with a negative isovector density,the r meson induces a repulsive potential for the Ky meson. This effect should certainly be taken into accountin investigations of the properties of neutron stars, and kaon flow in heavy ion collisions, where it has so far

0 yw xbeen omitted 22 . Indeed, it may be possible to test our estimate of this effect by calculating the K and Kflow in heavy ion collisions. In the present study, we have not included any effects which are expected to lower

w xthe chemical potential of the electron such as a non-zero hyperon density 5,9,40 . For a more realistic study, itwill be necessary to include self-consistently the effect of the hyperons in calculating scalar and vector fields. Inthat case, it is possible that the quark structure of the hadrons, which appears mainly in a non-linear variation oftheir effective masses, may give nontrivial effects. In particular, our further interest is whether the QMC modelcan avoid the negative effective mass problem for the nucleon, that was discussed by Schaffner and Mishustinw x9 .

Acknowledgements

The authors would like to thank G.Q. Li, T. Tatsumi and A.G. Williams for helpful discussions. This work issupported by the Australian Research Council and the Japan Society for the Promotion of Science.

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18 June 1998


Isomeric states in 66As

R. Grzywacz a,b, S. Andriamonje c, B. Blank c, F. Boue c, S. Czajkowski c, F. Davi c,´R. Del Moral c, C. Donzaud d, J.P. Dufour c, A. Fleury c, H. Grawe e, A. Grewe f,

A. Heinz e, Z. Janas a, A.R. Junghans f, M. Karny a, M. Lewitowicz g,A. Musquere c, M. Pfutzner a, M.-G. Porquet h, M.S. Pravikoff c, J.-E. Sauvestre i,` ¨

K. Summerer e¨a IFD, Warsaw UniÕersity, Pl-00681 Warsaw, Hoza 69, Poland

b UniÕersity of Tennessee, KnoxÕille, TN 37996, USAc CEN Bordeaux-Gradignan, Le Haut-Vigneau, F-33175 Gradignan Cedex, France

d IPN, 91406 Orsay Cedex, Francee GSI, Postfach 110552, 64220 Darmstadt, Germany

f Institut fur Kernphysik, TU Darmstadt, Schloßgartenstraße 9, D-64289 Darmstadt, Germany¨g GANIL, BP 5027, 14021 Caen Cedex, France

h CSNSM, Bat. 104 & 108, F-91405 Orsay Cedex, Francei CE Bruyeres-le-Chatel, BP 12, F-91680 Bruyeres-le-Chatel, France` ˆ ` ˆ

Received 17 February 1998; revised 7 April 1998Editor: R.H. Siemssen

Abstract

Two new isomeric states in the self-conjugate nucleus 66As have been observed at the final focus of the LISE3spectrometer at GANIL. The nucleus has been produced in a fragmentation reaction of 78Kr at 73A MeV. The 66m1,m2Asisomeric levels are placed at excitation energies of E) m1 s3023.9 keV and E) m2 s1356.7 keV. Half-lives of T m1 s1r2

Ž . m2 Ž .17.5 15 ms and T s1.9 5 ms have been measured for the isomers and a decay scheme has been established. q 19981r2

Published by Elsevier Science B.V. All rights reserved.

PACS: 25.70.Mn; 23.20.Lv; 27.50.qe

A study of excited states for NfZ nuclei in theintermediate mass region is a challenging experimen-tal task. The in-beam techniques reach their limits, b

decay studies are even more difficult. Both have toovercome experimental difficulties connected withlow production cross-sections and isotope identifica-tion. On the other hand, there is a need to study such

exotic nuclei for a better understanding of the nu-clear properties close to the drip-line or at lowisospin, such as isospin mixing, which is supposed to

w xincrease with decreasing T 1 . As pointed out inzw xRef. 2 , the NsZ region from germanium to zirco-

nium is characterized by rapid shape changes due tothe occupancy of the same orbitals by protons and


( )R. Grzywacz et al.rPhysics Letters B 429 1998 247–253248

neutrons which results in a strong proton-neutroninteraction and coherent superposition of respectiveshell gaps. For the nuclei in that region, any shell-model calculation has to overcome difficulties re-lated to the large model space due to the fact thatmany nucleons can be distributed over the fpg or-bitals. Various theoretical approaches have been pro-posed to explain the structure of excited states innuclei close to 66As. The microscopic calculations of

Žthe advanced variational method EXCITED VAM-. w xPIR 3 can be compared to calculations within the

Ž .interacting boson-fermion-fermion model IBFFMw x 684 for the T s1 odd-odd arsenic isotope As.z

The measurements which gave the basic informa-tion for the above mentioned theoretical studies havebeen performed using in-beam techniques. Other ex-perimental investigations in this region, which weremostly concentrated on the spectroscopy of theeven-even 64 Ge nucleus, using the recoil mass sepa-

w xrator technique 2,5,6 or recently the in-beam meth-w xods 7 , resulted in relatively detailed spectroscopic

information. Up to now, no experimental informationexisted on excited states in the odd-odd NsZneighbour 66As. The only measured property charac-terizing this nucleus was the b-decay half-life of

w x Ž .T s95.77"0.28 ms 8 . The deduced log ft1r2

value of about 3.5 points out that the 66As nucleus isa good candidate to study the superallowed b decay.

w xAs has been shown recently 9–12 , g-decay spec-troscopic studies may be efficiently performed em-ploying high- and intermediate-energy fragmentationreactions of heavy ions and projectile fragment sepa-

Žrators. The advantage of this method despite produc-tion cross-sections for NsZ nuclei being lower than

.in fusion-evaporation reactions is the in-flight iden-Ž . Ž .tification of atomic mass A , charge q , and atomic

Ž .number Z of ions reaching the implantation detec-tors. This method, however, is applicable only tonuclear states which have half-lives comparable withor longer than the time of flight through the separa-tor system, which is of the order of 1ms. Thein-flight identification can also be applied forfusion-evaporation reactions at recoil mass separa-

w xtors 6 but never gives such undisturbed informationof the mass and charge.

A search method sensitive to microsecond iso-mers was applied for the first time in a series of

w x 112experiments 9,10 with a Sn 63A MeV beam,

resulting in the detection of over forty known iso-mers. The principle of this experimental technique isbased on the time correlation between implantationof the identified fragment and the detection of g

w xradiation as described in 9 .Among other results, the signature for the decay

of 66 mAs was observed in an experiment using a112 w xSn beam 9 . The decay properties were measuredin a similar experiment but with a better suited,

78 Ž .neutron deficient Kr 73A MeV beam. Althoughthe main objective of this experiment was to search

w xfor new proton-rich isotopes 13 and to perform thew xspectroscopy of b-delayed protons 14 , the use of g

detectors allowed the simultaneous measurement ofisomeric decays with the same collection of im-planted nuclei. The description of the experimentaldetails concerning spectrometer settings and physics

w xresults are given in Refs. 13,14 .For the g-decay studies, there was a set of five

Žhigh-efficiency germanium detectors four in crossgeometry, fifth perpendicular to the others placed in

.the beam line axis mounted around the implantationsilicon-detector telescope. The full-energy-peak effi-ciency as a function of photon energy was deter-mined with standard calibration sources and reachedfor the whole detection system es 7% in maximumat 100 keV and amounted to es 2.7% at 1 MeV.

Five time-amplitude converters were used to mea-sure the time between a heavy-ion signal from the

Ž .first telescope detector DE1 and a g-ray signalfrom any of the germanium detectors allowing half-life measurements of isomeric states, but also provid-ing a relative timing between the g detectors. An

Fig. 1. g-ray energy spectrum registered in coincidence withimplanted 66As fragments.

( )R. Grzywacz et al.rPhysics Letters B 429 1998 247–253 249

important role of the g-detection setup was to con-firm the identification of the implanted nuclei usingknown isomers in 69,71Se.

For most of the counting time, the ALPHArLISE3spectrometers were optimized for the transmission ofNsZ and more exotic nuclei around Zs32. Forthis reason, 66As was one of the most frequently

w x Žimplanted fragments 13 173000 ions for 96 hoursat an average current of about 33 pnA primary

.beam . This was sufficient for a detailed g spec-troscopy of isomers discovered in this nucleus, bymeans of gyg coincidence measurements and anal-ysis of the time correlations.

The measured energy spectrum of the g radiationcorrelated with implanted 66As33q ions within a 50

ms time range is presented in Fig. 1. It contains afew strong and weak g lines, which belong toseveral decay branches. Possibly the strong ones,including the 115, 125, 394, 837, and 1552 keV linesestablish one cascade. The other lines, the 267, 670,963, and 1007 keV lines, belong to the satellitetransitions. As explained further in text all of theseg-energies can be arranged in a decay scheme, whichis proposed on Fig. 2a. There are also 511 keV and1460 keV lines in the spectrum, which are recordedin coincidence with 66As ions. They represent ran-dom coincidence events with background radiation.The decay spectra of some of the above-mentionedlines exhibit two different patterns, which show theexistence of metastable states with different half-

66 m Ž .Fig. 2. Proposed decay scheme of As, with level order, spins and parities deduced from experimental data a. , and with assignmentsŽ .based on the interpretation of the isomers as a 2qp proton-neutron excitation b. . Due to the fact that the order of the 837-394, 963-267 and

670-1007 cascades has not been determined experimentally, the respective levels are given as dashed lines. All intensities are normalized toŽ .that of the 837 keV transition. On the left-hand side of scheme a. , the isomeric ratio in % is given.


66 .Fig. 3. Determination of the half-lives of the two isomers of As. a Decay of the 1552 keV line: The value from the least-squares fit of anŽ . .exponential function is T s 17.5"1.5 ms. b Decay of the 125, 394, 837 keV lines gated by the 1552 keV line: A value of T s1r2 1r2

Ž . .1.6"0.5 ms is obtained from the fit. c Decay of the 125, 837, 394 keV lines: The value for the shorter-lived isomer, deduced from a twoŽ . Ž .component least-squares fit, is T s 2.1"0.6 ms see text .1r2

lives. Examples of time distributions of detected g

rays after fragment implantation are given in Fig. 3.Ž .For the 1552 keV line Fig. 3a but also for the 115,

670, and 1007 keV transitions, one observes a distri-bution which may be well described by a one-com-ponent exponential function, while for the others,

Žfast and slow decaying components are visible see.Fig. 3c . This is interpreted as the decay of an

isomeric state, which is populated independently inŽ .the reaction fast decay component and by the decay

of an isomer lying above this state and having alonger half-life. The least-squares fit of an exponen-

m1 Ž .tial curve gave a half-life of T s 17.5"1.5 ms1r2

for the one-component decay. The 1552 keV line hasbeen used for the fit, having the high intensity, zerobackground contribution and no time-energy depen-

Ž .dance ‘‘walk’’ .

The half-life of the shorter-lived isomer may bedetermined in two different ways. One is to evaluateat the distribution of time differences between the

Ž125, 394, 837 keV lines with two decay compo-. Ž .nents and the 1552 keV one component decay ,

which feeds the isomeric level deexcited by theabove lines. The least-squares fit resulted in a half-life

Ž . Ž .of 1.6"0.5 ms Fig. 3b . Another way is to anal-yse the time distribution between the implanted 66Asions and the 125, 394, 837 keV lines. Then, one hasto fit a function describing the decay of a state whichis fed directly and from the decay of a long livedone:

y1 yl t yl t1 2F t sA) l yl ) l ) e yeŽ . Ž . Ž .2 1 1

qB) eyl 2 t 1Ž .


Here l , l correspond to the decay constants of the1 2

higher- and lower-lying isomer, respectively. In thisŽ m1 .case the half-life of the long-lived state T was1r2

kept as a constant parameter. This procedure gave am2 Ž .half-life of T s 2.1"0.6 ms. The weighted av-1r2

Ž .erage of both values is 1.9"0.5 ms.The presented on Fig. 2a g-decay sequence is also

supported by the analysis of the gyg coincidencesŽ .Table 1 . From prompt and delayed coincidencesobserved between the lines, we propose the maindecay sequence consisting of two groups of transi-

Žtions: The first one 125, 394, and 837 keV transi-. Žtions depopulating the first isomeric level T s1r2.1.9 ms located at an excitation energy of 1358 keV,

Ž .the second one 115 and 1552 keV transitions de-Žcaying from the second isomeric level T s 17.51r2

.ms located at 3024 keV. The four weaker g linescan be arranged into two cascades. One can remarkthat the sum of the 1007 keV and 670 keV transi-tions is equal to the one of 1552 and 125 keVtransitions. Likewise, the 267 keV and 963 keV canbe placed in parallel to the main cascade formed bythe 394 keV and 837 keV g rays. However, at thisstage we cannot establish the exact order of sometransitions. For the full reconstruction of the feedingpattern and the decay sequence, the information onthe intensities of detected lines have to be used. Theintensity balances have to be preserved within theexperimental uncertainties, which yields informationon the internal conversion of the 115 and 125 keVtransitions which are considered as the primary tran-sitions depopulating the respective isomer. Values of

Table 1Experimental results for energies and relative intensities of g linesŽ . Ž .first two columns , decay half-lifes third column , and coinci-

Ž .dence relations last two columns . The half-life of 1.9r17.5 msŽ .means that a two-component decay has been observed see text

Energy Intensity Half-life Prompt DelayedŽ . Ž . Ž .keV % ms

Ž . Ž .1552.0 2 67 6 17.5 115.2 125.4, 837.1, 394.2Ž . Ž .1007.0 3 27 5 17.5Ž . Ž .963.3 3 14 4 1.9r17.5Ž . Ž .837.1 1 100 7 1.9r17.5 125.4, 394.2 1552.0, 115.2Ž . Ž .670.3 3 22 4 17.5Ž . Ž .394.2 1 98 7 1.9r17.5 125.4, 837.1 1552.0, 115.2Ž . Ž .267.3 3 10 3 1.9r17.5Ž . Ž .125.4 1 51 7 1.9r17.5 837.1, 394.2 1552.0, 115.2Ž . Ž .115.2 1 41 7 17.5 1552.2 125.4, 837.1, 394.2

the total internal conversion of the transitions wereŽ . Ž .determined to be 1.3 4 and 0.7 3 , respectively.

These a values exclude E1 and M1 multipolarity,tot

but do not discriminate definitely between E2, M2,E3 and M3. The deduced strengths for M2, E3 andM3 clearly exceed the recommended upper limits of1, 100 and 10 W.u., respectively. Hence, we assignmultipolarity E2 to both the 125 keV and the 115

Ž .keV transitions. The resulting B E2 values are veryŽ . 2 4 Ž . 2 4small 5.4 14 e fm and 0.7 1 e fm , respectively.

The decay scheme presented in Fig. 2a is notunique, alternative solutions have a different order ofthe 837 - 394, 963 - 267, and 670 - 1007 keVtransitions.

As we did not measure the angular correlations ofthe g rays, it is impossible to determine precisely thespin values of the excited states. The 0q assignment

Ž . Ž .to the ground state comes from the log ft s3.49 1 ,Žcalculated using the measured half-life T s95.771r2

w x."0.28 ms 8 and the Q s9.592"0.050 MeVECw xvalue deduced from the Coulomb energy shift 15 ,

which is close to the value of Q s9.550"0.050ECw x Ž .MeV given by the NDS evaluator 16 . The log ft

s3.49 is characteristic for a superallowed 0q™0q

w xFermi transition 17 . As the daughter nucleus is theeven-even nucleus 66 Ge, a Jp s0q was assumed forthe 66As ground state.

ŽThe spin values reported in the level scheme Fig..3a have been determined assuming that transitions

Žcan be E1, M1 or E2 except the 115 keV and 125.keV transition which have been found to be E2 and

that spin values increase with the excitation energy.The existence of isomeric states cannot be under-stood if states with same spin values exist at lowerexcitation energy.

Based on the absolute efficiency calibration andafter correction for the in-flight losses the populationof the isomer at the target relative to the totalproduction of 66As was found to be 21"3% and8"4% for the 3024 keV and 1357 keV states,

q Ž q.respectively. The ionic half-lives T qs33 used1r2

for the determination of the losses due to the decayin flight were calculated by using the measured total

Ž .conversion coefficients a and the half-livestotŽ .T , with the formula valid for the fully stripped1r2

ions:

T q sT 1qa . 2Ž . Ž .1r2 1r2 tot


From the proposed level sequence, spin and parityŽ .values Fig. 3a , we can speculate about the possible

configurations of the states of the odd-odd NsZnucleus 66As. We have no experimental evidence for

Ž qa strong deformation of this nucleus, the B E2 ; 2 -q. 660 value measured in the even-even Ge is only 12

w xW.u. 18 . In a first attempt, one can consider theparticle excitations in the spherical f , p , p5r2 3r2 1r2

and g subshells, which have been observed in the9r2

neighbouring odd-A nuclei. This approach takes intoŽ .account i the excitation energy of one-particle exci-

tation observed in the neighbouring odd-N and odd-ZŽnuclei f and p states are located within 1005r2 3r2

keV from each other; the excitation energies of p1r2

and g states are larger around 1 MeV and 1.59r2. Ž .MeV, respectively , ii the coupling rules of the

neutron and proton angular momenta in odd-oddnuclei, which say that the state having the parallelcoupling of the intrinsic spins of proton and neutronis lowered in energy by the residual n-p interactionw x Ž .19,20 , iii the additional pairing energy expected

w xin the NsZ odd-odd nuclei 21 . We can thuspropose the following configurations to the stateswhose energy and spin are reported in Fig. 3b:

– The Ip s0q,Ts1 ground state is mixed fromthe p f n f , p p n p and p p n p con-5r2 5r2 3r2 3r2 1r2 1r2

figurations.– The Ips2q,Ts1 isobaric analogue state of

66 Ge is expected at 960 keV. We therefore assign the963 keV state to be Ips2q, Ts1, which fixes thesequence in the 963 - 267 keV cascade. The highermembers of the Ts1 ground state band, expected at

Ž q. Ž q.2175 keV 4 and 3655 keV 6 , are non-yrastand therefore not populated in the decay of a high-spin isomer.

– The 1231.3 and 1356.7 keV levels are assignedto be 3q with predominant configurationp p n p and admixtures from pn p f and3r2 3r2 3r2 5r2

p f n f , and to be 5q with predominant config-5r2 5r2

uration p f n f , respectively. The large hin-5r2 5r2

drance of the isomeric E2 transition is then under-stood from its two-particle character and theparticle-hole character of a f ™p transition.5r2 3r2

– The mixed 1q state, obtained from thep p n p and p f n f configurations, is ex-3r2 3r2 5r2 5r2

pected in the vicinity of the 3q and 5q states. Theintermediate state in the 394 - 837 keV cascade fromthe predicted Ip s3q state would be a good candi-

date, and this would locate it at 837 keV excitationenergy. However, the unusual large E2rM1 branch-ing ratio would favour a placement of this state at394 keV. This possibility must be also taken intoaccount, but reaches beyond the scope of our discus-sion in terms of simple particle excitation. In such acase, a 394.2 keV level is expected to have a similarconfiguration as the Ip s1q,Ts0 state, which isknown to be the ground state in 58 Cu. It is highlymixed with the proton-neutron configurations pn

p f , p2 , f 2 and p p . This choice is3r2 5r2 3r2 5r2 3r2 1r2

supported by the fact that the placemement of the1qstate at 837.1 keV would yield only a strength of0.003 W.u. for the 267.3 keV M1 branch, assuming10 W.u. for the 394 keV E2 branch.

– The second isomeric state at 3024 keV can beinterpreted in two ways. An Ip s9q assignmentwith configuration pn g2 implies Ip s7q for the9r2

state at 2909 keV, which in turn cannot belong to thesame multiplet but rather must have 4-quasiparticlecharacter in agreement with the observed E2 retarda-tion. This is due to the fact that the pn g2 two-body9r2

matrix element is strongly attractive for the stretchedIp s9q state, but about 1 MeV less bound forp q w x p yI s7 22 . An alternative I s7 assignment for

the isomer based on the pn g f configuration9r2 5r2

and consequently Ip s5y for the 2909 keV stateŽ .with a mixed pn g = f ,p ,p configura-9r2 5r2 3r2 1r2

tion would also account for the retarded E2 strengthand the observed decay pattern.

– The intermediate state in the 1007 - 670 keVp Ž q.cascade in either case could have I s 4,5 of

mixed configuration. Its position, however, cannotbe deduced safely from the available experimentalevidence.

To a large extent, the observed structure of ex-cited states in 66As can be explained in terms ofsimple two-quasiparticle excitations. The abovespeculations have to be confirmed or refined using

Žnew experimental information such as the definitive.multipolarity assignments . A comparison with shell

model calculations would then give quantitative re-sults of the residual proton-neutron interaction in thisNsZ nucleus.

It is worth mentioning that the 3024 keV isomerhas an excitation energy larger than the proton sepa-

Ž . w xration energy S s 2700"226 keV 23 . It seems,p

however, that the energy window is not large enough


to make a proton emission competing with the g

decay in view of the large angular momentum differ-ence.

To summarize, two isomers in 66As, E) s3024Ž . )keV with T s 17.5"1.5 ms, and E s13571r2Ž .keV with T s 1.9 "0.5 ms, were found and1r2

studied by means of g spectroscopy using a frag-mentation reaction at intermediate energy. The g

spectrum of the isomeric decays has been measured.From the analysis of this spectrum as a function oftime and from the coincidence relationships betweenthe g transitions, we have identified states involvedin the decay of the two isomeric levels.

The presented results prove that a high-resolutionspectroscopic measurement can be done at a frag-ment separator with a relatively small number ofnuclei and using minimum equipment. Using a moresophisticated g-array it would be possible to measureangular correlations in order to deduce spins of thelevels. By using the now known g rays one couldrelatively easily tag the recoiling fusion-evaporationproducts at the final focus of the currently operatingrecoil spectrometers and obtain the information aboutexcited states lying above the isomers. This is proba-bly also an efficient way to determine spins ofexcited states by means of gyg angular correla-tions. In turn, information on higher levels, and inparticular the yrast states, would be very helpful inunderstanding the relative population of the isomericstates in the fragmentation reaction. It does nothappen so often, that there are two microsecondisomers in one nucleus. The analysis of the relativepopulation of both, measured as a function of inci-dent projectile type and energy or with differenttargets can reveal information about the spin andexcitation of the fragmentation reaction products inthe phase after particle evaporation. 66As is an exotic

nucleus but using the presented method and results,such a study can be rather easily performed.

We acknowledge the efforts of the cyclotron staffand technical support of GANIL. The authors would

˙like to thank professor Jan Zylicz and dr. FredericNowacki for the very helpfull discussions during thepreparation of the article. This work has been par-tially supported by IN2P3 and by the Polish Commit-

Ž .tee for Scientific Research KBN within the project2 P03B 039 13.

References

w x Ž .1 G. Colo et al., Phys. Rev. C 52 1995 R1175.w x Ž .2 C.J. Lister et al., Phys. Rev. C 42 1990 R1191.w x Ž .3 A. Petrovici et al., Phys. Rev. C 53 1996 2134.w x4 D. Sohler et al., to be published.w x Ž .5 S.S.L. Ooi et al., Phys. Rev. C 34 1986 R1153.w x Ž .6 P.J. Ennis et al., Nucl. Phys. A 535 1991 392; A 560

Ž . Ž .1993 1079 E .w x7 G. de Angelis et al., Int. Symposium on Exotic Nuclear

Shapes, Debrecen, Hungary, May 1997.w x Ž .8 R.H. Burch Jr. et al., Phys. Rev. C 38 1988 1365.w x Ž .9 R. Grzywacz et al., Phys. Lett. B 355 1995 439.

w x Ž .10 R. Grzywacz et al., Phys. Rev. C 55 1997 1126.w x Ž .11 P.H. Regan et al., Ac. Phys. Pol. 28 1997 431; C. Chandler

Ž .et al., Phys. Rev. C 56 1997 R2924.w x12 M. Pfutzner et al., Nucl. Phys. A, in print.¨w x Ž .13 B. Blank et al., Phys. Rev. Lett. 74 1995 4611.w x Ž .14 B. Blank et al., Phys. Lett. B 364 1995 8.w x Ž .15 W.E. Ormand, Phys Rev. C 55 1997 2407.w x Ž .16 M.R. Bhat, Nucl. Data Sheets 61 1990 461.w x17 A. Bohr, B.R. Mottelson, Nuclear Structure, vol. 1, W.A.

Benjamin Inc., 1969, p. 52.w x Ž .18 S. Raman et al., At. Data Nucl. Data Tables 36 1987 1.w x Ž .19 L.W. Nordheim, Phys. Rev. 78 1950 294.w x Ž .20 M.H. Brennan, A.M. Bernstein, Phys. Rev. 120 1960 927.w x Ž .21 A.L. Goodman, Adv. Nucl. Phys. 11 1979 263.w x Ž .22 H. Grawe et al., Z. Phys. A 358 1997 185.w x Ž .23 G. Audi, A.H. Wapstra, Nucl. Phys. A 595 1995 409.

18 June 1998


Rotating vacuum wormhole

V.M. Khatsymovsky 1

Budker Institute of Nuclear Physics, NoÕosibirsk 630090, Russia

Received 15 January 1998; revised 1 April 1998Editor: P.V. Landshoff

Abstract

We investigate whether self-maintained vacuum traversible wormhole can exist described by stationary but nonstaticmetric. We consider metric being the sum of static spherically symmetric one and a small nondiagonal component whichdescribes rotation sufficiently slow to be taken into account in the linear approximation. We study semiclassical Einsteinequations for this metric with vacuum expectation value of stress-energy of physical fields as the source. In suggestion thatthe static traversible wormhole solution exists we reveal possible azimuthal angle dependence of angular velocity of the

Ž .rotation angular velocity of the local inertial frame that solves semiclassical Einstein equations. We find that in theŽ .macroscopic in the Planck scale wormhole case a rotational solution exists but only such that, first, angular velocity

depends on radial coordinate only and, second, the wormhole connects the two asymptotically flat spacetimes rotating withangular velocities different in asymptotic regions. q 1998 Published by Elsevier Science B.V. All rights reserved.

1. Introduction

The possibility of existence of static spherically-symmetrical traversible wormhole as topology-nontrivialw xsolution to the Einstein equations has been first studied by Morris and Thorne in 1988 1 . Since that time much

Ž w x.activity has been developed in studying the wormhole subject see, e.g., review by Visser 2 . Rather interestingis the possibility of existence of self-consistent wormhole solutions to semiclassical Einstein equations.Checking this possibility requires finding vacuum expectation value of the stress-energy tensor as functional ofgeometry and solving the Einstein equations with quantum backreaction, i.e. with such the induced stress-energyas a source.

w xRecently some arguments in favour of this possibility has been given. In Refs. 3–5 the gravity inducedvacuum stress-energy tensor in the wormhole background has been found to violate energy conditions just as it

w x Žis required for this tensor itself be the source for such the wormhole metric 1,6 . We consider physical vacuum. w xof spin 1 and 1r2 massless fields in these papers . In Ref. 7 self-consistent spherically-symmetrical wormhole

solution has been found numerically for the quantised scalar field vacuum playing the role of a source forgravitation.


0370-2693r98r$19.00 q 1998 Published by Elsevier Science B.V All rights reserved.Ž .PII: S0370-2693 98 00448-1

( )V.M. KhatsymoÕskyrPhysics Letters B 429 1998 254–262 255

ŽIn the case if the wormhole connects the two distinct regions of the same Universe evidently, in this case the.wormhole cannot be spherically symmetric some manipulation with this wormhole leads to arising the closed

w xtimelike curves 6 i.e. converting it into time machine. We, however, consider the wormhole joining the twodistinct asymptotically flat spacetimes. This system still possesses rather unusual property: a material that

w xthreads it should violate weak energy condition averaged along radial null geodesic 6 . The only known suchthe material is the vacuum of physical fields in the wormhole background like Casimir vacuum between the

w xconducting plates 6 . Thus, motivated by the above works, let us assume that spherically symmetric staticwormhole maintained by physical vacuum in it’s own background does exist, and ask about small modificationsof such the solution.

The problem of existence of self-consistent static wormhole is a particular case of the more fundamentalproblem of self-consistent solutions to semiclassical Einstein equations with vacuum expectations of stress-en-

w xergy tensor of physical fields as a source. In Ref. 8 it has been found that such the problem linearised overŽ .metric perturbations off Minkowski spacetime gives solutions apart from some unphysical ones coinciding

with those for classical gravity wave problem. If, however, topology is not Minkowski one as in the wormholecase at hand, new solutions can appear such as static wormhole itself. The natural next step may be the searchfor stationary but nonstatic topology nontrivial solution, namely rotating vacuum wormhole. In the case of slowrotation one simply adds small nondiagonal polar-angle-time component of metric to the self-maintained staticwormhole metric:

2 2 2 2 2 2 2ds sexp 2F dt ydr yr du qsin u df q2hdf dt , 1Ž . Ž .Ž .Ž .where h r,u has the sense of angular velocity of the local inertial frame and will be called the angular velocity

Ž . Ž .of rotation in what follows. The static metric F r , r r is assumed to exist as a self-consistent solution ofstatic semiclassical Einstein equations, and solution for h is to be found. One can take h arbitrarily small inorder to limit oneself to the theory linearised in h. In practice, in order that the quadratic in h terms in theRiemann tensor could be disregarded, the space derivatives of h should be negligible as compared to thederivatives of the static part of metric in the scale of typical wormhole size. Note that in the linearapproximation the only component of Einstein equations being new compared to the static case is the tf one, asit follows from symmetry considerations. The role of the source is played there by the induced vacuum energy

Ž .flow angle-time stress-energy component .In the given note we show that the rotational solution exists at least in the case when the coefficient at the

w xWeyl term in the induced stress-energy is large and if one can neglect other terms. As noted in Refs. 4,5 , thisŽ .corresponds to the wormhole of macroscopic size that is, large in Planck units . The rotational solution which

we argue should exist is such that h depends on radial distance r only and has different asymptotic limits in thetwo asymptotically flat spacetimes connected by wormhole. This, in particular, means that these two spacetimescannot be ‘‘glued’’ together in asymptotic region, so that this wormhole cannot be considered as that connectingthe two regions of the same asymptotically flat spacetime.

As for the general case when we do not assume the Weyl term to dominate, we find the two kinds ofazimuthal angle u dependence of angular velocity h for which the radial and angular variables r, u areseparated and the Einstein equation with quantum backreaction for h reduces to that for the function of purelyr. One possibility is the above mentioned angle-independent angle velocity; another one is proportional to cosu

velocity.

2. Classical rotation

By classical we mean rotation considered without taking into account corresponding quantum backreaction,Ži.e. induced vacuum energy flow. However, it is implied that the static wormhole problem is already solved the

. Žstatic metric is found with taking into account corresponding backreaction. We show in the linearised in h

( )V.M. KhatsymoÕskyrPhysics Letters B 429 1998 254–262256

.theory used throughout the paper that only solution for h not depending on u is physically acceptable such thatit has different finite limits at r™q` and at r™y`.

The tf Einstein equation of interest can be conveniently written using the tetrad components introduced likea a m w xthe following basic 1-forms v se dx 9 :m

v 0 sexp F dt , v1 s dfqhdt rsinu , v 2 sdr , v 3 srdu . 2Ž . Ž . Ž .w xTaking the expressions for the Riemann tensor presented in Ref. 9 we find for the equation of interest:

exp F 1 E E h 1 1 E E hŽ .t 4 3yR sy R s exp yF r sinuq exp yF sin u s0. 3Ž . Ž . Ž .10 f 3 2ž /rsinu Er Er 2 r Eu Eu2 r sin u

Remind that we neglect the induced vacuum energy flow T t in this section. Separating the variables,f

Ž . Ž .hs f r Y u , we find hypergeometric function for the angle dependence:

1ycosuY;F a,3ya;2; , 4Ž .ž /2

asconst. This function diverges at usp unless asyk, ks0,1,2, . . . ; in the latter case it reduces to theŽ3r2.Ž .Gegenbauer polynomial C cosu . Then the radial part satisfies the equationk

XX4 2r exp yF f sk kq3 r exp yF f . 5Ž . Ž . Ž . Ž .Ž .

By asymptotical flatness F™0, r™r at r™"` and thus f;ryky3 at large distances. Another solution,f;r k, should be discarded at kG1 as unphysical one. Therefore if we choose some large L)0, we shall have

< < Ž yky3 .solutions at r )L parametrised by two constants C f™C r at r™"` , whereas in the intermedi-" "

< < Ž .ate region r -L the Eq. 5 is regular in the wormhole geometry and has a solution parametrised by twoconstants C , C . The four constants C , C , C , C are subject to four uniform equations which are matching1 2 1 2 q yconditions for f and it’s derivative f X at rsqL and at rsyL. Requiring for this system to have nonzero

Ž . Ž .solution, i.e. zero determinant, we get some constraint on the already known metric functions r r , F r .Ž .Therefore the set of possible solutions for f r at kG1 has zero measure as compared to the set of possible

Ž .static solutions r, F . At ks0 the Eq. 5 can be easily integrated and leads to physically admissible h notdepending on u and being monotonic function of r which has finite but different limits at r™"`:Ž . Ž .h y` /h q` . Because of the latter circumstance the two asymptotically flat spacetimes connected by the

Žwormhole channel cannot be ‘‘glued’’ together in asymptotic region glueing at a shorter distance would spoil.spherical symmetry of the background static solution described by two functions r, F so we cannot derive

from such the rotating wormhole the wormhole connecting the two distinct regions of the same asymptoticallyflat spacetime.

3. Quantum backreaction and angle dependence of rotation

Now consider possible dependence of the angle velocity h on the azimuthal angle u that could solve theŽ .semiclassical Einstein equation with backreaction . Here we show that the only two versions of the angle

Ž .dependence for this equation to be solved by separation of variables are the following ones: hs f r orŽ .hs f r cosu .

Evidently, the problem reduces to studying the angle dependence of T for a given angle dependence of h.10

Given any physical field, we should solve equations of motion for this field in curved spacetime and sumvacuum contributions into stress-energy from all the eigenmodes. The resulting expression can be regularisedby, e.g., covariant geodesic point separation and renormalised by subtracting the divergent parts known for

w xphysical fields 10 . Choosing such separation in the radial direction we avoid discussing the renormalisationissue as far as the angle dependence is concerned.


Let us consider general structure of the equations of motion and stress-energy for arbitrary field in the metricŽ . Ž . Ž .1 and illustrate this by the case of massless fields of spin 1 electromagnetic and 1r2 neutrino . Most naturalto display effect of rotation in the stationary axisymmetrical nonstatic metric is to use the complex Newman-

w xPenrose formalism in analogy with that applied to Kerr metric 9 . In particular, up to the linear order in h, wem m m ) m Ž .can choose isotropic tetrad of real l , n and complex m , m asterics means complex conjugation of

m m m Ž ) m. m mwhich l , n are tangential to some geodesics and m and thus m are orthogonal to l , n . Besides that,normalisation can be chosen such that l n m s1, m mm) sy1. These are the distinctive properties of them m

Newman-Penrose tetrad which can be written with the help of the derivatives over directions as

1m ml E sexp y2F E qexp F E yhE , n E s E yexp F E yhE ,Ž . Ž . Ž .m t r f m t r f2

y1 y1y1 y1m ) m' 'm E s r 2 E q i sinu E , m E s r 2 E y i sinu E . 6Ž . Ž . Ž .Ž . Ž .m u f m u f

Calculation gives the following values for the standard 12 complex spin-connection coefficients k , n , s , l, ´ ,w xD , m, t , p , a , b , g 9 :

i r X

ksns0, ss2exp y2F lsy2´s exp y2F h sinu , Ds2msy exp yF ,Ž . Ž . Ž .u2 r

i 1 cotu iytsps exp yF h rsinu , yasbs y exp yF h rsinu ,Ž . Ž .r r' ' 'r2 2 2 2 4 2

iX1

gs F exp F y h sinu 7Ž . Ž .u2 8

where subscript on h means corresponding derivative; prime on r, F means derivative over r. The mainfeature of the covariant equations of motion for an arbitrary physical field is therefore occurence of h in theform h sinu , h sinu and hE there. In the diagrammatic language, h-field-field vertex is combination of theser u f

expressions. Besides that, in the Newman-Penrose formalism operators acting on the angle variables appear inthe form of spin raising and lowering operators

iE iEf fqLL sE y qscotu , LL sE q qscotu , 8Ž .s u s usinu sinu

0FsFs , s being spin of the field. These operators simplify in the basis of spin spherical harmonics0 0l Ž . w x q l lproportional to the elements of rotation matrix D f,u ,0 11 : LL and LL transform D to D andsm s s sm sq1,m

l ŽD , respectively. In this basis the expression for T turns out to be a value of spin weight ss"1 andsy1,m 10. l Ž l .ms0 , that is, combination of D or D for different l. Quantum contribution to T can be viewed as10 y1,0 10

some loop diagram with external T - and h-legs. Since h can be expanded in the Legendre polynomials10

P ;Dk , we can take Dk as probe function for the angle dependence of h. Then the expressions h sinu andk 00 00 r

h sinu appearing in the vertex on h-line are combinations of the values Dkq1 and Dky1. This corresponds tou 10 10

the angular momenta kq1 and ky1 flowing through the h-line. It is less evident but shown at the end of thissection that the vertex hE corresponds to the combination of angular momenta kq1, ky1, ky3, . . . . Byf

conservation of angular momentum the T also should be combination of Dkq1y2 n, ns0,1,2, . . . . In10 10Ž 1 2 .particular, at ks0,1 the only term D or D remains and T factorises into the functions of r and of u .10 10 10

Moreover, since Dkq1 ;C3r2 just for ks0,1, the same u-dependence also factors out in the LHS of Einstein10 kŽ . Ž .equation. Therefore we conclude: hs f r or hs f r cosu solves for the u-dependence of semiclassical

Einstein equation.Finally, let us illustrate the above said by the examples of electromagnetic and neutrino fields; for more detail

w xon the Newman-Penrose description of these fields see Ref. 9 . Electromagnetic field is described by three


Ž .complex functions f , f , f ; for our choice of complex tetrad 6 these are related to the electromagnetic field0 1 2

strength tensor F as follows:mn

i2 f sF qhF qexp F F q F qexp F F ,Ž . Ž .0 tu uf ru tf rf sinu

i22 f syr exp yF F qhF qF ,Ž . Ž .1 tr rf uf sinu

i2 f sy F qhF qexp F F q F yexp F F . 9Ž . Ž . Ž .Ž .2 tu uf ru tf rf sinu

The eight real Maxwell equations can be recast into the following four complex ones:

E f yLL f shE f , E f qLLqf syhE f ,y 1 1 0 f 1 q 1 1 0 f 1

exp 2F f y fŽ . 0 2qE f q LL f s i h sinuy ih exp F sinu f yhE f ,Ž .q 0 0 1 u r 1 f 02 2r

exp 2F f y fŽ . 0 2E f y LL f syi h sinuq ih exp F sinu f qhE f , 10Ž . Ž .y 2 0 1 u r 1 f 22 2r

Ž .where E 'exp F E .E , iE sv being energy. Each mode should be normalised so that its full energy" r t t

T tr 2exp F drsinu du dfŽ .H t

exp 2FŽ .)

) ) )ssinu du dfexp yF dr f f q f f q2 f f y2hsinu Im f y f f 11Ž . Ž . Ž .0 0 2 2 1 1 0 2 12½ 5r

be equal to the vacuum value vr2 and then substituted into the expression for the stress-energy componentstudied,

exp F exp yFŽ . Ž .)tT s T sy2 Im f y f f . 12Ž . Ž .10 f 0 2 13ž /rsinu r

Analogously, massless fermion field is described by two complex values g , g obeying the field equations1 2

exp F iŽ .E g q LL g s sinu g h yrh g qhE g ,Ž .y 1 1r2 2 1 u r 2 f 1r 4

exp F iŽ .qLL g yE g s sinu yrh g yg h qhE g . 13Ž .Ž .1r2 1 q 2 r 1 2 u f 2r 4

Ž .Expression for the energy of each mode takes the form on the field equations

T tr 2exp F drsinu du dfs sinu du dfexp yF drP4v g ) g qg ) g , 14Ž . Ž . Ž .Ž .H Ht 1 1 2 2

while the stress-energy component of interest is

exp yFŽ .) )

) q q ) ) )T s g LL g q LL g g yg LL g y LL g g yE g g yg gŽ .Ž .Ž .½10 1 1r2 1 1r2 1 1 2 1r2 2 1r2 2 2 u 1 1 2 23r

X X ) ) ) )q rF yr g g qg g q2 rexp yF g E yhE g yg E yhE g . 15Ž . Ž . Ž .Ž . Ž . Ž . 52 1 1 2 2 t f 1 1 t f 2


Solutions to the above equations of motion can be constructed iteratively. In zero order one takes

Ž .0 lE RDf - y1,m0lf (y l lq1 RDs , 16Ž . Ž .1 0 m� 0 � 0lf2 E RDq q1,m

Žwhich are the well-known TE-modes for the electromagnetic field TM-modes follow by multiplying this by' .is y1 and

Ž . l0 Z D1 q1r2,mg1 s , 17Ž .lgž /2 ž /Z D2 y1r2,m

Ž . Ž .for the neutrino field. The R and Z , Z are some radial functions. Substituting Eqs. 16 and 17 into the RHS1 2Ž .of the equations of motion we find for the first O h correction an expression of the type

Ž .1 j j j. . . D . . . D . . . Df y1 ,m y1,m y1,m0l ky1 j l kq1 j l k jm j j jf . . . D . . . D . . . DŽ .s i y1 q q m ,1 Ý 0 m 0 m 0 mž / ž / ž /m 0 y m m 0 y m m 0 y mž / � 0 � 0 � 0j j j jf2 . . . D . . . D . . . Dq1 ,m q1,m q1,m

18Ž .

Ž .and quite analogous one for the fermion field. The dots mean real factors which do not depend on u , f and m.Ž .The linear order in h of interest comes from interplay in the bilinear T between zero order 16 and first10

Ž . lcorrection 18 . By properties of 3j-symbols and rotation matrix elements D summation over m just yieldssmkqp Ž .combination of the harmonics D , ps1,y1,y3, . . . . Important is that the last term in 18 which stems10

from hE operator in the equations of motion is representable as combination of 3j-symbols asf

l k j l kq1y2 n jm s A , 19Ž .Ý nž / ž /m 0 ym m 0 ymnG0

l Ž j .)where A does not depend on m. In T this and other terms enter multiplied by D D and summedn 10 0 m "1,m

over m thus giving just combination of Dkq1y2 n, nG0.10

4. Macroscopic wormhole and radial dependence of rotation

Usually, if one does not assume existence of fundamental scales in the theory other than the Planck scale oneexpects the typical wormhole size be of the Planck scale too. However, a new scale can exist connected withcoefficient of the Weyl term in the effective action. This coefficient is subject to renormalisation in both infraredŽ .if massless fields are present in the theory and ultraviolet regions. Possible large value of this coefficient canenable existence of the wormhole of macroscopic size.

Here we argue that if the Weyl term coefficient is large and one can disregard other terms in the effectiveaction then the conclusion concerning the existence of rotating wormholes resembles that for the case ofclassical rotation in Section 2.

The effective gravity Lagrangian density with taking into account the Weyl term can be written asŽ 2 .y1 mnlr w x mnlrproportional to Rq 2m C C 12 . Up to the full derivative, the Weyl term C C is equivalentmnlr mnl r

1mn 2Ž . Žto 2 R R y R . We calculate the latter up to the second order in h required to get the first order in themn 3

. w xequations of motion using Riemann tensor given in Ref. 9 in the tetrad components. Varying in h gives thedesired tf-component of the Einstein equations. Consider both versions of u-dependence of h found in Section


Ž . Ž .3, h;1 and h;cosu , and introduce new variable z via dzsexp yF dr and the function rsrexp yF .˜Ž . Ž Ž ..For h not depending on u , hs f r s f r z , the result reads

1 110 24 4 4 2 2r f exp 2F s r r f q r y1 q r r r f , 20Ž . Ž .˜ ˜ ˜ ˜ ˜ ˜ ˜Ž . Ž . Ž .z z z z z z3 3z z2 4½ 5m r z z

Ž . Ž .subscript z means differentiation over z . For hs f r cosu we find

r 4 f exp 2F y4exp 2F r 2 fŽ . Ž .˜ ˜Ž .z z

1 110 2 34 84 4 2 2 2s r r f q r q r ry r f q yr rqr q8 f . 21Ž .˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜Ž . Ž . Ž .z z z z z z z z3 3 3 3z2 4½ 5ž /m r z z

The infrared contribution to the coefficient my2 goes from the massless fields. In the considered case of my2

2 y2 2 Ž 2 .y1 w xlarge in the Planck scale the typical wormhole size r is defined just by m as r s 3m 4,5 . For0 0

example, in the vacuum of N spin 1 and N spin 1r2 massless fields we have1 1r2

G 120p L2y12 23m sr s 4N qN ln , 22Ž .Ž . Ž .0 1 1r2 ž /120p G 4N qN1 1r2

L being infrared cut off.Ž .Consider first Eq. 20 which upon integrating both parts and denoting f 'g reduces to the second orderz

one:

110 22 4 4 4 2 2m exp 2F r gqCsr r g q r y1 q r r r g , 23Ž . Ž .Ž .˜ ˜ ˜ ˜ ˜ ˜ ˜Ž .z z z3 3z4r z

where Csconst. Asymptotical r™"` form of this equation reads

d 1 d C4 2r gym gs . 24Ž .4 4dr drr r

y4 Ž y6 .The general solution is the sum of a particular one which behaves as g;r qO r at r™"` andarbitrary combination of the two independent solutions to the uniform equation. Of the latter two oneexponentially grows at r™q` or at r™y` and should be omitted as unphysical solution while another one

Ž . Ž . Ž . Ž .proportional to exp ymr at r™q` or exp mr at r™y` should be kept. Therefore, if we choose, as< <in Section 2, some large L)0 we shall have physically acceptable solutions at r )L parametrised by three

< < Ž .constants, one of which is C. Meanwhile, in the intermediate region r -L the Eq. 23 is regular in thewormhole geometry and has solution parametrised by maximal set of three constants, one of which is C. Theoverall set of five constants is subject to four uniform equations which are matching conditions for g and for it’sderivative at rsqL and at rsyL. This defines all five constants up to an overall factor. Note that imposing

q` Ž Ž . Ž ..additional condition H gdzs0 that is, h y` sh q` is, generally speaking, contradictory since it wouldy`

Ž . Ž .be condition not on a freely chosen constant, but on the already defined static metric F r , r r .Ž . Ž .Next consider Eq. 21 which has asymptotic form at Fs0, rsr

2d 1 d 1 d 1 d4 4 4 4r r fs r r f . 25Ž .4 2 4ž /dr dr dr drr m r

< <Assuming this form at r )L we find

d 1 d C "q44 2 "r fym fsC rq , 26Ž .y14 4dr drr r


q q Ž y y . Ž .where C , C C , C are some constants which parametrise the solution at r)L at r-yL . To gety1 q4 y1 q4

physical solution we put C " s0. Of the two solutions of uniform equation we discard exponentially growingy1

one in each region r)L or r-yL and retain exponentially falling off another solution. Thus, in each regionr)L or r-yL the solution is specified by two constants. At the same time, the regular fourth order

< <differential equation has solution parametrised by four constants at r -L. The overall number of constants iseight. These should ensure validity of eight matching conditions for f , f X, f XX and f XXX at rs"L. Thedeterminant of this uniform system should be zero. This imposes a constraint on the already known static metric

Ž .F , r. Therefore the subset of rotating wormhole solutions with hs f r cosu should have zero measure w.r.t.Ž . Ž .the set of spherically symmetrical static wormhole solutions F r , r r .

5. Conclusion

Ž .We have shown that if the coefficient at the Weyl term is large infrared cut off is large and one can discardŽ . Ž .other terms in the effective action then the rotation existing for any static wormhole background F r , r r is

that which proceeds with the angular velocity h not depending on the azimuthal angle u and having theŽ . Ž .different finite limits h q` and h y` in the asymptotic region r™"`. We see that despite that the

structure of the equations is drastically changed because of enhancing the maximal order of derivatives upontaking into account vacuum polarisation, the result practically does not differ from that for the classical case ofSection 2.

Note that the sign of infrared divergent coefficient my2 at the Weyl tensor squared in the effective action isŽ . y2crucial for existence or, rather, nonexistence of more rotational solutions in our case. Were m substituted by

negative value, the equations above would have oscillating instead of monotonic exponential solutions, and wewould not have to omit some of them as unphysical ones. Then the general solution of interest would beparametrised by more constants, and the set of such solutions would be larger.

Also we can say that if we denote m2 'x and extend the equations to arbitrary real x, the problem will besingular at xs0.

In the case if the infrared logarithm is not large we do not have macroscopic vacuum wormhole, and thefollowing interesting question arises: whether microscopic wormhole can rotate so that it would have the

Ž .macroscopic ‘‘tail’’ of rotation when h falls off in power law in asymptotic region . Answering this questionimplies rather complicated problem of calculation and analysis of the terms in stress-energy other than the Weylterm.

The solution for h described is defined up to a constant factor which can be taken arbitrarily small: we canrotate the wormhole arbitrarily slowly so that quadratic in h terms are negligible. That is, linear approximationused for h is physically attainable. In this framework the diagonal components of Einstein equations whichdescribe spherically symmetric static wormhole remain unchanged so we can rely upon the already studiedconditions of existence of the static wormhole. To resume, in the assumption of existence of the spherically

Ž .symmetric vacuum-supported static traversible wormhole we establish possibility of some at least slowrotation of the latter. One can say that, probably, vacuum can rotate! What may be the consequences of suchrotation for the wormhole spacetime structure remains to be studied.

w xWhen this paper was under consideration for publication the paper Ref. 13 appeared which considersspacetime structure properties of general rotating traversible wormholes.

Acknowledgements

This work was supported in part by the President Council for Grants through grant No. 96-15-96317.


References

w x Ž .1 M.S. Morris, K.S. Thorne, Amer. J. Phys. 56 1988 395.w x2 M. Visser, Lorentzian Wormholes: from Einstein to Hawking, American Institute of Physics, Woodbury, 1995.w x Ž .3 V.M. Khatsymovsky, Phys. Lett. B 320 1994 234.w x Ž .4 V.M. Khatsymovsky, Phys. Lett. B 399 1997 215.w x Ž .5 V.M. Khatsymovsky, Phys. Lett. B 403 1997 203.w x Ž .6 M.S. Morris, K.S. Thorne, U. Yurtsever, Phys. Rev. Lett. 61 1988 1446.w x Ž7 D. Hochberg, A. Popov, S.V. Sushkov, Self-consistent Wormhole Solutions of Semiclassical Gravity Preprint LAEFF 96r25,

. Ž .KSPU-96-03, gr-qcr9701064 , Phys. Rev. Lett. 78 1997 2050.w x Ž .8 G.T. Horowitz, Phys. Rev. D 21 1980 1445.w x9 S. Chandrasekhar, The mathematical theory of black holes, Clarendon Press, Oxford, 1983.

w x Ž .10 S.M. Christensen, Phys. Rev. D 17 1978 946.w x Ž .11 J.N. Goldberg, A.J. Macfarlane, E.T. Newman, F. Rohrlich, E.C.G. Sudarshan, J. Math. Phys. 8 1967 2155.w x Ž .12 B.S. DeWitt, Phys. Rep. C 19 1975 295.w x13 E. Teo, Rotating traversable wormholes, Preprint DAMTP R98r17, gr-qcr9803098.

18 June 1998


The hierarchy problem and new dimensions at a millimeter

Nima Arkani–Hamed a, Savas Dimopoulos b, Gia Dvali c

a SLAC, Stanford UniÕersity, Stanford, CA 94309, USAb Physics Department, Stanford UniÕersity, Stanford, CA 94305, USA

c ICTP, Trieste 34100, Italy

Received 12 March 1998; revised 8 April 1998Editor: H. Georgi

Abstract

We propose a new framework for solving the hierarchy problem which does not rely on either supersymmetry ortechnicolor. In this framework, the gravitational and gauge interactions become united at the weak scale, which we take asthe only fundamental short distance scale in nature. The observed weakness of gravity on distances R 1 mm is due to theexistence of nG2 new compact spatial dimensions large compared to the weak scale. The Planck scale M ;Gy1r2 is notPl N

a fundamental scale; its enormity is simply a consequence of the large size of the new dimensions. While gravitons canŽ .freely propagate in the new dimensions, at sub-weak energies the Standard Model SM fields must be localized to a

4-dimensional manifold of weak scale ‘‘thickness’’ in the extra dimensions. This picture leads to a number of strikingsignals for accelerator and laboratory experiments. For the case of ns2 new dimensions, planned sub-millimetermeasurements of gravity may observe the transition from 1rr 2 ™1rr 4 Newtonian gravitation. For any number of newdimensions, the LHC and NLC could observe strong quantum gravitational interactions. Furthermore, SM particles can bekicked off our 4 dimensional manifold into the new dimensions, carrying away energy, and leading to an abrupt decrease inevents with high transverse momentum p R TeV. For certain compact manifolds, such particles will keep circling in theT

extra dimensions, periodically returning, colliding with and depositing energy to our four dimensional vacuum withfrequencies of ;1012 Hz or larger. As a concrete illustration, we construct a model with SM fields localized on the

Ž . Ž . Ž .4-dimensional throat of a vortex in 6 dimensions, with a Pati-Salam gauge symmetry SU 4 =SU 2 =SU 2 in the bulk.q 1998 Published by Elsevier Science B.V. All rights reserved.

1. Introduction

There are at least two seemingly fundamentalenergy scales in nature, the electroweak scale mEW

;103 GeV and the Planck scale M sGy1r2 ;Pl N

1018 GeV. Explaining the enormity of the ratioM rm has been the prime motivation for con-Pl EW

structing extensions of the Standard Model such as

models with technicolor or low-energy supersymme-try. It is remarkable that these rich theoretical struc-tures have been built on the assumption of theexistence of two very disparate fundamental energyscales. However, there is an important differencebetween these scales. While electroweak interactionshave been probed at distances approaching ;my1 ,EW

gravitational forces have not remotely been probed at


( )N. Arkani–Hamed et al.rPhysics Letters B 429 1998 263–272264

distances ;My1 : gravity has only been accuratelyPl

measured in the ;1 cm range. Our interpretation ofŽM as a fundamental energy scale where gravita-Pl

.tional interactions become strong is based on theassumption that gravity is unmodified over the 33orders of magnitude between where it is measured at; 1 cm down to the Planck length ;10y33 cm.Given the crucial way in which the fundamental roleattributed to M affects our current thinking, it isPl

worthwhile questioning this extrapolation and seek-ing new alternatives to the standard picture of physicsbeyond the SM.

Given that the fundamental nature of the weakscale is an experimental certainty, we wish to takethe philosophy that m is the only fundamentalEW

short distance scale in nature, even setting the scalefor the strength of the gravitational interaction. Inthis approach, the usual problem with the radiativestability of the weak scale is trivially resolved: theultraviolet cutoff of the theory is m . How can theEW

Ž .usual 1rM strength of gravitation arise in such aPl

picture? A very simple idea is to suppose that thereare n extra compact spatial dimensions of radius

Ž .;R. The Planck scale M of this 4qnPlŽ4qn.dimensional theory is taken to be ;m accordingEW

to our philosophy. Two test masses of mass m ,m1 2

placed within a distance r<R will feel a gravita-Ž .tional potential dictated by Gauss’s law in 4qn

dimensions

m m 11 2V r ; r<R . 1Ž . Ž . Ž .nq2 nq1M rPlŽ4qn.

On the other hand, if the masses are placed atdistances r4R, their gravitational flux lines cannot continue to penetrate in the extra dimensions,and the usual 1rr potential is obtained,

m m 11 2V r ; r4R , 2Ž . Ž . Ž .nq2 n rM RPlŽ4qn.

so our effective 4 dimensional M isPl

M 2 ;M 2qn Rn . 3Ž .Pl PlŽ4qn.

Putting M ;m and demanding that R bePlŽ4qn. EW

chosen to reproduce the observed M yieldsPl

230 1q

y17 n1 TeVnR;10 cm= . 4Ž .ž /mEW

For ns1, R;1013 cm implying deviations fromNewtonian gravity over solar system distances, sothis case is empirically excluded. For all nG2,however, the modification of gravity only becomesnoticeable at distances smaller than those currently

Žprobed by experiment. The case n s 2 R ;.100 mm–1 mm is particularly exciting, since new

experiments will be performed in the very nearfuture, looking for deviations from gravity in pre-

w xcisely this range of distances 11 .While gravity has not been probed at distances

smaller than a millimeter, the SM gauge forces havecertainly been accurately measured at weak scaledistances. Therefore, the SM particles cannot freelypropagate in the extra n dimension, but must belocalized to a 4 dimensional submanifold. Since weassume that m is the only short-distance scale inEW

the theory, our 4-dimensional world should have a‘‘thickness’’ ;my1 in the extra n dimensions. TheEW

Ž .only fields propagating in the 4qn dimensionalŽ .bulk are the 4qn dimensional graviton, with cou-

Ž .plings suppressed by the 4qn dimensional Planckmass ;m .EW

As within any extension of the standard model atthe weak scale, some mechanism is needed in thetheory above m to forbid dangerous higher di-EW

Ž .mension operators suppressed only by m whichEW

lead to proton decay, neutral meson mixing etc. Inour case, the theory above m is unknown, beingEW

whatever gives a sensible quantum theory of gravityŽ .in 4qn dimensions! We therefore simply assume

that these dangerous operators are not induced. Anyextension of the SM at the weak scale must also notgive dangerously large corrections to precision elec-troweak observables. Again, there could be unknowncontributions from the physics above m . How-EW

ever, at least the purely gravitational corrections donot introduce any new electroweak breakings beyondthe W,Z masses, and therefore should decouple as

Ž .2loop factor = m rm , which is already quiteW ,Z EW

small even for m ;1 TeV.EW

( )N. Arkani–Hamed et al.rPhysics Letters B 429 1998 263–272 265

Summarizing the framework, we are imaginingthat the space-time is R4 =M for nG2, where Mn n

is an n dimensional compact manifold of volumen Ž . Ž .R , with R given by Eq. 4 . The 4qn dimen-

sional Planck mass is ;m , the only short-dis-EW

tance scale in the theory. Therefore the gravitationalforce becomes comparable to the gauge forces at theweak scale. The usual 4 dimensional M is not aPl

fundamental scale at all, rather, the effective 4 di-mensional gravity is weakly coupled due to the largesize R of the extra dimensions relative to the weakscale. While the graviton is free to propagate in allŽ .4qn dimensions, the SM fields must be localizedon a 4-dimensional submanifold of thickness my1 inEW

the extra n dimensions.Of course, the non-trivial task in any explicit

realization of this framework is localization of theSM fields. A number of ideas for such localizationshave been proposed in the literature, both in thecontext of trapping zero modes on topological de-

w xfects 7 and within string theory. In Section 3, wewill construct models of the first type, in which thereare two extra dimensions and, given a dynamicalassumption, the SM fields are localized within thethroat of a weak scale vortex in the 6 dimensionaltheory. We want to stress, however, that this particu-lar construction must be viewed at best as an ‘‘ex-istence proof’’ and there certainly are other possibleways for realizing our proposal, without affecting itsmost important consequences.

It is interesting that in our framework supersym-metry is no longer needed from the low energy pointof view for stabilizing the hierarchy, however, it maystill be crucial for the self-consistency of the theoryof quantum gravity above the m scale; indeed, theEW

theory above m may be a superstring theory.EW

Independently of any specific realization, thereare a number of dramatic experimental consequencesof our framework. First, as already mentioned, grav-ity becomes comparable in strength to the gaugeinteractions at energies m ; TeV. The LHC andEW

NLC would then not only probe the mechanism ofelectroweak symmetry breaking, they would probethe true quantum theory of gravity!

Second, for the case of 2 extra dimensions, thegravitational force law should change from 1rr 2 to1rr 4 on distances ;100 mm–1 mm, and this devia-tion could be observed in the next few years by the

new experiments measuring gravity at sub-millimeterw xdistances 11 .

Third, since the SM fields are only localizedwithin my1 in the extra n dimensions, in suffi-EW

ciently hard collisions of energy E Rm , theyesc EW

can acquire momentum in the extra dimensions andescape from our 4-d world, carrying away energy. 1

In fact, for energies above the threshold E , escapeesc

into the extra dimensions is enormously favored byphase space. This implies a sharp upper limit to thetransverse momentum which can be seen in 4 dimen-sions at p sE , which may be seen at the LHC orT esc

NLC if the beam energies are high enough to yieldcollisions with c.o.m. energies greater than E .esc

Notice that while energy can be lost into the extraŽdimensions, electric charge or any other unbroken

.gauge charge cannot be lost. This is because themassless photon is localized in our Universe and anisolated charge can not exist in the region whereelectric field cannot penetrate, so charges cannotfreely escape into the bulk. In light of this fact, let usexamine the fate of a charged particle kicked into theextra dimensions in more detail. On very general

Žgrounds which we will discuss in more detail in. ŽSection 3 , the photon or any other massless gauge

.field can be localized in our Universe, provided itcan only propagate in the bulk in the form of amassive state with mass ;m , my1 setting theEW EW

penetration depth of the electric flux lines into theextra dimensions. In order for the localized photon tobe massless it is necessary that the gauge symmetrybe unbroken at least within a distance 4my1 fromEW

Žour four-dimensional surface otherwise the photonwill get mass through the ‘‘charge screening’’, see

.Section 3 . As long as this condition is satisfied, thefour-dimensional observer will see an unbrokengauge symmetry with the right 4-d Coulomb law.

1 Usually in theories with extra compact dimensions of size R,states with momentum in the compact dimensions are interpretedfrom the 4-dimensional point of view particles of mass 1rR, butstill localized in the 4-d world. This is because the at the energiesrequired to excite these particles, there wavelength and the size ofthe compact dimension are comparable. In our case the situation iscompletely different: the particles which can acquire momentumin the extra dimensions have TeV energies, and therefore havewavelengths much smaller than the size of the extra dimensions.Thus, they simply escape into the extra dimensions.


ŽNow, consider a particle with nonzero charge or any.other unbroken gauge quantum number kicked into

the extra dimensions. Due to the conservation offlux, an electric flux tube of the width my1 must beEW

stretched between the escaping particle and our Uni-verse. Such a string has a tension ;m2 per unitEW

length. Depending on the energy available in thecollision, the charged particle will be either be pulledback to our Universe, or the flux tube will break intopieces with opposite charges at their ends. In eithercase, charge is conserved in the 4-dimensional world,although energy may be lost in the form of neutralparticles propagating in the bulk. Similar conclusionscan be reached by considering a soft photon emis-

w xsion process 8 .Once the particles escape into the extra dimen-

sions, they may or may not return to the 4-dimen-sional world, depending on the shape andror thetopology of the n dimensional compact manifoldM . In the most interesting case, the particles orbitn

around the extra dimensions, periodically returning,colliding with and depositing energy to our 4 dimen-sional space with frequency Ry1 ;1027y30r n Hz.This will lead to continuous ‘‘fireworks’’, which inthe case of ns2 can give rise to ; mm displacedvertices.

2. Phenomenological and astrophysical con-straints

In our framework physics below a TeV is verysimple: It consists of the Standard Model togetherwith a graviton propagating in 4qn dimensions.Equivalently – in four dimensional language – ourtheory consists of the Standard model together with

Ž .the graviton and all its Kaluza-Klein KK excita-tions recurring once every 1rR, per extra dimensionn. We shall refer to all of them collectively as the‘‘gravitons’’, independent of their mass. Since eachgraviton couples with normal gravitational strength;1rM to matter, its effect on particle physics andPl

astrophysical processes is negligible. Nevertheless,since the multiplicity of gravitons beneath any rele-

Ž .nvant energy scale E is ER can be large, thecombined effect of all the gravitons is not alwaysnegligible and may lead to observable effects andconstraints. In this section we will very roughly

estimate the most stringent of these constraints,mainly to show that our framework is not grosslyexcluded by current lab and astrophysical bounds.Clearly, a much more detailed study must be done tomore precisely determine the constraints on n andm in our framework.EW

Consider any physical process involving the emis-sion of a graviton. The amplitude of this process isproportional to 1rM and the rate to 1rM 2 . Conse-Pl Pl

quently, the total combined rate for emitting any oneof the available gravitons is

1 n; DER 5Ž . Ž .2MPl

where DE is the energy available to the graviton andthe last term counts the KK gravitons’ multiplicity

Ž .for n extra dimensions. Using eq 3 we can rewritethis as

DEn

; . 6Ž .2qnmEW

Note that the same result can be seen from the 4qndimensional point of view. The m suppressions ofEW

the couplings of the 4qn dimensional graviton are2qn(determined by expanding g sh qh r m ,A B A B A B EW

where h is the canonically normalized graviton inA B

4qn dimensions. Squaring this amplitude to obtainthe rate yields precisely the m dependence foundEW

above. As a result, the branching ratio for emitting agraviton in any process is

2qn; DErm . 7Ž . Ž .EW

ŽThe experimentally most exciting and most danger-.ous case has m ; TeV and ns2. Of course, weEW

must assume that weak-scale suppressed operatorsgiving proton decay, large K-K mixing etc. areforbidden. Of the remaining lab constraints, the ones

Žinvolving the largest energy transfers DE such as F.and Z decays are most constrained. The branching

ratio for graviton emission in Upsilon decays isunobservable ;10y8. For Z Z™XqgraÕiton thebranching ratio goes up to ;10y5. Absence of suchdecay modes puts the strongest laboratory constraintsto the scale m andror n. Nevertheless, they areEW

easy to satisfy, in part because of their sensitivity tosmall changes in the value of m . Production ofEW

gravitons in very high energy collisions will give the


same characteristic signatures as the missing energysearches, except for one difference: the missing en-ergy is now being carried by massless particles.

Next we consider astrophysical constraints. Thegravitons are similar to goldstone bosons, axions andneutrinos in at least one respect. They can carryaway bulk energy from a star and accelerate itscooling dynamics. For this reason their properties areconstrained by the sun, red giants and SN 1987A.The simplest way to estimate these constraints is totranslate from the known limits on goldstone parti-cles. The dictionary that allows us to do that follows

Ž .from Eq. 6 :

1rF 2 §™DEnrm2qn 8Ž .EW

relating the emission rate of goldstones and gravi-tons. Here F is the goldstone boson’s decay con-stant. For the sun the available energy DE is only akeV. Therefore, even for the maximally dangerouscase m s1 TeV and ns2, the effective F isEW

1012 GeV, large enough to be totally safe for thesun; the largest F that is probed by the sun is ;107

GeV.For red giants the available energy is ;100 keV

and the effective F;1010 GeV. This value is anorder of magnitude higher than the lower limit fromred giants. Finally we consider the supernova 1987A.There, the maximum available energy per particle ispresumed to be between 20 and 70 MeV . Choosingthe more favorable 20 MeV we find an effectiveF;108 GeV, which is smaller than the lower limitof 1010 GeV claimed from SN 1987A. Therefore, theastrophysical theory of SN 1987 A places an interest-ing constraint on the fundamental scale m orrandEW

the number of extra dimensions n. The constraint iseasily satisfied if n)2 or if m )10 TeV. OfEW

course, when the number of dimensions gets largeŽenough so that 1rRR100 MeV, corresponding to

.nR7 , none of the astrophysical bounds apply, sinceall the relevant temperatures would be too low toproduce even the lowest KK excitation of the gravi-ton.

Finally, although accelerators have not achievedcollisions in the TeV energy range where the mostexotic aspects of the extra dimensions are revealed,one may wonder whether very high energy cosmic

15 19 Žrays of energies ;10 –10 eV which in collidingwith protons correspond to c.o.m. energies ; 1–100

.TeV have already probed such physics. However,the cosmic rays are smoothly accelerated to theirhigh energies without any ‘‘hard’’ interactions, andthey have dominantly soft QCD interactions with theprotons they collide with. Therefore, there are nosignificant constraints from very high energy cosmicray physics on our framework.

Having outlined our general ideas, some dramaticexperimental consequences and being reassured thatexisting data do not significantly constrain theframework, we turn to constructing an explicit modelrealizing our picture, with SM fields localized on thefour-dimensional throat of a vortex in 6 dimensions.

3. Construction of a realistic model

In this section we construct a realistic modelincorporating the ideas of this paper. As stressed inthe introduction, this should be viewed as an exam-ple or an ‘‘existence proof’’, since similar construc-tions are possible in the context of field theory aswell as string theory. In particular one can changethe structure and dimensionality of the manifold, thelocalization mechanism, the gauge group and theparticle content of the theory without affecting thekey ideas of our paper. Furthermore, many of thephenomenological consequences are robust and donot depend on such details.

The space time is 6-dimensional with a signatureŽ .g s y1,1,1,1,1,1 . The two extra dimensions areA B

compactified on a manifold with a radius R;1 mm.We will discuss two possible topologies: a two-sphereand a two-torus with the zero inner radius. In bothcases the key point is that the observable particlesŽ .quarks, leptons, Higgs and gauge bosons are local-ized inside a small region of weak-scale size equal tothe inverse cutoff length ;Ly1 and can penetrate inthe bulk only in form of the heavy modes of mass;L. Thus for the energies -L ordinary matter getsconfined to a four-dimensional hypersurface, ouruniverse. The transverse x , x dimensions can be5 6

probed only through the gravitational force, which isthe only long-range interaction in the bulk.

There are several ways to localize the StandardModel particles in our four-dimensional space-time.Here we consider the possibility that localization is


dynamical and the ordinary particles are ‘‘zeromodes’’ trapped in the core of a four-dimensionalvortex. This topological defect, in its ground state, is

Ž .independent of four coordinates x and thusm

carves-out the four-dimensional hypersurface whichconstitutes our universe.

Consider first x , x to be compactified on a5 6

two-sphere. Define a six-dimensional scalar fieldŽ . Ž .F x transforming under some U 1 symmetry.A V

We assume that F gets a nonzero VEV ;L andŽ .breaks U 1 spontaneously. The vortex configura-V

tion is independent of the four coordinates x andm

can be set up through winding the phase by 2p

around the equator of the sphere: Fsf eiu wherebulk

2p)u)0 is an azimuthal angle on the sphere andf is the constant expectation value that mini-bulk

Žmizes a potential energy modulo the small curvature.corrections . Such a configuration obviously implies

two zeros of the absolute value F on the both sidesof the equator, which can be placed at the north andthe south poles respectively. These zeros representthe vortex–anti-vortex pair of characteristic thick-ness ;Ly1. Since this size is much smaller than theseparation length ;1 mm, vortex can be approxi-

w xmated by the Nielsen-Olesen solution 3

iu < y1Fs f r e , f 0 s0, f r ™fŽ . Ž . Ž . r 4 L bulk

9Ž .

where 0-r-2p R is a radial coordinate on thesphere, and an anti-vortex corresponds to the change

Ž .u™yu ,r™2p Ryr. If U 1 is gauged the mag-V

netic flux will be entering the south pole and comingout from the north one.

Alternatively, compactification on a torus can leadto a single vortex. This is because a torus containsmany non-contractible loops, and the phase of F

winding along such a loop is topologically stable.Such a configuration is obtained from the previouslydiscussed two-sphere by identifying its poles with asingle point and subsequently removing this pointfrom the manifold. This manifold is then equivalentto a two-torus with zero inner radius; it can carrytopological charge and accommodate a single vortexon it. The magnetic flux goes through the point thatwas removed from the manifold. An observer look-ing at the south pole will see the vortex with incom-ing flux. If he travels towards the north pole along

the meridian he will arrive to the same vortex, sincethe poles have been identified, but will see it as ananti-vortex since he will now be looking at the fluxup-side down.

Next, we come to the localization of the standardmodel particles on a vortex. We discuss the localiza-tion of different spins separately.

3.1. Localization of fermions and Higgs scalars

Fermions can be trapped on the vortex as ‘‘zerow x w xmodes’’ 1 due to the index theorem 2 . Consider a

pair of six-dimensional left-handed Weyl spinorsG c ,jsc ,j , which can be written in terms of the7

four-dimensional chiral Weyl spinors as c sŽ . Ž .c ,c , js j ,j . This pair obtains a mass fromL R L R

coupling to the vortex field as hFcjqh.c.. Thesix-dimensional Dirac equation in the vortex back-ground

G E Acqshf eiuj 10Ž .A bulk

Ž q.similarly for j has solutions with localized mass-Ž . Ž . Ž . Ž .less fermions csc x b r and jsj x b r ,m m

Ž .where ms1,..4, b r is a radial scalar function inthe 2 dimensional compact space of x and x ,5 6

Ž . Ž .provided that the spinors c x and j x satisfym m

G eiu Žy i G5 G6 .cqE b r shf eiuj 11Ž . Ž .5 r bulk

Ž q. qand similarly for j . Since c and j must beŽ .eigenvalues of the yiG G operator, they automat-5 6

Žically have definite four-dimensional chirality say.left for the vortex and right for the anti-vortex . In

this case the normalizable wavefunction has the lo-Ž . yhH0

r f Ž rX .drX

calized radial dependence b r se . Thusthe vortex supports a single four-dimensional mass-less chiral mode which can be c qjq. In this way,L R

through the couplings to the vortex field one canreproduce the whole set of the four-dimensionalchiral fermions – quarks and leptons – localized onthe vortex. These localized modes can get nonzeromasses through the usual Higgs mechanism. Let c

X Žand c be the six-dimensional chiral spinors from.different pairs that deposit two different zero modes

on the vortex. These zero modes can get massesthrough the couplings to a scalar field H Hcc

X,provided H has a nonzero expectation value in thecore of the vortex but vanishes in the bulk. The


index theorem argument is unaffected by the exis-tence of such a scalar since it has a zero VEVoutside the core. Now let us consider how the Higgsfields with non-zero VEVs can be localized on thevortex. A massive scalar field can be easily localizedprovided it has a suitable sign coupling to the vortexfield in the potential

22X 2 q q< <h F ym HH qc HH ,Ž .Ž .with m2 ,hX ,c)0. 12Ž .

If hXF 2 ym2 )0, H gets a positive mass2 in thebulk

bulk and will see the vortex as an attractive potential.For a certain range of parameters, this potential canlead to a bound-state solution with an effective nega-tive mass2 on the vortex. The analysis is similar to

w xthe one of the superconducting cosmic string 4 . Thelinearized equation for small excitations is the twodimensional Schroedinger equation

X2 2 2yE Hq h fym Hsv H 13Ž .5,6

which certainly has a normalizable boundstate solu-tion with v 2 -0 in a range of parameters. As aresult H develops an expectation value in the throatof the vortex which decays exponentially for large r.We can identify H as the six-dimensional progenitorof the Weinberg-Salam Higgs particle. Aside fromthe three massless Goldstone modes localized on thevortex, which get eaten up by W and Z bosonsthrough the usual Higgs effect, there is also a local-ized massive mode, an ordinary Higgs scalar, whichcorresponds to the small vibrations of the expecta-

Ž . Ž . Ž .tion value in the core H 0 ™H 0 qh x .m

3.2. Localization of gauge fields

There are several possible mechanisms for localiz-Žing gauge fields on a vortex or on any other topo-

. 2logical defect through the coupling to the vortexscalar. In general, a particle localized in such a waywill not be massless, unless there is a special reasonsuch as the index theorem for fermions and theGoldstone theorem or supersymmetry for bosons.Here, we propose to localize massless gauge fieldsby generalizing the four-dimensional confinement

2 An alternate way to localize massless gauge fields involvesw xD-brane constructions 10

w x Ž w x.mechanism of ref 5 see also 8 . The simplest ideaŽ . Ž .for localizing say a U 1 gauge field on a defect is

Ž .to arrange for the U 1 to be broken off but unbrokenon the defect. Since the ‘‘photon’’ becomes massiveoff the defect, one might expect to find a localizedphoton on the defect. Unfortunately, the four-dimen-sional photon trapped in this way does not remain

Ž .massless. Since the U 1 is broken off the defect, thevacuum is superconducting everywhere except onthe defect. Two test charges placed at different pointsx and xX in the defect will polarize the surroundingm m

medium, and their electric field lines will end on thesuperconductor. As a result, the electric flux alongthe vortex dies-off exponentially with distance, withina characteristic length given by the width of thevortex.

It is clear that for the localized gauge field to bemassless the surrounding medium must repel theelectric field lines and should therefore not containany charge condensate, otherwise all the field linescan be absorbed by the medium. To construct suchan example, consider a thin planar-layer between twoinfinite superconductors. Two magnetic monopoleslocated inside the layer interact through a long rangemagnetic field. This is because the magnetic flux is

Ž .repelled or ‘‘totally reflected’’ from the supercon-ductor, since it contains no magnetic charges onwhich the magnetic field lines can end. Conse-quently, the magnetic flux is entirely contained in-side the layer and, as a result of flux conservation,the field lines spread according to Coulomb’s law.

w xIn ref 5 a dual version of this mechanism – inwhich the superconductor is replaced by a confiningmedium with monopole condensation – was sug-gested as a way to obtain massless gauge bosonslocalized on a sub-manifold. Suppose that away from

Ž .the vortex U 1 becomes a part of a confiningEM

group which develops a mass gap ;L. Then theelectric flux lines will be repelled by monopolecondensation in the dual Meisner effect; no imagesare created since there is no charged condensate inthe medium.

It is not difficult to construct an explicit four-di-mensional prototype model of this sort. It includes an

Ž .SU 2 Yang-Mills theory with a scalar field x in theadjoint representation, plus a vortex field F , which

Ž .breaks some additional U 1 symmetry and formsVŽthe string for the present discussion it is inessential


Ž . .whether U 1 is global or gauged . The LagrangianVŽ Ž . .has the form SU 2 indices are suppressed

1 22 2LLsy TrG G q D x yl xŽ .Ž .mn mn m24 g

2 < < 2 2 < < 2yx h F yM q E FŽ . m

22X 2< <yl F yf 14Ž .Ž .bulk

where G is the ‘‘gluon’’ field strength tensor, andmn

h,f 2 , M 2,l,lX are the positive parameters and webulk

assume hf 2 )M 2. In a certain range of parame-bulk

ters the absolute minimum of the theory is achievedfor xs0. In this vacuum, F develops the VEV² :F sf and forms the vortex. Although x isbulk

zero in the vacuum, it can acquire an expectationvalue inside the vortex, where its mass2 becomesnegative, just like in the example with the Higgs

Ž .doublet considered above. In this case SU 2 isŽ .broken to U 1 on the string, but is restoredEM

outside. Inside the string, two out of the three gluonsacquire large masses of order of M. The third gluonbecomes a photon. Two degrees of freedom in the x

field are eaten up by the Higgs mechanism, theremaining degree of freedom is neutral. The masslessdegree of freedom in the effective 1 q 1 dimen-sional theory on the string is a photon. It is massless,

Ž .since the U 1 gauge symmetry is unbroken ev-EM

erywhere. On the other hand outside the vortex thephoton becomes a member of the non-Abelian gaugetheory, which confines and develops a mass gap.Thus the photon can only escape from the string inthe form of a composite heavy ‘‘glueball’’ with amass of the order of L which we take to be the UVcutoff ;m . This guarantees that at low energiesEW

the massless photon will be trapped on the string.The theory inside the string is in the Abelian 1 q 1dimensional ‘‘Coulomb’’ phase.

How is this mechanism generalized to our six-di-mensional case? Of course, we do not know howconfinement works in a higher dimensional theory.Nevertheless, we believe and will postulate that thehigher dimensional theory in the bulk will posses amass gap ;L provided that:.1 Outside the vortex the standard model gauge

group, in particular electromagnetism and strong in-teractions, are extended into a larger non-Abeliangauge theory.

. Ž2 There is no light with a mass below the cut-off.scale L matter in the bulk enforced by general

principles, such as Goldstone’s theorem..3 The tree-level gauge coupling blows up away

w xfrom the vortex 8 .The latter condition can be satisfied e.g. if the valueof the gauge coupling is set by an expectation value

Ž .of the higgs field or any function of it whichvanishes away from the vortex. For instance, in theprevious four-dimensional toy model such a couplingis Ly2 Trx 2 TrG G mn.mn

3.3. A realistic theory

In this section we assemble the above ingredientsto construct a prototype model incorporating theideas of this paper. We embed the Standard Model in

Ž . Ž . Ž .the Pati-Salam group GsSU 4 mSU 2 mSU 2R L

which is the unbroken gauge group in the bulk. InŽ .addition, we introduce a U 1 factor and a singletV

scalar field F charged under it. F develops anexpectation value and forms a vortex of thickness;Ly1 in the compact 2-D submanifold spanned byx , x . The interior of the vortex is our 4-dimensional5 6

space-time with all the light matter confined to it.The only light particle propagating in the bulk is thesix-dimensional graviton.

The gauge group is spontaneously broken toŽ . Ž .SU 3 mU 1 inside the vortex, by a set of six-di-EM

Ž . X Ž .mensional scalar fields xs 15.1.1 , x s 4.2.1 andŽ .Hs 1.2.2 which develop nonzero VEVs only in

the core of the vortex due to their interactions withthe F field. We assume a soft hierarchy x

X;x;L

;10H;m . The crucial assumption is that in theEW

bulk the gauge group is strongly coupled and devel-ops a mass gap of the order of the cut-off. This,

Ž . Ž .together with the fact that SU 3 mU 1 is unbro-EM

ken everywhere guarantees that the gluons and thephoton are massless and trapped in our four-dimen-sional manifold. W " and a Z bosons are localizedas massive states.

The matter fermions are assumed to originatefrom the following six-dimensional chiral spinors pergeneration:

Qs 4,1,2 , Qs 4,1,2 ,Ž . Ž .Q s 4,2,1 , Q s 4,2,1 15Ž . Ž . Ž .c c

which get their bulk masses through the coupling to


X) Žthe vortex field hF QQqhF Q Q where h andc c

X .h are parameters of the inverse cut-off size . Theindex theorem ensures that each pair deposits asingle chiral zero mode which can be chosen as

q qQ qQ and Q qQ . These states get theirL R c L c R

masses through the couplings to the Higgs doubletfield which condenses in the core of the vortexgHQQ qgHQQ .c c

To avoid unacceptable flavor violations, the firstset of couplings should be flavor-universal. This canbe guaranteed by some flavor symmetry. Flavorviolations must come from the ordinary Yukawacouplings in order to be under control.

4. Summary and outlook

The conventional paradigm for High EnergyPhysics – which dates back to at least 1974 –postulates that there are two fundamental scales, theweak interaction and the Planck scale. The largedisparity between these scales has been the majorforce driving most attempts to go beyond the Stan-dard Model, such as supersymmetry and technicolor.In this paper we propose an alternate framework inwhich gravity and the gauge forces are united at theweak scale. As a consequence, gravity lives in morethan four dimensions at macroscopic distances –leading to potentially measurable deviations fromNewton’s inverse square law at sub-mm distances.The LHC and NLC are now even more interestingmachines. In addition to their traditional role ofprobing the electroweak scale, they are quantum-gravity machines, which can also look into extradimensions of space via exotic phenomena such asapparent violations of energy, sharp high-p cutoffssT

and the disappearance and reappearance of particlesfrom extra dimensions.

The framework that we are proposing changes theway we think about some fundamental issues inparticle physics and cosmology. The first and mostobvious change in particle physics occurs in ourview of the hierarchy problem. Postulating that thecutoff is at the weak scale nullifies the usual argu-ment about ultraviolet sensitivity, since the weakscale now becomes the ultraviolet! The new hierar-chy that we now have to face, in the six dimensionalcase, is that between the millimeter and the weak

scales. This hierarchy is stable in the sense that smallchanges of parameters have small effects on thephysics – so there is no fine tuning problem. Thereis also no issue of radiatively destabilizing the mmscale by physics at the weak cutoff. In this respect,our proposal shares the same ‘‘set it and forget it’’philosophy of the original proposal supersymmetric

w xstandard model 12 . An important and favorabledifference is that the mm scale is not a Lagrangeanparameter that needs to be stabilized by a symmetry,such as supersymmetry. It is a parameter characteriz-ing a solution, the size of the two extra dimensions.It is not uncommon to have solutions much largerthan Lagrangean parameters; the world around usabounds with solutions that are much larger than theelectron’s Compton-wavelength. A related secondaryquestion is whether the magnitude of the mm scalemay be calculated in a theory whose fundamentallength is the weak scale. We have not addressed thisquestion which is embedded in the higher dimen-sional theory. It is amusing to note that if there are

Ž .many new dimensions, their size – given by Eq. 4– approaches the weak scale and there is no largehierarchy.

Finally we come to the early universe. The mostsolid aspect of early cosmology, namely primordialnucleosynthesis, remains intact in our framework.The reason is simple: The energy per particle duringnucleosynthesis is at most a few MeV, too small tosignificantly excite gravitons. Furthermore, the hori-zon size is much larger than a mm so that theexpansion of the universe is given by the usual4-dimensional Robertson-Walker equations. Issuesconcerning very early cosmology, such as inflationand baryogenesis may change. This, however, is notnecessary since there may be just enough space toaccommodate weak-scale inflation and baryogenesis.

In summary, there are many new interesting is-sues that emerge in our framework. Our old ideasabout unification, inflation, naturalness, the hierar-chy problem and the need for supersymmetry areabandoned, together with the successful supersym-metric prediction of coupling constant unificationw x12 . Instead, we gain a fresh framework whichallows us to look at old problems in new ways.Lagrangean parameters become parameters of solu-tions and the phenomena that await us at LHC, NLCand beyond are even more exciting and unforeseen.


Acknowledgements

We would like to thank I. Antoniadis, M. Dine, L.Dixon, N. Kaloper, A. Kapitulnik, A. Linde, M.Peskin, S. Thomas and R. Wagoner for useful dis-cussions. G. Dvali would like to thank the Instituteof Theoretical Physics of Stanford University fortheir hospitality. N.A.H. is supported by the Depart-ment of Energy under contract DE-AC03-76SF00515. S.D. is supported by NSF grant PHY-9219345-004.

References

w x Ž .1 R. Jackiw, P. Rossi, Nucl. Phys. B 190 1981 681.w x Ž .2 R. Jackiw, C. Rebbi, Phys. Rev. D 13 1976 3398; E.

Ž .Weinberg, Phys. Rev. D 24 1981 2669.

w x Ž .3 H.B. Nielsen, P. Olesen, Nucl. Phys. B 61 1973 45.w x Ž .4 E. Witten, Nucl. Phys. B 249 1985 557.w x Ž .5 G. Dvali, M. Shifman, Phys. Lett. B 396 1997 64.w x6 L. Dixon, J.A. Harvey, C. Vafa, E. Witten, Nucl. Phys. B

Ž .261 1985 678.w x7 See, e.g., V. Rubakov, M. Shaposhnikov, Phys. Lett. B 125

Ž . Ž .1983 136; G. Dvali, M. Shifman, Nucl. Phys. B 504 1997127.

w x Ž .8 See A. Barnaveli, O. Kancheli, Sov. J. Nucl. Phys. 51 3Ž .1990 .

w x Ž .9 For a review see, e.g. J.E. Kim, Phys. Rep. 150 1987 1;Ž .M.S. Turner, Phys. Rep. 197 1990 67.

w x10 J. Polchinski, TASI Lectures on D-Branes, hep-thr9611050.w x Ž .11 J.C. Price, in: P.F. Michelson Ed. , Proc. Int. Symp. on

Experimental Gravitational Physics, Guangzhou, China,World Scientific, Singapore, 1988; J.C. Price et. al., NSFproposal 1996; A. Kapitulnik, T. Kenny, NSF proposal,1997.

w x Ž .12 S. Dimopoulos, H. Georgi, Nucl. Phys. B 193 1981 150.

18 June 1998


Open superbranes

C.S. Chu a, P.S. Howe b, E. Sezgin c,1, P.C. West b

a ( )International School for AdÕanced Studies SISSA , Via Beirut 2, 34014 Trieste, Italyb Department of Mathematics, King’s College, London, UK

c Center for Theoretical Physics, Texas A&M UniÕersity, College Station, TX 77843, USA

Received 30 March 1998Editor: M. Cvetic

Abstract

Open branes ending on other branes, which may be referred to as the host branes, are studied in the superembeddingformalism. The open brane, host brane and the target space in which they are both embedded are all taken to besupermanifolds. It is shown that the superspace constraints satisfied by the open brane are sufficient to determine thecorresponding superspace constraints for the host branes, whose dynamics are determined by these constraints. As abyproduct, one also obtains information about the boundary of the open brane propagating in the host brane. q 1998Published by Elsevier Science B.V. All rights reserved.

1. Introduction

It is well-known that certain open branes can endon other branes provided that the dimensionalities of

w xthe branes are chosen correctly 1–16 . These config-urations are intimately connected with the specialcases of intersecting branes where the dimension ofthe intersection manifold coincides with one of theintersecting brane boundary dimension. This is

Ž .known to occur for fundamental string or D py2 -branes ending on Dp-branes and Dp-branes ending

Ž . Ž w x w xon NS 5-branes 1FpF6 . See 13 and 14 for an.extensive analysis .

Recently, such systems have been discussed in ahybrid Green-Schwarz formalism in M-theory and in

w xthe context of D-branes in ten dimensions 15,16 .Ž w x.For earlier related work, see 8–11 . The basic idea

1 Research supported in part by NSF Grant PHY-9722090.

w x w xof Refs. 15 and 16 is to write down the Green-Ž .Schwarz GS -action for a brane, for example the

M2-brane, with a boundary. The worldvolume forthe GS action is a bosonic manifold S while thetarget space in this case is the superspace foreleven-dimensional supergravity, M. The boundaryof the worldvolume, ES, is taken to be embedded ina supermanifold M which is also embedded in M.The supermanifold M is taken to be the worldsur-face of the M5-brane, and it was found that k-sym-metry of the M2-brane action with boundary con-tained in M imply the equations of motion for theM5-brane. These results were extended to strings andŽ . w xD py2 branes ending on Dp-branes in 16 . The

formalism used in these papers is therefore a hybridone, since the first brane is treated from the GS pointof view while the second one is treated using thesuperembedding formalism. Since k-symmetry is arelic of local worldsurface supersymmetry in the


( )C.S. Chu et al.rPhysics Letters B 429 1998 273–280274

superembedding formalism it is to be expected thatthe same results can be obtained by working entirelyin the superembedding formalism. This is indeed thecase as we shall show in the current paper. We shallagain focus on M-branes and D-branes.

In the approach presented here, it is not necessaryto exhibit the open-host brane system as a classicalsolution of the target space theory, nor is it necessaryto make assumption on the topology of the hostbrane. For example, the host brane itself may beclosed or open. An open host brane, can in turn endon another suitable secondary host brane and so on.Thus one can obtain a brain chain. A special case ofthis arises when all members of the chain have thesame dimension, thus forming a brane network. Thetwo possibilities allowed are those which use thestring ending on D1-brane junction as building blockw x17–19 , and those which use D5-branes ending on

w xNS 5-branes as building blocks 20 . Here, for thepurposes of the present paper, we shall focus on thebasic building blocks of chain or network configura-tions, namely an open brane ending on a host brane.

The next three sections are devoted to the discus-sion of an M2-brane ending on an M5-brane, funda-mental strings ending on Dp-branes and Dp-branes

Ž .ending on D pq2 -branes, respectively. A sum-mary of our results and further comments about themare provided in the Conclusions.

2. Open branes in M-theory

We consider the following picture: a 2-braneŽ < . Ž < .worldsurface M , with even odd dimension 3 162

Žcan end on an M5-brane worldsurface M dimen-5Ž < ..sion 6 16 , via a boundary M which is a super-1

Ž < .manifold of dimension 2 8 . Thus M sE M and1 2

M ;M while both M and M are embedded in an1 5 2 5Ž < .11 32 -dimensional target space M. We thereforehave embeddings

f :M ®M ; is1,2,5 1Ž .i i

as well as an embedding

f 5 :M ®M . 2Ž .1 1 5

Clearly

f s f ( f 5 . 3Ž .1 5 1

The fact that the 2-brane can end on a supermanifoldwhich has bosonic dimension two is related to thefact that the 5-brane admits stringlike soliton solu-

w xtions to its equations of motion 21 . We shall com-ment on the possibility of host branes other than theM5-brane in the Conclusions.

The boundary of the open membrane, in general,may consist of an arbitrary number of closed strings.However, to keep the discussion as simple as possi-ble, we shall consider an open membrane that has thetopology of a disk, and hence a single componentboundary, a closed string. Our analysis can easily beextended for multi-component boundaries, with es-sentially same results.

We shall now demonstrate that if the standardembedding condition is assumed for the 2-brane thenthe standard embedding condition for the 5-brane isimplied. We shall also show, although it is notessential for the derivation of the M5-brane equa-tions of motion, that this picture requires for itsconsistency a 2-form gauge potential A on M whose5Ž .modified 3-form field strength FF is only non-vanishing when all of its indices are bosonic. This isnot essential for the derivations of the M5-braneequations of motion because, as has been shown

w xelsewhere 22 , this result actually follows from theembedding condition. However, it is useful to intro-duce this discussion here as it will play a moresignificant role in the analysis of D-branes.ˆ

To make the analysis of the embedding conditionswe introduce the embedding matrices E which aresimply the derivatives of the embeddings given abovereferred to standard bases. Thus we have the follow-ing set of embedding matrices,

A A A A5E , E , E and E , 4Ž .A A A A1 2 5 1

corresponding to the derivatives of the embeddingsf , f , f , and f 5 respectively. The notation here is1 2 5 1

that underlined indices refer to the target space Mwhile indices for each of the manifolds M , is1,2,5i

are distinguished by appending to them the corre-sponding numerical subscripts. As usual, indices fromthe beginning of the alphabet are preferred basisindices while indices from the middle of the alphabetdenote coordinate indices. Capital indices run over

Ž .both bosonic and fermionic indices while latin greekŽ .letters are used for bosonic fermionic indices sepa-

( )C.S. Chu et al.rPhysics Letters B 429 1998 273–280 275

rately, for example As a,a . We shall denoteŽ .normal indices by primes; it should be clear from thecontext which embedding is being referred to when anormal index is employed. A more explicit formulafor the embedding matrix is, using the 2-brane as anexample,

A M M A2E sE E z E , 5Ž .A A M M2 2 2

where we have introduced the vielbein matrices,AE etc., which relate the preferred bases to theM

coordinate bases.The basic embedding condition for the 2-brane is

aE s0 . 6Ž .a2

This equation simply states that the odd tangentspace to M is a subspace of the odd tangent space2

to M at each point in M ; M. It can be shown that2

this condition determines the equations of motion ofthe membrane and also that the geometry of thetarget space is required to be that of on-shell eleven-dimensional supergravity. On the boundary M s1

E M we therefore have2

aE s0 . 7Ž .a1

By the chain rule, we have

A A A5E sE E 8Ž .A A A1 1 5

and so

a a a a a5 50sE sE E qE E . 9Ž .a a a a a1 1 5 1 5

We now introduce a complementary normal matrixa

Xin the bosonic sector for M in M denoted by E .5 a5a aŽ .XThe inverse of the pair E , E is denoted bya a5 5

ŽŽ y1 . a5 Ž y1 . aX5. Ž . Ž y1 . aX

5E , E . Multiplying 9 by Ea a a

we find

a a y1 aX5 5E E E s0 ; 10Ž . Ž .aa a1 5

Ž y1 . a5while multiplying the same equation by E a

we get

a a y1 a a5 5 5E E E qE s0 . 11Ž . Ž .aa a a1 5 1

Now for any superembedding it is always possible tochoose the odd tangent space of the embedded sub-

Ž .manifold such that taking M ; M as an example5

a y1 a5E E s0 . 12Ž . Ž .aa 5

To see this we observe that the odd tangent spacebasis E for M can be written quite generally asa 55

a aE sE E qE E 13Ž .a a a a a5 5 5

while for the even subspace we can write

a aE sE E qE E 14Ž .a a a a a5 5 5

Hence if we redefine E bya5

˜ a5E ™E sE qL E 15Ž .a a a a a5 5 5 5 5

for some fermionic superfield L a5 we finda5

a a a a5E sE qL E . 16Ž .a a a a5 5 5 5

Ž y1 . a5Multiplying this equation with E , we observeaa y1 a5Ž .that the quantity E E can always be madea a5

to vanish by choosing L a5 appropriately. Thus,a5

Ž .the result 12 is proved. Using this result in conjunc-Ž .tion with 10 , M being arbitrary, we see that the1

odd tangent space of M can always be chosen such5

that

aE s0 17Ž .a5

Ž .and this implies, from 11 , that

E a5 s0 18Ž .a1

as well.Ž .Eq. 17 is the standard embedding condition for

the M5-brane. It has been shown that this equationdetermines the equations of motion for the 5-branew x22,23 . Furthermore, the boundary brane M also1

obeys the standard embedding condition as a subsu-permanifold of M . We therefore conclude that the5

5-brane equations of motion are implied by requiringthe consistency of 2-branes ending on 5-branes.

Ž .The embedding constraint 18 for the boundarystring confined to propagate within the 5-brane israther interesting. We shall comment further on thispoint in the Conclusions, but here we shall focus onthe derivation of the 5-brane equations of motion.

Let us now consider the Wess-Zumino form W4

for the 2-brane. As this brane is type I, i.e. itsworldsurface multiplet contains only scalars asbosonic degrees of freedom, W is simply the pull-4

back of the target-space 4-form field strength ofeleven-dimensional supergravity

G sdC . 19Ž .4 3


Since W is a closed form on a supermanifold with4

bosonic dimension three, it must be exact becausethe de Rham cohomology of a supermanifold coin-cides with that of its body. Thus we can writew x24–26

W s f ) G sdK 20Ž .4 2 4 3

for some globally defined 3-form K on M . We3 2

therefore have

K sdY q f ) C on M . 21Ž .3 2 2 3 2

Since both of the potentials Y and C are, in2 3

general, locally defined, the fact that K is globally3

defined dictates the transformation rule for Y . On2

the boundary M we identify Y as the pull-back of1 2

a 2-form potential on M ,5

)5Y s f A on M . 22Ž .Ž .2 1 2 1

This implies that the corresponding field strength3-form on M should be defined by5

FF sdA q f ) C on M . 23Ž .3 2 5 3 5

The associated Bianchi identity is

dFF s f ) G . 24Ž .3 5 4

It is straightforward to demonstrate that the onlynon-vanishing component of K is the one with3

purely bosonic indices, so that it must vanish whenrestricted to the boundary M . This implies in turn1

that FF must vanish on M ;M . The pull-back of3 1 5

FF to M is given by3 1

)5 C B A5 5 5f FF sE E E FF , 25Ž .Ž .1 A B C C B A A B C1 1 1 1 1 1 5 5 5

up to Grassmann sign factors which we suppress.Since the left-hand side vanishes, and since E a5 sa1

0, we conclude, M being arbitrary, that this can1

only be satisfied if

FF s0 , 26Ž .a B C5 5 5

that is, FF must be purely even on M .3 5

To summarise then, we have shown that the con-sistency of the picture of a 2-brane with a boundaryending on a 5-brane in which the boundary is em-bedded requires that, if the standard embedding con-

Ž .dition 6 for the 2-brane is imposed, the standardembedding condition for the 5-brane should alsohold and that there is a 2-form gauge potential onM whose 3-form field strength should satisfy equa-5

Ž .tion 26 above. This is in perfect agreement with thew xresults of 15 obtained from a hybrid approach

mentioned in the introduction. As we have remarkedearlier, the second of these equations actually fol-lows from the first in the case of M-branes, but this

w xis not true for all D-branes 27 .

3. Fundamental strings ending on D-branes

The discussion of the previous section can becarried over straightforwardly to fundamental type IIstrings in ten dimensions ending on D-branes. Ingeneral, the end points of the open string may lie ontwo different Dp-branes or one end-point of a semi-infinite open string may lie on a Dp-brane while theother end is freely moving. It is sufficient to considerthe case where both end-points are ending on asingle Dp-brane for the purpose of deriving theconstraints that govern the dynamics of the Dp-brane.It is straightforward to generalize the discussion forthe other two cases. Thus we have the following

Žsupermanifolds: the string manifold, M dimension1Ž < .. Ž Ž < ..2 16 , its boundary M sE M dimension 1 8 ,0 1

Žthe worldvolume of the Dp-brane M dimensionpŽ < ..pq1 16 and the target space M which is eithertype IIA or type IIB superspace and which has

Ž < .dimension 10 32 . The associated embeddings are

f :M ® M, is0,1, p 27Ž .i i

and

f p :M ®M 28Ž .0 0 p

with

f s f ( f p . 29Ž .0 p 0

For either of the fundamental strings the embeddingcondition

aE s0 30Ž .a1

implies the equations of motion for the string. Byusing exactly the same procedure as in the previoussection we conclude that the embedding conditionwill also hold for the Dp-brane on which it ends, sothat

aE s0 31Ž .ap


In addition the standard embedding condition willalso hold for the worldvolume of the 0-brane bound-ary considered as a subsupermanifold of M :p

E a p s0 32Ž .a0

Each string has a Wess-Zumino 3-form W sdZ3 2

which is simply the pull-back of the NS 3-form

H sdB , 33Ž .3 2

on the target space, so we have

W s f )H . 34Ž .3 1 3

The Wess-Zumino form is exact on M so1

W sdK 35Ž .3 2

from which we conclude that

K sdY q f )B on M . 36Ž .2 1 1 2 1

On the boundary we identify Y with the pull-back1

of a 1-form gauge potential A on M :1 p

)pY s f A on M . 37Ž .Ž .1 1 1 0

We can thus identify the modified field strength2-form FF on M as2 p

FF sdA q f )B on M , 38Ž .2 1 p 2 p

with the associated Bianchi identity

dFF s f )H . 39Ž .2 p 3

It follows from the constraints on the fundamentalstring that the only non-vanishing component of thetwo-form K is the one with purely bosonic indices2

and so it vanishes on the boundary. By a similarargument to that given in the preceeding section, wetherefore conclude that

FF s0 . 40Ž .a Bp p

Ž .The embedding condition 31 and the FF-constraintŽ .40 are together sufficient to imply the equations ofmotion for the Dp-brane in all cases. They are notnecessarily necessary, however. In type IIA one canhave ps2,4,6,8. For ps2,4, the embedding condi-tion is already enough to imply the equations ofmotion while for ps6,8, the FF-constraint is re-quired as well. It is clear that an additional constraintis required for ps8 since the brane has co-dimen-sion one. In this case the worldvolume multipletdetermined by the embedding condition is an entirescalar superfield. That an additional constraint is

required in the case of ps6 is less obvious sincethe worldvolume multiplet is in this case a ds7‘‘linear multiplet’’, i.e. a superfield whose leadingcomponent is three scalars and whose next leadingcomponent is a spin-half field in seven dimensions.There is also a 0-brane in type IIA for which theembedding condition alone is sufficient to give thedynamics. In type IIB there are D-branes for all oddp. For ps1,3,5 the embedding condition gives theequations of motion while for ps7,9 the FF-con-straint is required as well. For ps7 the worldvol-ume multiplet determined by the embedding condi-tion is a chiral scalar superfield, which is otherwiseunconstrained, and for ps9 there are no scalars sothat it is clear that an additional constraint is re-quired. Further details of D-brane embeddings will

w x Žbe found elsewhere 27 . The case of D9-brane hasbeen recently treated from the superembedding point

w x.of view in 28 .

4. D-branes ending on D-branes

In this section we shall consider Dp-branes end-Ž .ing on D pq2 -branes of the Type IIArB super-

string theories in ten dimensions. The discussion isvery similar to the preceding two cases. It will

w xenable us to recover the results of 16 in a super-space approach.

The relevant supermanifolds are: the Dp-braneŽ Ž < ..manifold, M dimension pq1 16 , its boundaryp

Ž Ž < ..M sE M dimension p 8 , the worldvolume ofpy1 pŽ . Ž Ž < ..the D pq2 -brane M dimension pq3 16pq2

and the target space M which is either type IIA ortype IIB superspace and which again has dimensionŽ < .10 32 . The associated embeddings are

f :M ® M, ispy1, p , pq2 41Ž .i i

and

f pq2 :M ®M 42Ž .py1 py1 pq2

with

f s f ( f pq2 . 43Ž .py1 pq1 py1

For the Dp-brane the embedding condition

aE s0 44Ž .ap


is assumed to hold, as well as the FF-constraint

FF s0 . 45Ž .a Bp p

Using the same argument as before we deduce thatthe standard embedding condition

aE s0 46Ž .apq 2

Ž .will hold for the D pq2 brane as well. In additionthe standard embedding condition will also hold for

Ž .the worldvolume of the py1 -brane boundary con-sidered as a subsupermanifold of M :pq2

E apq 2 s0 . 47Ž .apy 1

The Wess-Zumino form for a Dp-brane is W spq2

dZ where the Wess-Zumino potential is given aspq1

the pq1-form component of an inhomogeneouspotential form

Z s f ) C e FF , 48Ž .Ž .pq1 p pq1

where C is the sum of the RR potential forms on thetarget space. In the case of massive type IIA Dp-branes, the term mf )v , where m is the massp pq1

Ž .parameter and v A,dA is the Chern-Simonspq1w xform, has to be added to the right hand side 29 .

W is a closed form on M and by the samepq2 p

arguments that were used before it must be exact, sothat we can write

W sdK 49Ž .pq2 pq1

Ž .for some globally defined pq1 -form K onpq1

M . Clearlyp

K sdY qZ on M . 50Ž .pq1 p pq1 p

On the boundary we identify Y with the pull-backp

of a p-form potential A on M :p pq2

)

pq2Y s f A on M . 51Ž .Ž .p py1 p py1

We also identify the 1-form potential of the Dp-braneon the boundary with the pull-back of the 1-form

Ž .potential of the D pq2 -brane and use the sameŽ .letter A . The field strength pq1 -form associated1

with A isp

FF sdA q f ) C e FF on M , 52Ž . Ž .pq1pq1 p pq2 pq2

which obeys the Biachi identity

dFF s GeFF , 53Ž . Ž .pq2pq1

with the definition

GsdCyCH . 54Ž .In the case of massive type IIA Dp-branes, theChern-Simons term mv needs to be added to thepq1

right hand side of this definition but the same BianchiŽ .identity 53 holds.

The equations for the p-brane imply that the onlynon-vanishing component of K is the one withpq1

purely bosonic indices and so we deduce that thepull-back of FF to M must vanish and thispq1 py1

implies the FF-constraint for FF , namelypq1

FF s0 . 55Ž .a B C . . .pq 2 pq2 pq2

Ž .The equations for the D pq2 -brane derived fromletting a Dp-brane end on it are therefore the stan-

Ž .dard embedding condition 46 together with FF-con-Ž .straints of the form of equation 55 for both a

Ž .2-form field strength FF and a pq1 -form field2

strength FF . These field strengths are essentiallypq1

duals of one another. To be more precise, at thelinearised level FF is the dual of FF , but inab a . . . a1 pq1

w xthe full theory there are non-linear corrections 16 .

5. Conclusions

In this paper we have seen yet again the power ofsuperembeddings in the description of superbranedynamics. Starting from simple geometrical consid-erations having to do with the way an open M2-braneis embedded in eleven dimensions or the way anopen string or Dp-brane is embedded in ten dimen-sions, we were able to derive the superfield con-straints that govern the dynamics of the host braneson which these open branes end. The constraintsconsist of the embedding condition of the host braneand a constraint on a suitable field strength living onthe host brane, namely FF for the M5-brane, FF for3 2

Ž .the Dp-brane and FF for the D pq2 branes.pq1

The first two case are the most familiar ones whilethe last case is somewhat novel in that it contains ap-form potential A as well as the usual Maxwellp

w xfield A , in accordance with the results of 16 . As1Žmentioned earlier, these are dual to each other in a


.highly nonlinear fashion and consequently we ex-pect to acquire new insights about duality symme-tries within this framework.

As a byproduct of our approach to open super-branes, we have obtained the boundary embedding

Ž . Ž . Ž .conditions 18 , 32 and 47 that characterize theembedding of the boundary manifolds within thehost branes, e.g. the closed string boundary of the

Ž .M2-brane within the host M5-brane, as given in 18 .These constraints are not sufficient, however, to putthe boundary brane on-shell.

Consider, for example, the case of Dp-branesŽ .ending on D pq2 -branes, where there are bound-

Ž . Ž .ary py1 -branes embedded in pq2 -branes.Note that 1FpF7. In the special case of the D7-brane ending on the D9-brane, we expect the bound-ary brane to be the familiar D6-brane moving inD9-brane, alias the ten dimensional spacetime.Therefore let us concentrate on the remaining cases

Ž .of Dp-branes ending on D pq2 -branes with 1FpF6. All of these are codimension 3 embeddings in

Ž .which the boundary D py1 -brane is propagatingŽ .in pq3 -dimensions. The 3,4,5 branes propagating

in 7,8,9 dimensional target spacetimes, respectively,have already been encountered in the context of

w xsuperembeddings in 30 , where they were called thew xL-branes. It was argued in 30 that the associated

superembedding constraints imply the equations ap-propriate to linear multiplets which happen to beoff-shell supermultiplets.

ŽConsider the L5-brane in 9 dimensions the de-scription of the other cases can be obtained by

.dimensional reduction . Its worldvolume multipletconsists of three scalars, a 4-form potential and aspinor with 8 real components. So the off-shelldegrees of freedom count is 3q5 bosons and 8fermions. Interestingly, there are no auxiliary fieldsin this multiplet. The existence of 4-form potentialson the worldvolume leads, by arguments similar tothose of the previous section, to the boundary FF-

w xconstraint 31

)8dFF s f G , 56Ž .Ž .5 5 6

where G is a closed super-form in 9 dimensions.6

However, the system remains off-shell even in pres-ence of this constraint. In order to put the systemon-shell, one has to construct an action that yields

the equations of motions. We will show elsewherew xthat this is indeed possible 31 .

In passing we note that putting the linear multipleton-shell means that the 4-form potential obeys aMaxwell type equation and therefore on-shell it isdual to a scalar field. Since the fermions describe 4degrees of freedom, one then obtains a 4q4 on-shellmultiplet which is essentially a hypermultiplet withone of the scalars dualized to a 4-form potential.This system is therefore intimately related to a verti-cal reduction of a 5-brane in 10 dimensions, fol-lowed by the dualization of the 10th coordinatescalar to a 4-form potential on the 5-brane worldvol-

w xume 30 .In this paper we focused on M2-brane ending on

Ž .M5-brane, Dp-branes ending on D pq2 -branesand fundamental string ending on Dp-branes. In thefirst two cases we assumed that the open branes havesingle component boundaries, while in the last casewe let the two ends of the open string lie on a singleDp-brane, for simplicity. Not all of these configura-tions are necessarily BPS saturated or anomaly free.While anomaly freedom is essential, the BPS satura-tion is less crucial property since the BPS statespresumably constitute only a tiny fraction of allpossible states.

The universal nature of the superembedding for-malism suggests that it can successfully be applied tomany other generalizations of the systems studied inthis paper. For example, it can be applied straightfor-wardly to Dp-branes ending on NS 5-branes. Onecan also treat configurations in which the open M2-brane ends on an M5-brane at one end and anM9-brane on the other, or both ends ending on

ŽM9-branes when we discuss M9-branes, we have inmind the Horawa-Witten picture of such objects asboundaries of the eleven dimensional spacetime with

w x.suitable topology 2 .One may also consider a system in which the

open M2-brane has multi-component boundarieswhich may end on any M5-branes or M9-branes inall possible ways. It is clear that there is a richspectrum of possibilities due to the fact that the basicbuilding blocks can have a large class of nontrivialtopologies. Further novel possibilities can also arisebecause brane theories are intrinsically nonlinear andconsequently the topology of branes can changethrough self-interactions.


Acknowledgements

It is a pleasure to thank M. Duff and P. Sundellfor stimulating discussions.

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dimensional field theories and grid diagrams, hep-thr9710116.

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18 June 1998


Gauged supergravity vacua from intersecting branes

P.M. Cowdall a, P.K. Townsend b,1

a DAMTP, UniÕersity of Cambridge, SilÕer Street, Cambridge, UKb Departament ECM, Facultat de Fısica, UniÕersitat de Barcelona and Institut de Fısica d’Altes Energies, Diagonal 647,´ ´

E-08028 Barcelona, Spain

Received 19 February 1998Editor: P.V. Landshoff

Abstract

Ž . Ž . Ž .Domain wall and electrovac solutions of gauged Ns4 Ds4 supergravity, with gauge group SU 2 or SU 2 =SU 2 ,are interpreted as supersymmetric Kaluza-Klein vacua of Ns1 Ds10 supergravity. These vacua are shown to be thenear-horizon geometries of certain intersecting brane solutions. q 1998 Elsevier Science B.V. All rights reserved.

1. Introduction

Gauged supergravity theories are those for whichŽa subgroup of the R-symmetry group alias the auto-

.morphism group of the supersymmetry algebra isgauged by vector potentials in the graviton supermul-tiplet. The gauging is invariably accompanied by acosmological constant of order g 2, where g is thegauge coupling constant, and in the simplest casesŽ .the ‘adS-supergravities’ there is a maximally super-

Ž .symmetric anti-de Sitter adS vacuum, maximal inthe sense that the adS vacuum of the gauged super-gravity theory preserves the same total number ofsupersymmetries as the Minkowski vacuum of theungauged theory. A classic example is gauged Ns8Ds4 supergravity, which can be obtained by gaug-

Ž . Ž .ing an SO 8 subgroup of the SU 8 R-symmetrygroup. This theory has an adS vacuum with the

Ž < .Ns8 adS supergroup OSp 8 4;R as its isometrysupergroup. It can also be obtained by S7 compacti-

1 On leave from DAMTP, University of Cambridge, U.K.

fication of Ds11 supergravity, in which case theŽ . Ž .SO 8 gauge group has a Kaluza-Klein KK origin

as the isometry group of S7. Another example is thegauged Ds5 maximal supergravity for which the

Ž . Ž . Žgauge group is an SU 4 subgroup of the Sp 4 aliasŽ ..USp 8 R-symmetry group. This too has an adS

vacuum and can be obtained from an S5 compactifi-cation of IIB supergravity. Yet another example isthe gauged maximal Ds7 supergravity, for which

Ž . Ž .the gauge group is the full Sp 2 (Spin 5 R-sym-metry group. This theory has the curious feature thatthe g™0 limit is singular, so it cannot be found bythe usual ‘Noether’ procedure, in which the un-gauged theory is taken as the starting point; it wasactually found from an S4 compactification of Ds

w x11 supergravity. We refer to 1,2 for reviews andreferences to work of this period.

w xIt was shown in 3 that all the above mentionedKK vacua associated with gauged maximal super-gravities arise as near-horizon geometries of 1r2supersymmetric p-brane solutions of Ds10 or Ds11 supergravity theories. The adS =S7 and adS =4 7


( )P.M. Cowdall, P.K. TownsendrPhysics Letters B 429 1998 281–288282

S4 vacua of Ds11 supergravity are the near-hori-zon geometries of the extreme membrane and five-brane solutions, respectively, while the adS =S5

5

vacuum of IIB supergravity is the near-horizon ge-ometry of the threebrane solution. In other words,these p-brane solutions interpolate between maxi-mally supersymmetric vacua of the respective super-gravity theories. More recently it has been shownthat some intersecting brane solutions have a similarproperty. For example, the extreme black hole andblack string solutions of Ds5 minimal supergravity

'w x Ž4 reducing to as1r 3 dilaton black holes in. 3Ds4 interpolate between either adS =S or adS2 3

2 w x=S , respectively 5–7 , but these are the reductionof Ds11 supergravity solutions that can be inter-

w xpreted 8 as, respectively, three intersecting M2-branes or three intersecting M5-branes. Other exam-

w xples have been given in 9,7 , and it was shown morew xgenerally in 7 that intersecting M-branes interpolate

between the Ds11 Minkowski vacuum and al m Žspacetime of the form adS =E =S with kq lqk

.ms11 .In this paper we shall explore similar issues in the

context of Ns1 Ds10 supergravity. Some obser-vations concerning this case have been made previ-

w xously in the context of black hole entropy 10,7,11 ,and the topic has been revitalized by recent conjec-tures relating near-horizon geometries to large rank

w xlimits of supersymmetric gauge theories 12 , but ourprincipal concern is to explore some connections tothe Ns4 Ds4 gauged supergravity theory of

w xFreedman and Schwarz 13 , which we call the ‘FSŽ . Ž .theory’. The gauge group is SU 2 =SU 2 with

gauge coupling constants e and e , unless e s0A B BŽ . Ž .3in which case the gauge group is SU 2 =U 1

Ž .with e 'g being the SU 2 gauge coupling con-A

stant; we shall call the latter theory the ‘half-gauged’FS model. There is another Ns4 Ds4 gauged

w x Ž .supergravity 14 , usually called the ‘SO 4 theory’,which has the same field content and gauge groupbut different interactions, except for the ‘half-gauged’case which coincides with the half-gauged FS model.Only the FS model, gauged or half-gauged, will beof relevance here. Its distinguishing feature is that

Ž .the single scalar field s the Ds4 dilaton has apotential

Vs2 e2 qe2 es , 1Ž .Ž .A B

so that there is no Minkowski vacuum. This featureis also shared by gauged Ds7 minimal supergrav-

w x Ž w x.ity 15 with vanishing topological mass term 16 .This is no coincidence as the ‘half-gauged’ FS modelis the dimensional reduction on T 3 of the Ds7

w xtheory 17 .It is natural to suppose that gauged Ds7 super-

gravity is an S3 compactification of Ds10 Ns1Ž .supergravity since the SU 2 gauge group would

Ž .then acquire a KK origin as one factor of the SU 2Ž . 3 Ž=SU 2 isometry group of S the gauge fields of

Ž .the other SU 2 factor would have to belong to threevector multiplets which could likely be consistently

.truncated . If so, the FS model would then have anatural KK interpretation as an S3 =S3 compactifi-

Žcation of Ds10 Ns1 supergravity followed by aŽ . Ž .truncation of an SU 2 =SU 2 Yang-Mills multi-

.plet . The ‘half-gauged’ FS model would then ac-quire a similar interpretation as an S3 =T 3 compact-ification. These suppositions are in fact correct, al-though it was a long time before this was appreciatedw x 318,19 . One reason for the delay is that the first Scompactification of Ds10 Ns1 supergravity to be

w x f Žfound 20 is such that F'e where f is the. 3Ds10 dilaton is not everywhere positive on S .

The analogous S3 =S3 compactification suffers fromthe same problem, and requiring positivity of F led,in 1983, to a ‘no-go’ theorem that apparently pre-cluded the existence of a physically acceptable S3 =

3 w xS compactification to Ds4 21 .There were no further attempts to provide a KK

origin for the FS model until 1990, when the FSmodel was identified as part of the effective Ds4field theory for the heterotic string theory in an

3 3 w xS =S vacuum 18 . The ‘no-go’ theorem is cir-cumvented by the fact that the Ds4 dilaton is notpresumed to be constant. In a subsequent indepen-

w xdent development, it was discovered 3 that theŽ .non-singular fivebrane solution of Ds10 super-

w xgravity 22,23 interpolates, in the string metric, be-tween the Minkowski vacuum and an S3 compactifi-cation to Ds7 Minkowski spacetime, and it wasnoted that an S3 =S3 compactification to Ds4Minkowski spacetime is also possible. Again, neitherthe Ds7 nor the Ds4 dilaton is constant in thesecompactifications but, rather, linear in one of theMinkowski coordinates. We shall show here that theDs7 linear dilaton vacuum is actually the 1r2

( )P.M. Cowdall, P.K. TownsendrPhysics Letters B 429 1998 281–288 283

supersymmetric domain wall solution of gauged Dw xs7 supergravity found in 24 . This reduces in

Ds4 to a 1r2 supersymmetric domain wall solu-tion of the half-gauged FS model, which is also a

Ž .1r2 supersymmetric solution of the full SU 2 =Ž . w xSU 2 FS model 17 . There is therefore no obstacle

to the identification of gauged Ds7 supergravity asa consistent truncation of S3 compactified Ds10

Ž . Ž .Ns1 supergravity, and of the SU 2 =SU 2 FSmodel as a consistent truncation of S 3 =

3 ŽS compactified Ds10 Ns1 supergravity thetruncations merely removing inessential matter mul-

.tiplets . The latter identification was made and veri-w x Žfied in 18,19 . The former identification conjec-w x. w xtured in 7 then follows from the results in 17 .

Here we consider further these S3 and S3 =S3

Žcompactifications of Ds10 supergravity Ns1 is.henceforth assumed . We take the Ds10 action to

be

2 110 y2f 2'Ss d x yg e Rq4 Ef y F , 2Ž . Ž .H 12

where f is the dilaton and F is the 3-form fieldstrength. The metric is thus the string-frame metric.We shall show that the S3 =S3 compactification ofthis theory is the near horizon geometry of a solutionrepresenting the intersection of two fivebranes on a

w xline. The latter solution 25 , which preserves 1r4supersymmetry, therefore interpolates between thefully supersymmetric Minkowski vacuum and the1r2 supersymmetric S3 =S3 compactification to thedomain wall of the FS model.

The domain wall is not the only 1r2 supersym-metric solution of the FS theory. There is a 1r2supersymmetric electrovac solution of the half-

w x w xgauged model 26 . This was shown in 17 to de-scend from an analogous ‘electrovac’ solution ofgauged Ds7 supergravity, for which the metric is

Ž . 4actually just adS =E , corresponding to an adS3 3

=E 4 =S3 solution of Ds10 supergravity. Unlikethe domain wall, the ‘electrovac’ solution has aconstant dilaton. Here we shall show that it is thenear-horizon geometry of a 1r4 supersymmetric in-tersecting brane solution of Ds10 supergravity inwhich a string lies inside a fivebrane 2. Alterna-tively, by replacing E 4 by T 4 in the Ds10 ‘elec-trovac’ solution, we can consider it as an adS =S3

3

solution of the dimensionally reduced Ds6 super-gravity. This solution is the near horizon geometry of

w xthe self-dual Ds6 string of 28 , which is thereduction to Ds6 of the Ds10 intersecting branesolution. Considered in the context of minimal Ds6supergravity, the self-dual Ds6 string is similar tothe M2-brane, M5-brane and D3-brane in that it is a1r2 supersymmetric solution that interpolates be-tween the fully supersymmetric Mink and adS =S3

6 3

vacua of this theory 3. Thus, surprisingly, the Gib-bonsrFreedman electrovac of gauged Ds4 super-gravity is directly related to the Ds6 self-dualstring.

The Ds10 string-in-fivebrane solution can begeneralized to a string in the intersection of two

Ž .fivebranes which we choose to be orthogonal ; this1r4 supersymmetric solution can be found by anapplication of a ‘generalized harmonic function rule’

w xof 27,30,31 . By an appropriate choice of the har-monic functions one can arrange for the dilaton to beconstant and for the metric to interpolate between theMinkowski vacuum and an S3 =S3 compactificationto adS =E1. This establishes the existence of a3

1 Ž .supersymmetric adS =E vacuum of the SU 2 =3Ž .SU 2 FS model with at least 1r4 supersymmetry. It

is presumably the 1r4 supersymmetric ‘axionic’ so-w xlution recently found in 32 .

2. Domain walls from intersecting fivebranes

The 1r4 supersymmetric solution of Ds10 su-Ž .pergravity in string frame representing two orthog-

onal fivebranes intersecting on a line is

ds2 sds2 E Ž1,1. qHdxPdxqH XdxXPdxX ,Ž .

e2f sHH X , FswdHqwXdH X , 3Ž .

Ž1,1. Ž .where E indicates a 1 q 1 -dimensionalMinkowski space, H and H X are harmonic functionson their respective 4-dimensional Euclidean spaceswith metrics dxPdx and dxX

PdxX, and w and wX are

2 w xThis fact was also noted in 27,7 , but without the connectionw xto the FS model. A related observation was made in 10 .

3 w xSee 29 for recent related observations.


the Hodge duals on these two spaces. We choose theharmonic functions to be

1 1XHs1q , H s1q , 4Ž .2 X 2r r

< < X < X <where rs x and r s x are the distances from theorigins of the two 4-dimensional Euclidean spaces.

Close to the first fivebrane, but far from thesecond one, we have H;1rr 2 and H X

;1. In thisw xcase, the asymptotic metric is 3

dr 2X X2 2 Ž1 ,1. 2 3ds ;ds E qdx Pdx q qds SŽ . Ž .2r

sds2 E Ž1,6. qds2 S3 , 5Ž . Ž . Ž .

while the dilaton is f;r, where rsylog r. Fromthe discussion in the introduction we now deducethat this S3 compactification of Ds10 supergravityimplies the existence of a solution of Ds7 gaugedsupergravity with dilaton fsr, Minkowski 7-met-ric and vanishing Ds7 gauge fields. Using therelation ds2 sey4f r5ds2 between the string-frameE

7-metric and the Einstein-frame 7-metric we findthat this solution has Einstein-frame 7-metric

42 y r 2 Ž1 ,5. 2ds se ds E qdr . 6Ž . Ž .5E

Ž . 2 rDefining ys 1r2 e we find the solution

2 122 y 2 Ž1 ,5. y 2 2fds sH ds E qH dy , e sH , 7Ž . Ž .5 5E

with Hs2 y.To compare with the solutions of gauged Ds7

supergravity we need to write the Ds10 dilaton f

in terms of the Ds7 dilaton f . Later we shallŽ7.need to do the same for the Ds4 dilaton f 's .Ž4.We therefore pause here to deduce the relation be-tween f and the D-dimensional dilaton f . SinceŽD.the 10-metrics we consider are direct products ofD-dimensional metrics with spheres, or products ofspheres, of constant radius, the Ds10 dilaton f

remains the only scalar field in the lower dimension,so that the effective D-dimensional lagrangian is stillof the form

2y2f'Ls yg e Rq4 =f q . . . . 8Ž . Ž .

This is equivalent to

21L s yg Ry =f q . . . , 9Ž .( Ž .E E ŽD.2 E

where

'2 2f s f , 10Ž .ŽD. 'Dy2

and the subscript ‘E’ indicates a D-dimensional Ein-stein-frame metric. Thus, for Ds7 we have

'2 2f s f . 11Ž .Ž7. '5

Ž .Using this relation one sees that the solution 7 isw xthe 1r2 supersymmetric domain wall solution of 24

Ž .for which, in general, H is piecewise linear in y .We now turn to the asymptotic metric in the

region near both fivebranes. In this case H;1rr 2

and H X;1rrX 2. Setting

1 1X X

rsy log rr , ls log rrr , 12Ž . Ž . Ž .' '2 2

we then find

ds2 ;ds2 E Ž1,1. qdr 2 qdl2 qds2 S3 =S3 ,Ž . Ž .'f; 2 r , 13Ž .

while the 3-form field strength is now the sum of thevolume forms of the two S3 factors. This resultimplies the existence of a supersymmetric solution to

Ž . Ž .the Ds4 SU 2 =SU 2 FS model with vanishingŽ .gauge fields. Passing to the Ds4 Einstein frame,

for which f ' s s 2f, and defining y sŽ4.'2 2 r'Ž .1r2 2 e , we find that

ds2 sHy1ds2 E Ž1,2. qHy3dy2 , eys sH , 14Ž . Ž .E

'with Hs 2 y. This is just the 1r2 supersymmetricw xdomain wall solution of 17 , shown there to be the

dimensional reduction of the Ds7 domain wallsolution.

Thus, the intersecting fivebrane solution interpo-lates between the Ds10 Minkowski vacuum and a1r2 supersymmetric domain wall solution of either

Ž . Ž .Ds7 gauged supergravity or SU 2 =SU 2 Ds4gauged supergravity, according to whether we areclose to just one of the fivebranes or both of them.


3. Electrovacs from string-in-fivebrane

The 1r4 supersymmetric string-in-fivebrane solu-tion of Ds10 supergravity is

ds2 sHy1ds2 E Ž1,1. qH dxPdxqds2 E 4 ,Ž . Ž .1 5

e2f sHy1H ,1 5

FsÕol E Ž1,1. ndHy1 qwdH , 15Ž . Ž .1 5

where H and H are both harmonic functions on1 5

the 4-space with Euclidean 4-metric dxPdx, and w

is the Hodge dual on this space. We shall choose

H sH sH x . 16Ž . Ž .1 5

This choice has the property that the dilaton isconstant, in fact zero. The metric and 3-form fieldstrength are now

ds2 sHy1ds2 E Ž1,1. qHdxPdxqds2 E 4 ,Ž . Ž .FsÕol E Ž1,1. ndHy1 qwdH . 17Ž . Ž .

4 ŽIgnoring the E factor which may be replaced by4.T , this field configuration is automatically a 1r4

Ž .supersymmetric solution of the 1,1 Ds6 super-gravity theory obtained by T 4 compactification ofDs10 supergravity. Because the dilaton vanishes itis also a 1r2 supersymmetric solution of the mini-

Ž .mal 1,0 Ds6 supergravity. In fact, it is just thew x Žself-dual string solution of 28 for which the singu-

w xlarities of H were shown in 4 to be horizons of the.geodesically complete maximal analytic extension .

It then follows from the analysis below that theself-dual string solution interpolates between themaximally supersymmetric Minkowski and adS =3

3 Ž .S vacua of 1,0 Ds6 supergravity. In this respectthe Ds6 self-dual string is smilar to the M2-brane,M5-brane and D3-brane.

Ž .We now return to 17 and choose

1H x s1q , 18Ž . Ž .2r

< <where rs x . Near the origin the asymptotic metricis

dr 22 2 2 Ž1 ,1. 2 3 2 4ds ;r ds E q qds S qds E ,Ž . Ž . Ž .2r

19Ž .which is adS =S3 =E 4. The 3-form field strength3

F is asymptotic to the sum of the volume forms on

the S3 and adS factors. By ignoring the E 4 factor,3

we deduce the result just claimed above for theDs6 self-dual string. Instead, we may ‘ignore’ theS3 factor, i.e. we may interpret the asymptotic solu-tion just found as a new S3 compactification ofDs10 supergravity preserving at least 1r4 super-symmetry. This implies the existence of an adS =E 4

3

vacuum of gauged Ds7 supergravity, again pre-serving at least 1r4 supersymmetry. It actually pre-serves 1r2 supersymmetry, so supersymmetry is par-tially restored near the horizon. This follows fromthe fact that the ‘new’ adS =E 4 vacuum of gauged3

Ds7 supergravity is actually the Ds7 ‘electrovac’w xfound in 17 . The Ds7 ‘electrovac’ metric is

essentially of the form 4

22 2 r 2 2 r'ds sye dt qdr q dyq2 2 e dtŽ .qds2 E 4 . 20Ž . Ž .

The 3-form field strength of Ds7 gauged super-gravity is dual to a 4-form field strength which, inthis solution, is proportional to the volume form on

4 Ž .E . The SU 2 gauge fields are zero. If we dimen-sionally reduce on y and two of the cartesian coordi-nates of E 4 then we recover the 1r2 supersymmetricGibbons-Freedman electrovac of the half-gauged FS

Ž 2 .model with adS =E 4-metric , hence the name2w xgiven to the Ds7 solution in 17 . However, the

3-metric obtained by ignoring the E 4 factor is justadS , as we have verified by a computation of the3

Ricci tensor. In fact, the Ds7 ‘electrovac’ is equiv-w xalent to the compactification to adS found in 33 .3

4. ‘Axiovac’ from string-in-two-fivebranes

We now turn our attention to the solution repre-senting a string in the common linear-intersection oftwo fivebranes. The solution is

ds2 sHy1ds2 E Ž1,1. qH dxPdxqH X dxXPdxX ,Ž .1 5 5

e2f sHy1H H X ,1 5 5

FsÕol E Ž1,1. ndHy1 qwdH qwXdH X , 21Ž . Ž .1 5 5

where H is a harmonic function on the Euclidean5

4-space with the x coordinates, H X is a harmonic5

4 'Here we set g s 2 , set f s0, and choose horosphericalcoordinates.


function on the Euclidean 4-space with the xX coor-w xdinates, and the function H satisfies 30,311

y1 y1X X 22H = q H = H s0 . 22Ž . Ž . Ž .5 5 1

w xIt was noted in 31 that this can be solved byadditive separation of variables. Of more relevancehere is the fact that it can also be solved by multi-plicative separation of variables. Specifically, it issolved by H s ff X where f is harmonic in x and f X

1

is harmonic in xX. In particular, we may choose

H sH H X . 23Ž .1 5 5

This choice has the property that the dilaton is againzero. The other fields are

y1X2 2 Ž1 ,1.ds s H H ds EŽ . Ž .5 5

qH dxPdxqH X dxXPdxX ,5 5

y1X X XŽ1 ,1.Fsvol E nd H H qwdH qw dH .Ž . Ž .5 5 5 5

24Ž .

We now choose

1 1XH s1q , H s1q . 25Ž .5 52 X 2r r

Far away from one fivebrane, but close to the otherone, we recover the previous adS =S3 =E 4 solu-3

Ž .tion. Near both fivebranes we use the coordinates 8to write the asymptotic metric as

'2 y2 2 r 2 Ž1 ,1.ds se ds EŽ .

qdr 2 qdl2 qds2 S3 =S3 . 26Ž . Ž .

We recognize this as adS =S3 =S3 =E1. The3

square of the radius of curvature of the adS factor is3

now half as large as before, as required by the3 Ž .presence of two S factors given constant dilaton .

The original intersecting brane solution of Ds10supergravity preserves 1r4 supersymmetry, so theasymptotic solution near the fivebranes must alsopreserve at least this fraction. It therefore corre-sponds to a solution of the Ds4 FS model thatpreserves at least 1r4 supersymmetry and has metricadS =E1. This is presumably the 1r4 supersym-3

metric ‘axionic’ vacuum solution, or ‘axiovac’, ofw x32 .

5. Discussion

We have shown that various supersymmetricvacua of the Ns4 Ds4 gauged supergravity modelof Freedman and Schwarz can be reinterpreted ascompactifications of Ds10 Ns1 supergravity, andthat these compactifications are the near-horizon ge-ometries of various intersecting brane solutions. TheFS vacua that we can interpret in this way include

Ž . Ž .3the domain wall, the SU 2 =U 1 electrovac, andthe adS =E1 ‘axiovac’.3

There are other supersymmetric solutions forwhich we have not yet found a similar interpretation.An example which we believe should have such aninterpretation is the 1r4 supersymmetric electrovac

Ž . Ž . w xof the SU 2 =SU 2 FS model 26 . Although wehave not seen how to interpret this solution in termsof intersecting branes its existence follows from the1r4 supersymmetric adS =E1 ‘axiovac’. To see3

this, one writes the adS =E1 metric in the form3Ž .20 and reduces to Ds3 in the y direction. Thisyields a Ds3 electrovac which can be lifted to theDs4 electrovac with adS =E 2 metric. Thus these2

Ž .two 1r4 supersymmetric solutions of the SU 2 =Ž .SU 2 FS model are dual to each other.There are also other gauged supergravities. Many

have now been provided with a KK interpretationand, given such an interpretation, it is often possibleto interpret the KK compactification as the near-horizon geometry of a p-brane or, as shown here, ofintersecting branes. An exception is the gauged Ds

w x7 supergravity with topological mass term 16 . Thistheory has an adS vacuum but no known KK inter-pretation, although it is tempting to suppose that it isobtainable by some modification of the S3 compacti-fication. Another outstanding exception is the Ds6

Ž . w xSU 2 gauged supergravity of Romans 34 . Thistheory has an adS vacuum with the exceptional

Ž .supergroup F 4 as its isometry supergroup, but ithas no known KK interpretation. It is natural tosuspect that it arises as the effective theory in somecompactification of Ds10 supergravity. If so onemight suppose that it is again the near-horizon geom-etry of some intersecting brane solution, but noobvious candidate presents itself. We should alsopoint out that there are non-compact gaugings ofDs4 Ns8 supergravity that arise from ‘non-com-

w xpact’ compactifications of Ds11 supergravity 35 ,


but the latter are not known to occur as near-horizongeometries of any brane, or intersecting brane, solu-tions.

Consideration of the near-horizon geometries ofbranes and their intersections has led to a number ofcompactifications of Ds10 and Ds11 supergrav-ity theories that were unknown in the heyday ofKaluza-Klein theory. The S3 and S3 =S3 compacti-fications of Ds10 supergravity to domain wallsand electrovacs are examples. Another example isthe S7 compactification of IIA supergravity to an

w x 3 3adS ‘linear dilaton’ vacuum 36 . The S =S com-3

pactification to the Ds4 ‘axiovac’ discussed heresimilarly establishes a new 1r4 supersymmetric S3

=S3 =S1 compactification of Ds10 Ns1 super-gravity to Ds3; the effective Ds3 field theory ispresumably a matter-coupled adS supergravity. Onemay wonder whether there are any new gaugedsupergravity theories that might be found in thisway. For example, the fact that the near horizongeometry of the linear intersection of an M2-brane

5 3 w xwith an M5-brane is adS =E =S 9,7 means that3

there is an S3 compactification of Ds11 supergrav-ity to Ds8 preserving at least 1r4 supersymmetry.

Ž .The effective Ds8 field theory will have an SO 4gauge group of KK origin. One might expect thiseffective theory to be a new maximally supersym-metric gauged Ds8 gauged supergravity with vacuathat preserve only 1r4 supersymmetry, but this is

Ž .impossible because SO 4 is not contained in theR-symmetry group of the maximal Ds8 supersym-metry algebra. Instead, the effective field theorymust be minimal Ds8 supergravity coupled to an

Ž .SO 4 Yang-Mills multiplet in such a way that itadmits a 1r2 supersymmetric adS =E 5 vacuum. It3

would be interesting to determine what this couplingis, but it would not be a new gauged supergravity.

Acknowledgements

P.K.T. gratefully acknowledges the support of theIberdrola Profesor Visitante program. P.M.C. thanksthe members of the Faculty of Physics at the Univer-sity of Barcelona for hospitality. We are also gratefulto A. Chamseddine, D.Z. Freedman and C.M. Hullfor helpful conversations.

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gravities, hep-thr9710214.w x18 I. Antoniadis, C. Bachas, A. Sagnotti, Phys. Lett. B 235

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18 June 1998


Higher-order black-hole solutions in Ns2 supergravityand Calabi-Yau string backgrounds

Klaus Behrndt a,1, Gabriel Lopes Cardoso b,2, Bernard de Wit b,3, Dieter Lust a,4,¨Thomas Mohaupt c,5, Wafic A. Sabra d,6

a Humboldt-UniÕersitat, Institut fur Physik, D-10115 Berlin, Germany¨ ¨b Institute for Theoretical Physics, Utrecht UniÕersity, 3508 TA Utrecht, The Netherlands

c Martin-Luther-UniÕersitat Halle-Wittenberg, Fachbereich Physik, D-06122 Halle, Germany¨d Physics Department, QMW, Mile End Road, London E1 4NS, United Kingdom

Received 15 January 1998; revised 31 March 1998Editor: P.V. Landshoff

Abstract

Based on special geometry, we consider corrections to Ns2 extremal black-hole solutions and their entropiesoriginating from higher-order derivative terms in Ns2 supergravity. These corrections are described by a holomorphic

Ž .function, and the higher-order black-hole solutions can be expressed in terms of symplectic Sp 2nq2 vectors. We applythe formalism to Ns2 type-IIA Calabi-Yau string compactifications and compare our results to recent related results in theliterature. q 1998 Elsevier Science B.V. All rights reserved.

Dedicated to the memory of Constance Caris

One of the celebrated successes within the recentnon-perturbative understanding of string theory andM-theory is the matching of the thermodynamicBekenstein-Hawking black-hole entropy with the mi-croscopic entropy based on the counting of the rele-vant D-brane configurations which carry the same

w xcharges as the black hole 1,2 . This comparisonworks nicely for type-II string, respectively, M-the-

1 E-mail: [email protected] E-mail: [email protected] E-mail: [email protected] E-mail: [email protected] E-mail: [email protected] E-mail: [email protected].

ory backgrounds which break half of the supersym-metries, i.e. exhibit Ns4 supersymmetry in fourdimensions. In this paper we will discuss chargedblack-hole solutions and their entropies in the con-text of Ns2 supergravity in four dimensions.Four-dimensional Ns2 supersymmetric vacua areobtained by compactifying the type-II string on aCalabi-Yau threefold CY or by M-theory compacti-3

fication on CY =S1. In addition, the heterotic string3

on K 3=T 2 also leads to Ns2 supersymmetry infour dimensions and the heterotic and the type-IIvacua are expected to be related by string-string

w xduality 3 .Extremal, charged, Ns2 black holes, their en-

tropies and also the corresponding brane configura-


( )K. Behrndt et al.rPhysics Letters B 429 1998 289–296290

w xtions were discussed in several recent papers 4–11 .The macroscopic Bekenstein-Hawking entropy SS ,BH

i.e. the area A of the black-hole horizon, can beobtained as the minimum of the graviphoton charge

w xZ 4 ,

A2< <SS s sp Z , 1Ž .BH min4

where, as a solution of the minimization procedure,Ž .the entropy as well as the scalar fields moduli at

the horizon depend only on the electric and magneticblack hole charges, but not on the asymptotic bound-ary values of the moduli fields. This procedure hasbeen used extensively to determine the entropy ofNs2 black holes for type-II compactifications on

w xCalabi-Yau threefolds 5 . One of the key features ofextremal Ns2 black-hole solutions is that the mod-uli depend in general on r, but show a fixed-pointbehaviour at the horizon. This fixed-point behaviouris implied by the fact that, at the horizon, full Ns2supersymmetry is restored; at the horizon the metricis equal to the Bertotti-Robinson metric, correspond-ing to the AdS =S2 geometry. This metric can be2

Ž .described by a 0,4 superconformal field theoryw x12 . The extremal black hole can be regarded as asoliton solution which interpolates between two fullyNs2 supersymmetric vacua, namely correspondingto AdS =S2 at the horizon and flat Minkowski2

spacetime at spatial infinity.The Ns2 black holes together with their en-

tropies which were considered so far, appeared assolutions of the equations of motion of Ns2Maxwell-Einstein supergravity action, where thebosonic part of the action contains terms with at

Žmost two space-time derivates i.e., the Einsteinaction, gauge kinetic terms and the scalar non-linear

.s-model . This part of the Ns2 supergravity actionŽ .can be encoded in a holomorphic prepotential F X ,

which is a function of the scalar fields X belongingto the vector multiplets. However, the Ns2 effec-tive action of strings and M-theory contains in addi-tion an infinite number of higher-derivative termsinvolving higher-order products of the Riemann ten-sor and the vector field strengths. A particularlyinteresting subset of these couplings in Ns2 super-gravity can be again described by a holomorphic

Ž 2 . w xfunction F X,W 13–18 , where the additionalchiral superfield W is the Weyl superfield, compris-

ing the covariant quantities of conformal supergrav-ity. Its lowest component is the graviphoton field

Ž y .strength in the form of an auxiliary tensor field T ,mn

while the Weyl tensor appears at the u 2 level. Theaim of this paper is, using the superconformal calcu-lus, to study the Ns2 black-hole solutions and thecorresponding entropies for higher order Ns2 su-pergravity based on the holomorphic functionŽ 2 . 2F X,W . We treat W as a new chiral background

and expand the black-hole solutions as a powerseries in W 2. As an interesting example we computethe entropy of the charged Ns2 black holes intype-II compactifications on a Calabi-Yau threefold.At the end we compare our results to a recentcomputation of the microscopic Calabi-Yau black

w xhole in M-theory 9,10 .Let us start by recalling the Ns2 black-hole

solutions and their entropies in the case where theŽ .holomorphic function F X does not depend on the

Weyl multiplet. The bosonic Ns2 supergravity ac-tion coupled to n vector multiplets is given by

14 m A B'S s d x yg y Rqg E z E zHNs2 A B m2

1yI yJmn qI qJmny i NN F F yNN F F ,ž /I J mn I J mn8

2Ž .A Ž .where the z with As1 . . . ,n denote complex

" I mn Ž .scalar fields, and F with Is0, . . . ,n are theŽ . Žanti- selfdual abelian field strengths including the

.graviphoton field strength . An intrinsic definition ofw xa special Kahler manifold 19 can be given in terms¨

Ž .of a flat 2nq2 -dimensional symplectic bundleŽ .over the 2n -dimensional Kahler-Hodge manifold,¨

with the covariantly holomorphic sections

X I

Vs , 3Ž .ž /FI

obeying the symplectic constraint

I I² :i V ,V s i X F yF X s1 . 4Ž .Ž .I I

Usually the F can be expressed in terms of aIŽ .holomorphic prepotential F X , homogeneous of

Ž . Idegree two, via F sE F X rE X . The field-depen-IŽ .dent gauge couplings in 2 can then also be ex-

pressed in terms of derivatives of F. The constraintŽ .4 can be solved by introducing the projective holo-

( )K. Behrndt et al.rPhysics Letters B 429 1998 289–296 291

IŽ . Imorphic sections X z , which are related to the Xaccording to

1Ž .K z , z

2I IX se X z ,Ž .I IK z , z sylog iX z F X zŽ . Ž . Ž .Ž .I

I IyiX z F X z . 5Ž . Ž . Ž .Ž .I

Ž .Here K z, z is the Kahler potential which gives rise¨to the metric g . The holomorphic sections trans-A B

IŽ .form under projective transformations X z ™w Ž .x IŽ .exp f z X z , which induce a Kahler transforma-¨

Ž .tion on the Kahler potential K and a U 1 transfor-¨mation on the section V,

K z , z ™K z , z y f z y f z ,Ž . Ž . Ž . Ž .1Ž Ž . Ž ..f z yf z

2V z , z ™e V z , z . 6Ž . Ž . Ž .Besides V, the magneticrelectric field strengthsŽ I .F ,G also constitute a symplectic vector. Heremn mn I

q Ž .G is generally defined by G x smn I m n I

y4 igy1r2 dSrdFqmn I. Consequently, also the cor-Ž I .responding magneticrelectric charges Qs p ,qI

transform as a symplectic vector.In terms of these symplectic vectors the stationary

w xsolutions have been discussed in 7 in a fixed Kahler¨gauge. The generalized Maxwell equations can besolved in terms of 2nq2 harmonic functions, which

Žtherefore also transform as a symplectic vector m,n.s1,2,3 ,

1 1I I˜F s e E H r , G s e E H r .Ž . Ž .m n m n p p m n I m n p p I2 2

7Ž .

Throughout this paper we assume that the metric canw xbe brought in the form 20

ds2 sye2Udt 2 qey2 Udx mdx m , 8Ž .where U is a function of the radial coordinate r

m m's x x . The harmonic functions can beparametrized as

p I qII I˜ ˜H r sh q , H r sh q , 9Ž . Ž . Ž .I Ir r

and we write the corresponding symplectic vector as˜ IŽ . Ž Ž . Ž ..H r s H r , H r shqQrr.I

Once one has identified the various symplecticvectors that play a role in the solutions, it followsfrom symplectic covariance that these vectors shouldsatisfy a certain proportionality relation. The sim-plest possibility is to assume that V and H aredirectly proportional to each other. Because H is

Ž .real and invariant under U 1 transformations, thereis a complex proportionality factor, which we denote

Ž .by Z. Hence we define a U 1 -invariant symplecticŽvector here we use the homogeneity property of the

.function F ,

IPsZVs Y ,F Y , 10Ž . Ž .Ž .I

I Iso that Y sZX , and assume

P r yP r s iH r . 11Ž . Ž . Ž . Ž .

This equation determines Z,

I ˜ IZ r syH r X qH r F X ,Ž . Ž . Ž . Ž .I I

2< < ² :Z r s i P r ,P r . 12Ž . Ž . Ž . Ž .

Ž .The first of these equations indicates that Z r isrelated to the auxiliary field Ty. This relation maymn

2 Ž .not hold when F depends on W . The Eqs. 11 ,which we call the stabilization equations, also govern

AŽ .the r-dependence of the scalar moduli fields: z rA 0 ˜ IŽ . Ž . Ž .sY r rY r . So the constants h ,h just deter-I

mine the asymptotic values of the scalars at rs`.In order to obtain an asymptotically flat metric withstandard normalization, these constants must fullfill

Ž . Ž .some constraints. Near the horizon rf0 , 11w xtakes the form used in 5 and Z becomes propor-

Ž .tional to the holomorphic BPS mass MM z sIŽ . I Ž Ž .q X z yp F X z .I I

When in addition we make the symplecticallyy2 Uinvariant ansatz e sZZ, it can be shown that the

solution preserves half the supersymmetries, exceptat the horizon and at spatial infinity, where super-symmetry is unbroken. From the form of the static

Ž .solution at the horizon r™0 we can easily deriveits macroscopic entropy. Specifically theBekenstein-Hawking entropy is given by

2 y2U 2SS sp r e sp r ZZŽ . Ž .rs0 rs0BH

I Is ip Y F Y yF Y Y , 13Ž . Ž .Ž .Ž .hor I hor I hor hor


where the symplectic vector P at the horizon,

P Y Ir™0 r™0hor horIP r f , Y r f , 14Ž . Ž . Ž .

r r

is determined by the following set of stabilizationequations:

P yP s iQ. 15Ž .hor hor

So we see that the entropy as well as the scalar fieldsz A sY A rY 0 depend only on the magneticrelec-hor hor hor

Ž I .tric charges p ,q . It is useful to note that the set ofIŽ .stabilization Eqs. 15 is equivalent to the minimiza-

w xtion of Z with respect to the moduli fields 4 . Asalready said, at the horizon rs0 full Ns2 super-symmetry is recovered. Also note that in the sameway one can construct more general stationary solu-tions, such as rotating Ns2 black holes, multi-

w xcentered black holes, TAUB-NUT spaces, etc. 7 .As an example, consider a type-IIA compactifica-

tion on a Calabi-Yau 3-fold. The number of vectorsuperfields is given as nshŽ1,1.. The prepotential,which is purely classical, contains the Calabi-Yauintersection numbers of the 4-cycles, C , and, asA BC

aX-corrections, the Euler number x and the rational

instanton numbers nr. Hence the black-hole solu-tions will depend in general on all these topological

w xquantities 5 . However, for a large Calabi-Yau vol-ume, only the part from the intersection numberssurvives and the Ns2 prepotential is given by

Y AY B Y C 1F Y sD , D sy C .Ž . A BC A BC A BC0 6Y

16Ž .Based on this prepotential we consider in the follow-

Žing a class of non-axionic black-hole solutions that.is, solutions with purely imaginary moduli fields

A Žwith only non-vanishing charges q and p As0Ž1,1.. Ž .1, . . . ,h . So only the harmonic functions H r0

˜ AŽ .and H r are nonvanishing. This charged configu-ration corresponds, in the type-IIA compactification,to the intersection of three D4-branes, wrapped overthe internal Calabi-Yau 4-cycles and hence carryingmagnetic charges p A, plus one D0-brane with elec-tric charge q . In the corresponding M-theory picture0

these black holes originate from the wrapping ofthree M5-branes, intersecting over a common string,

Žplus an M-theory wave solution momentum along.the common string . For the solution indicated above,

the four-dimensional metric of the extremal black-w xhole solutions is given by 7

y2 UŽ r . A B C˜ ˜ ˜(e s2 H r D H r H r H r .Ž . Ž . Ž . Ž .0 A BC

17Ž .The scalars at the horizon are determined as

AY 1 1 DhorA A A 0z s , Y s ip , Y s ,hor hor hor0 (2 2 qY 0hor

DsD p Ap BpC . 18Ž .A BC

Finally, the corresponding macroscopic entropy takesw xthe form 5,21

SS s2p q D . 19Ž .(BH 0

Let us now turn to the discussion of Ns2black-hole solutions in the presence of higher-deriva-tive terms in the Ns2 supergravity action. In thatsituation the function F depends on both X and W 2

and is still holomorphic and homogeneous of seconddegree. Just like the chiral superfield, whose lowestcomponent is X I, W is also a reduced chiralmn

superfield. Therefore it has the same chiral and Weylweights as X I. Thus, W 2 sW W mn is a scalarmn

chiral multiplet with Weyl and chiral weight equal totwice that of the X I. Hence the homogeneity of Fimplies

E F X ,W 2Ž .I 2 2 2X F X ,W q2 W s2 F X ,W .Ž . Ž .I 2E W

20Ž .The lowest-u component of the superfield W con-tains the auxiliary tensor field that previously took

Žthe form of the graviphoton field strength up to.fermionic terms . However, in the case at hand the

Lagrangian is more complicated, so that this tensorcan only be evaluated as a power series in terms ofexternal momenta divided by the Planck mass. Simi-lar comments apply to all the auxiliary fields. Never-theless we can still use the superconformal Ns2

w xmultiplet calculus 13 . Obviously, this case is muchmore complicated and we do not attempt to give afull treatment of the solutions here. A detailed dis-cussion of these solutions will appear elsewhere. Inthe following we will instead rely on symplecticcovariance to analyze the immediate consequencesof the W-dependence of F and discuss its implica-tion for the black-hole entropy.


Ž 2 .Our strategy will be to expand F X,W as apower series in W 2 as follows,

`

I 2 Ž g . I 2 gF X ,W s F X W , 21Ž . Ž . Ž .Ýgs0

where F Ž0. is nothing else than the prepotentialdiscussed before. This expansion will allow us to

w xmake contact with the microscopic results of 9,10 ,where a suitable expansion of the microscopic en-tropy was proposed. From the fact that the superfieldW contains the Weyl tensor at order u 2, the W 2

dependence leads, for instance, to terms proportionalto the square of the Weyl tensor and quadratic in theabelian field strengths, times powers of the tensorfield T. Nevertheless, these terms are all precisely

Ž .encoded in 21 .For type-II compactifications on a Calabi-Yau

Ž g .Ž .space, the terms involving F X arise at g-looporder, whereas in the dual heterotic vacua the F Ž g .

appear at one loop and also contain non-perturbativecorrections. In passing we note that the physicalcouplings are in general non-holomorphic, where theholomorphic anomalies are governed by a set of

w xrecursive holomorphic anomaly equations 14,15,17 .The presence of the chiral background W 2 will

not modify the special geometry features that wediscussed before, provided one now uses the fullfunction F with the W 2 dependence included. So,the section V, which transforms as a vector under

Ž .Sp 2nq2 , now takes the form

X IIX

Ž g . 2 gVs s . 22Ž .F X WŽ .2 Ý Iž /F X ,WŽ .I � 0gs0

In order for the Einstein term to be canonical, V hasŽ .to obey again the symplectic constraint 4 , so that

I 2 I 2Ž . Ž . Ž .X F X,W yX F X,W syi. Using the U 1 -I II I 2 2invariant combinations Y sZX and Z W we can

Ž .define a U 1 -invariant symplectic vector as

IYPsZVs 2 2ž /F Y ,Z WŽ .I

Y I

gŽ g . 2 2s , 23Ž .F Y Z WŽ . Ž .Ý I� 0gs0

2 2Ž .where we used the expansion F Y,Z W s

Ž g . 2 2 gŽ .Ž .Ý F Y Z W . The factor Z will again begs0

determined by the stabilization equations and willthus depend on W 2.

Second, the field strength G , which togethermn I

with F I forms a symplectic pair, is still defined bymn

the derivative with respect to the abelian field strengthof the full action, and therefore modified by theW-dependence of F. Prior to eliminating the auxil-iary fields, this action is at most quadratic in the fieldstrengths, and G" is generally parametrized asmn I

q 2 qJ qG sF X ,W F qOO ,Ž .mn I I J mn mn I

Gy sF X ,W 2 FyJ qOOy , 24Ž . Ž .mn I I J mn mn I

where OO " represents bosonic and fermionic mo-mn I

ment couplings to the vector fields, such that theBianchi identities and the field equations read

n Ž q y. I n Ž q y.E F yF sE G yG s0. Again, it ismn mn I

crucial to include the full dependence on the Weylmultiplet, also in the tensors G" and OO " . Themn I mn I

reason is that the symplectic reparametrizations arelinked to the full equations of motion for the vector

Ž .fields which involve the Weyl multiplet and not toŽ .parts of the Lagrangian. The modification of thefield strength G can be interpreted as having themn I

Žeffect that the electric charges q which, togetherI

with the magnetic charges p I form the symplectic. 2charge vector Q get modified in the chiral W

‘‘medium’’, compared to their original ‘‘micro-scopic’’ values qŽ0.. Below we will comment on theI

relevance of this observation.Extremal Ns2 black-hole solutions in the pres-

ence of higher-derivative interactions must againpreserve half the supersymmetries, except for rs0,`, but obviously this condition is now much harderto solve. Nevertheless it is possible to show that theŽ .tangent-space derivative of the moduli with respectto the radial variable is still vanishing at the horizon,indicating the expected fixed-point behaviour. Ratherthan solving the equations for the full black-holesolution, we impose the stabilization equations. For

Ž . y2 Uthe metric we make again the ansatz 8 where etakes the same form in terms of P , which nowincorporates the modifications due to the back-ground,

y2 U I 2 2 I 2 2e sZZs i Y F Y ,Z W yY F Y ,Z W .Ž . Ž .Ž .I I

25Ž .


The stabilization equations now read

I I I˜Y -Y H rŽ .s i

2 2 2 2 ž /ž / H rF Y ,Z W yF Y ,Z W Ž .Ž . Ž . II I

I i I˜ phs i q , 26Ž .qž /ž /h r II

where the harmonic functions characterize the valuesŽ .of the field strengths according to 7 , just as before,

except that the field strength G incorporates themn I

modifications due to the background.We are now particularly interested in the behavior

of our solution at the horizon r™0, that is wewould like to compute the corrected expression forthe black-hole entropy. Without any W-dependencein F, previous calculations show that the quantity

2 2 2 y ymnŽ .Z W sZ T T has the following behaviourmn

near the horizon:

12 2 0Z W s qOO r . 27Ž . Ž .2r

Ž .Eq. 27 could get modified when F depends on W.These corrections should then be viewed as the backreaction of the non-trivial W 2-background on theblack-hole solution. We will work to leading orderand therefore assume that possible corrections toŽ .27 can be neglected to that order. Thus, we will

Ž .simply use Eq. 27 at the horizon, so that themodified Ns2 black-hole entropy is then given by:

2 y2U 2SS sp r e sp r ZZŽ . Ž .rs0 rs0BH

I Is ip Y F Y ,1 yY F Y ,1 .Ž . Ž .Ž .hor I hor hor I hor

28Ž .

As before, the symplectic vector P is given as inhorŽ .Eq. 15 , where now, however, P has a non-triv-hor

ial dependence on the W 2-background. So we have

I I IY yY s ip , F Y ,1 yF Y ,1 s iq .Ž . Ž .hor hor I hor I hor I

29Ž .

Consequently, the modified black hole entropy onlydepends again on the magneticrelectric charges.

For concreteness, let us again discuss the type-IIAcompactification on a Calabi-Yau 3-fold in the limitof large radii, which amounts to suppressing alla

X-corrections. We are, in particular, interested in thecontribution from F Ž1., which arises at one loop in

the type-IIA string. This term is of topological ori-gin; it is related to a one-loop R4 term in the

w xten-dimensional effective IIA action 22 . For the2 2Ž .function F Y,Z W we thus take

2 2 Ž0. Ž1. 2 2F Y ,Z W sF Y qF Y Z WŽ . Ž . Ž .Y AY B Y C 1 Y A

2 2sD y c Z W . 30Ž .A BC 2 A0 024Y Y

Here the c are the second Chern class numbers of2 A

the Calabi-Yau 3-fold. For simplicity we will con-sider again axion-free black holes with p0 s0,q sA

Ž .0. Then the stabilization Eqs. 29 have the solution

A1 1 D c p2 AA A 0Y s ip , Y s 1q .(hor hor (2 2 q 6D0

31Ž .Ž . Ž .Inserting 31 into 28 yields

1ADq c p2 A12

SS s2p q . 32Ž .(BH 0 1ADq c p( 2 A6

Ž .Eq. 32 is only to be trusted to linear order in c ,2 A

because we have not included the back reaction.Ž .Expanding 32 to lowest order in c yields2 A

1 1 q0ASS s2p q D q2p y c p( (BH 0 2 Až /12 12 D

q PPP . 33Ž .The terms linear in c thus cancel out! Thus, when2 A

Ž I .expressed in terms of the charges Qs p ,q , thereI

is no correction to SS to lowest order in c !.BH 2 A

Although this is a rather striking result, whose signif-icance is not quite clear to us at the moment, itseems to disagree with the microscopic findings ofw x9,10 , as we will now discuss.

We would like to compare the macroscopic en-Ž .tropy formula 33 with the microscopic entropy

w xformula recently computed in 9,10 for certain com-pactifications of M-theory and type-IIA theory on

w xCalabi-Yau 3-folds. In 9,10 the microscopic en-tropy was, to all orders in c , found to be2 A

Ac p2 ASS s2p q D 1q . 34Ž .(micro 0 ž /6D


Ž .Expanding 34 to lowest order in c yields2 A

1 q0ASS s2p q D q2p c p q PPP .( (micro 0 2 A12 D35Ž .

Ž . w xThe correction in 35 was then matched 9,10 witha correction to the effective action involving R2-typeterms with a coefficient function F Ž1..

The approach used above to obtain the macro-Ž .scopic entropy 33 is different from the approach

w xused in 9,10 for obtaining the macroscopic formula.A comparison of the results of both approaches, that

Ž . Ž .is of 35 and 33 , indicates that in order to obtainmatching of both results, the macroscopic electric

Ž .charge q appearing in 33 cannot be identical to0Ž .the electric q appearing in 35 . This is one way of0

explaining the discrepancy. If we denote the chargeŽ . Ž0. Ž .q appearing in 35 by q , then matching of 350 0

Ž .and 33 can be achieved provided that the q ap-0Ž . Ž0.pearing in 33 is related to q as follows:0

c p A2 AŽ0.q sq 1q . 36Ž .0 0 ž /6D

As alluded to earlier, this can be interpreted as amodification of the ‘‘microscopic’’ charge qŽ0. due0

to the W 2 medium. This interpretation can be furthermotivated by noting that if one defines qŽ0. to be0

given by

Ž0. Ž0. Ž0.q 'yi F Y yF YŽ . Ž .Ž .I I hor I hor

I Ž g . Ž g .sq q i F Y yF Y , 37Ž . Ž .Ž .Ž .Ý I hor I horgG1

Ž . Ž . Ž .then insertion of 31 into 37 precisely yields 36 .Ž . 2Eq. 37 makes it clear that, in the presence of a W

medium, the charges q and qŽ0. cannot be identical.0 0

Since the ‘‘microscopic’’ charge qŽ0. has the inter-0w xpretation of quantized momentum around a circle 9 ,

it is integer valued. We thus note that the macro-Ž .scopic charge q given in 36 , which is measured at0

spatial infinity, is not integer valued in the presenceof a W 2 medium.

Ž . Ž . ŽGiven that the Eqs. 35 and 33 agree providedŽ . .the relation 36 holds , it is then conceivable that

the full macroscopic entropy, derived from the met-Ž .ric given in 25 , also agrees with the full micro-

Ž .scopic entropy 34 . In order to calculate the full

Ž .macroscopic entropy from 25 , one will have to takeinto account a possible back reaction of the non-triv-ial W 2 background on the black hole solution, in

2 2Ž .particular the corrections to the quantity Z W atŽ .the horizon 27 . In the presence of such corrections,

2 2Ž .the quantity Z W will presumably also dependhor2 2Ž .on the magneticrelectric charges, that is Z W hor

Ž .f p ,q Ž g .2 Žs . In addition the higher functions F g)r.1 might also contribute to the entropy. Finally, even

non-holomorphic corrections to the higher-derivativeeffective action might play a role. At the end, let us

Ž . Ž .note that if one inserts 36 into 34 , the micro-scopic entropy takes again the very simple form:

SS s2p q D . 38Ž .(micro 0

So with this charge ‘‘renormalization’’ one rediscov-Ž .ers the zero-th order entropy 19 . It is tempting to

speculate that this feature is related to the enhance-ment of the supersymmetry at the horizon, namely tothe fact that at the horizon we still have an AdS =S2

2

geometry, even after including all higher-derivativeterms.

This work is supported by the DeutscheŽ .Forschungsgemeinschaft DFG and by the European

Commission TMR programme ERBFMRX-CT96-0045 in which Humboldt-University Berlin andUtrecht University participate. W.A.S is partiallysupported by DESY-Zeuthen.

w xNote Added. In 23 it has been pointed out thatthere are modifications to the Bekenstein-Hawkingentropy formula in the presence of higher curvatureterms. In this paper we have not considered suchmodifications. It is not clear at present what theircontribution to our result is. We would like to thankSerge Massar for raising this issue and for bringing

w xRefs. 23 to our attention. B. d. W. thanks RobertMyers for useful discussions regarding this topic.

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18 June 1998


Ns2 supersymmetric quantum black holesin five dimensional heterotic string vacua

Ingo GaidaDepartment of Applied Mathematics and Theoretical Physics, UniÕersity of Cambridge, SilÕer Street, Cambridge CB3 9EW, UK

Received 23 February 1998Editor: P.V. Landshoff

Abstract

Exact black hole solutions of the five dimensional heterotic S-T-U model including all perturbative quantum correctionsand preserving 1r2 of Ns2 supersymmetry are studied. It is shown that the quantum corrections yield a bound on electriccharges and harmonic functions of the solutions. q 1998 Elsevier Science B.V. All rights reserved.

Keywords: String theory; Ns2 supergravity; Black holes

w xIn 1 Strominger and Vafa considered five dimensional string theory with Ns4 supersymmetry to derivew xthe Bekenstein-Hawking entropy 2 by counting black hole microstates. In this letter the low-energy effective

action of the five dimensional S-T-U model in heterotic string vacua with Ns2 supersymmetry is studied. Thismodel yields the Strominger-Vafa black hole including, in addition, perturbative quantum corrections.

The action of five dimensional Ns2 supergravity coupled to Ns2 vector multiplets has been constructedw x w xin 3 and the compactification of Ns1, Ds11 supergravity M-theory down to five dimensions on

Ž . Ž .Calabi-Yau 3-folds CY with Hodge numbers h ,h and topological intersection numbers C has been3 1,1 2,1 L SD

w x Ž .given in 5,6 : The N -dimensional space MM N sh y1 of scalar components of Ns2 abelian vectorV V 1,1

multiplets coupled to supergravity can be regarded as a hypersurface of a h -dimensional manifold whose1,1Ž . Ž .coordinates X f are in correspondence with the vector bosons including the graviphoton . The definining

Ž .equation of the hypersurface is VV X s1 and the prepotential VV is a homogeneous cubic polynomial in theŽ .coordinates X f :

1 L S DVV X s C X X X , L, S ,Ds1, . . . h . 1Ž . Ž .LSD 1,16

In five dimensions the Ns2 vector multiplet has a single scalar and MM is therefore real. Moreover, if theprepotential is factorizable, it is generically symmetric and of the form

VV X sX 1 Q X Lq1 , Ls1, . . . N , 2Ž . Ž . Ž .V


( )I. GaidarPhysics Letters B 429 1998 297–303298

where Q denotes a quadratic form. It follows that the scalar fields parametrize the coset space

SO N y1,1Ž .VMMsSO 1,1 = . 3Ž . Ž .

SO N y1Ž .V

ŽThe bosonic action of Ns2 supergravity coupled to N vector multiplets is given by omitting LorentzV.indices

ey11 1 1y1 i j L S L S De LLsy Ry g EfEf y G F F q C eF F A . 4Ž .i j L S L SD2 2 4 48

The corresponding vector and scalar metrics are encoded in the function VV completely1 L SG sy E E lnVV X , g sG E X f E X f . 5Ž . Ž . Ž . Ž .< <VVs1 VVs1L S L S i j L S i j2

LŽ .Here the derivatives in the scalar metric are with respect to the h coordinates X f and the h y1 scalar1,1 1,1i L w xfields f , respectively. It is useful to introduce special coordinates t and their duals t 4 :L

t L f s6y1r3X L f sC LS f t f , t f sC t S f t D f sC f t S f . 6Ž . Ž . Ž . Ž . Ž . Ž . Ž . Ž . Ž . Ž .S L L SD L S

From these definitions follows t Lt s1 and C C SD sd D. In these special coordinates one finds for theL L S L

gauge coupling matrix

61r3 23 LS L S L SG sy C y t t , G sy C y3t t , 7Ž . Ž .Ž .LS L S L S2 1r32 6

with G G SD sd D and g sy3C E t LE t S.LS L i j L S i jw xIt has been shown in 7,4 that the supersymmetry transformations of the gaugino and the gravitino vanish if

Ž . Lthe electric central charge Zs t q , appearing in the supersymmetry algebra, has been minimized in moduliL

Ž .space E Zs0 . This minimization procedure yields the fixed values of the moduli on the black hole horizoniw x7,8 . Equivalently one may use the ‘‘stabilisation equations’’

q s t Z , Z 2 sC LSq q . 8Ž .L L fix fix fix L S

The geometry of the corresponding extreme Ds5 black holes is determined by the following metric:

ds2 syey4 V Ž r . dt 2 qe2V Ž r . dr 2 qr 2 dV 2 . 9Ž .Ž .3

where the metric function e2V Ž r . is a function of harmonic functions. The moduli for so-called double-extremew xblack holes are constant and given by their fixed values throughout the entire space-time 9 . For these

S w x'double-extreme black holes the gauge fields satisfy 2 yg G F sq . Moreover, the entropy 2 of extremeLS L

w xblack holes in five dimensions is given by 73r22A p Z

S s s . 10Ž .BH 4G 2G 3 < fixN N

In Ds5 point-like objects are dual to string-like objects. Thus, corresponding to the electric central charge Zexist the dual magnetic central charge Z s t p L with magnetic charges p L. The electric and magnetic chargesm L

w xarise in M-theory from two- and five-brane solitons which wrap even cycles in the CY-space 5,15 .

q s G , p L s F . 11Ž .H HL 7 44L LC =S C =S3 2 2

d LL 4L Lw xHere, F is the field-strength of the three-form in Ds11 supergravity while G s is its dual; C C4 7 2dF4

w xdenotes a four- two- cycle in CY . From the point of view of the heterotic string q correspond to3 2,3

perturbative electric charges of Kaluza-Klein excitations and winding modes, p1 is the charge of the

( )I. GaidarPhysics Letters B 429 1998 297–303 299

2,3 w x w xfundamental string and p q arise from Ds10 solitonic five-branes wrapping around K K =S . The1 3 3 1

magnetic central charge Z determines the tension of magnetic string states as a function of the moduli. Thus,mw xanalogous to the fixed value of the electric central charge, there exist a fixed value for the string tension 12,10 .

p L s t L Z , Z3 s27C p Lp Sp D. 12Ž .m ,fix m ,fix L SD

w xIt follows that the Ds5 entropy-density of the magnetic string is given by 12

2r32 L S D< <S ; Z ; C p p p . 13Ž .Ž .S m < fix L SD

Compactifying the Ds10 effective heterotic string on K 3=S one can construct the Ds5, Ns2 S-T-U1w xmodel 14 . This model contains 244 neutral hypermultiplets, which we will ignore in the following. Moreover it

contains three vector moduli S, T and U, where S denotes the heterotic dilaton and T ,U are associated to theŽ .graviphoton and the additional U 1 gauge boson of the S compactification. The Ds5 heterotic S-T-U model1

w xis dual to M-theory compactified on a Calabi-Yau threefold 5 . Further compactification on S yields the rank 41Ž .S-T-U model in Ds4, which is dual to the X 1,1,2,8,12 model of the type II string compactified on a24

w xCalabi-Yau 14 . In special coordinates the prepotential reads

VV S,T ,U sSTUqh T ,U . 14Ž . Ž . Ž .Ž . w xThe function h T ,U denotes perturbative quantum corrections, which have been determined in 6

a a3 3h T ,U s U u TyU q T u UyT . 15Ž . Ž . Ž . Ž .

3 3

Here we have introduced the parameter as1 in order to discuss the classical limit a™0 in the followingŽ .explicitly. In the classical limit the scalar fields parametrize the coset 3 with N s2. Using very specialV

geometry the dilaton field S can be eliminated through the algebraic equation

1yh T ,UŽ .Ss . 16Ž .

TU

For convenience we define the functions

2 a a a a3 3 3 3f x , y s x u yyx y y u xyy , g x , y s x d yyx y y d xyy . 17Ž . Ž . Ž . Ž . Ž . Ž . Ž .

3 3 3 3

It follows

1q f T ,U g U,T 1q f U,T g T ,UŽ . Ž . Ž . Ž .E Ssy y , E Ssy y . 18Ž .T U2 2TU TUT U U T

1,2,3 Ž .If we take t s S,T ,U , we find for the dual coordinates

a a1 1 12 2t s TU , t s SUq T u UyT , t s STq U u TyU . 19Ž . Ž . Ž .1 2 33 3 33 3

Ž .Thus, for the matrix C with components C we obtainLS

0 U T1yh T ,UŽ .

U 2 aTu UyTŽ .1 TUCs . 20Ž .6

1yh T ,UŽ .� 0T 2 aUu TyUŽ .TU


Hence, the gauge coupling matrix reads

T 2U 2 Uf T ,U Tf U,TŽ . Ž .1 1yh T ,UŽ .

21 Uf T ,U 1y2h T ,U q f T ,U 2h T ,UŽ . Ž . Ž . Ž .2Gs . 21Ž .TUT2r32P61yh T ,U 1Ž .

2� 0Tf U,T 2h T ,U 1y2h T ,U q f U,TŽ . Ž . Ž . Ž .2TU U

Moreover, it is straightforward to compute the metric g of the scalar fieldsi j

1 11yh T ,U qTg U,T 1q2h T ,U qTg U,T qUg T ,UŽ . Ž . Ž . Ž . Ž .2 2TUT

gs .1 1� 01q2h T ,U qTg U,T qUg T ,U 1yh T ,U qUg T ,UŽ . Ž . Ž . Ž . Ž .22TU U

22Ž .

It follows in the weak coupling regime S)T)U)0

3 1 Udet gs ya , 23Ž .2 2 24 T U T

a a a31 3 3 2 6 9detGs 1y U 1y U ya U q U . 24Ž .288 ž / ž /3 3 27

Note that the gauge coupling matrix depends only on U. Thus, one obtains for the boundaries of theŽ .Weyl-chamber S)T)U Here the boundaries are regular up to the critical points with det g s 0,` . Thecrit.

chamber S)T)U)0 has three boundaries. The lines SsT and TsU are generically regular. These twoŽ .lines intersect at one point in moduli space SsTsU . Classically this intersection point is a ‘‘double

Ž . Ž .self-dual point’’, i.e. this point is self-dual with respect to T-duality Rs1 and S-duality g s1 . Including5

quantum corrections one obtainsy1r3a

U s 1q 'U S™T 'U T™U 25Ž . Ž . Ž .0 crit . crit .ž /3

at this point. Thus, the scalar metric degenerates at this point and, therefore, the moduli space simply ends herew x11 in this limit. However, strictly speaking one leaves the perturbative regime with abelian gauge groups, only,

Ž .in this limit and additional non- perturbative effects must be taken into account, too.For convenience we will restrict ourselves now to the fundamental Weyl chamber T)U. Moreover, we will

consider first of all double-extreme black hole solutions before studying the bigger class of extreme solutionsw x Ž . Ž . 1given in 16 . Starting with the prepotential 14 and the constraint VV X s1 one obtains from the electric

stabilisation equations

3q sZTU , 3q sZSU , 3q sZSTqaZU 2 . 26Ž .1 2 3

It follows

2 aU 3 q3 Zy9q Us0 , aZ 2U 4 y3q ZU 2 q9q q s0 . 27Ž . Ž .3 3 1 2

1 In this double-extreme context all the operators take their fixed values in moduli space.


Ž . w xIn the classical limit as0 one obtains for the fixed values of the fields 101r3 1r3 1r3q q q q q q2 3 1 3 1 2

Ss , Ts , Us . 28Ž .2 2 2ž / ž / ž /q q q1 2 3

Ž .1r3 w xand the central charge Zs3 q q q . Thus, we obtain the Strominger-Vafa black hole 1 with entropy1 2 3

2pS s q q q . 29Ž .(BH 1 2 32GN

Ž . 3Including the quantum corrections as1 one obtains a quadratic equation in U with solution23 4aq q y3q 9q q1 2 3 1 23 2(U syg 1y 1ydrg , gs , ds 30Ž .ž / 2 2 2ž /2 a 4aq q q3q 4a q q q3aq1 2 3 1 2 3

Since U is real we obtain a bound g 2 ydG0, which becomes, in terms of the charges,

q2 G4q q . 31Ž .3 1 2

The appearance of this bound is a true quantum effect. The corresponding fixed values of the moduli S,T andthe central charge follow from the solution straightforward. Note that the solution also has to satisfy theinequality S)T)U in terms of the charges. In the classical limit this condition is satisfied if q )q )q . It3 2 1

follows q2 )q q and, therefore, the quantum bound is stronger 2. If we consider, for convenience, the case3 1 2Ž .where 31 is saturated, we obtain for the fixed values of the fields

q q1r3 1r3 1r32 13 3 3Ss , Ts , Us . 32Ž .Ž . Ž . Ž .4 4 4( (q q1 2

It follows that the black hole entropy is given by

p 23r2S s q . 33Ž . Ž .BH 36GN

Ž . 2Clearly this result does not coincide with the classical entropy 29 in the limit q s4q q . Note that the metric3 1 2Z2V w xfunction is always given by e s1q in the double extreme limit 16 . Moreover, the entropy vanishes if one3r

of the electric charges vanishes. The dual string solution as been extensively discussed in the literaturew x 1,2,313,12,10 . The fixed values of the scalar fields are given by S,T ,Usp rZ and the fixed value of them

magnetic central charge reads1r3a 31 2 3 3Z s3 p p p q p . 34Ž .Ž .m ž /3

In the classical limit the electric and magnetic central charge are dual to each other, if one exchanges electricand magnetic charges. This property does not hold at the quantum level. It follows that some of the magneticcharges can vanish to give a non-trivial entropy-density of the dual magnetic string.

w xNow we will consider the more general class of black hole solutions of 16 . The static, sphericallyw x Ž .symmetric BPS black hole solution of 16 has metric 9 and

qLS y4V Ž r . nm2G F se E H , n ,ms1,2,3,4 , h E E H r s0´H sh q . 35Ž . Ž .LS 0 m m L n m L L L 2r

Here the five-dimensional harmonic functions H are characterized by the electric charge q of the threeL L

2 I thank M. Green for a discussion on this point.


Ž .abelian gauge fields including the graviphoton and the arbitrary constants h . For special values of h weL L

obtain the double-extreme solution discussed above. Moreover, the solution satisfies1'yg t s H . 36Ž .L L3

Ž .From 36 follows

ey2 V H sTU , ey2 V H sSU , ey2 V H sSTqaU 2 . 37Ž .1 2 3

Thus, analogous to the double-extreme black hole solution we obtain

2 aU 3 q3 e2V y3H Us0 , ae4V U 4 yH e2V U 2 qH H s0. 38Ž . Ž .3 3 1 2

Ž . w xIn the classical limit as0 one finds 161r3 1r3 1r3H H H H H H2 3 1 3 1 2

Ss , Ts , Us . 39Ž .2 2 2ž / ž / ž /H H H1 2 3

Ž . 3Including the quantum corrections as1 one obtains again a quadratic equation in U with solution23 4aH H y3H 9H H1 2 3 1 23 2(U syg 1y 1ydrg , gs , ds . 40Ž .ž / 2 2 2ž /2 a 4aH H q3H 4a H H q3aH1 2 3 1 2 3

Since U is real we obtain the bound g 2 ydG0. If we take, for instance, 4H H q3H 2 )0 we obtain, in terms1 2 3

of the harmonic functions,

H 2 G4H H . 41Ž .3 1 2

The corresponding values for the moduli S,T and the metric function e2V in terms of harmonic functions followstraightforward. Note that this black hole configuration exhibits a Z symmetry: H ™einp H for integer n.2 L L

The corresponding black hole entropy of this extreme black hole solution is by definition the same as for thedouble-extreme solution. Although we can compute now the full quantum solution, i.e. the values of the modulion the horizon, the entropy and the metric, these expressions are not very illuminating for the exact solution.Instead we give here the first order quantum corrections to various quantities to give a qualitative discussion, i.e.

Ž 2 .we omitt contribution of order OO a . The corresponding fixed values of the moduli on the horizon are1r3 1r3 1r3q q q q q q2 3 1 3 1 2

S s 1ya , T s 1ya , U s 1q2a , 42Ž . Ž . Ž . Ž .< fix < fix < fix2 2 2ž / ž / ž /q q q1 2 32 aq q 1r31 2 Ž .with as . It follows for the central charge Z s3 q q q 1ya . The corresponding black hole2 Ž .< fix 1 2 39q3

entropy is2p

2S s q q q 1y a . 43Ž .( Ž .BH 1 2 3 32GN

Moreover, the leading order correction for the metric function e2V is given by2 a H H1 21r32Ve s H H H 1yD , Ds . 44Ž . Ž . Ž .1 2 3 29 H3

Ž .Near the horizon rs0 the metric becomes approximately

r 4 l21r32 2 2 2 2 2ds sy dt q dr ql dV , l s q q q 1ya . 45Ž . Ž . Ž .3 1 2 32 2l r

It follows that the five-dimensional space-time manifold MM is a product space near the horizon MM sAdS = S35 5 2

Ž . Ž .with symmetry group SO 2,1 =SO 3 . It is straightforward to obtain the leading order quantum correction tothe ADM-mass of this extreme black hole. Using diffeomorphism invariance the metric can always be broughtinto the following form:

8G MN ADM2 2ds sy 1y q PPP dt q PPP . 46Ž .2ž /3p r


Introducing ‘‘dressed charges’’ q sq rh and expanding the metric function one obtainsˆL L L

p a h h h h1 2 1 2M s 1q q ya q . 47Ž .ˆ ˆÝADM L 32 2½ 5ž /4G 3 h hN 3 3Ls1,2,3

w xIn the classical limit we obtain the results of 16 . Moreover, we find that there are no leading order quantumcorrections to the ADM-mass if

q qqˆ ˆ1 2s2. 48Ž .

q3

In addition, the extreme black hole solution has vanishing ADM-mass if

h2 a 2 q yq yqˆ ˆ ˆ3 3 1 2s . 49Ž .

h h 3 q qq qqˆ ˆ ˆ1 2 1 2 3

Although this result only holds to the leading order one expects a similar condition for the massless black holeconfiguration including all quantum corrections.

To conclude, exact black hole solutions preserving 1r2 of Ns2 supersymmetry in the five dimensionalS-T-U model including all perturbative quantum corrections have been studied. It has been shown that thequantum corrections yield a new bound on electric charges and harmonic functions of the solutions. Theappearence of bounds of this kind in Ns2 supersymmetric models in five and four dimensions has been

w xpreviously studied in 17,10 . It would be very interesting to find the corresponding statistical mechanicalw xinterpretation of the black hole entropy analogous to the analysis of Strominger and Vafa 1 including this

quantum bound.

Acknowledgements

I would like to thank T. Mohaupt for discussions. This work is supported by DFG.

References

w x Ž .1 A. Strominger, C. Vafa, Phys. Lett. B 379 1996 99.w x Ž . Ž . Ž .2 J. Bekenstein, Lett. Nuov. Cim. 4 1972 737; Phys. Rev. D 7 1973 2333; S. Hawking, Nature 248 1974 30; Comm. Math. Phys. 43

Ž .1975 199.w x Ž .3 M. Gunaydin, G. Sierra, P.K. Townsend, Nucl. Phys. B 242 1984 244.¨w x Ž .4 A. Chamseddine, S. Ferrara, G.W. Gibbons, R. Kallosh, Phys. Rev. D 55 1997 3647.w x Ž .5 A.C. Cadavid, A. Ceresole, R. D’Auria, S. Ferrara, Phys. Lett. B 357 1995 76; G. Papadopoulos, P.K. Townsend, Phys. Lett. B 357

Ž . Ž .1995 300; S. Ferrara, R.R. Khuri, R. Minasian, Phys. Lett. B 375 1996 81.w x Ž .6 I. Antoniadis, S. Ferrara, T.R. Taylor, Nucl. Phys. B 460 1996 489.w x Ž . Ž . Ž .7 S. Ferrara, R. Kallosh, A. Strominger, Phys. Rev. D 52 1995 5412; S. Ferrara, R. Kallosh, Phys. Rev. D 54 1996 1514; D 54 1996

Ž .1525; S. Ferrara, G.W. Gibbons, R. Kallosh, Nucl. Phys. B 500 1997 75.w x Ž .8 F. Larsen, F. Wilczek, Phys. Lett. B 375 1996 37.w x Ž .9 R. Kallosh, M. Shmakova, W.K. Wong, Phys. Rev. D 54 1996 6284.

w x Ž .10 A. Chou, R. Kallosh, J. Rahmfeld, S.-J. Rey, M. Shmakova, W.K. Wong, Nucl. Phys. B 508 1997 147.w x Ž .11 E. Witten, Nucl. Phys. B 471 1996 195.w x Ž .12 K. Behrndt, G. Lopes Cardoso, I. Gaida, Nucl. Phys. B 506 1997 267.w x Ž .13 K. Behrndt, T. Mohaupt, Phys. Rev. D 56 1997 2206.w x Ž .14 S. Kachru, C. Vafa, Nucl. Phys. B 450 1995 69.w x Ž .15 K. Becker, M. Becker, A. Strominger, Nucl. Phys. B 456 1995 130.w x16 W.A. Sabra, hep-thr9708103; A. Chamseddine, W.A. Sabra, hep-thr9801161.w x Ž .17 K. Behrndt, Phys. Lett. B 396 1996 77.

18 June 1998


New supersymmetric vacua for Ds4, Ns4gauged supergravity 1

Harvendra Singh 2

INFN Sezione di PadoÕa, Departimento di Fisica ‘Galileo Galilei’, Via F. Marzolo 8, 35131 PadoÕa, Italy

Received 14 January 1998; revised 17 March 1998Editor: L. Alvarez-Gaume

Abstract

Ž . Ž .In this paper we obtain supersymmetric brane-like configurations in the vacuum of Ns4 gauged SU 2 =SU 2supergravity theory in four spacetime dimensions. Almost all of these vacuum solutions preserve either half or one quarter ofthe supersymmetry in the theory. We also study these solutions in presence of nontrivial axionic and gauge fieldbackgrounds. In the case of pure gravity with axionic charge the geometry of the spacetime is AdS =R1 with Ns13

supersymmetry. An interesting observation is that the domain walls of this theory cannot be given an interpretation of a2-brane in four dimensions. But it still exists as a stable vacuum. This feature is quite distinct from the domain-wallconfiguration in massive type IIA supergravity in ten dimensions. q 1998 Published by Elsevier Science B.V. All rightsreserved.

1. Introduction

w xRecent understanding of the nonperturbative aspects in string theories and its duality symmetries 1 hastriggered a series of new developments in the study of massive supergravity theories in various dimensionsw x2–10 . It is viewed that the massive supergravities are required for the spacetime interpretations of higher

Ž . Ž .dimensional world volume Dirichlet objects D-p-branes of type II superstrings. For example, massive gaugedw x w xtype IIA supergravity 2 is a candidate theory for type IIA D-8-brane 11,3,12 whose field strength is a

10-form. The dual of a 10-form in ten dimensions is a constant which can be identified as a mass termŽ .cosmological constant of massive type IIA supergravity. Next, the strong coupling limit of type IIA strings is

w xM-theory whose spacetime theory is sought to be the eleven dimensional supergravity 13 . String dualitiesrequire that all even branes of type IIA and odd branes of IIB should have an eleven-dimensional interpretationw x14 . In this picture a D-8-brane can be obtained from a double dimensional reduction of M-9-brane or toroidalreduction of an M-8-brane. Therefore 11-dimensional supergravity should also have massive analog of type IIA

w xsupergravity. There has been some progress recently in finding massive 11-dimensional supergravity 9,10 .

1 Research supported by INFN fellowship.2 E-mail: [email protected].


( )H. SinghrPhysics Letters B 429 1998 304–312 305

There are various ways to get massive supergravities starting from the massless ones. For example thew xgeneralised Scherk and Schwarz dimensional reduction scheme 15 do give rise to massive supergravities.

Under generalised reduction scheme some of the higher dimensional fields are given linear dependences alongw x w xthe directions to be compactified 3,4 . This procedure has been used in 3 to demonstrate the type IIA and IIB

duality in Ds9 for the massive case. Other methods require the gauging of internal global symmetries in thew xsupergravity theory 16,17 . An interesting feature of massive or gauged supergravity theories is that, in general,

Ž .they possess original amount of supersymmetry even in the presence of mass gauge parameters. If supersym-metry is only the criterion then massive supergravities are as consistent as massless ones. However, distinctionappears in terms of internal symmetries. The price we pay is that the internal global symmetries including

Ždualities are generally reduced to smaller symmetry groups during the process of gauging. Once the mass or. Ž .gauge parameters are set to zero the ‘standard’ massless supergravities are recovered.

Ž . Ž . w xIn this work we will study Ns4 gauged SU 2 =SU 2 supergravity in four dimensions 17 . It has beenrecently established that this gauged supergravity can be embedded into Ns1 supergravity in ten dimensions as

3 3 w x 3 Ž . Ž . Ž .an S =S compactification 22 . Previously also, a Kaluza-Klein KK interpretation for the SU 2 =SU 2w xgauged supergravity was given in 23 where this model was identified as part of the effective Ds4 field

theory for the heterotic string theory in an S3 =S3 vacuum. These two KK interpretations are essentially thesame upto consistent truncations. Thus it is worthwhile to study above Ns4, Ds4 gauged supergravitykeeping in mind the recent developments in massive supergravities.

It is well known that the dimensional reduction of ten-dimensional supersymmetric Yang-Mills theory to 4Ž . w x Ž .dimensions gives us SU 4 invariant supersymmetric theory 18 . An off-shell supergravity with SU 4 global

w xsymmetry and Ns4 local supersymmetry was constructed in four dimensions long ago 19 . The spectrum ofthis theory in the gravity multiplet consists; the graviton E M, 4 Majorana spin-3r2 gravitinos C I, 3 vectorm m

fields Aa , 3 axial-vector fields B a, 4 spin-1r2 Majorana fields x I, the dilaton field f, and a pseudo-scalarm m

Ž . w xfield h. There do exist other versions of Ns4 supergravity in Ds4, e.g. SO 4 supergravity theory 20 . Itturns out that at the equation of motion level both the above supergravities are equivalent upto certain field

Ž . Ž . Ž .redefinitions and duality relations. The internal subgroup SU 2 =SU 2 of SU 4 supergravity can be gauged,Ž . Ž . w xgiving rise to Ns4 supergravity with local SU 2 =SU 2 symmetry 17 . However, imposition of local

Žsupersymmetry in presence of the local internal symmetry requires introduction of mass like terms or. w xcosmological constant in the theory in addition to certain bilinear fermionic terms 17 . The mass term appears

w xas an effective dilaton potential in the theory. It was noted 17 that the dilaton potential leads to energy densitywhich is unbounded from below which is physically nonacceptable 4. But the question can be asked whetherthis theory yields some stable vacuum configurations. It has been noted that theory allows supersymmetric

w xsolutions like ‘electro-vac’ solutions 21 and more recently non-Abelian solitons as stable vacuum configura-w x w xtions were also shown to exist 24,22 . In related work on strings in curved backgrounds 25 , exact

supersymmetric solutions of four-dimensional gauged supergravities have been constructed using the powerfulŽ .techniques of super conformal field theory by exploiting a connection between gauged supergravities and

non-critical strings.In this work we will show that extended objects like strings, domain walls, pure axionic gravity,

dilaton-axion gravity and maximally symmetric point-like configurations do exist in the vacuum of theNs4, Ds4 gauged theory. All these backgrounds preserve half or one-fourth of the supersymmetries exceptfor the point-like configuration for which we have not been able to find supersymmetric case. We also presentthe generalisation of the fundamental string solution as a vacuum solution of the massive supergravity. Thepaper has been organised as follows. In the next section we briefly describe the gauged model. In the subsequent

3 I thank Prof. P.K. Townsend bringing this fact into my knowledge4 Ž . w xSimilar unbounded potential also arises in gauged SO 4 supergravity theory 16

( )H. SinghrPhysics Letters B 429 1998 304–312306

Sections 3, 4, and 5 we obtain various background solutions and discuss there supersymmetric properties in eachcase. In the last Section 6 we summarise our results.

2. The gauged model

Ž . Ž . w xWe consider the truncated version of the Ns4 gauged SU 2 =SU 2 supergravity model 17 given byA B

the condition B a s0, that is half of the gauge fields are vanishing.. In addition we shall only study them

configurations which are either purely electric type or magnetic type with respect to remaining 2-form fieldw xstrength. We also set all the spinor fields to zero in our study. Under these specifications the action 17

becomes

21 1 L14 m 2f m yf a amn a amn f˜'Ss d x yg Rq E fE fqe E hE h y e F F y h F F q e , 1Ž .Ž .H m m mn mn2 2P2! 2P2! 2

2 2 2 Ž .where L se qe , e and e being the gauge couplings of respective SU 2 groups andA B A B

1a a a b c a sr a˜F sE A yE A qe e A A , F s e F .mn m n n m A abc m n mn mn sr2

w xThe corresponding supersymmetry transformations are 17 ,

ffi

1 1I I f m y I a a mn I2dx s e E fq ig e E h g y e e a F G q e e e q ig e ,Ž .2Ž .m 5 m mn A 5 B4 4'2 2

fi i§ 1 1I I m n a a f y I a a nrdC se E y v G q e a A y e eg E hy e e a F g G2ž /m m m ,m n A m 5 m nr m2 2 '4 4 2

fi

I2q e e e q ig e g ,Ž .A 5 B m'4 2

d bosons s0, 2Ž . Ž .where e I are four spacetime dependent Majorana spinors. Since fermionic backgrounds are absent therefore thebosonic fields do not vary under supersymmetry. Our convention for the metric is with mostly minus signs

1m nŽ . � 4 Ž . w xqyyy , v , are the spin connections, g ,g s2 g , ms0,1,2,3 , and G s G ,G , where m, nm m n mn m n m n4a Ž . Ž .are tangent space indices. a are three 4=4 SU 2 matrices and along with other three matrices of SU 2A B

1 1Ž . w xgenerate the , representation of the gauged model 17 .2 2

3. Domain wallsrrrrrmembranes

3.1. Pure dilaton graÕity

The first solution we obtain is a domain wall in four dimensions with the following configuration satisfyingŽ .the field equations derived from the action 1 ,

y12 2 2 2 2 a< <ds sU y dt ydx ydx yU y dy , fsylnU, Usm yyy , A s0, hs0, 3Ž . Ž . Ž .Ž .1 2 0 m

where m2 sL2r2. This background is singular at ysy which is the position of the tyx hyperplane or0

domain wall. Since no matter fields are present other than the dilaton this background represents pure dilatonf Ž .gravity. Note that e , analog of string coupling, vanishes at asymptotic infinity y™"` and so also the


Ž .curvature scalar. But both are divergent at ysy . The isometry group for this background is P 1,2 =ZZ ,0 3 2

where P represents three dimensional Poincare group and ZZ is the reflection symmetry of the dilaton´3 2

potential around ysy .0Ž .We now study the supersymmetric properties of this background. If we substitute 3 in the supersymmetry

Ž .Eqs. 2 we find that the fermionic variations vanish provided the supersymmetry parameters satisfyŽ .a when e s0, LseB A

1

4I I I I< <e G s i e , for Usm yyy , e sU e 4Ž .3 0 0

Ž .b and when e s0, LseA B

1

4I I I Ie G g sye , e sU e , 5Ž .3 5 0

I Ž . Ž .where e is a constant spinor. The conditions 4 and 5 can be satisfied because for spatial indices all G ’s0

have imaginary eigen values "i, while g 2 s1. These conditions break half of the supersymmetries in either5

case. Thus we see that there exist nontrivial Killing spinors preserving Ns2 supersymmetry for pure dilatonicdomain wall backgrounds which is sufficient for the stability of the solution against quantum fluctuations.

3.2. Pure axionic graÕity

A nontrivial axionic field configuration can also be obtained in the vacuum of this theory. Let us first presentŽ .a much familiar geometry which is the solution of the field equations derived from 1 ,

2dr2 2 2 2 2 2 a 'ds s 1qQr dt y yr du ydy , fs0, A s0, hs"2 Q y , 6Ž .Ž . m2ž /1qQrŽ .

2 Ž . 5such that 4QsL r2. The geometry of the four manifold is an anti-de Sitter AdS space AdS =R . This3Ž . w xsolution is analogous to the Freedman-Gibbons FG pure electro-vac solution 21 which has the geometry of

AdS =R2, we will discuss it in the next section. We expect this background to be stable if the stability analysis2w x Ž .is done based on the methods of anti-de Sitter spaces 26 . Instead we try to show here that the solution 6 will

be stable from the supersymmetry arguments. We present another solution below which has similar asymptoticŽ . Ž .properties as the background in 6 . We write down a new axionic gravity vacuum solution of 1 ,

2dr2 2 2 2 2 2 a' 'ds s f dt y y f du ydy , fs0, f r s1q Q r , A s0, hs1"2 Q y , 7Ž . Ž .m2ž /f

2 'with 4QsL r2 where 2 Q is the measure of the axion charge. Note that for Qs1 the asymptotic propertiesŽ . Ž . Ž .of both the solutions 6 and 7 are the same. But the background in 7 is much transparent from the

Ž . Ž .supersymmetry point of view. If we substitute 7 in Eq. 2 we find that the background preserves 1r4 of thesupersymmetries. Corresponding Killing spinors are for e s0A

1

2I I I I I Ie s f r e , e G s"i e , e G g sye . 8Ž . Ž .0 3 1 5

Ž .The twin conditions in 8 on the Killing spinors break the supersymmetry to 1r4th. Thus pure axionic gravityin the gauged model has Ns1 supersymmetry intact. Other conditions when e s0, Lse can also beB A

similarly derived.

5 When y-coordinate is a closed circle the geometry will be AdS =S1.3


3.3. NonÕanishing dilaton and axion

Now we generalise above two special cases and show that both dilaton and axion can exist together in thevacuum of the gauged supergravity in a supersymmetric fashion. We obtain correspondingly a solution

dr 2y12 2 2 2 2 2 < <ds sU y f dt y y f du yU y dy , fsylnU, U y s1qm yyy ,Ž . Ž . Ž . 02ž /f

a' 'hs1q2 Q y , f r s1q Q r , A s0, 9Ž . Ž .m

2 2 Ž . 6such that m sy4QqL r2. This solution is regular at ysy as compared to domain wall solution in 3 .0Ž .The geometry of the manifold is AdS =R. Thus the Minkowskian flat geometry in 3 has turned into an3

Ž .anti-deSitter geometry in presence of nontrivial axion flux. The corresponding isometry group now is O 2,2 ofthe AdS , Z symmetry is broken due to nontrivial axion field. This background preserves 1r4-th of the3 2

supersymmetries. We find that most general Killing spinors can be written when m2 se2r2 and 4Qse2r2,A B

1 1

4 2I I I I I Ie sU y f r e , e G s i e , e G g sye . 10Ž . Ž . Ž .0 3 1 5

Ž . Ž .It is clear from 9 that pure dilatonic and pure axionic solutions are special cases of 9 when Qs0 and ms0Ž . 'respectively. Note that in 9 only positive values for the axionic charges Q are allowed. This restriction

comes from the supersymmetry in presence of a regular dilaton field which is nontrivial now. While in theŽ .previous case of pure axionic gravity no dilaton both plus and minus values for the axion charge were allowed.

4. Vorticesrrrrrstrings

4.1. Pure dilatonic case

Ž .We also obtain vortex or string-like solution of the equations of motion derived from action 1 . We look forŽ . Ž .the solutions such that the isometry group is P 1,1 =SO 2 . Again we consider pure dilaton gravity. The2

background solution is22 2 2 2 2 2 ads sU y dt ydx ydy yy du , fsy2lnU, Usm y , A s0, hs0, 11Ž . Ž . Ž .m

2 2 2 2(where the parameter m satisfies m sL r8. Here ys y qy is the radial distance in the transverse1 2Ž . Ž .2-dimensional Euclidean plane and u is the azimuthal angle. It should be noted that the function U y in 11 is

of very specific nature which vanishes at ys0. It indicates that the volume of the tyx space shrinks to zero atys0 much like the size of the 2-sphere shrinking to zero at the centre of the Cartesian three-space. For thisbackground the curvature scalar is singular at the origin ys0 and also the string coupling while at asymptoticinfinity both are vanishing.

Ž .Background in 11 preserves half of the supersymmetries of the theory. There exist a nontrivial Killingspinor for e s0B

1

2I I I Ie G s i e , e sU e , for Usm y. 12Ž .2 0

where G corresponds to the direction tangent to y. Note that these Killing spinors vanish at ys0. Other2

Killing spinors when e s0 can also similarly be derived.A

6 < < Ž .We could have as well chosen Us1q m yy y in Eq. 3 instead of taking a divergent solution at ys y .0 0


4.2. Presence of nonÕanishing electric field

As in the case of Domain-walls, we find here that a nontrivial electric flux can be switched on in theŽ .supersymmetric string solution 11 . The electric field is given by a constant electric field tangent to the

x-direction. Before showing that we assume that the singularity at the origin is modified in presence of electric<field. We consider that at the origin U s1 so that string coupling is unity at the origin. Advantage of this isys0

that we can take the smooth limit m™0 in the background below. The limit correponds to constant dilatonŽ .field. It will be clear soon. Under these specifications the background solution of action 1 in presence of

electric field is

q2 dx 22 2 2 2 2 2ds sU y 1q x dt y ydy ydy , fsylnU, Us1qm y ,Ž . 1 22ž /2 q� 021q xž /2

Aa s q x ,0,0,0 d a3 , hs0, 13Ž . Ž .m

L2 yq 2 7 Ž .such that ms , where q is the measure of the electric field strength . From 13 it is clear that we have8

Ž . Ž .chosen an specific direction in the isospin space thus breaking SU 2 to U 1 . Note that now there is noA

singularity at ys0 and asymptotically string coupling and curvature scalar both vanish. Though backgroundŽ .13 does not preserve any supersymmetry but we are free to consider the limit m™0, q™L, e s0. This isA

2 Ž .the advantage of choosing the potential Us1qm r . Now we can set ms0 in 13 . The vacuum solution inŽ .13 then reduces to Freedman-Gibbons purely electric ‘electro-vac’ solutions. It is easy to follow this bydefining the following relations:

1 t p pxs tan r , ts , y FrF . 14Ž .

q q 2 2

Ž .Eq. 13 becomes

12 2 2 2 2 a a3ds s dt ydr ydy ydy , fs0, Us1, A s q x ,0,0,0 d , hs0, 15Ž . Ž .Ž . 1 2 m2 2L cos r

w x awhich is the electro-vac solution of 21 corresponding to case e s0, B s0. This background has Ns2A m

w x Ž .supersymmetry 21 and can be described as the extremal or supersymmetric limit m™0 of the background inŽ .13 .

4.3. Presence of axion charges

Ž .In the presence of axion field we obtain a vacuum solution for the action 1 which is a generalisation of thew x Ž . Ž . Ž .fundamental string solution 27 . Such solutions non-static with an isometry group O 1,1 =SO 2 have

w xalready been obtained in a previous work 7 by the author. But here we are going to present the background1 Ž .solution which is static and has the symmetry R =Z =SO 2 . This background is2

dx 22 2 2 2 2ds s f x dt y yU y , x dy qy du , fsylnU y , x ,Ž . Ž . Ž .Ž .ž /f xŽ .

< < < < < < ) y2f y1Us1yQln yyy qm xyx , f x s1qm xyx , Ehs e H , B sU , 16Ž . Ž .0 0 0 01

where H represents antisymmetric field strength of B and m2 sL2r2. Note that this background hasmn

7 Ž .When we take q™0 the background 13 is still a vacuum solution


invariances under time translations, reflections about xsx plane and rotations in the two dimensional y-plane.0Ž .This background, however, does not preserve supersymmetries in 2 . But it has two special limits; one when

Ž . Ž . Ž .Qs0 16 reduces to the supersymmetric domain wall solution in 3 , other when ms0 background 16w x Ž .becomes a fundamental string 27 . Thus we claim that the background 16 is the most simple generalisation of

Ž . Ž .string geometry, though unstable, in presence of dilatonic potential cosmological constant as in 1 . That is toŽ .say, background 16 represents a fundamental string whose vacuum is a domain-wall instead of a Minkowski

Ž . Žspace. The string vacuum domain-wall is stable but not the strings geometric excitations which carry axionic.charges in this vacuum. The stable geometries in the domain-wall vacuum which carry axionic charges have

already been presented in Section 3.

5. Point-like solutions

Ž .To complete the analysis we also obtain maximally symmetric isotropic background solutions of 1 . Wehave not been able to find a supersymmetric case for them. The solution we are going to present below is ofvery specific type that it allows only unit value of the magnetic charge. The solution is given by

dr 2 r2 2 2 2 2ds s f dt y yU du qsin u dF , fsylnU, Us1qm r , f r s "1,Ž . Ž .

f r0

Aa s 0,0,0,Q 1ycosu d a3 , hs0, 17Ž . Ž .Ž .m m

L2 qQ 2mwith ms r such that Q s1. Thus this solution exists only in the presence unit value of magnetic0 m2

Ž .charge. For plus sign in f r the solution is smooth every where and is asymptotically flat. But for minus signthere appears a nonsingular horizon at rsr . From the nonsingular horizon we mean by the finite size.0

However, this does not hide any singularity.

6. Summary

We summarise our results as follows:In this work we have obtained a stable domain-wall solution in the purely dilatonic vacuum of gauged Ns4

supergravity. The solution preserves half of the supersymmetries. Then we obtained pure axionic gravityvacuum which preserves 1r4 of the supersymmetries of the theory. We also have obtained domain-walls in thepresence of nontrivial axionic background. To our surprise these backgrounds also preserve one fourth of thesupersymmetries and are therefore stable. As we find that the domain wall solution cannot be given an

Ž .interpretation of a dual of a 4-form field strength in the action 1 because the dilatonic potential is unboundedw xfrom below. Though, in the case of type IIA gauged supergravity 2 the domain walls have an interpretation as

a dual of a 10-form field strength of an 8-D-brane.Ž .Next, we have obtained vortex string-like solutions which preserve half of the supersymmetries and

therefore are stable solutions. Then we have generalised these solutions in presence of constant electric flux andshowed that these solutions are generalisation of Freedman-Gibbons electro-vac background. Though these

Ž .solutions do not preserve any supersymmetry but can be smoothly taken to their extremal supersymmetriclimit which is an electro-vac background preserving Ns2 supersymmetries.

In last we obtained the maximally symmetric point-like solutions also. We haven’t found any fraction ofsupersymmetry being left unbroken for point case. It should be explored further if black holes could be found in

Ž . Ž .gauged SU 2 =SU 2 supergravity.


In conclusion we have seen that the vacuum of the gauged supergravity is quite interesting and is rich withstable brane-like configurations.

7. Note added

When this work was communicated I came to know from Prof. P. K. Townsend that another domain-wallw x Ž .solution has been obtained in 28 . We can see that this domain-wall is related to the domain-wall solution in 3

Ž . Ž .by going to a frame where the potential U y ™1rU y . However, the string coupling diverges at asymptoticŽ . 2 finfinity in that case. One thing can also be seen from 3 that the dilaton potential yL r2 e is bounded

because ef ™0 as y™"`.

Acknowledgements

I would like to thank Mario Tonin for many useful discussions. I also thank anonymous referee for bringingw xup the previous works 23,25 .

References

w x Ž . Ž .1 C.M. Hull, P.K. Townsend, Nucl. Phys. B 438 1995 109, hep-thr9410167; E. Witten, Nucl. Phys. B 443 1995 85, hep-thr9503124;Ž . Ž .J. Polchinski, Phys. Rev. Lett. 75 1995 4724; A. Strominger, C. Vafa, Phys. Lett. B 379 1996 99, hep-thr9601029; T. Banks, W.

Ž .Fischler, S. Shenker, L. Susskind, Phys. Rev. D 55 1997 5112, hep-thr9610043; for more details see the reviews and lectures; A.Sen, Unification of String Dualities, hep-thr9609176; P.K. Townsend, Four Lectures on M Theory, hep-thr9612121; J. Polchinski,

Ž . Ž .Lectures on D-branes, in: C. Efthimiou, B. Greene Eds. , Fields, Strings, and Duality TASI 96 , World Scientific, 1997,Ž .hep-thr9611050; J. Schwarz, Lectures on Superstrings and M-Theory Dualities, in: C. Efthimiou, B. Greene Eds. , Fields, Strings,

Ž .and Duality TASI 96 , World Scientific, 1997, hep-thr9607201; C. Vafa, Lectures on Strings and Dualities, hep-thr9702201; J.Maldacena, Black Holes in String Theory, hep-thr9607235; Black Holes and D-branes, hep-thr9705078; T. Banks, Matrix Theory,hep-thr9710231. D. Bigatti, L. Susskind, Review of Matrix Theory, preprint SU-ITP-97r51, hep-thr9712072.

w x Ž . Ž .2 L.J. Romans, Phys. Lett. B 169 1986 374; J.L. Carr, S.J. Gates, R.N. Oerter, Phys. Lett. B 189 1987 68.w x Ž .3 E. Bergshoeff, M. de Roo, M. Green, G. Papadopoulos, P. Townsend, Nucl. Phys. B 470 1996 113, hep-thr9601150.w x Ž .4 P. Cowdall, H. Lu, C.N. Pope, K.S. Stelle, P.K. Townsend, Nucl. Phys. B 486 1997 49; I.V. Lavrinneko, H. Lu, C.N. Pope, From

topology to generalised dimensional reduction, CTP-TAMU-59r96, hep-thr9611134.w x Ž .5 J. Maharana, H. Singh, Phys. Lett. B 408 1997 164, hep-thr9505058.w x6 E. Bergshoeff, M. de Roo, E. Eyras, Gauged supergravity from dimensional reduction, preprint UG-7r97, hep-thr9707130.w x7 H. Singh, Macroscopic String-like Solutions in Massive Supergravity, preprint DFPDr97rTHr48, Phys. Lett. B, in press, hep-

thr9710189.w x8 E. Bergshoeff, P.M. Cowdall, P.K. Townsend, Massive type IIA supergravity from the topologically massive D-2-brane, preprint

UG-6r97, hep-thr9707139.w x9 P.S. Howe, N.D. Lambart, P.C. West, A New Massive type IIA supergravity from Compactification, hep-thr9707139.

w x10 E. Bergshoeff, Y. Lozano, T. Ortin, Massive Branes, preprint UG-8r97, hep-thr9712115.w x Ž .11 J. Polchinski, E. Witten, Nucl. Phys. B 460 1996 525, hep-thr9510169.w x12 E. Bergshoeff, M.B. Green, The type IIA super-eight brane, preprint VG-12r95, M.B. Green, C.M. Hull, P.K. Townsend,

hep-thr9604119.w x Ž .13 E. Cremmer, B. Julia, J. Scherk, Phys. Lett. B 76 1978 409.w x Ž .14 P.K. Townsend, M-Theory from its Superalgebra Cargese Lectures 97 , hep-thr9712004.w x Ž . Ž .15 J. Scherk, J.H. Schwarz, Nucl. Phys. B 153 1979 61; Phys. Lett. B 82 1979 60.w x Ž .16 A. Das, M. Fischler, M. Rocek, ITP-SB-77-15, ITP-SB-77-38; Phys. Rev. D 16 1977 3427.w x Ž .17 D.Z. Freedman, J.H. Schwarz, Nucl. Phys. B 137 1978 333.w x Ž . Ž .18 F. Gliozzi, J. Scherk, D. Olive, Nucl. Phys. B 122 1977 253. L. Brink, J. Schwarz, J. Scherk, Nucl. Phys. B 121 1977 253.w x Ž .19 E. Cremmer, J. Scherk, S. Ferrara, Phys. Lett. B 74 1978 61.w x Ž . Ž .20 A. Das, Phys. Rev. D 15 1977 259; E. Cremmer, J. Scherk, Nucl. Phys. B 127 1977 253.


w x Ž .21 D.Z. Freedman, G.W. Gibbons, Nucl. Phys. B 233 1984 24.w x22 A.H. Chamseddine, M.S. Volkov, Non-Abelian Solitons in Ns4 Gauged Supergravity and Leading Order String Theory, preprint

RUr97-5-B, hep-thr9711181.w x Ž .23 I. Antoniadis, C. Bachas, A. Sagnotti, Phys. Lett. B 235 1990 255.w x Ž .24 A.H. Chamseddine, M.S. Volkov, Phys. Rev. Lett. 79 1997 3343.w x Ž .25 I. Antoniadis, S. Ferrara, C. Kounnas, Nucl. Phys. B 421 1994 343.w x Ž . Ž .26 P. Breitenlohner, D. Freedman, Phys. Lett. B 115 1982 197; Ann. of Phys. 144 1982 249; G.W. Gibbons, C.M. Hull, P. Warner,

Ž . Ž .Nucl. Phys. B 218 1983 173; E. Witten, Comm. Math. Phys. 80 1981 381.w x Ž .27 A. Dabholkar, G. Gibbons, J. Harvey, F. Ruiz-Ruiz, Nucl. Phys. B 340 1990 53.w x28 P.M. Cowdall, Supersymmetric Electrovacs in Gauged Supergravities, preprint Rr97-51, hep-thr9710214.

18 June 1998


A non-perturbative non-renormalization theoremfor the Wilsonian gauge couplings in supersymmetric theories

Michael Graesser a,1,b, Bogdan Morariu a,2,b

a Department of Physics, UniÕersity of California, Berkeley, CA 94720, USAb Theoretical Physics Group, Lawrence Berkeley National Laboratory, 1 Cyclotron Rd., 50A-5101, Berkeley, CA 94720, USA

Received 21 December 1997Editor: H. Georgi

Abstract

We present a direct proof that the holomorphic Wilsonian beta-function of a renormalizable asymptotically freesupersymmetric gauge theory with an arbitrary semi-simple gauge group, matter content, and renormalizable superpotentialis exhausted at 1-loop with no higher loops and no non-perturbative contributions. This is a non-perturbative extension of thewell known result of Shifman and Vainshtein. q 1998 Published by Elsevier Science B.V. All rights reserved.

1. Introduction

w xIn their 1986 paper 1 Shifman and Vainshteinsolved the anomaly puzzle in supersymmetric gaugetheories. They argued that the supersymmetric exten-sion of the anomaly equation should be written inoperator form and then showed that the coefficient ofthe trace anomaly involves the Wilsonian gaugebeta-function rather than the exact Gell-Mann and

w xLow function 2 . The puzzle is resolved if it can beshowed that the Wilsonian gauge beta-function is1-loop exact. A perturbative proof of the above

w xstatement was presented in 1 where it was arguedthat all possible operators that could, in principle,contribute to the gauge beta-function beyond 1-loopare necessarily of infra-red origin, and should notappear in the Wilsonian effective action.

1 E-mail address: [email protected] E-mail address: [email protected].

In this Letter we present a direct proof that thereare no further non-perturbative violations. Morespecifically, we prove that the holomorphic Wilso-nian beta-function of an arbitrary renormalizableasymptotically-free supersymmetric gauge theorywith matter is exhausted at 1-loop with no higherloops and no non-perturbative contributions.

The technique we employ to prove the theoremw xwas introduced by Seiberg 3 and it is briefly re-

viewed here. To obtain the beta-function we com-pare two versions of the theory with different cutoffsand coupling constants and the same low energyphysics. The couplings of the theory with the lowercutoff can be expressed in terms of the couplings ofthe theory with the higher cutoff and the ratio of thetwo cutoffs. We can restrict their functional depen-dence on the high cutoff couplings using holomor-phy of the superpotential and gauge kinetic termsand selection rules. Holomorphy is a consequence ofsupersymmetry. To see this, elevate the couplings to


( )M. Graesser, B. MorariurPhysics Letters B 429 1998 313–318314

background chiral superfields. They must appearholomorphically in the superpotential in order topreserve supersymmetry. Selection rules generalizeglobal symmetries in the sense that we allow thecouplings in the superpotential to transform underthese symmetries. Non-zero vacuum values of thesecouplings then spontaneously break these symme-

Ž . Ž .tries. Here we only consider U 1 and U 1 symme-R

tries. In the quantum theory they are generallyanomalous, but we can use the same technique weused for the coupling in the superpotential. Weassume that the u-angle is a background field andtransform it non-linearly to make the full quantumeffective action invariant.

w xThen, following a method used in 4 we translatethese conditions on the functional relations betweenthe couplings of the theories at different cutoffs intorestrictions of the functional form of the gaugebeta-function. We can show that the gauge beta-function is a function of the holomorphic invariantsallowed by selection rules. Then we can restrictfurther the functional dependence of the beta-func-tion by varying the couplings while keeping theinvariants fixed. This allows us to relate the beta-function of the original theory to the beta-function ofa theory with vanishing superpotential. In addition,we also obtain a strong restriction of the functionaldependence of the beta-function on the gauge cou-pling. It has exactly the form of a one-loop beta-function. The only ambiguity left is a numericalcoefficient which can be calculated in perturbationtheory.

Next we make a short detour to explain what wew xmean by the Wilsonian beta-function 5 . The Wilso-

nian beta-function describes the renormalizationgroup flow of the bare couplings of the theory sothat the low energy theory is cutoff invariant. Addi-tionally, we do not renormalize the vector and chiralsuperfields, i.e. we do not require canonical normal-

w xization of the kinetic terms 1 . The usual conventionin particle physics is to canonically normalize thekinetic term. It is obtained by using the covariantderivative EqgA. Instead, here we allow non-canon-ical normalization of the kinetic term. The normal-ization of the gauge fields is such that the covariantderivative has the form EqA. The gauge couplingonly appears in front of the gauge kinetic term. Inthis case it is convenient to combine the u-angle and

gauge coupling constant g into the complex variabletsur2pq4p irg 2. In supersymmetric gauge the-ories the beta-function is holomorphic in the barecouplings only if we do not renormalize the fields.Even if we start with canonical normalization at ahigher cutoff, the Kahler potential will not be canon-¨ical at the lower cutoff. The rescaling of the chiral or

w xgauge superfields is an anomalous transformation 1that destroys the holomorphy of the superpotentialand the beta-function 3. The relation between thebeta-functions in the two normalizations was first

w xdiscussed in 1 . The beta-function for canonicallyw xnormalized fields is known exactly 2 and receives

contributions to all orders in perturbation theory. Forw xa recent discussion of these issues see also 4 . Again

we emphasize that here we are only concerned withthe holomorphic Wilsonian beta-function.

We should also clearly state that the proof is notvalid if any one of the one-loop gauge beta-functionsis not asymptotically free. This includes the casewhen the one-loop beta-function vanishes. As we

Ž .will see, exactly in this case the U 1 symmetry isR

non-anomalous. This makes it difficult to control thedependence of the beta-function on the gauge cou-pling.

Various partial versions of this result alreadyexisted. As we already mentioned, the perturbative

w xnon-renormalization theorem was proven in 1 . Ananalysis of possible non-perturbative violations tothis theorem in the case of a simple gauge group

w xwith a vanishing superpotential could be found in 4 .It was also known that in the case of a simple gaugegroup with only Yukawa interactions present in thesuperpotential, possible non-perturbative correctionsto the Wilsonian beta-function are independent of

w xthe gauge coupling 6 . We should also mention thatfor some supersymmetric gauge theories it is possi-ble to determine the exact beta-function for thecanonically normalized fields, including all the non-

w xperturbative terms 7 . The exact Wilsonian beta-function for these theories could then be obtained ifthe rescaling anomaly relating the different normal-

3 For some special theories like Ns2 SUSY Yang-Mills therescaling anomaly of the chiral superfields cancels the rescaling

w xanomaly of the vector superfield 4 . For these theories we canmake stronger statements since the canonical and holomorphicWilsonian couplings coincide.

( )M. Graesser, B. MorariurPhysics Letters B 429 1998 313–318 315

izations were known exactly, both perturbatively andnon-perturbatively. While the rescaling anomaly is

w xused in various places in the literature 1,7 to relatethe two gauge couplings, it was not clear to uswhether for these theories the exact form of theanomaly, including non-perturbative terms is known.A perturbative calculation of the anomaly was pre-

w xsented in 4 .Finally, we note that the theorem is valid in

theories where no mass terms are allowed by thesymmetries of the theory. This is of phenomenologi-cal interest as many supersymmetric extensions ofthe Standard Model share this characteristic.

2. Simple gauge group

We will consider first the case of a simple gauge˜group G. Let the generalized superpotential W be

defined to include the kinetic term for the gaugefields

taWs tr W W qW , 1Ž .Ž .R a64p itR

where

Ws l F F F qM m F F qM 2 c FÝ Ý Ýi jk i j k i j i j i i

2Ž .is the usual superpotential and tr T aT b s t d ab. HereR R

M is the cutoff mass and was factored out so that allthe couplings are dimensionless. The gauge couplingg and u-angle are combined in the complex variable

u 4p it' q . 3Ž .22p g

Note that unitarity requires t to be valued in theupper half plane. Since u is a periodic variable it isconvenient to introduce a new variable q'e2p it. Itis valued in the complex plane and transforms lin-early under the anomalous transformations to bediscussed below. Weak coupling is at qs0.

Consider now a theory with a different cutoff M X

and with the same low energy physics. The La-grangian at the new cutoff is

2 2 † 2VhLLs d u d uZ F e FÝH i i ii

2 ˜ X X X X Xq d u W t ,l ,m ,c , M qh.c. 4Ž .Ž .H i jk i j iž /

where in particular, the fields F are not renormal-i

ized to canonical normalization. The Z dependsi

non-holomorphically on the couplings, so renormal-izing the chiral superfields would destroy the holo-

˜ Xmorphic form of W. The new coupling t is afunction of the old dimensionless couplings and theratio MrM X. For later convenience we write this as

tX st

Xt ,l ,m ,c ;ln MrM X . 5Ž . Ž .Ž .i jk i j i

Supersymmetry requires a holomorphic dependenceof t on the first four arguments. To see this, notethat the couplings in the generalized superpotentialcan be considered as vacuum values of backgroundchiral superfields. Invariance of the action undersupersymmetry transformations requires holomorphyof the superpotential.

To prove the non-renormalization theorem wewill use selection rules. These are global symmetriesof the superpotential with all couplings considered aschiral superfields. We assign them non-trivial trans-formation properties under the symmetry group.These symmetries will be spontaneously broken bynon-zero vacuum values of the couplings. In generalthey are also anomalous. We will make them non-anomalous by assigning a charge to q, i.e. transform-ing u to compensate for the anomaly. Consider theŽ . Ž .U 1 =U 1 global symmetry with the followingR

charge assignment:

W F l m c qa i i jk i j i

Ž .U 1 1 2r3 0 2r3 4r3 2b r3R 0Ž . Ž .U 1 0 1 y3 y2 y1 2Ý t Ri i

Ž .The quantity b is given by b s3t yÝ t R ,0 0 adj i iŽ .where t R is the normalization of the generatorsi

for the representation of the chiral superfield F . ForiŽ .example, ts1r2 for a fundamental of SU N . De-

fine the gauge b-function by

dX

X<b s 2p it M sMX2p it dln MrMŽ .

sb t ,l ,m ,c . 6Ž .Ž .i jk i j i

Ž Ž ..The holomorphy of t in Eq. 5 translates intoholomorphy of the beta-function.


Since t™tq1 is a symmetry of the theory, b isa single valued function of q

b s f q ,l ,m ,c . 7Ž .Ž .2p it i jk i j i

First, consider the case when at least one mass term,let us call it m , can be non-zero. If any c could be

) i

non-zero, then there is a gauge singlet field whichcould be given a Majorana mass, so this is the samecase as above.

Ž . Ž .The gauge beta-function is U 1 =U 1 invari-R

ant. This statement is non-trivial and requires someexplanation. Consider some arbitrary coupling l that

Ž . Ž .transforms linearly under some U 1 or U 1 sym-R

metry. Its beta-function b must also transforml

linearly with the same charge as l

eiQlab l, . . . sb eiQlal, . . . 8Ž . Ž . Ž .l l

where Q is the charge of l. This is true in particu-l

lar for the beta-function of q. However when we goto the t variable we have

d dX X

b s 2p it s 2p it bX2p it qdln MrM dqŽ .

sqy1 b . 9Ž .q

The additional q factor makes the t beta-functioninvariant. In what follows we only consider thegauge beta-function since all the others are trivial,

w xi.e. there are no perturbative 8 or non-perturbativew x3 corrections to the usual superpotential. We willdrop the subscript and denote it b.

Ž .First consider U 1 invariance. It requires thatR

q m ci j i˜bs f , , ,l . 10Ž .i jkb 20ž /mm m)) )

˜ Ž .However, the variables of f are not U 1 invariant.They have charges 6T ,0,3,y 3, respectively. In-adj

Ž . Ž .variance under U 1 =U 1 requires that b is a yetR

another function

l2 tadj c2 tadj mi jk i i jy1bsF q ,q , . 11Ž .b0 t q tž /mm adj iÝ)) m

)

We next take the limit m ™0 keeping q and all the)

arguments of F constant. If b )0, this corresponds0

to taking all couplings except t to zero. AssumingŽ .that b is continuous we see that b q,l ,m ,c si jk i j i

Ž .b q,l sm sc s0 and thus it is independenti jk i j i

of all the couplings in the superpotential. In factw xwhen the superpotential vanishes it is known 4 that

there are no non-perturbative corrections to thebeta-function and the gauge coupling only runs at

4 Ž .1-loop . This just reflects the fact that no U 1 =RŽ .U 1 holomorphic invariant can be constructed solely

in terms of q. Note the importance of holomorphy inthese arguments. For example, if we do not require

Ž .holomorphy qq is invariant under an arbitrary U 1Ž .and U 1 symmetry. No higher loops or non-per-R

turbative corrections are present and we concludethat

bsb , 12Ž .0

thus extending the perturbative result of Shifman andw xVainshtein 1 .

An exception to the previous argument occurswhen the gauge and global symmetries of the theoryallow only Yukawa couplings to be present in thesuperpotential. For these theories

bs f q ,l . 13Ž .Ž .i jk

Ž .The beta-function must be U 1 invariant. ThisR

requires

f e2 b0 a i r3q ,l s f q ,l .Ž .Ž .i jk i jk

Then by holomorphy b is independent of q. Further,Ž .invariance of b under the U 1 symmetry requires

that f is a function of ratios of l only. We mayi jk

choose one of the non-zero l , l say, and dividei jk )

through by l . Then)

li jkbs f l sF . 14Ž .Ž .i jk ž /l

)

Consider the limit l ™0 while keeping the ratiosi jk

l rl constant. We know that in this limit bi jk )

reduces to the one-loop result. So assuming that b isŽ . Ž .continuous, we find b l sb l s0 sb , i.e.i jk i jk 0

it is independent of the Yukawa couplings.To conclude this Section, we note that our discus-

sion of the proof of the theorem was divided intotwo cases requiring separate proofs. Here we presenta short argument that extends the proof of the theo-

4 d b 30Note that this result can also be written as g sy g2dt 16p

which is just the standard 1-loop beta-function.

( )M. Graesser, B. MorariurPhysics Letters B 429 1998 313–318 317

rem, valid when at least one mass term is allowed, totheories which do not admit any bare mass terms.Consider a theory with Lagrangian LL for which thesymmetries of the theory forbid the presence of anymass terms. To this theory, add a non-interactinggauge-singlet field with mass m . More concretely,

)

the new theory defined at M is described by theLagrangian

2 2 †LL sLLq d u d uF FHnew 0 0

q d2u Mm F 2 qh.c. .H) 0ž /

This new theory satisfies the conditions of the theo-rem proven when at least one mass term is allowed,so the beta-function of the new theory, b , isnew

exhausted at one-loop. But we can conclude onphysical grounds that b is identical to b , thenew

beta-function of the original theory, since in integrat-ing over momentum modes M to M X the contribu-tion from the gauge singlet completely factors outsince it is non-interacting. So by this argument theproof of the theorem for theories with mass termscan be extended to theories for which mass terms areforbidden by the symmetries of the model.

The results of this section are also valid for asemisimple gauge group. We shall sketch the proofin the next section.

3. Extension to a semi-simple gauge group

Assume that the gauge group is GsP G withA A

each G a simple group. Also assume that the super-A

potential has the form given in Section 2. Then if allthe simple gauge groups are asymptotically-free theWilsonian beta-functions of all the gauge couplingsare one-loop exact.

For each simple gauge group G defineA

u 4p iAt s q 15Ž .A 22p g A

and introduce q 'e2p itA as in Section 2. We ex-AŽ . Ž .tend the U 1 =U 1 selection rules of Section 2 byR

assigning all gauge chiral multiplets W chargea , AŽ . Ž A Ž ..1,0 . Then q has charge 2b r3,2Ý t R . ItA 0 i A i

Ž .1r b A0will be convenient to define k ' q . Then kA A A

Ž Ž . A.has charge 2r3,2Ý t R rb . Weak coupling isi A i 0

at k s0 since b A is positive.A 0

The beta-functions for each simple gauge groupare defined as in Section 2, so that

b s f q ,l ,m ,c 16Ž .Ž .A A B i jk i j i

is a function of holomorphic invariants and invariantŽ . Ž .under the U 1 =U 1 symmetry.R

We do the proof for two cases:1. Only Yukawa couplings are allowed.2. At least one m /0 is allowed.i j

Ž .In the first case invariance of b under U 1A R

requires that b is a function of ratios of k only.A B

That is,

b sF k rk ,l . 17Ž .Ž .A A B B i jk)

We have divided through by an arbitrarily chosenk , so that each k other than k appears in theB B B

) )

argument of F only once. Now consider the weakcoupling limit k ™0 for all the gauge couplings.B

The argument of the beta-functions is

k rk sexp2p i t rb B yt rb B) . 18Ž .Ž .B B B 0 B 0

) )

Since by assumption the one-loop beta-functions allhave the same sign it is possible to take this limitwhile keeping the ratios k rk fixed. In this limitB B

)

the beta-function is a function of the Yukawa cou-plings only. So assuming that the beta-functions are

Ž .continuous in this limit, we find that b k ,l sA B i jkŽ . Ž .b k s0,l sF l . But now we may use theA B i jk A i jkŽ .U 1 symmetry to conclude that b is a function ofA

l rl . The argument of Section 2 may now bei jk )

Ž .repeated and we conclude that b q ,l sA B i jk

constant.For the second case a straightforward generaliza-

tion of the argument of Section 2 may be repeatedand we conclude that

kBb sF . 19Ž .A A ž /kB

)

Then the argument used in the first case of thisSection is used to conclude that F is independent ofA

all of the q and superpotential couplings.B

4. Note

The statement of this theorem for the case of asimple gauge group was also made in the lecture


w xnotes 9 . In that proof the author considers a super-potential containing no composite operators, i.e. onlyoperators linear in the fundamental fields. Of coursesuch superpotential is not gauge invariant. Howeverit is is only used in an intermediate step to simplifythe study the charge assignment for the couplings in

Ž .the physical gauge invariant superpotential. The U 1charge of the coupling of a composite operator equalsthe sum of the charges of the couplings of thefundamental fields entering the composite. However

w x Ž .in 9 it is also assumed that the U 1 charge of theR

couplings of composite gauge invariant operators inthe superpotential equals the sum of the charges ofthe couplings of fundamental fields forming the

Ž .composite. While this is true for usual U 1 symme-tries since the superpotential has charge zero and thesum of charges of the couplings must equal minusthe sum of charges of the fields entering the compos-

Ž .ite, for U 1 symmetries the superpotential hasR

charge two and the arithmetic is more complicated.w xBecause of this, the proof in 9 only works for a

superpotential linear in matter fields, i.e. when onlygauge singlet chiral superfields are present. We alsogeneralized the theorem to a semi-simple gaugegroup.

Acknowledgements

The authors would like to thank Nima Arkani-Hamed, Hitoshi Murayama, Mahiko Suzuki andBruno Zumino for many useful discussions and valu-able comments. Our paper also benefitted from pri-

vate communications with Mikhail Shifman. Thiswork was supported in part by the Director, Office ofEnergy Research, Office of High Energy and Nu-clear Physics, Division of High Energy Physics ofthe US Department of Energy under Contract DE-AC03-76SF00098 and in part by the National Sci-ence Foundation under grant PHY-90-14797. M.G.was also supported by NSERC.

References

w x Ž .1 M.A. Shifman, A.I. Vainshtein, Nucl. Phys. B 277 1986 456.w x2 V.A. Novikov, M.A. Shifman, A.I. Vainshtein, V.I. Zakharov,

Ž .Nucl. Phys. B 229 1983 381; M.A. Shifman, A.I. Vain-Ž .shtein, V.I. Zakharov, Phys. Lett. B 166 1986 334.

w x Ž .3 N. Seiberg, Phys. Lett. B 318 1993 469.w x4 N. Arkani-Hamed, H. Murayama, Renormalization Group In-

variance of Exact Results in Supersymmetric Gauge Theories,LBNL preprint LBNL-40346, 1997, hep-thr9705189; N.Arkani-Hamed, H. Murayama, Holomorphy, RescalingAnomalies and Exact Beta Functions in Supersymmetric GaugeTheories, LBNL preprint LBNL-40460, 1997, hep-thr9707133.

w x Ž .5 K.G. Wilson, J. Kogut, Phys. Rep. 12 1974 75; J. Polchinski,Ž .Nucl. Phys. B 231 1984 269.

w x6 N. Arkani-Hamed, H. Murayama, private communication.w x Ž .7 M. Shifman, A. Vainshtein, Nucl. Phys. B 296 1988 445; I.I.

Ž .Kogan, M. Shifman, Phys. Rev. Lett. 75 1995 2085; I.I.Ž .Kogan, M. Shifman, A. Vainshtein, Phys. Rev. D 53 1996

4526.w x Ž .8 B. Zumino, Nucl. Phys. B 89 1975 535; P. West, Nucl.

Ž .Phys. B 106 1976 219; M. Grisaru, W. Siegel, M. Rocek,ˇŽ .Nucl. Phys. B 159 1979 429.

w x9 P. Argyres, Introduction to Supersymmetry Lecture Notes,unpublished.

18 June 1998


Misleading anomaly matchings?

John Brodie a,1, Peter Cho b,2, Kenneth Intriligator c,3

a Department of Physics, Princeton UniÕersity, Princeton, NJ 08540, USAb Lyman Laboratory, HarÕard UniÕersity, Cambridge, MA 02138, USA

c Department of Physics, UniÕersity of California at San Diego, 9500 Gilman DriÕe, La Jolla, CA 92093, USA

Received 2 March 1998Editor: H. Georgi

Abstract

Ž .We investigate the low energy dynamics of NNs1 supersymmetric SO N gauge theories with a single symmetric tensormatter field. These theories exhibit non-trivial matching of global ’t Hooft anomalies at the origin of moduli space. We arguethat their quantum moduli spaces possess distinct Higgs and confining branches which touch at the origin in an interactingnon-Abelian Coulomb phase. The matching of anomalies between microscopic degrees of freedom and colorless moduli

Ž .therefore appears to be coincidental. We discuss a formal mathematical relation between the SO N model and an analogousŽ .Sp 2 N theory with a single antisymmetric matter field which provides an explanation for the anomaly matching

coincidence. q 1998 Elsevier Science B.V. All rights reserved.

Recent advances in analyzing the strong couplingdynamics of supersymmetric gauge theories haveopened up several new directions for model buildingw x1 . One interesting application has been to the studyof dynamical supersymmetry breaking. A number oftheories utilizing novel dynamical mechanisms tobreak SUSY have been constructed during the past

w xfew years 2 . One especially simple proposal isŽ .based upon an SU 2 model with a single isospin-3r2

w xmatter field 3 . Nontrivial ’t Hooft anomaly match-ing in this chiral model suggests that it confines with

1 E-mail: [email protected] E-mail: [email protected] E-mail: [email protected].

a smooth quantum moduli space. Supersymmetrybreaking then results upon adding a tree level super-potential. However, there is another possibility for

Ž . w xthe SU 2 model’s low energy dynamics 3 : the ’tHooft anomaly matching may be coincidental andthe theory may have a non-trivial RG fixed point atthe origin. In this case, supersymmetry need not bebroken upon adding the tree level superpotential.Which scenario is correct remains an unsettled ques-tion.

In this note, we examine a simple class of NNs1Ž .supersymmetric SO N models with a single matter

chiral superfield S in the two-index, symmetric,traceless, tensor representation . As we shall see,

Ž .these models are similar to the SU 2 theory inas-much as they exhibit non-trivial ’t Hooft anomalymatching, which suggests confinement with a smooth


( )J. Brodie et al.rPhysics Letters B 429 1998 319–326320

moduli space. 4 We will argue, however, that themoduli space for these theories must have a moreintricate structure, with various branches and a non-trivial RG fixed point at the origin. The anomalymatching then appears to be coincidental. A skeptic

Ž .might view the SO N models as casting doubt onthe proposed confinement and supersymmetry break-

w xing of the model of 3 . At the very least, theydemonstrate that anomaly matching can be mislead-ing.

It is important to recall that the S field does nottransform according to a faithful representation of

Ž .the SO N gauge group’s center. Test charges inspinor or vector representations cannot be screenedby either gluons or dynamical S matter fields. There-

Ž .fore, the SO N model’s moduli space can a priorihave distinct Higgs, confining and a variety ofoblique confining phases, associated with the full

Ž .center of SO N , where Wilson and ’t Hooft loopsexhibit various possible area and perimeter law scal-

Ž .ings. This feature of the SO N model represents aŽ .clear qualitative difference with the SU 2 theory of

w x3 whose Is3r2 field does provide a faithfulŽ .representation of the Z center of SU 2 .2

In the absence of any tree level superpotential, theŽ . Ž .SO N model has an anomaly free U 1 symmetryR

Ž . Ž . 5with R S s4r Nq2 . Its one loop beta functionŽ .is b s2 Ny4 , so the NG5 theories are asymp-0

Ž .totically free. Although b s0 for the SO 4 (0Ž . Ž . Ž .SU 2 =SU 2 model with S; 3,3 , this theory is

not asymptotically free at two loop order. It thusŽ .flows to a free theory in the IR. Similarly, the SO 2

Ž .and SO 3 theories are not asymptotically free andflow to free theories in the IR.

Ž .The SO N model possesses a moduli space ofclassical vacua given by solutions to the D-flatnesscondition modulo gauge transformations. D saŽ †.Tr T SS s0 implies that the real and imaginarya

4 w xThese models were recently considered in 4 as part of acomplete classification of all theories based on simple gaugegroups with a freely generated moduli space and matching ’tHooft anomalies. Some of our previously unpublished observa-

Ž .tions on these models were cited in that work. The SO Nw xtheories were also recently constructed via branes in 5 .

5 Ž . Ž . Ž .We adopt the SO N index values m I s2, m Adjs sŽ .2 Ny4 and m s2 Nq4 which count numbers of fermion

zero modes in a single instanton background.

parts of S commute and can be simultaneously diag-Ž .onalized by an SO N rotation. The moduli space is

consequently Ny1 complex dimensional. Through-out its bulk, the gauge group is generically com-pletely broken by the Higgs mechanism. But thereexist subspaces of enhanced gauge symmetry wheresome diagonal expectation values of S are equal. Onthe subspace where

z 11 m =m1 1

z 12 m =m2 2² :S s ,. . .� 0z 1ll m =mll ll

1Ž .

with Ý ll m sN and Ý ll m z s0, the low energyis1 i is1 i iŽ .theory reduces to l decoupled SO m models, eachi

with a traceless two index symmetric tensor ,iand ly1 singlet moduli corresponding to the z .i

The Ny1 dimensional classical moduli space isfreely generated by arbitrary expectation values ofthe gauge invariant operators

O sTrSn , ns2,3, PPP , N. 2Ž .n

One can also form the additional composite BsdetS.But it is linearly related by the trace of the charac-teristic polynomial for matrix S

N1 1O y O O y O O q PPP q y1 NBŽ .N 2 Ny2 3 Ny32 3

s0 3Ž .Ž .to the operators in 2 . This last expression can be

used to eliminate B regardless of any possible quan-tum corrections.

In the quantum theory, any dynamically generatedsuperpotential which could lift the classical modulispace degeneracy is determined by holomorphy andsymmetry considerations to be of the form

12 Nq4 erS 4

W sC , 4Ž .dyn 2 Ny8L

Ž . 2 Nq4where L denotes the SO N scale and S standsfor some function of the O operators which has Sn

number equal to 2 Nq4. Asymptotic freedom re-quires the classical moduli space to be recovered in

( )J. Brodie et al.rPhysics Letters B 429 1998 319–326 321

² :the weak coupling S rL™` limit. This conditionis incompatible with the form of W which yields adyn

potential that grows with S. Consequently, the con-stant C must vanish. We will refer to this part of themoduli space where W sW qW s0 as thetot dyn tree

‘‘Higgs branch.’’ Shortly, we will argue that theŽ .W in 4 with C/0 must be generated on anotherdyn

‘‘confining branch’’ of the theory when W /0.tree

Although strong dynamics do not generate anysuperpotential on the Higgs branch, they can stillalter the theory’s vacuum structure and lead to inter-esting phenomena near the origin. Classically, themoduli space metric for the O fields has singulari-n

ties on subspaces of the vacuum manifold where thegauge group is not completely broken. In the classi-

Ž .cal theory, massless matter fields and SO mi i

gluons must be included in order to obtain a non-sin-gular description. In the quantum theory, the modulispace singularities are either smoothed out or elsereflect possibly different massless fields.

Ž .Since the global U 1 symmetry remains unbro-R

ken at the origin, it is possible to check ’t Hooftanomalies in order to constrain the massless spec-trum at this point. In the microscopic theory, theŽ . Ž .3U 1 and U 1 anomalies assume the valuesR R

Ny2Nq1 N w xA s y1 y q 1UŽ1.R ž / ž /2 2Nq2

sNy1,3Ny2 3Nq1 N w x3A s y1 y q 1UŽ1.R ž / ž /2 2Nq2

Ny1 5N 2 y4Nq4Ž . Ž .s . 5Ž .2Nq2Ž .

In the effective theory, we find the following contri-butions from the fermionic components of the On

moduli superfields:

N 4nA s y1 sNy1,ÝUŽ1.R Nq2ns2

3N 4n3A s y1ÝUŽ1.R Nq2ns2

Ny1 5N 2 y4Nq4Ž . Ž .s . 6Ž .2Nq2Ž .

The anomalies precisely match! This nontrivialagreement suggests that the O composites saturaten

the massless spectrum of the quantum theory. If so,the quantum Kahler metric for the O moduli should¨ n

be flat near the origin and non-singular throughoutthe moduli space. This anomaly matching representscircumstantial evidence for confinement in the

Ž . Ž .SO N model in the same way as for the SU 2w xmodel of 3 .

It is interesting to further consider discrete anoma-w xlies 6–8 . Global anomalies for Z groups shouldN

match between high and low energy descriptions ofw xany gauge theory 9 . In particular, the Z -gravity-N

gravity coefficients are supposed to agree modulo NŽ . Ž .modulo Nr2 for N odd even . On the other hand,

3 2 Ž .other anomalies such as Z and Z U 1 can beN N R

corrupted by unknown massive state contributions.Therefore, they cannot be used to definitively ruleout a proposed massless confining phase spectrum.

Ž .In the SO N model, instantons break the classi-cal S-number symmetry down to a Z subgroup2 Nq4w x 610 . After assigning the S field charge 1 under this

3 2 Ž .discrete group, we find that the Z , Z U 12 Nq4 2 Nq4 RŽ .2and Z U 1 anomalies do not match for general2 Nq4 R

N. However, the Z anomalies calculated in the2 Nq4

microscopic gauge theory and low energy sigmamodel are equal:

N Nq1Ž .A s y1 1Ž .Z2 Nq4 2

N Ny1 Ny2Ž . Ž .s ns . 7Ž .Ý

2ns2

These discrete anomaly matching results are there-Ž .fore consistent with the hypothesis that the SO N

model confines and yields a free field theory at theorigin.

Although the picture of simple confinement at theorigin passes all non-trivial anomaly tests, it can notprovide a complete description of the moduli space.The first problem arises along subspaces of enhancedgauge symmetry within the Higgs branch where

Ž . Ž .SO N ™Ł SO m with some m s2, 3 or 4. Oni i i

6 Ž .A Z subgroup of Z is contained within U 1 .ŽNq2.r4 2 Nq4 R

But there is no loss in working with the larger Z symmetry2 Nq4Ž .since U 1 anomalies are already known to match.R


these subspaces, the low energy theory is not asymp-totically free. The massless spectrum must therefore

Ž .include SO m gauge fields and matter fields.i i

Since these subspaces intersect the origin, the mass-² :less spectrum at S s0 can not simply consist of

the confined O moduli. We note that no suchnŽ .free-electric subspaces enter into the SU 2 model of

w x3 or other theories which are believed to confine.Another way in which there could exist a free

theory at the origin would be to have the same² :massless spectrum at S s0 as that which exists on

the free-electric subspace with maximal unbrokengauge group. For example when N is a multiple offour, the massless spectrum on the free-electric sub-space with maximal unbroken gauge group consists

Ž .Nr4of free-electric SO 4 gauge fields and symmet-ric tensors, along with a subset of the O operatorsn

which parameterize this subspace. This masslessspectrum could conceivably extend down to the ori-gin of moduli space. However, there seems to be noway to make this scenario compatible with the re-quired ’t Hooft anomaly matching at the origin. Itthus appears that the theory at the origin can not befree and that the ’t Hooft matching observed above issimply a misleading coincidence.

Another reason why confined O fields cannotn

represent the complete massless spectrum at the ori-gin of moduli space may be seen by turning on a treelevel superpotential

N

W s g O . 8Ž .Ýtree n nns2

In the presence of these classical terms, the eigenval-ues z of S represent different solutions ofÝN ng z ny1 s0. 7 Various vacua exist where dif-ns1 n

ferent numbers of eigenvalues are equal. The generallow energy theory is NNs1 pure Yang-Mills with a

Ž . Ž .Ł SO m gauge group. For m G3, SO m con-i i i i

fines and yields m y2 supersymmetric vacua wheni

m G5, four vacua when m s4 and two vacuai i

when m s3. The low energy spectrum contains ai

photon for each m s2.i

For simplicity, consider the special case where

7 The coupling g is a Lagrange multiplier which enforces the1

tracelessness condition for S.

1 Ž .only the mass coupling g s m in 8 is nonzero.2 2

For m4L, the heavy S can be integrated out. TheŽ .resulting low energy SO N Yang-Mills theory is

known to confine with a mass gap and to have Ny2Ž .supersymmetric vacua. If the original SO N model

simply had a moduli space with confinement in1terms of the O fields, turning on W s mOn tree 22

would lead to dynamical SUSY breaking in a similarw xfashion to the proposed mechanism in 3 . This

would contradict the fact that the low energy theoryhas Ny2 supersymmetric vacua.

Ž .Gluino condensation in the low energy SO NYang-Mills theory generates the superpotential

Ž .1r Ny21 3Ž Ny2.W s Ny2 16L , 9Ž . Ž .lo lo2

where the low energy scale is fixed by the matchingrelation L3Ž Ny2.smNq2L2Ž Ny4.. This result can belo

recovered from the effective superpotential1r4Nq2O2

W sy 10Ž .conf Nq2 2 Ny8Nq2 LŽ .1in the upstairs theory after adding W s mO andtree 22

integrating out O . This superpotential is of the form2Ž .4 . Since we have already argued that it is absent onthe Higgs branch, we conclude that there must existanother ‘‘confining branch’’ of the theory on whichW is generated when m/0. 8

confŽ .The multi-branch structure of the SO N model’s

Ž .moduli space is reminiscent of that for SU 2 withw xtwo adjoints 11,12 . We sketch its basic features in

Fig. 1. The plane in the figure represents the Higgsbranch which is strongly coupled inside the shadedregion nearby the origin and weakly coupled in the

² :domain far away from O s0. Partial free electricn

subspaces within the Higgs branch which intersectthe origin are illustrated by the diagonal line in Fig.1. When m/0, the Higgs branch is lifted, and thetheory resides on the confining branch representedby the cone. For any fixed value of m, the vacuum

8 1When other source terms besides g s m in the tree level2 2

Ž .superpotential in 8 are nonzero, more general confining phasesuperpotentials involving O must be generated. The confiningn) 2

branch must also include the vacua with different unbrokenŽ .SO m , including the ones with m s2 which have a masslessi i

photon.


on the confining branch consists of Ny2 pointswith a mass gap. As a result, there are no masslessmoduli associated with the confining branch. As can

² : Žbe seen from the expectation value O s Nq2.Ž 4 2 Ny8.1rŽNy2.2 16m L , small values for m yield

vacua which lie within the strongly coupled regionnearby the moduli space origin. The spreading of thecone with increasing m mimics the behavior of² :O . Points located on the confining branch at2

m4L are still strongly coupled, for the low energyŽ .limit of the SO N model at such points is a super

Yang-Mills theory. The entire cone is thus shadedgrey.

In Fig. 1, the Higgs and confining branches areŽ .shown touching at the origin much as in SU 2

w xtheory with two adjoints 11,12 . The origin mustthen reside in an interacting non-Abelian Coulombphase, for no free field spectrum could incorporateboth the Higgs and confining branches at this point.

It is also possible, though unlikely, that the dis-Ž .tinct Higgs and confining branches for the SO N

model do not meet. Such a disconnected branchŽ .structure actually does occur in SO N theory withc

w xN sN y4 vector flavors 11 . Two inequivalentf c

branches arise from aligned and misaligned gauginoŽ .condensates within the low energy unbroken SO 4

Ž . Ž .(SU 2 =SU 2 gauge group. On the Higgs branch,W vanishes as a result of cancellation between thedyn

two gaugino condensates, ’t Hooft anomalies match

Ž .Fig. 1. A schematic picture of the SO N model’s quantummoduli space.

at the origin and the low energy spectrum containsonly the massless moduli. On the confining branch, adestabilizing superpotential is dynamically gener-ated. When W /0, the Higgs branch ceases totree

yield supersymmetric vacua and is eliminated, andthe low energy theory lives on the confining branch.Since gaugino condensates do not appear at generic

Ž .points within our SO N model’s moduli space wherethe gauge group is completely broken, we do notexpect the theory with a symmetric matter field toexhibit such a disconnected branch structure.

More generally, we expect that the theories re-w xcently classified in 4 for which the gauge group can

Ž . Ž .can not be completely broken do not do possessdisconnected Higgs and confining branches associ-ated with aligned and misaligned gaugino conden-sates in the unbroken product gauge group. Thegeneral distinction between the two cases may beseen by turning on a mass term m. On the confiningbranch, the expectation value of a matter field Qwith index m is fixed by symmetries and holomor-

² : Ž myG 3Gym.1r2Gphy to be Q ; m L , where G de-notes the adjoint index. If m-G, the gauge groupgenerally breaks to a non-trivial subgroup and themoduli run off to infinity as m™0. The confiningbranch can then be thought of as a hyperbolic coneturned upside-down relative to that in Fig. 1. Thecone approaches, but never touches, the plane atinfinite moduli values. The Higgs and confiningbranches are therefore disconnected when m-G. Incontrast, it seems more likely that the Higgs andconfining branches intersect at the origin in a non-Abelian Coulomb phase for m)G theories wherethe gauge group can be completely broken. The

Ž .SO N model with a single symmetric matter fieldfalls into the latter category.

The structure of the theory at the origin can beŽ .probed by perturbing the SO N model with W stree

lTr Skq1 for kG2. The equations of motion for theŽ . keigenvalues of S are l kq1 z qg s0, where g1 1

is a Lagrange multiplier that enforces the traceless-ness condition. These equations generally only havethe trivial solution zsg s0. The superpotential1

thus lifts the moduli space. When Nsmk with man integer, there is a one complex dimensional mod-

² :uli space of vacua which looks like S sŽ 2 p i ll r k .zdiag e m 1 where z g C and ll sm= m

Ž . Ž .k1, PPP ,k. This vev breaks SO N ™SO m , and all


massless matter fields are eaten. For all values of Nand k, the theories with W slTrSkq1 can betree

Ž .regarded as the N s0 limit of a class of SO Nfw xmodels analyzed in Ref. 13 with a single symmetric

matter field and N vectors. It has a dual descriptionfŽ .in terms of an SO 4kyN gauge group, a traceless˜symmetric tensor S and a tree level superpotential

˜ ˜ ˜kq1W slTr S . This duality follows from the towertreew xof dual pairs found in 13 for general N after allf

vector matter fields are given masses and integratedout.

A special case occurs when Ns2k. W thentreeŽ .respects the anomaly free U 1 symmetry, and theR

electric and magnetic theories are identical. A onecomplex dimensional moduli space remains unliftedby W along which the gauge group is broken totree

Ž .kSO 2 . The massless spectrum consequently con-Ž .tains k photons. In the particular case of SO 4 (

Ž . Ž . Ž .SU 2 =SU 2 theory with S; 3,3 and W stree

lS3, a two-dimensional manifold of non-trivial NNs1 renormalization group fixed points emanates out

w xfrom the origin as a function of g , g , and l 14 .1 2

For Ns2k)4, there can not be an analogous non-trivial RG fixed point at the origin of the modulispace since the theories are asymptotically free witha dynamical scale L. However, there could be non-trivial RG fixed points for Ns2k)4 away fromthe origin, with a line of fixed points as a function of

² :l and S rL. This scenario requires the sponta-² :neous breaking of scale invariance for S /0 and

the explicit breaking due to L to cancel. The dualityw xof 13 would then provide a dual description of this

fixed point.˜Ž . Ž .For general N and k, W W breaks U 1tree tree R

Ž .in the electric magnetic theories. In the far infrared,Ž .an accidental U 1 symmetry must be restored asR

Žpart of the super-conformal algebra of the possibly.free IR fixed point. When the gauge coupling is

Ž .strong, the appropriate U 1 symmetry in the sameR

super-multiplet as the stress tensor should be close tothe anomaly free one. The statement that the super-

Ž .potential generally violates this U 1 symmetry isR

then equivalent to saying that it is generally relevantor irrelevant, rather than marginal. In cases where

Ž .the dual SO 4kyN gauge group is not asymptoti-cally free and the magnetic superpotential also ap-

3˜ ˜Ž Ž . Ž .pears irrelevant D W s R W )3´2kFtree tree2

.N , it seems likely that the dual becomes free in the

infrared. The requirement that these dual or freemagnetic theories be recovered when the original

Ž .SO N model is perturbed again favors the hypothe-sis that various moduli space branches meet at theorigin in a non-Abelian Coulomb phase, which al-lows for nontrivial low energy dynamics.

To recapitulate, we have given three argumentsŽ .for why the SO N model’s moduli space origin

exists in an interacting non-Abelian Coulomb phase.Firstly, it is difficult to reconcile the ’t Hooft anomalymatching results with the existence of free electricfields along subspaces that intersect the origin. Sec-ondly, the moduli space must have a confining branchwhen the symmetric matter field is given a nonzeromass. This confining branch most likely touches theHiggs branch at the origin. Finally, a nontrivialphase and branch structure must arise when the

Ž .original SO N model is perturbed with a generaltree level superpotential. Perhaps a dual descriptionof the RG fixed point at the origin can be foundwhich would provide a weak coupling picture for theconfining branch and other phenomena associatedwith W /0. At present, no such dual is known.tree

In light of all these findings, we conclude that theŽ .SO N model represents a rare example where global

anomaly matching does not signal simple confine-ment. The existence of an infinite chain of theorieswhere anomaly matching appears to be misleadingrepresents a worthwhile point to bear in mind whenanalyzing the infrared behavior of other supersym-metric and nonsupersymmetric models.

Ž .If the SO N theory does not confine, why do theanomalies match? Some insight into this coincidencecan be gained from the negative dimensional grouptheory relationship

SO 2 N ,Sp y2 N , 11Ž . Ž . Ž .

where the overbar indicates interchange of sym-w xmetrization and antisymmetrization 15–20 . This

formal expression links orthogonal groups acting onconventional bosonic tensor spaces to negative di-mensional symplectic groups defined by their actionon tensors in Grassmann vector spaces. For instance,

Ž .the dimension of an SO 2 N irrep labeled by aYoung tableau l can be obtained up to a sign from

Ž .that for the corresponding Sp 2 N irrep with thetransposed tableau lT by simply setting N™yN.


Ž .Similarly, every SO 2 N invariant scalar is relatedŽ .to an Sp 2 N counterpart by replacing the symmet-

ric g metric with its antisymmetric J analogueab a b

and swapping N™yN.Ž .The group theory relation in 11 suggests that the

Ž .SO N model is mathematically similar to the theorywith symmetry group

GsSp 2 N =U 1 12aŽ . Ž . Ž .local R

and matter content

4A; ;y , 12bŽ .ž /2 Ny2

w xwhich was studied in Refs. 21,22 . Like its orthogo-nal counterpart, this symplectic supersymmetricmodel possesses Ny1 complex flat directions whichare labeled by the operators

nO sTr AJ , ns2,3, PPP , N. 13Ž . Ž .n

At generic points in moduli space, the expectationŽ . Ž .Nvalue for A breaks Sp 2 N ™SU 2 . The quantum

moduli space consequently has a variety of discon-nected branches which are associated with the differ-ent possible signs for the gaugino condensates in

Ž .each of the SU 2 factors of the unbroken gaugegroup. The condensate sum generally yields non-vanishing dynamical superpotentials, which lift theclassical vacuum degeneracy. But on one branch, thedifferent condensate contributions precisely canceland W s0. 9 The moduli space on this Higgsdyn

˜branch is smooth in terms of the O fields, and the ’tn

Hooft anomalies

A sy2 Ny1,UŽ1.R

y2 Ny1 5N 2 q2 Nq1Ž . Ž .3A s 14Ž .UŽ1. 2R yNq1Ž .

match at the microscopic and macroscopic levels.

9 After taking into account scale matching relations, one cancheck that there is a Higgs branch with such a cancellation for allN.

Comparing these anomaly expressions with theirŽ .analogues in 6 , we observe

AS pŽ2 N . ™ ASOŽN . ,UŽ1. UŽ1.R R2 N™yN

A 3S pŽ2 N . ™ A 3

SOŽN . . 15Ž .UŽ1. UŽ1.R R2 N™yN

Ž . Ž .3So matching of the U 1 and U 1 anomalies inR RŽ .the SO N theory with a single symmetric matter

field appears to be an automatic mathematical conse-Ž .quence of the same anomaly matchings in the Sp 2 N

theory with an antisymmetric field. In the symplecticmodel, the matching is physically significant andsignals genuine confinement.

Given the connection between the SO and Spanomalies, it is amusing to note that Sp analog of the

Ž .discrete anomaly matching in the SO N model doesŽ .not generally work. In the Sp 2 N theory, a Z2 Ny2

Ž .subgroup of the classical U 1 group, which assignsA

the A field charge 1, remains intact at the quantum3 2 Ž . Ž .2level. The Z , Z U 1 and Z U 12 Ny2 2 Ny2 R 2 Ny2 R

anomalies, which need not match since they can becorrupted by the massive spectrum, indeed do notmatch. Furthermore, the difference between the mi-croscopic and macroscopic values for the Z -2 Ny2

gravity-gravity anomalies

AŽparton .s 2 N 2 yNy1 1 s Ny1 2 Nq1 ,Ž . Ž . Ž . Ž .Z2 Ny2

N1Žhadron .A s ns Ny1 Nq2 16Ž . Ž . Ž .ÝZ 22 Ny2

ns2

3 Ž .is Ds N Ny1 . So the Z anomalies match2 Ny22

modulo Ny1 if N is even, but they fail to match ifN is odd.

Instances where naive discrete anomaly matchingw xarguments fail have previously been noted 9 . For

example, the nonvanishing vev for the baryonic glue-ball operator Bse W m1 m2W m3 m4V m5 PPP V mNc

m PPP m1 NcŽ .within SO N theory with N sN y4 vectorsc f c

leaves intact all continuous global symmetries butbreaks the instanton induced Z down to Z .2 N y8 N y4c c

Anomalies therefore match only for the latter dis-crete subgroup and not for its larger progenitor. Asimilar phenomenon may resolve the discreteanomaly mismatch in the symplectic model. 10 The

10 We thank Csaba Csaki for discussions on this point.`


Ž .N glueballs which must emerge when Sp 2 N ™Ž .NSU 2 come from linear combinations of B snŽ .2Ž .nTr WJ AJ where 0FnFNy1. The only one

Ž .of these operators whose vev does not break U 1 R

is B . If this glueball composite develops aNy1

nonzero expectation value, all anomalies involvingthe unbroken discrete group Z match betweenNy1

the microscopic and macroscopic theories for Neven. Moreover, anomalies also match for N oddprovided the low energy sigma model contains anodd number of massive Majorana fermions withcharge Ny1. It is important to note that only aneven number of such Majorana fermions can existwhen N is even if the prior anomaly results are notto be disrupted. This rather involved scenario ap-pears to yield viable anomaly matching results withinthe symplectic model.

Acknowledgements

J.B. thanks Lance Dixon and Lisa Randall fordiscussions. P.C. thanks Per Kraus for collaboratingat an early stage on this work and Philip Argyres,Howard Georgi, Martin Schmaltz, Nathan Seiberg,Matt Strassler and Sandip Trivedi for sharing theirinsights. K.I. thanks Nathan Seiberg for useful dis-cussions and Gustavo Dotti and Aneesh Manohar forrekindling his interest in these models. We alsoacknowledge support from the Dept. of Energy under

Ž .Grant DOE-FG02-91ER40671 J.B. , the NationalScience Foundation under Grant aPHY-9218167Ž .P.C. , and the Dept. of Energy under Grant DOE-FG03-97ER40506 and the Alfred Sloan Fellowship

Ž .Foundation K.I. .

References

w x1 For reviews, see K. Intriligator, N. Seiberg, hep-thr9509066,Ž . Ž .Nucl. Phys. Proc. Suppl. 45BC 1996 1; M.E. Peskin,

hep-thr9702094; M. Shifman, hep-thr9704114, Prog. Part.Ž .Nucl. Phys. 39 1997 1.

w x2 For recent reviews of dynamical supersymmetry breaking,see W. Skiba, hep-thr9703159; A. Nelson, hep-phr9707442;E. Poppitz, hep-phr9710274; S. Thomas, hep-thr9801007.

w x3 K. Intriligator, N. Seiberg, S. Shenker, hep-thr9410203,Ž .Phys. Lett. B 342 1995 152.

w x4 G. Dotti, A.V. Manohar, hep-thr9712010.w x5 C. Csaki, M. Schmaltz, W. Skiba, J. Terning, hep-

thr9801207.w x6 J. Preskill, S. Trivedi, F. Wilczek, M. Wise, Nucl. Phys. B

Ž .363 1991 207.w x Ž .7 L. Ibanez, G. Ross, Phys. Lett. B 260 1991 291; Nucl.´˜

Ž . Ž .Phys. B 368 1992 3; L. Ibanez, Nucl. Phys. B 398 1993´˜301.

w x Ž .8 T. Banks, M. Dine, Phys. Rev. D 45 1992 1424.w x9 C. Csaki, H. Murayama, hep-thr9710105.`

w x Ž .10 G.’t Hooft, Phys. Rev. Lett. 37 1976 8; Phys. Rev. D 14Ž .1976 3432.

w x11 K. Intriligator, N. Seiberg, hep-thr9503179, Nucl. Phys. BŽ .444 1995 125.

w x12 K. Intriligator, N. Seiberg, hep-thr9506084, in: I. Bars et al.Ž .Eds. , Proc. of Strings’95, World Scientific, 1996.

w x Ž .13 K. Intriligator, hep-thr9505051, Nucl. Phys. B 448 1995187.

w x14 R.G. Leigh, M.J. Strassler, hep-thr9503121, Nucl. Phys. BŽ .447 1995 95.

w x Ž .15 G. Parisi, N. Sourlas, Phys. Rev. Lett. 43 1979 744.w x Ž .16 R. Penrose, in: D.J.A. Welsh Ed. , Combinatorial Mathemat-

ics and its Applications, Academic Press, New York, 1971.w x17 P. Cvitanovic, Group Theory, Nordita Notes, 1984, p. 136.w x Ž .18 R.C. King, Can. J. Math. 23 1971 176.w x Ž .19 P. Cvitanovic, A.D. Kennedy, Phys. Scr. 26 1982 5.w x Ž .20 G.V. Dunne, J. Phys. A: Math. Gen. 22 1989 1719.w x Ž .21 P. Cho, P. Kraus, hep-thr9607200, Phys. Rev. D 54 1996

7640.w x22 C. Csaki, M. Schmaltz, W. Skiba, hep-thr9607210, Nucl.`

Ž .Phys. B 487 1997 128.

18 June 1998


Bound states of dimensionally reduced SYM at finite N2q1

Francesco Antonuccio, Oleg Lunin, Stephen S. PinskyDepartment of Physics, The Ohio State UniÕersity, Columbus, OH 43210, USA

Received 25 March 1998Editor: M. Cvetic

Abstract

We consider the dimensional reduction of NNs1 SYM to 1q1 dimensions. The gauge groups we consider are2q1Ž . Ž .U N and SU N , where N is finite. We formulate the continuum bound state problem in the light-cone formalism, and

Ž .show that any normalizable SU N bound state must be a superposition of an infinite number of Fock states. We alsodiscuss how massless states arise in the DLCQ formulation for certain discretizations. q 1998 Published by Elsevier ScienceB.V. All rights reserved.

1. introduction

Solving for the non-perturbative properties of quantum field theories – such as QCD – is typically anintractable problem. In order to gain some insights, however, a number of lower dimensional models have been

Ž .investigated in the large N planar approximation, with a plethora of examples emerging in the last few yearsŽ w x.for a review see 1 .

In this work, we will pursue the same theme of studying a lower dimensional field theory, but unlike manyprevious investigations, we will allow the number of gauge colors, N, to be finite. In particular, we willconsider the 1q1 dimensional theory that is obtained by dimensionally reducing 2q1 dimensional NNs1

w xsupersymmetric Yang Mills theory. The large N limit of this theory has already been investigated in 2 , andw xwas shown to exhibit the phenomena of screening 3,4,9 . In this work, we will find it advantageous to quantize

the theory on the light-cone, and to adopt the light-cone gauge. Since the light-cone Hamiltonian is proportionalŽ .to the square of the supercharge from supersymmetry , one may formulate the bound state problem in terms of

w xthe supercharge 2,5 .A particular motivation for studying 1q1 dimensional field theories in the light-cone formalism is the

simplicity of certain bound states – the ’t Hooft pion and Schwinger particle being well known examples of this.Ž .Analogs of the ’t Hooft pion in a non-supersymmetric theory involving complex adjoint fermions have also

w xbeen discovered 8,10 . All of these bound states are characterized by relatively simple Fock state expansions,and in particular, there is an upper bound on the allowed number of partons appearing in each Fock state. It is

w xtherefore of interest to see whether the massless states in the DLCQ 6 formulation of the model studied herealso admit simple Fock state expansions in the continuum limit. We will be addressing that question in detail

w xhere, while providing a more thorough discussion of our numerical results elsewhere 7 .


( )F. Antonuccio et al.rPhysics Letters B 429 1998 327–335328

The organization of the paper proceeds as follows; in Section 2 we formulate the bound state problem for atwo dimensional matrix model in the light-cone formalism. In Section 4 we discuss in detail the massless

Ž .solutions that appear in the DLCQ bound state equations and conclude that there can be no normalizable SU Nbound state with an upper limit on the number of partons in its Fock state expansion. Finally, in Section 5, wereview some of the implications of our results.

2. formulation of the bound state problem.

The light-cone formulation of the supersymmetric matrix model obtained by dimensionally reducing NNs1w xSYM to 1q1 dimensions has already appeared in 2 , to which we refer the reader for explicit derivations.2q1

We simply note here that the light-cone Hamiltonian Py is given in terms of the supercharge Qy via they y y q y q y'� 4 Ž . Ž .supersymmetry relation Q ,Q s2 2 P . f sf x , x and c sc x , x are N=N Hermitiani j i j i j i j

Ž .matrix fields representing the physical boson and fermion degrees of freedom respectively of the theory, andare remnants of the physical transverse degrees of freedom of the original 2q1 dimensional theory. On thelight-cone, the theory is quantized by introducing Fourier expansions of these fields at fixed light-cone time

q Ž q.x s0, and imposing appropriate commutation relations for the bosonic Fourier modes a k , and anti-com-i jŽ q.mutation relations for the fermionic modes b k .i j

y y < : ySince P is proportional to the square of the supercharge Q , any eigenstate C of P with mass squaredM 2 gives rise to a natural four-fold degeneracy in the spectrum because of the supersymmetry algebra.

Focusing attention on zero mass eigenstates, we simply note that a massless eigenstate of Py must also bey y Ž y. 2annihilated by the supercharge Q , since P is proportional to Q . Thus the relevant eigen-equation is

y< : yQ C s0. We wish to study this equation. However, first we need to state the explicit equation for Q ,y1r4

`i2 gyQ s dk dk dk d k qk ykŽ .H 1 2 3 1 2 3'p 0

=1 k yk2 1 † † †q a k a k b k yb k a k a kŽ . Ž . Ž . Ž . Ž . Ž .i k 1 k j 2 i j 3 i j 3 i k 1 k j 2½ k2 k k( 31 2

1 k qk1 3 † † †q a k a k b k ya k b k a kŽ . Ž . Ž . Ž . Ž . Ž .i k 3 k j 1 i j 2 i k 1 k j 2 i j 3k2 k k( 21 3

1 k qk2 3 † † †q b k a k a k ya k b k a kŽ . Ž . Ž . Ž . Ž . Ž .i k 1 k j 2 i j 3 i j 3 i k 1 k j 2k2 k k( 12 3

1 1 1† † †q q y b k b k b k qb k b k b k . 2.1Ž . Ž . Ž . Ž . Ž . Ž . Ž .i k 1 k j 2 i j 3 i j 3 i k 1 k j 2 5ž /k k k1 2 3

w xIn order to implement the DLCQ formulation 6 of the theory, we simply restrict the momenta k ,k and k1 2 3Pq 2 Pq 3Pq

� 4appearing in the above equation to the following set of allowed momenta: , , , . . . . The integer K isK K K

called the harmonic resolution, and 1rK measures the coarseness of our discretization Physically, 1rKrepresents the smallest unit of longitudinal momentum fraction allowed for each parton. A detailed discussion of

w xthe DLCQ analytical and numerical results of this work will appear elsewhere 7 .For now, we concentrate on the structure of the zero mass eigenstates for the continuum theory. Firstly, note

Ž .that for the U N bound state problem, massless states appear automatically because of the decoupling of theŽ . Ž . Ž . Ž .U 1 and SU N degrees of freedom that constitute U N . The U 1 creation operators only introduce

Ž .degeneracies in the SU N spectrum. The non-trivial problem here is to determine whether there are masslessŽ .states for the SU N sector.

( )F. Antonuccio et al.rPhysics Letters B 429 1998 327–335 329

3. The proof

It was pointed out in the previous section that a zero mass eigenstate is annihilated by the light-coneŽ . Ž .supercharge 2.1 . We wish to show that if such an SU N eigenstate is normalizable, then it must involve a

superposition of an infinite number of Fock states. The basic strategy is quite simple; normalizability willimpose certain conditions on the light-cone wave functions as one or several momentum variables vanish.

< : y< :Moreover, if we assume a given eigenstate C has at most n partons, then the terms in Q C consisting ofnq1 partons must sum to zero, providing relations between the n parton wave functions only. We then showthese wave functions must all vanish by studying various zero momentum limits of these relations. Interestingly,the utility of studying light-cone wave functions at small momenta also appears in the context of light-front

w xQCD 11 .3q1

In order to proceed with a systematic presentation of the proof, we start by considering the large N limitcase. This simply means that we consider Fock states that are made from a single trace of a product of boson or

< :fermion creation operators acting on the light-cone Fock vacuum 0 . Multiple trace states correspond to 1rN< :corrections to the theory, and are therefore ignored. In this limit, a general state C is a superposition of Fock

states of any length, and may be written in the form` n q dq . . . dqP 1 n q Žn ,r . † †< : < :C s d q q PPP qq yP f q , . . . ,q tr c q . . . c q 0 ,Ž . Ž . Ž .Ž .Ý Ý ÝH 1 n P 1 n 1 nq ...q(0 1 nns2 rs0 P

3.1Ž .†Ž q. qwhere c q represents either a boson or fermion creation operator carrying light-cone momentum q , and

f Žn,r . denotes the wave function of an n parton Fock state containing r fermions in a particular arrangement P.P

It is implied that we sum over all such arrangements, which may not necessarily be distinct with respect tocyclic symmetry of the trace.

< :At this point, we simply remark that normalizability of a general state C above impliesq dq . . . dqP 1 n 2q Žn ,r .< <d q q PPP qq yP f q , . . . ,q -` 3.2Ž . Ž .Ž .H 1 n P 1 nq . . . q0 1 n

for any particular wave function f Žn,r .. Therefore, any wave function vanishes if one or several of its momentaP

are made to vanish.< :We are now ready to carry out the details of the proof. But first a little notation. We will write C toŽn,m.

Ž .denote a superposition of all Fock states – as in 3.1 – with precisely n partons, m of which are fermions.Such a Fock expansion involves only the wave functions f Žn,m., and the number of them is enumerated by theP

< : Ž .index P. For the special case C i.e. no fermions , there is only one wave function, which we denote byŽn,0.f Žn,0. for brevity:

q dq . . . dqP 1 n q Žn ,0. † †< : < :C s d q q PPP qq yP f q , . . . ,q tr a q . . . a q 0 . 3.3Ž . Ž . Ž . Ž .Ž .HŽn ,0. 1 n 1 n 1 nq ...q(0 1 n

< :There is another special case we wish to consider; namely, the state C consisting of n parton Fock statesŽn,2.with precisely two fermions. If we place one of the fermions at the beginning of the trace, then there are ny1ways of positioning the second fermion, yielding ny1 possible wave functions. We will enumerate such wavefunctions by the subscript index k, as in f Žn,2., where ks2,3, . . . ,n. The subscript k denotes the location of thek

second fermion. Explicitly, we haven q dq . . . dqP 1 n q Žn ,2.< :C s d q q PPP qq yP f q , . . . ,q , . . . ,qŽ .Ž .Ý HŽn ,2. 1 n k 1 k nq ...q(0 1 nks2

= † † † † < :tr b q a q . . . b q . . . a q 0 . 3.4Ž . Ž . Ž . Ž . Ž .1 2 k n

Of course, depending upon the symmetry, the ny1 Fock states enumerated in this way need not be distinct


with respect to the cyclic properties of the trace. This provides us with additional relations between wavefunctions – a fact we will make use of later on.

< : Ž .Now let us assume that C is a normalizable SU N zero mass eigenstate with at most n partons. GlancingŽ .at the form of 2.1 , we see that the nq1 parton Fock states containing a single fermion in each of they< : y< :combinations Q C and Q C must cancel each other to guarantee a massless eigenstate. ThisŽn,0. Žn,2.

immediately gives rise to the following wave function relation:

q q2 q q q2 q1 2 1 nq1Žn ,0. Žn ,0.f q qq ,q , . . . ,q y f q qq ,q , . . . ,qŽ . Ž .1 2 3 nq1 nq1 1 2 nq qq q qq1 2 1 nq1

nq q yq( 1 kq1 k Žn ,2.s2 f q , . . . ,q ,q qq ,q , . . . ,q . 3.5Ž . Ž .Ý k 1 ky1 k kq1 kq2 nq13r2n q qqŽ .ks2 kq1 k

In the limit q ™0, for 3F iFn, this last equation is reduced toi

1 1Žn ,2. Žn ,2.f q , . . . ,q ,q , . . . ,q y f q , . . . ,q ,q , . . . ,q s0. 3.6Ž . Ž . Ž .i 1 iy1 iq1 nq1 iy1 1 iy1 iq1 nq1q q( (iq1 iy1

An immediate consequence is that any wave function f Žn,2. for is3,4, . . . ,n, may be expressed in terms ofi

f Žn,2.. Explicitly, we have2

qiŽn ,2. Žn ,2.f q ,q , . . . ,q s f q ,q , . . . ,q , is3,4, . . . ,n. 3.7Ž . Ž . Ž .i 1 2 n 2 1 2 n( q2

Ž .Moreover, the limit q ™0 of Eq. 3.5 yields the further relation after a suitable change of variables:2

2 q1Žn ,0. Žn ,2.f q ,q ,q , . . . ,q s f q ,q ,q , . . . ,q . 3.8Ž . Ž . Ž .1 2 3 n 2 1 2 3 n(n q2

Finally, because of the cyclic properties of the trace, there is an additional relation between wave functions:

f Žn ,2. q ,q , . . . ,q , . . . ,q syf Žn ,2. q ,q , . . . ,q ,q ,q , . . . ,q . 3.9Ž . Ž . Ž .i 1 2 i n nyiq2 i iq1 n 1 2 iy1

Ž .Setting is2 in the above equation, and isn in Eq. 3.7 , we deduce

q1Žn ,2. Žn ,2.f q ,q , . . . ,q sy f q ,q , . . . ,q ,q . 3.10Ž . Ž . Ž .2 1 2 n 2 2 3 n 1( q2

q q' '2 3 Žn,0.Ž . Ž . Ž .Combining this with Eq. 3.8 , we conclude q f q , . . . ,q s0, where we use the fact that the1 nq q1 2

wave functions f Žn,0. are cyclically symmetric. Thus f Žn,0. must vanish. It immediately follows that f Žn,2.i

vanish for all i as well.< :To summarize, we have shown that if C is a normalizable zero mass eigenstate, where each Fock state in

< : < :its Fock state expansion has no more than n partons, the contributions C and C in this Fock stateŽn,0. Žn,2.< :expansion must vanish. Since we may assume C is bosonic, the only other contributions involve Fock states

< : < :with an even number of fermions: C , C , and so on. We claim that all such contributions vanish. ToŽn,4. Žn,6.y< :see this, first observe that the nq1 parton Fock states with three fermions in the combinations Q C andŽn,2.

y< :Q C must cancel each other, in order to guarantee a zero eigenstate mass. But our previous analysisŽn,4.< : y< :demonstrated that C '0, and thus the nq1 parton Fock states with three fermions in Q C aloneŽn,2. Žn,4.

must sum to zero.We are now ready to perform an induction procedure. Namely, we assume that for some positive integer k

< : y< :the state C vanishes. Then the nq1 parton Fock states in Q C which contain 2ky1 fermionsŽn,2w ky1x.y< :receive contributions only from Q C in which a fermion is replaced by two bosons. This has to sum toŽn,2 k .

zero. We therefore obtain a relation among the wave functions f Žn,2 k . by considering the action of theP


Ž .supercharge 2.1 in which a fermion is replaced by two bosons. Keeping in mind that we are free torenormalize any wave function by a constant, we end up with the following relation:

s ysiq1 iŽn ,2 k .f s , . . . ,s ,s qs ,s , . . . ,s s0. 3.11Ž . Ž .Ý P 1 iy1 i iq1 iq2 nq1 3r2s qsŽ .P iq1 iŽn,2 k . Ž .It is now an easy task to show that the wave functions f appearing in Eq. 3.11 must vanish; one simplyP

considers various limits s ™0 as we did before. This completes our proof by induction. Namely, there can bej

no non-trivial normalizable massless state with an upper limit on the number of allowed partons. Of course, thisproof is valid only in the large N limit. We now turn our attention to the finite N case.

Let us define Qy to be that part of the supercharge Qy that replaces a fermion with two bosons, or replaceslead

a boson with a boson and fermion pair. As in the large N case we begin by assuming that we have a< :normalizable zero mass eigenstate C which is a sum of Fock states that have at most n partons. The proof for

finite N consists of two parts. First, we consider bosonic states consisting of only n parton Fock states that haveat most two fermions, and show the wave functions must vanish. We then invoke an induction argument toconsider n parton wave functions involving an even number of fermions, and show they must vanish as well.

The additional complication introduced by the assumption that N is finite is that a given Fock state mayinvolve more than just a single trace. However, note that Qy cannot decrease the number of traces; it canlead

either increase the number of traces by one, or leave the number unchanged. Thus we have a natural inductionprocedure in the number of traces as well. Since the terms in Qy have only one annihilation operator, it actslead

on a given product of traces according to the Leibniz rule.Schematically, the general structure of an arbitrary Fock state with k traces has the form

i i1 kŽn , i , i , . . . , i . † † † † † †1 2 k < :f tr b a . . . a . . . tr b a . . . a 0 , 3.12Ž . Ž . Ž .P

where n denotes the total number of partons in each Fock state, and the integers i ,i , . . . denote the number of1 2

fermions in the first trace, second trace, and so on. We will always order the traces so that the number offermions in each trace decreases to the right. The index P labels a particular arrangement of fermions.

y < :We now consider the nq1 parton Fock states of Q C that have precisely one fermion. The onlyleadŽn,0. Žn,2. Žn,1,1. Žpossible contributions involve three types of wave functions; f , f and f we only include theP.permutation index P if there is more than one distinct arrangement . If these three wave functions contribute to

the same one fermion Fock state, then the distribution of bosons in the Fock state corresponding to f Žn,2.P

determines the distribution of bosons for f Žn,0. and f Žn,1,1.. We allow Qy to act only on the first trace in bothlead

f Žn,0. and f Žn,2., and only on the second one in f Žn,1,1.. If there are more than two traces in these states they mustP

be identical in all the components, and so don’t play a role in the calculation. Thus, it is sufficient to considerstates with two traces only. Such a state has the form

q dmq nqP q Žnqm ,0.< : <F s d q q PPP qq yP f q , . . . ,q q , . . . ,qŽ . Ž .H 1 nqm 1 m mq1 mqnq . . . q(0 1 nqm

= † † † † < :tr a q . . . a q tr a q . . . a q 0Ž . Ž . Ž . Ž .1 m mq1 mqn

q dmq ny2qdp dpP 1 2 qq d q q PPP qq qp qp yPŽ .H 1 nqmy2 1 2q . . . q p p(0 1 nqmy2 1 2

= Žnqm ,1 ,1. † † †<f p ,q , . . . ,q p ,q , . . . ,q tr b p a q . . . a qŽ . Ž . Ž .Ž .1 1 m 2 mq3 mqn 1 1 m½† † †=tr b p a q . . . a qŽ . Ž . Ž .2 mq3 mqn

Žnqm ,2. <w xq f p , P q . . . q ; p q . . . qŽ .Ý P 1 1 my2 2 mq1 mqnP

† † † † † † < :=tr b p P a q . . . a q ;b p tr a q . . . a q 0 , 3.13Ž . Ž . Ž . Ž . Ž . Ž . Ž .Ž .1 1 my2 2 mq1 mqn 5


where we have summed over the index P representing all possible permutation arrangements of bosons andfermions that contribute. We then find,

<F p ,q , . . . ,q q ,q , . . . ,qŽ .1 m mq1 mq2 mqn

q yqmq 2 mq1 Žnqm ,1 ,1. <q f p ,q , . . . ,q q qq ,q , . . . ,q s0, 3.14Ž .Ž .1 m mq1 mq2 mq3 mqn3r2q qqŽ .mq 2 mq1

where F is the contribution from f Žnqm ,0. and f Žnqm ,2.. Now we see that the limit q ™0 gives:P mq1Žnqm ,1,1. Ž . < : < :f '0. Thus if 3.13 represents a contribution to the massless eigenstate state C , then F takes the

form

q dmq ny2qdKqP q q< :F s d q q PPP qq y P yKŽ .Ž .H 1 nqmy2q . . . q(0 1 nqmy2

=q dq dqP my 1 m q Žnqm ,0. † †<d q qq yK f q , . . . ,q q , . . . ,q tr a q . . . a qŽ . Ž .Ž . Ž . Ž .H my 1 m 1 m mq1 mqn 1 mq q(0 my 1 m

q dp dpP 1 1 q Žnqm ,2. <w xq d p qp yK f p , P q , . . . ,q ; p q , . . . ,qŽ . Ž .ÝH 1 2 P 1 1 my2 2 mq1 mqnp p(0 1 2 P

= † † † † † † < :tr b p P a q . . . a q ;b p tr a q . . . a q 0 3.15Ž . Ž . Ž . Ž . Ž . Ž . Ž .Ž .Ž .1 1 my2 2 mq1 mqn

and Qy acts only on the terms in the square brackets. All these terms have only one trace, which is a scenariolead

we already encountered in the large N limit case. Using the results of that discussion, we find that the onlyŽ .massless solution of the form 3.15 is the trivial one. This is the starting point of the induction procedure for

finite N.< :As explained earlier, we look for n parton Fock states in the expansion for C that have 2k fermions

Ž .k)1 , To finish the proof we need to show that for any k the only allowed wave function is the trival one.From the large N result we know there are no such one trace states. We now consider the state with an arbitrarynumber of traces,

q ds . . . dsP 1 n q Žn ,2 k .< : < <C s d s q PPP qs yP f s . . . s . . . . . . sŽ . Ž .ÝHŽn ,2 k . 1 n P 1 i n1s . . . s(0 1 nP

= † † † < :tr c s . . . c s tr . . . tr . . . c s 0 , 3.16Ž . Ž . Ž . Ž . Ž .Ž .Ž .1 i n1

Ž .then the analog of 3.11 for such states reads:s ysX iq1 iŽn ,2 k . < <f s . . . s . . . s ,s qs ,s . . . s . . . s s0. 3.17Ž .Ž .Ý P 1 j iy1 i iq1 iq2 j qk nq1 3r2i a a a s qsŽ .i iq1 i

Here, ÝX means that for each trace we should include one additional term with ’’i’’s j qk , ’’iq1’’s j if ci a a a

corresponding to both j qk and j is a. If the number of traces is a, we introduce j sÝay1 k . If any of thea a a a bs1 bŽ . Ž .blocks tr . . . in the state for which 3.17 is written contains two or more fermions, then, as in the large N

case, all the corresponding wave functions f Žn,2 k . vanish. So we only need to consider the states of the form:P

< : Žn ,k1q1 , . . . . < <C s dpdqf p ,q , . . . ,q p ,q , . . . ,q . . .Ž .ÝHŽn ,k q1, . . . . P 1 1 k 2 k q1 k qk1 1 1 1 2P

= † † † † † † < :tr b p a q . . . a q tr b p a q . . . a q . . . 0 . 3.18Ž . Ž . Ž . Ž . Ž . Ž . Ž .Ž . Ž .1 1 k 2 k q1 k qk1 1 1 2


˜ yLet Q denote that part of the supercharge Q which replaces a fermion with two bosons. Let us consider theresult of such a change in the first trace. Suppose there are a traces having the same form as the first trace. Thenwithout loss of generality, we may assume they are the first a traces. Then using the symmetries of the wavefunctions we find:

1 qP Žn ,k q1, . . . .1˜ < : < <QC sy dkdpdqf p ,q , . . . ,q p ,q , . . . ,q . . .Ž .ÝHŽn ,k q1, . . . . P 1 1 k 2 k q1 2 k1 1 1 1'2 2p 0P

a p y2k 1b bq1= y1Ž .Ý

p k p yk( Ž .bbs1 b

= † † † † † † † < :tr b p a q . . . a q . . . tr a k a p yk a q . . . a q . . . 0Ž . Ž . Ž . Ž . Ž . Ž . Ž .Ž . Ž .1 1 k b Žby1.k q1 bk1 1 1

1 q p y2k 1P 1sy dkdpdqÝH' p2 2p k p yk0 ( Ž .1P 1

† † † † † † † < := tr a k a p yk a q . . . a q tr b p a q . . . a q . . . 0Ž . Ž . Ž . Ž . Ž . Ž . Ž .Ž . Ž .1 1 k 2 k q1 k qk1 1 1 2

abq1 bq1 Žn ,k q1, . . . .1 < <= y1 y1 f p ,q , . . . ,q p ,q , . . . ,q . . . .Ž . Ž . Ž .Ý P 1 1 k 2 k q1 k qk1 1 1 2

bs1

If the above expression vanishes then the only solution is the trivial one in which all wave functions vanish.This finishes the proof of the induction procedure for the finite N case.

< :The extension of the proof to massive bound states is straightforward. Firstly, assume C is a normalizable1q y 2 y y 2 y˜Ž . < : < : < :eigenstate of 2 P P with mass squared M /0. Then, since P s Q , the state C ' C qa Q C'2

2 q 2 y'for a s 2 P rM is a normalizable eigenstate of the supercharge Q , with eigenvalue 1ra . We therefore1y ˜ ˜< : < :study the eigen-problem Q C s C . The resulting constraints on the wave functions may be obtained bya

modifying our original expressions by including a wave function multiplied by a finite constant. However, inour analysis, we always need to let some of the momenta vanish, and therefore this additional contribution

Ž .vanishes. The analysis and therefore the conclusions remains unchanged.Ž . Ž .We therefore conclude that any normalizable SU N bound state massless or massive that exists in the

model must be a superposition of an infinite number of Fock states.

4. Bound states in DLCQ

In the previous section we proved that the continuum formulation of the theory does not have anynormalizable bound states with a finite number of partons. Our proof used the behavior of wave functions atsmall momenta arising from the normalizability assumption. Neither of these properties can be used in DLCQ,however. Here we consider some simple examples of massless DLCQ solutions with n bosons to help shedsome light on the relation between DLCQ solutions and the solutions of the continuum theory. For simplicity,we work in the large N limit case.

r qiWe write the momentum of a state in DLCQ in terms of the momentum fraction q where q s P , and thei i rŽn,0.Ž .r are positive integers. The wave function of such a state is f r , . . . ,r . There are two conditions thati 1 n

must be satisfied to show that it is massless. One is the that the coefficient of the term with one additionalfermion that is producted by the action of Qy is zero. This condition gives the relation,

2 r q t 2 r q tn ny1Žn ,0. Žn ,0.f r , . . . ,r ,r q t y f r , . . . ,r q t ,r s0. 4.1Ž . Ž . Ž .1 ny1 n 1 ny1 nr q t r q tn ny1


where t correspond to the momentum fraction of the one fermion. The second is that the coefficient of the statewith two fewer bosons and one additional fermion which is also produced by the action of Qy is zero. Thiscondition gives the relation,

ty2kŽn ,0.f r , . . . ,r ,k ,tyk d s0. 4.2Ž . Ž .Ý 1 ny2 Ž r qr , t .ny 1 nk tykŽ .k , t

Ž .For the case where all r s1, and the total harmonic resolution is n, it is trivial that Eq. 4.1 is satisfiedi

since there is not enough resolution to increase the number of particles in the state. It is also easy to see fromŽ . Žn,0.Ž .Eq. 4.2 since the coefficient of the one term in the sum is zero. Thus the wave function f 1,1,....1 is a

massless state for every resolutionŽ .To discuss additional solutions it is useful to start by considering Eq. 4.1 . The case ts1 is trivial to satisfy

Ž .if r s1 for all i. The contributions in Eq. 4.2 come from the two terms in the sum, ks1, ts3 and ks2,iŽn,0.Ž .ts3. Each term has the same coefficient but of opposite sign and cancel. Therefore the state f 1,...1,2 is a

massless state for all resolutionsŽ .From Eq. 4.1 with ts2 and with ts1 used twice we find:

2 2r q1 r q1Ž . Ž .n ny1Žn ,0.f r ...r ,r q2,r y s0. 4.3Ž . Ž .1 ny2 ny1 n ž /2 r q3 2 r q1 2 r q3 2 r q1Ž . Ž . Ž . Ž .n n ny1 ny1

Ž .Using relation 4.1 several times we can always express an arbitrary wave function in the following form:

f Žn ,0. r ...r sC r ...r f Žn ,0. 1...1, Lq1,1 4.4Ž . Ž . Ž . Ž .1 n 1 n

Ž .where Lsr q ...q r yn and C r ...r is some nonzero coefficient.The two massless states we found above1 n 1 nŽ . Žn,0.Ž Ž . .correspond to Ls0 and Ls1. Choosing r s ...sr sr s1 in 4.3 we find: f 1...1, Ly1 q2,1 s01 ny2 n

Ž .for L)2 due to monotonic behavior of the function in the parenthesis. Then using 4.4 we conclude that all theŽ .wave functions with L)2 vanish. So the only case we need consider is Ls2. In this case 4.1 has only two

nontrivial cases: ts1 and ts2. In the case where ts2 we can only have r s . . . sr s1 so it is trivially1 nŽn,0.Ž .satisfied. The case where ts1 however gives a nontrivial relation for the wave function: f 1, . . . ,1,2,2 s

10Žn,0. Žn,0. Žn,0.Ž . Ž . Ž . Ž .f 1, . . . ,2,1,2 s . . . s f 2, . . . ,1,1,2 s f 1, . . . ,1,3 . finally we must show that Eq. 4.2 is satis-9

fied which is straight forward.w xThese are only a few examples of massless states, and there are in fact many more in DLCQ 7 . In the

continuum limit we have proven that there are no massless normalizable states with a finite number of particles.However, there is the possibility that at each finite value of the harmonic resolution, one obtains an exactlymassless bound state, but as the harmonic resolution is sent to infinity, the number of Fock states required tokeep the bound state massless must also be infinite.

5. Conclusions

In this work we considered the dimensional reduction of NNs1 SYM to 1q1 dimensions, and at finite2q1

N. Our main objective was to analyze the structure of bound states both in the continuum and in the DLCQŽformulation. We discovered many massless states in the DLCQ formulation, but showed that any massless or

.massive normalizable bound states in the continuum theory must be a superposition of an infinite number ofFock states. Our work therefore shows that any exact analytical treatment of the continuum bound state problemis probably a too ambitious objective for the near future. This scenario is to be contrasted with the simple boundstates discovered in a number of 1q1 dimensional theories with complex fermions, such as the Schwinger

w xmodel, the ’t Hooft model, and a dimensionally reduced theory with complex adjoint fermions 8,10 . While


these theories offered hope that bound states viewed in the light-cone formalism might be much simpler than inthe equal-time quantization approach, we see here that this is not the case. Numerical DLCQ studies of themodel are, of course, insensitive to the complexity of the bound state problem, and extensive numerical results

w xon the model studied here will appear elsewhere 7 .

References

w x1 S.J. Brodsky, H.C. Pauli, S.S. Pinsky, Quantum Chromodynamics and Other Field Theories on the Light Cone, to appear in Phys.Rept., hep-phr9705477.

w x Ž .2 Y. Matsumura, N. Sakai, T. Sakai, Phys. Rev. D 52 1995 2446, hep-thr9504150.w x Ž .3 D.J. Gross, I.R. Klebanov, A.V. Matytsin, A.V. Smilga, Nucl. Phys. B 461 1996 109, hep-thr9511104.w x4 A. Armoni, J. Sonnenschein, Screening Confinement in Large N QCD and in Ns1 SYM , TAUP-2412-97, hep-thr9703114.f 2 2w x Ž .5 A. Hashimoto, I.R. Klebanov, Nucl. Phys. B 434 1995 264, hep-thr9409064.w x Ž .6 H.-C. Pauli, S.J. Brodsky, Phys. Lett. D 32 1985 1993, 2001.w x7 F. Antonuccio, S.S. Pinsky, O. Lunin, to appear.w x Ž .8 F. Antonuccio, S.S. Pinsky, Phys. Lett. B 397 1997 42, hep-thr9612021.w x9 D.J. Gross, A. Hashimoto, I.R. Klebanov, The Spectrum of a Large N Gauge Theory Near Transition From Confinement to Screening,

NSF-ITP-97-133, October 1997, p. 16, hep-thr9710240.w x10 S. Pinsky, The Analog of the ’t Hooft Pion with Adjoint Fermions, Invited talk at New Nonperturbative Methods and Quantization of

the Light Cone, Les Houches, France, 24 February - 7 March 1997, hep-thr9705242.w x Ž .11 F. Antonuccio, S.J. Brodsky, S. Dalley, Phys. Lett. B 412 1997 104.

18 June 1998


Novel integrable spin-particle modelsfrom gauge theories on a cylinder

Jonas Blom, Edwin LangmannTheoretical Physics, Royal Institute of Technology, S-10044, Stockholm, Sweden

Received 6 April 1998; revised 13 April 1998Editor: P.V. Landshoff

Abstract

We find and solve a large class of integrable dynamical systems which includes Calogero-Sutherland models and variousnovel generalizations thereof. In general they describe N interacting particles moving on a circle and coupled to an arbitrary

Ž .number, m, of su N spin degrees of freedom with interactions which depend on arbitrary real parameters x , js1,2, . . . ,m.jŽ .We derive these models from SU N Yang-Mills gauge theory coupled to non-dynamic matter and on spacetime which is a

cylinder. This relation to gauge theories is used to prove integrability, to construct conservation laws, and solve thesemodels. q 1998 Elsevier Science B.V. All rights reserved.

Integrable models have always played a centralrole in classical and quantum mechanics. Mostprominent examples, like the Kepler problem, are

Ž .systems with few F3 degrees of freedom. Animportant exception is a class of integrable N-par-ticle models associated with the names Calogero,

w x Ž w x.Moser and Sutherland 1,2 for review see 3 .These are models for identical particles moving onone dimensional space and interacting via certain

Ž .repulsive two-body potentials Õ r . A well-knownŽ . 2 2Ž . Ž Ž .example is Õ r Ag rsin gr which includes Õ r

2 .A1rr in the limit g™0 , and we refer to thecorresponding model as CS model. It is known thatthese models allow for interesting generalizationswhich also have dynamic spin degrees of freedomw x4,5 . The CS model and its generalizations haverecently received quite some attention in differentcontexts. Here we only mention their relation to

w xgauged matrix models 6 and gauge theories on aw x 1cylinder 7 which will be relevant for us .

In this article we find and solve a new class ofintegrable systems containing the CS model and their

w xspin-generalizations 4 as limiting cases. Our methodis to extend and exploit the relation of the CS modelto gauge theories on a cylinder, as will be explainedin detail below. For simplicity our discussion here isrestricted to classical models, and we only consider aspecial type of gauge theory. Our method is simple,and it should be possible to generalize it in severaldifferent directions: We conjecture that the corre-sponding quantum models are also integrable, andthat our method to prove integrability should apply

1 The latter relation is implicit already in earlier work; see e.g.w x8 .


( )J. Blom, E. LangmannrPhysics Letters B 429 1998 336–342 337

Žto the quantum case, too. In this context it is worthnoting that the quantum-analog of the gauge theorywe consider is closely related to QCD on a cylinderand in the limit where the masses of the quarksbecomes infinite; a good starting point to the litera-

w x.ture on this is Ref. 9 . Moreover, it would beinteresting to extend our method to other gauge

Žtheories e.g. with supersymmetry andror more gen-eral types of matter fields than the one considered

.here and thus try to find and solve other integrablemodels.

The models we solve are given by a Hamiltonian

2aN N mp 1Ž .a bHs q Õ q yqŽ .Ý Ý Ý jk2 2

as1 j ,ks1a/b

a ,bs1

N m1ab ba a a a ar r q c r r , 1Ž .Ý Ýk j jk k j2

as1 j ,ks1

where q a and pa are particle coordinates and mo-� a b 4menta with the usual Poisson brackets q , p s

ab a b bad etc., and r s r are complex valuedj jŽ . N a asu N -spins, i.e. Ý r s0 andas0 j

r ab ,r aXb

X

s ig2pd d baX

r abX

yd bXar a

Xb .� 4 Ž .j k jk j j

2Ž .

Ž .The other Poisson brackets vanish. The interactionpotentials are given by

1 yi g r x jkÕ r s eŽ .jk 4

=< <1 ix xjk jk

q cot p gr y ,Ž .2ž /p psin p grŽ .

3Ž .where

x s x yx , 4Ž . Ž .jk j k 2p

with s :ssy2p n for the integer n such that2p

ypFsy2p n-p , and

2 < <x x 1jk jkc s y q . 5Ž .jk 2 4p 128p

Ž . ŽThe parameters g real positive , N and m positive.integers are arbitrary, and ypFx Fx F . . . F1 2

x -p . Furthermore we have the following con-m

straint on the possible initial conditions,m

aar s0 ;a . 6Ž .Ý jjs1

Ž . Ž .Note that Õ r sÕ yr sÕ yr , which im-Ž .jk jk k j

plies that the Hamiltonian is real. Moreover, since

nyi nŽ x yx .j kÕ rq se Õ r , 7Ž . Ž .jk jkž /g

the Hamiltonian is invariant under the followingtransformations,

na

a aq ™q q ,g

pa ™pa , r ab ™r abeyi x jŽnayn b . 8Ž .j j

for all integers na. Thus these models describe parti-cles moving on a circle of length 1rg and interact-ing with a potential whose strength depends ondynamic spins. We note that the particles repel eachother, and we have a further natural restriction onphase space,

q a /q b;a/b . 9Ž .

The main result of this article is a proof of integrabil-ity and the explicit solution of all these models.

Next the relation of our particle-spin models togauge theories is discussed. It is known that the CSmodel can be obtained from a gauged one dimen-

w xsional matrix model 6 . More recently a relation togauge theories in 1q1 dimensions was pointed outw x7 . In this article we explore this relation further anduse it to find and solve new integrable models. We

Ž .present a simple argument that SU N Yang-Millsgauge theory on a cylinder coupled to certain non-dynamic matter is equivalent to a model of interact-ing particles and spins. We then show that thisequivalence can be used as a powerful tool to ana-lyze and solve these models: integrability is mani-fest, the construction of conservation laws trivial,and a simple solution method is obtained by exploit-ing gauge invariance, i.e. the possibility to changefrom the Weyl gauge 2 A s0 to what we call the0

diagonal Coulomb gauge, i.e. the condition that the

2 A and A will be defined further below.0 1

( )J. Blom, E. LangmannrPhysics Letters B 429 1998 336–342338

Ž .spatial component of the Yang-Mills field, A t, x ,1

is independent of x and diagonal in color space,

A t , x sQ t sdiag q1 t ,q2 t , . . . ,q N t .Ž . Ž . Ž . Ž . Ž .Ž .1

10Ž .This is due to the fact that the gauge theory model inthe Weyl gauge is free and can be solved trivially,whereas in the diagonal Coulomb gauge the timeevolution equations are non-linear and, in a specialcase, equal to the Hamilton equations of the model

Ž .given by Eq. 1 . To be more specific: We restrictourselves to gauge theory models with matter fieldslocalized at a finite number of points x , jsj

Ž Ž . .1,2, . . . ,m for simplicity see Eq. 25 below . Wefind that the dynamics of such a model in thediagonal Coulomb gauge is governed by the equa-

˙ a a a b� 4tions of motion Xs X, H , Xsq , p ,r , whichjŽ .follow from the Hamiltonian Eq. 1 and the Poisson

� 4brackets P,P given above. These observations al-lows us to derive the full solution of the initial valueproblem for all these models generalizing the known

w xsolution of the CS model 3 .We now consider 1 q 1 dimensional Yang-

Mills theory, i.e. the differential equationsw mn x nÝ D ,F sJ where D sE q igA , A arems0,1 m m m m m

the Yang-Mills fields, J m the matter currents, g theYang-Mills coupling strength, E sErE x m, and m,nm

s0,1. Spacetime is a cylinder i.e. tsx 0 gR is1 w x Ž . 3time, and xsx g yp ,p s circle . Moreover ,

w x ŽF s D , D rig, and our metric tensor is diag 1,mn m n

. Ž .y1 . As gauge group we take SU N in the funda-mental representation 4.

We restrict ourselves to non-dynamic matter, i.e.J 1 s0, and J 0 'r. We denote r as charge. Note

w xthat we have to impose D ,r s0 for consistency.0

Setting E:sF , we can write these equations as01

follows,w xE A sEqE A q ig A , A , 11Ž .0 1 1 0 1 0

w xE Eq ig A , E s0 , 12Ž .0 0

w xE rq ig A ,r s0 , 13Ž .0 0

w xE Eq ig A , E sr . 14Ž .1 1

Ž .Eq. 14 is called Gauss’ law and is a constraint onpossible initial data for the system of time evolution

3 w xa,b :s aby ba.4 I.e. A etc. are functions with values in the traceless, com-m

plex N = N matrices.

Ž . Ž .Eqs. 11 – 13 . We now exploit gauge invariance:Ž . Ž .Eqs. 11 – 14 are obviously invariant under gauge

transformations

1y1 y1A ™U A Uq U E U ,m m mig

E™Uy1EU , r™Uy1rU , 15Ž .Ž .where UsU t, x is an arbitrary differentiable

Ž .SU N -valued function on spacetime. To eliminatethe gauge degrees of freedom one has to fix a gauge.

Ž .One convenient choice is the Weyl gauge A t, x s05 Ž . Ž .0 . Then the Eqs. 11 – 13 can be solved trivially:

E t , x sE 0, x ,Ž . Ž .A t , x sA 0, x qE 0, x t ,Ž . Ž . Ž .1 1

r t , x sr 0, x , 16Ž . Ž . Ž .Ž . Ž . Ž .with the initial data E 0, x , A 0, x and r 0, x1

Ž . Žsatisfying the Gauss’ law Eq. 14 note that ourŽ . Ž .solution Eq. 16 satisfies Eq. 14 for all t if it

.satisfies it for ts0 .As mentioned, E, A and r are functions withm

values in the traceless N=N matrices. In the fol-lowing we write the matrix elements of MsE, Am

or r as M ab, a ,bs1,2, . . . , N. Note that, sinceÝN M aa is zero, the independent components areas0

M ab for a/b , and M aa yM aq1,aq1 for as1,2, . . . , Ny1.

We now show that one can also impose theŽ . Ž .diagonal Coulomb gauge 10 , i.e. for each generic

Ž .Yang-Mills configurations A t, x one can find a1

gauge transformation U such that AU 'Uy1A Uq1 1

Uy1E Urig is a diagonal matrix Q independent of x1w x10 . For that we construct such a U explicitly. Wefirst note that a solution to the equation E Sq igA S1 1

Ž .s0 with S t,yp s1 is the parallel transporter

xS t , x sPPexp yig dy A t , y , 17Ž . Ž . Ž .H 1ž /yp

where PPexp is the path ordered exponential. NoteŽ .that S t, x is not a gauge transformation since it is

Žnot periodic in x its values at xsyp and p are.different in general . To construct a gauge transfor-

5 I.e. to consider the model in terms of the gauge transformedŽ .fields on the r.h.s. of Eq. 15 , which by abuse of notation we

denote by the same symbol, and with a gauge transformation Uwhich is a solution of E Uq igA Us0.0 0


Ž . Ž .mation, we introduce the SU N -matrix V t diago-Ž . Ž .nalizing the SU N -matrix S t,p ,

y1 yi g 2p QŽ t .V t S t ,p V t se 18Ž . Ž . Ž . Ž .Ž .for some diagonal matrix Q t . This implies that

U t , x sS t , x V t ei g Ž xqp .QŽ t . 19Ž . Ž . Ž . Ž .

is periodic in x, and it satisfies E Uq igA Us igUQ1 1U Ž .equivalent to A sQ. Moreover, if A t, x is a1 1

Ž .generic differentiable map on spacetime, then Q tŽ . w xand V t can be chosen to be differentiable in t 11 ,Ž . Ž .and U t, x Eq. 19 is indeed a differentiable func-

tion on space-time i.e. a gauge transformation.‘Generic’ here means that the latter is only true if

a Ž . bŽ .q t /q t for all t and a/b since otherwiseŽ . w xdiscontinuous functions V t can occur 11 . Due to

Ž . Ž .Eq. 9 , gauge field configurations A t, x where1

this condition fails are irrelevant for us. Note that oura Ž .discussion here implies that the q t can be ob-

Ž .tained as eigenvalues of the Wilson line S t,p . Thisobservation will allow us to determine the explicitsolution of the Hamilton equations following from

Ž . Ž .Eqs. 1 and 3 .We now determine the time evolution equations

a Ž .for the variables q defined in Eq. 10 . We useFourier transformation,

pab yi n x a bE t ,n s dx e E t , x , ngZ 20Ž . Ž . Ž .H

yp

Ž .and similarly for A and r. Then Eq. 11 gives0

âaE t ,0Ž .a aE q t sp t ' . 21Ž . Ž . Ž .0 2p

Note that this and the following equations all areN a N a Žconsistent with Ý q sÝ p s0 this corre-as1 as1

sponds to translation invariance of the mechanicalŽ ..system defined in Eq. 1 . The time evolution of the

a Ž .p follows from Eq. 12 ,

E pa tŽ .0

Nigab baˆ ˆsy A t ,n E t ,ynŽ . Ž .ŽÝ Ý 022pŽ . ngZ b/a

bs1

âb ˆbayE t ,n A t ,yn . 22Ž . Ž . Ž ..0

The r.h.s. of this equation can be evaluated usingŽ . Ž .Eqs. 11 and 14

a b a bˆyi nqg q t yq t A t ,nŽ . Ž . Ž .Ž . 0

âbsE t ,n ,Ž .a b a bî nqg q t yq t E t ,nŽ . Ž . Ž .Ž .

sr ab t ,n 23Ž . Ž .ˆŽ .note that this holds true even for asb if n/0 .Inserting this we get

E pa tŽ .0

N ab ba2 g r t ,n r t ,ynŽ . Ž .ˆ ˆs .Ý Ý2 3

a b2pŽ . nqg q t yq tngZ Ž . Ž .b/a Ž .bs1

24Ž .Note that up to now no specific choice for the

charges was made. To proceed, we restrict ourselvesto charges of the following form for simplicity,

mab a br t , x ' r t d xyx , 25Ž . Ž . Ž . Ž .Ý j j

js1

which describe matter localized at the points x , asj

mentioned above. Thenm

ab a b yi n x kr t ,n s r t e , 26Ž . Ž . Ž .ˆ Ý kks1

Ž .and we can then write Eq. 24 asN mE

a a bE p t sy r tŽ . Ž .Ý Ý0 kaE q tŽ . j,ks1b/a

bs1

=r ba t Õ q a t yq b t , 27Ž . Ž . Ž . Ž .Ž .j jk

with

einŽ x jyx k .

Õ r s , 28Ž . Ž .Ýjk 2 22p nqgrŽ . Ž .n

where the summation is over all ngZ. This equalsŽ .Eq. 3 , as can be seen by a simple computation

using the identity

ein syi r s2pseÝ 2nqrŽ .ngZ

=p 2

< <q ip s cot p r yp s .Ž .2p 2p2ž /sin p rŽ .29Ž .


Note also that

ein s s2 p 22p

< <s yp s q . 30Ž .Ý 2p2 2 3nn/0

Ž . Ž .Eqs. 21 and 27 are precisely the Hamilton equa-a � a 4 a � a 4tions q s q , H and p s p , H . We are left to˙ ˙

determine the time evolution of the r ab. From Eq.jŽ .13 we get

Nab ag gbE r t syig A t , x r tŽ . Ž .Ž .ŽÝ0 j 0 j j

gs1

yr ag t Agb t , x . 31Ž . Ž .Ž . .j 0 j

a b Ž .M oreover, w e can com pute A t , x0 j1 ab i n x jˆ Ž . Ž . Ž .s Ý A t,n e from Eqs. 23 and 26 . Noten 02p

âaŽ . Ž .that Eq. 23 also determines A t,n if n is non-0âa 6Ž .zero, and we can set A t,0 s0 . Then a simple0

computation givesm

ab a b a bA t , x s2p Õ q t yq t r t ,Ž . Ž . Ž .Ž . Ž .Ý0 j jk kks1

32Ž .Ž . Ž . Ž .with Õ r given by Eq. 3 and Õ 0 sc by Eq.jk jk jk

Ž . Ž Ž . Ž ..5 . We used Eqs. 28 – 30 . Note that for asb

Ž .the summation in Eq. 28 is restricted to the non-zeroŽ . .integers n, which implies Õ 0 sc . With that Eq.jk jk

Ž . ab31 becomes equal to the Hamilton eq. r s˙j� ab 4 Ž . Ž .r , H following from Eqs. 1 and 2 .j

We finally note that the ns0 components ofŽ .Gauss’ law Eq. 14 for asb reads

r aa t ,0 s0 ;a . 33Ž . Ž .ˆThis is a consistency requirement. Our argumentsabove show that this condition is fulfilled for r in

Ž . Ž .Eq. 25 if and only if Eq. 6 holds for all t, whichis true if it holds for ts0.

Ž . Ž . Ž . Ž .We now solve the Eqs. 21 , 27 , 31 and 32with the initial conditions

q a 0 sq a , pa 0 spa , r ab 0 sr ab .Ž . Ž . Ž .0 0 j j ,0

34Ž .

Our discussion above implies that we can obtain thea Ž .solution q t of this initial value problem by solv-

6 âaŽ .A non-zero A t,0 can be removed by a gauge transforma-0

tion compatible with the DCG.

Ž . Ž .ing the Yang-Mills Eqs. 11 – 14 with the initialconditions

Aab ts0, x sd q a ,Ž .1 a b 0

p dxaa aE ts0, x sp ,Ž .H 02pyp

Nab a br ts0, x s r d xyx . 35Ž . Ž . Ž .Ý j ,0 j

js1

ab Ž .We first have to determine E ts0, x for a/b

Ž . Ž .from Gauss’ law Eq. 14 . The solution A t, x of1

the gauge theory in the Weyl gauge A s0 is then0Ž . a Ž .given in Eq. 16 . To obtain the q t , we only need

to evaluate the corresponding parallel transporterŽ . Ž .S t,p Eq. 17 : as discussed, the eigenvalues ofŽ . y2 p i g q aŽ t .S t,p are equal to e . Moreover,

y1r t sU t , x r U t , x , 36Ž . Ž .Ž . Ž .j j j ,0 j

Ž . Ž .with U t, x given by Eq. 19 . Here and in thefollowing we use an obvious matrix notation.

Ž .For ts0 we can write Gauss’ law Eq. 14 asfollows 7

E ei g Q0 xE 0, x eyi g Q0 x sei g Q0 xr 0, x eyi g Q0 x ,Ž . Ž .Ž .1

37Ž .Ž . m Ž .with r 0, x sÝ r d xyx andjs1 j,0 j

Q sdiag q1 ,q2 , . . . ,q N . 38Ž .Ž .0 0 0 0

Ž .Since r 0, x s0 except for xsx , we obtainjŽ . yi g Q0 x i g Q0 xE 0, x se B e , where B is some con-j j

stant matrix, for x -x-x , js0, . . . ,m, x sj jq1 0

yp and x sp . To determine the matrices Bmq 1 jŽ . q qwe integrate Eq. 37 from x y0 to x q0 . Thisj j

gives the recursion relations B y B sj jy 1i g Q0 x j yi g Q0 x j Ž .e r e , and the condition E 0,yp sj,0Ž . 2 i g Q0p y2 i g Q0pE 0,p implies e B e sB . Putting this0 m

Ž .together and using the second relation in Eq. 35 weobtain after a straightforward calculation,

j m ix yxiq1 ia b a a a a aB sd p q r y rÝ Ý Ýj a b 0 ll ,0 ll ,0ž /2pis1lls1 lls1

m ab i g w q ayq b xw x qp sgnŽ x yx .x0 0 ll j llr ell ,0q 1yd ,Ž . Ýab a b2 isin gp q yqŽ .0 0lls1

39Ž .Ž .with sgn x s1 for xG0 and y1 for x-0. With

7 Note that the same argument applies for times t)0.


Ž . Ž . y i g Q 0 xthat, Eq. 16 gives A t , x s e1

= Q qB t ei g Q0 x for x -x-x . This is theŽ .0 j j jq1

solution of the Yang-Mills equations in the Weylgauge.

Ž . Ž . Ž .We now solve E S t, x q igA t, x S t, x s01 1

which is equivalent to

˜ ˜E S t , x q igB tS t , x s0 for x -x-xŽ . Ž .1 j j jq1

40Ž .˜ i g Q0 x ˜Ž . Ž . Ž .for S t, x se S t, x . This implies S t, x s

yi g Bj tŽ xyx j. Ž .e S t, x for x -x-x , thusj j jq1

yi g Q0 x jq 1 yi g Bj tŽ x jq 1yx j. ˜S t , x se e S t , x s . . .Ž . Ž .jq1 j

seyi g Q0 x jq 1 eyi g Bj tŽ x jq 1yx j.eyi g Bjy 1 tŽ x jyx jy 1. PPP

=eyi g B0 tŽ x1qp .eyi g Q0p , 41Ž .Ž . Ž .where we used S t,yp s1. Especially for jsm ,

S t ,p seyi g Q0peyi g Bm tŽpyx m. PPPŽ .=eyi g Bmy 1 tŽ x myx my 1. PPP eyi g B0 tŽ x1qp .eyi g Q0p .

42Ž .

We thus obtain our main result:

The Hamiltonian equations of the the dynamicalŽ . Ž .system defined in Eqs. 1 – 6 are given in Eqs.

Ž . Ž . Ž .21 , 27 and 31 . The solutions of these equationsŽ .with the initial conditions Eq. 34 can be obtained

Ž .from the eigenvalues of the matrix S t,p given inŽ . Ž . Ž .Eqs. 42 and 39 according to Eq. 18 . Moreover,

Ž . Ž . Ž .r t is given by Eq. 36 with U t, x defined inj jŽ . Ž . Ž .Eq. 19 and S t, x in Eq. 41 .j

Note that for ms1, the dynamics of the spin andthe particles decouple, and our result reduces to the

w xknown solution of the CS model; see e.g. 3 .It is now also easy to construct conservation laws

Ž .for our dynamical systems: Eq. 16 implies thatw Ž .n xtr E t, x , where tr is the N=N matrix trace, is

time independent for all ypFx-p and all posi-tive integers n. Since these quantities are gaugeindependent, they are time independent also in thediagonal Coulomb gauge. In this latter gauge, we can

Ž . Ž .evaluate E t, x as above and obtain E t, x syi g QŽ t . x Ž . i g QŽ t . x Ž .e B t e for x -x-x where Q tj j jq1

Ž . Ž . Ž . aand B t are as in Eqs. 38 and 39 but with q ,j 0a a b a Ž . a Ž . ab Ž .p , and r replaced by q t , p t , and r t ,0 j,0 j

i.e. the solution of the initial value problem whichwe solved above. Using cyclicity of the trace, we

w Ž .n xconclude that tr B t for an arbitrary positive inte-j

ger n and js1, . . . ,m are time independent: Each ofthem is a conservation law. For ms1 these are the

w xknown conservation laws for the CS model 3 . It isalso worth noting the corresponding Lax-type equa-

Ž . w Ž . Ž .xtions E B t q ig M t , B t s0 where0 j j j

M t sei g QŽ t . x j A t , x eyi g QŽ t . x j yx P t ,Ž . Ž .Ž .j 0 j j

43Ž .

Ž .which are obtained from Eq. 12 setting xsx andjŽ . Ž . Ž Ž . Ž . Ž ..using E Q t sP t :sdiag p t , p t , . . . , p t0 1 2 N

Ž Ž . Ž ..A t, x is given by Eq. 32 .0 j

As discussed above, the models we find general-ize the CS models with the interaction potentialŽ . 2 2Ž .Õ r Ag rsin gr which describe particle moving

on a circle of length 1rg. There is a an integrableCS-type model of particles moving on the real line

Ž . 2 2Ž .and interacting with a potential Õ r Ag rsinh gr ,w xsee e.g. 3 . The sinh-model and its solution can be

obtained from the sin-model and its solution bya a a a w xreplacing q ™ iq and p ™ ip 3 . This replace-

Ž . Žment in our Hamiltonian, Eq. 1 together with.H™yH , leads to spin-generalizations of the sinh-

model. It is natural to conjecture that this veryreplacement allows to obtain the solution of the latterfrom our solution of the former model.

We would like to thank A. Polychronakos forw xpointing out Ref. 4 to us. His explanation of the

results of these papers helped us to resolve a diffi-culty which had prevented us from completing thiswork earlier. We also thank him for useful commentson the manuscript.

References

w x Ž . Ž .1 F. Calogero, Lett. Nuovo Cimento 13 1975 411; 16 1976Ž .77; J. Moser, Adv. Math. 16 1976 1.

w x Ž .2 F. Calogero, Jour. Math. Phys. 10 1969 2197; B. Suther-Ž . Ž .land, J. Math. Phys. 12 1970 246; Phys. Rev. A 4 1971

Ž .2019; A 5 1972 1372.w x Ž .3 M.A. Olshanetsky, A.M. Perelomov, Phys. Rep. 71 1981

313; see also A.M. Perelomov, Integrable Systems of Classi-cal Mechanics and Lie Algebras, vol. I, Birkhauser, 1990.¨

w x Ž .4 J. Gibbons, T. Hermsen, Physica D 11 1984 337; S.Ž .Wojciechowski, Phys. Lett. A 111 1985 101.

w x Ž .5 J.A. Minahan, A.P. Polychronakos, Phys. Lett. B 302 1993265.


w x Ž .6 A.P. Polychronakos, Phys. Lett. B 266 1991 29.w x Ž .7 A. Gorskii, N. Nekrasov, Nucl. Phys. B 414 1994 213; E.

Langmann, M. Salmhofer, A. Kovner, Mod. Phys. Lett. A 9Ž .1994 2913; A.J. Niemi, P. Pasanen, Phys. Lett. B 323Ž .1994 46; A.P. Polychronakos, J.A. Minahan, Phys. Lett. B

Ž .326 1994 288.w x Ž .8 F. Lenz, M. Thies, S. Levit, K. Yazaki, Ann. Phys. NY 208

Ž .1991 1; E. Langmann, G.W. Semenoff, Phys. Lett. B 296Ž .1992 117.

w x9 G. Grignani, L. Paniak, G.W. Semenoff, P. Sodano, Ann.Ž . Ž .Phys. NY 260 1997 275.

w x10 See e.g. E. Langmann, G.W. Semenoff, Phys. Lett. B 303Ž . Ž .1993 303; J.E. Hetrick, Nucl. Phys. Proc. Suppl. 228Ž .1993 30.

w x Ž .11 M. Blau, G. Thompson, Commun. Math. Phys. 171 1995639.

18 June 1998


Vacuum oscillations of solar neutrinos:correlation between spectrum distortion and seasonal variations

S.P. Mikheyev a,c, A.Yu. Smirnov b,c

a ( )Laboratory Nazionale del Gran Sasso dell’INFN, I-67010 Assergi L’Aquilla , Italyb International Center for Theoretical Physics, P.O. Box 586, 34100 Trieste, Italy

c Institute for Nuclear Research, Russian Academy of Sciences, 117312 Moscow, Russia

Received 23 October 1997; revised 27 January 1998Editor: R. Gatto

Abstract

The experimental signatures of vacuum oscillations solution of the solar neutrino problem are discussed. We show thatthere is a strict correlation between a distortion of the neutrino energy spectrum and an amplitude of seasonal variations ofthe neutrino flux. The slope parameter which characterizes a distortion of the recoil electron energy spectrum in theSuper-Kamiokande experiment and the seasonal asymmetry of the signal have been calculated in a wide range of oscillationparameters. The correlation of the slope and asymmetry gives crucial criteria for identification or exclusion of this solution.

Ž .For the positive slope indicated by preliminary Super-Kamiokande data we predict 40–60 % enhancement of the seasonalvariations. q 1998 Published by Elsevier Science B.V. All rights reserved.

1. Long length vacuum oscillations of neutrinoson the way from the Sun to the Earth are consideredas a viable solution of the solar neutrino problemw x Ž w x . 11–5 see 6–9 for latest analysis . There are two

Ž .key signatures of this solution: i distortion of thew x Ž .neutrino energy spectrum 4,5 , and ii seasonal

w xvariations of the flux 3 .Ž .i . The distortion follows from a dependence of

1 Large mixing angles implied by this solution are howeverw xdisfavored by the data from the SN87A 10 .

the oscillation survival probability, P, on the neu-trino energy E:

Dm2 L2 2Ps1ysin 2u sin . 1Ž .ž /4 E

Ž 2 . 2We consider mixing of two neutrinos . Here Dm

2 In the case of three neutrino mixing results are the same as inthe two neutrino case, provided the electron neutrino has small

Ž .admixture in the third the heaviest neutrino state. The n and nm t

are practically indistinguishable as far as the solar neutrinos areconcerned. Therefore results for the solar neutrinos are unchangedif n and n mix strongly as is required by the atmosphericm t

neutrino problem.


( )S.P. MikheyeÕ, A.Yu. SmirnoÕrPhysics Letters B 429 1998 343–348344

'm2 ym2 is the neutrino mass squared difference,2 1

u is the mixing angle and L is the distance betweenthe Sun and the Earth. A variety of distortions of theboron neutrino spectrum is expected depending onvalues of Dm2 and sin2 2u .

Ž .ii . Seasonal variations are stipulated by elliptic-ity of the Earth’s orbit. The flux of neutrinos at theEarth, F, can be written as

2L0FsF PP L, E , 2Ž . Ž .0 ž /L

where F is the flux at the astronomical unit L . The0 0

distance between the Sun and the Earth at a givenmoment t equals:

y12p tLfL 1qecos , 3Ž .0 ž /T

where es0.0167 is the eccentricity and T'1 year.The variations are expected both due to the geomet-rical factor, 1rL2, and due to change of the oscilla-

w xtion probability 3,5 . Depending on values of theoscillation parameters Dm2 and sin2 2u one may getan enhancement or damping of the geometrical effectw x11–13 or even more complicated variations of sig-nals.

In this Letter we point out that there is a strictcorrelation between a distortion of the boron neu-trino energy spectrum and seasonal variations of theboron neutrino flux. This gives crucial criteria foridentification or discrimination of vacuum oscilla-tions solution.

2. The correlation between the distortion and timevariations can be immediately seen from the expres-

Ž .sion for the oscillation probability. Indeed, Eq. 1gives

dP dP E PEsy P sys P , 4Ž .ndL dE L L

where

1 dPs E ' 5Ž . Ž .n P dE

is the slope parameter which characterizes the distor-tion of the neutrino spectrum at a given energy E. Ifspectrum distortion has a positive slope, s )0, thenn

Ž .according to Eq. 4 one gets dPrdL-0, i.e., withincrease of the distance the survival probability de-creases. The change of the flux is in the samedirection as due to the geometrical factor. Therefore,for a positive slope the vacuum oscillations enhance

Ž .seasonal variations. For a negative slope s -0 then

Ž .Eq. 4 gives dPrdL)0, that is, the probabilityincreases with distance. In this case the oscillationsweaken seasonal variations due to pure geometricalfactor.

Let us find the correlation explicitly. From Eqs.Ž . Ž .1 , 2 we get the total change of the neutrino fluxwith distance:

dF F dP Esy 2q P . 6Ž .ž /dL L dE P

Here the first term is due to the geometrical effectand the second one corresponds to a change of theprobability with distance. For a positive slope bothterms have the same sign in accordance with previ-ous discussion. Introducing the seasonal asymmetry:

DFA s2 , 7Ž .n F

where DF is the difference of the averaged fluxesduring the winter and the summer and F is theaveraged flux during the year, we can define thequantity

A yA0n n

r ' , 8Ž .n 0An

0 Ž .with A 'A Ps1 being the asymmetry withoutn n

oscillations. The asymmetry r describes variationsn

due to the oscillations in the units of pure geometri-Ž .cal effect. From 6 we get final relation between the

slope parameter, s , and the oscillation asymmetry,n

r :n

2s s r . 9Ž .n nE

Let us stress that this relation does not depend onoscillation parameters Dm2 and sin2 2u at least inthe lowest order on the eccentricity e .

Ž . Ž .3. The correlation described by 6 , 9 holds forfixed neutrino energy. In real experiments the energy

( )S.P. MikheyeÕ, A.Yu. SmirnoÕrPhysics Letters B 429 1998 343–348 345

Ž .spectrum of the recoil produced electrons is mea-sured, and moreover, the integration over certainenergy intervals of neutrinos as well as electronsŽ .due to finite energy resolution takes place. Thismodifies the correlation.

Let us consider a manifestation of the correlationw xin the Super-Kamiokande experiment 14,15 . For a

majority of relevant values of Dm2 and sin2 2u thewhole observable part of the boron neutrino spec-

Ž .trum 6–14 MeV is in regions with definite sign ofdPrdE. Integrations over the neutrino energy and onthe electron energy weighted by the energy resolu-tion function lead to a strong smoothing of theobservable distortion of the recoil electron spectrum.The distortion can be well characterized by a single

w xslope parameter s 16 defined ase

NoscfR qs T , 10Ž .0 e eN0

where N and N are the numbers of events withosc 0

and without oscillations correspondingly, R is a0

constant, T is the recoil electron energy in MeV, se e

is in the units MeVy1. There is some change of thespectrum distortion during the year due to variations

w xof distance L 11 . However, this effect is small andcan be neglected in the first approximation.

The slope parameter is determined in the follow-ing way:

Ž . 2 2i For different values sin 2u and Dm wecalculate expected numbers of events, N i , in theosc

energy bins with DT s0.5 MeV from 6.5 MeV toeŽ15 MeV as in the Super-Kamiokande presentation.of the data :

2L i0 T qDT X Xe ei iN L s dT dT f T ,TŽ . Ž .H Hosc e e e ež / iL Te

=

Xds E,TŽ .e e 2dEF E P E, L,Dm ,uŽ . Ž .H XX dTT ee

Xds E,TŽ .m e 2q 1yP E, L,Dm ,u , 11Ž . Ž .Ž .XdTe

Ž X.where f T ,T is the energy resolution function,e eŽ X. X Ž X. Xds E,T rdT and ds E,T rdT are the differen-e e e m e e

Ž . Ž y1 .Fig. 1. The dependence of a the slope parameter s in MeVeŽ . 2and b the signal asymmetry r on Dm for different values ofe

sin2 2u . Solid, dashed, dash-dotted, and dotted lines correspond tosin2 2u s 1.0, 0.75, 0.5, and 0.25 respectively.

tial cross-section of the n e- and n e- scatteringe m

correspondingly.Ž . iii Similar numbers of events, N , have been0

i iŽcalculated in absence of oscillations: N sN Ps0. i i1 . Then ratios N rN have been found for eachosc 0

bin.Ž . 2 i iiii The x fit of the histogram N rN by theosc 0

Ž . Ž .function 10 gives s Fig. 1a .e

We will characterize the seasonal variations bythe summer-winter asymmetry:

N yNW SA '2 . 12Ž .e N qNSP A

Here N , N , N , N are the numbers of eventsW S SP A

detected from November 20 to February 19, fromMay 22 to August 20, February 20 to May 21, fromAugust 21 to November 19 respectively:

N s dt N i L t V'W ,S,SP , A . 13Ž . Ž . Ž .Ž .ÝHV oscV i


Similar asymmetry is calculated due to geometrical0 Ž . 3factor in absence of oscillations: A sA Ps1 .e e

Ž Ž ..Finally, we introduce as in 8 the signal asym-metry

Aer ' y1 14Ž .e 0Ae

which characterizes the asymmetry due to oscilla-tions in the units of geometrical asymmetry. Positiver corresponds to an enhancement of the geometricale

effect.

4. The correlation between the spectrum distortionand seasonal variations appears as the correlationbetween the slope parameter s and the signal asym-e

Ž .metry r Fig. 1–4 .e

In Fig. 1 we show the dependence of the slopeand asymmetry on Dm2 for different values ofsin2 2u . Both the slope and the asymmetry increasewith mixing angle, whereas positions of maxima and

Ž 2 .zeros in Dm scale do not depend on u . Note thatzeros of the asymmetry are shifted with respect tozeros of the slope by approximately 8% towardsbigger values of Dm2. Maximal positive asymmetry

2 Žis also shifted towards bigger Dm about 25% for.the first maximum , however this difference dimin-

ishes for next maxima. In contrast, first negativemaximum of asymmetry is shifted towards smallerDm2. These features are related to the integrationover the neutrino and electron energies. The asym-metry due to oscillations can exceed the geometrical

Ž < < .asymmetry r )1 .eŽ .In the slope–asymmetry plot Fig. 2 the points

correspond to different values of Dm2 and sin2 2u .Scattering of the points is related to the integrationover the neutrino and electron energies. The solidline shows changes of the slope and asymmetry withDm2 for maximal mixing. One can understand a

3 Equivalently the seasonal asymmetry can be introduced byŽ . w xsplit of the data into two bins rather than four . In 12 the

Ž . Žnear-far asymmetry has been defined as A ' Ny F r NqNF.F , where N and F are the numbers of events detected during

Ž .‘‘near’’ half of the year centered at January 4th and ‘‘far’’ halfŽ .of year centered at July 4th correspondingly. Numerically, the

'Ž .asymmetry 12 is larger: A s2 2 A . However, statisticallye NF

both methods are equivalent, and equally sensitive to the seasonalvariations.

Ž .Fig. 2. The slope - asymmetry plot. The points s y r corre-e e2 Ž y1 1spond to different values of Dm 200 points between 10 and

y9 2 . 2 Ž .10 eV , and sin 2u 40 points between 0.025 and 1.0 .Calculations have been done on grid 200=40. The solid lineshows changes of the slope and asymmetry with Dm2 for maxi-mal mixing.

behavior of the curve using results of Fig. 1. For2 Ž y11 2 .very small Dm F10 eV a point is in the

center: s sr s0. With increase of Dm2 it de-e e

scribes the ‘‘petals’’ in the positive and negatives yr quarters. Different petals correspond to dif-e e

ferent peaks in the s and r dependences on Dm2e e

Ž .see Fig. 1 . Thus the narrow petal in the positivequarter corresponds to the first positive peak at

2 Ž . y11 2Dm s 4–6 P10 eV . Large negative petal is dueŽ 2 Ž .to the first negative peak Dm s 0.8–1.5 P

y10 2 .10 eV , Next wide petal in the positive quartercorresponds to the second positive peak Dm2 sŽ . y10 22.0–2.2 P10 eV , etc. The sizes of followingpetals decrease quickly due to the averaging effect inaccordance with Fig. 1. For non-maximal mixingsimilar petals are inside those described above andtheir sizes decrease with the mixing angle.

According to Fig. 2 the correlation between theslope and asymmetry can be approximated by

s s 0.03–0.10 MeVy1 r . 15Ž . Ž .e e

Note that for the neutrino spectrum the slope isŽ . y1 Ž .larger: according to 9 s s0.2 MeV r r ;rn n n e

for Es10 MeV.

( )S.P. MikheyeÕ, A.Yu. SmirnoÕrPhysics Letters B 429 1998 343–348 347

Ž . Ž . 2Iso-asymmetry a and iso-slope b lines in Dm ,sin2 2u plot are shown in Fig. 3. The iso-asymmetry

Ž .plot Fig. 3a is similar to the iso-plot of the near-farw xasymmetry found in 12 .

Preliminary Super-Kamiokande data for 306 daysw x Ž15 show a positiÕe slope: s ; 0.013–e

. y10.017 MeV . According to Figs. 1,3 for this slopeone expects positive asymmetry r s0.4–0.6 in thee

Ž 2 Žregion of the best fit of the integral data Dm s 5–. y11 2 .8 P10 eV . That is, an enhancement of the

Ž .seasonal variations by 40–60 % should be ob-served.

In Fig. 4 we show real time seasonal variations ofthe flux for different values of the neutrino parame-ters. The dashed-dotted line corresponds to very

. 2 2Fig. 3. a . Iso-asymmetry lines in Dm , sin 2u plot. Solid linescorrespond to positive asymmetries: r s 0.19, 0.38, 0.57, 0.76,e

Ž .0.95 from outer to inner lines . Dashed lines correspond tonegative asymmetries: y r s 0.28, 0.56, 0.84, 1.12, 1.4.e. 2 2b . Iso-slope lines in Dm , sin 2u plot. Solid lines correspond to

positive values of the slope parameter: s s 0.011, 0.022, 0.033,eŽ .0.044, 0.055 from the outer to inner lines . Dashed lines corre-

spond to negative slopes: y s s 0.013, 0.026, 0.039, 0.052,e

0.065.

Fig. 4. Real time seasonal variations of the flux for differentvalues of the neutrino parameters: Dm2 s5.82P10y11 eV 2,

2 Ž . 2 y10 2 2sin 2u s0.90 solid line ; Dm s1.06P10 eV , sin 2u sŽ . 2 y11 2 2 Ž0.95 dashed line ; Dm s7.85P10 eV , sin 2u s0.75 dash-

.dotted line . Dashed-dotted line coincides practically with the onedue to pure geometrical effect. Also shown are preliminary results

Ž . w xfrom the Super-Kamiokande 306 days experiment 14 .

weak variations due to oscillation effect and there-fore practically coincides with geometrical effect.Two other lines correspond to enhancement andsuppression of the geometrical effect by oscillations.Solid line shows variations expected from the ob-served slope. Also shown are the preliminary results

Ž .from the Super-Kamiokande 306 days experimentw x15 . Obviously, present statistics does not allow oneto make any conclusion. Significant result can beobtained after 4–5 years of the detector operation.

In larger interval of the recoil electron energiesŽ .from 5 MeV to 15 MeV which will be accessiblesoon a description of the distortions by only oneparameter may not be precise. Moreover, seasonalvariations differ at different energies. In this case one

Ž .can divide the interval into two parts: e.g., 5–8Ž .MeV and 8–15 MeV and study the correlations in

these intervals separately. A comparison of the ef-fects will give the cross-check of the results.

Similar correlation can be obtained for SNO ex-w xperiment 17 where the effect is expected to be even

more profound due to weaker averaging.


Acknowledgements

Authors are grateful to E. Kh. Akhmedov and E.Kearns for discussion.

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18 June 1998


Lattice gauge fixing, Gribov copies and BRST symmetry

M. Testa a,b

a Theory DiÕision, CERN, 1211 GeneÕa 23, Switzerland 1

b Dipartimento di Fisica, UniÕersita di Roma ‘‘La Sapienza’’, Sezione INFN di Roma, P.le A. Moro 2, 00185 Roma, Italy 2`

Received 30 March 1998Editor: L. Alvarez-Gaume

Abstract

We show that a modification of the BRST lattice quantization allows to circumvent an old paradox, formulated byNeuberger, related to lattice Gribov copies and non-perturbative BRST invariance. In the continuum limit the usual BRSTformulation is recovered. q 1998 Elsevier Science B.V. All rights reserved.

1. Introduction

Besides being a landmark in our understanding ofnon-perturbative hadron dynamics, the lattice regu-

w xlarization of gauge theories 1 is an important labo-ratory where formal field theoretical ideas can becross-checked for their validity or possible flawsmay be identified.

One of these ideas is the gauge fixing. In fact weare in the fortunate position that lattice regularizationdoes not require any gauge-fixing, so that the stepstowards a gauge-fixed version of the theory arecompletely non formal. In particular a non formalmeaning could be attached to the Faddeev-Popov

w xprocedure 2 , so essential for many applications.The Faddeev-Popov procedure has afterwards beenreplaced by the more formal apparatus of the so-

w xcalled BRST symmetry 3 . Despite these importantdevelopments, some problems are still open. It is in

w xfact known 4,5 that compact gauge fixing is af-

1 Address until August 31st, 1998.2 Permanent address.

fected by the presence of Gribov copies and one istherefore faced to the problem of dealing, in thefunctional integral, with several gauge equivalentcopies of the same configuration. An appealing solu-

w xtion 6 , naturally suggested by the BRST formula-tion, is that the various Gribov copies, weighted bythe Faddeev-Popov determinant, contribute to thefunctional integral with alternating signs. This opensthe way to a cancellation in which, finally, only thecontribution from a single copy should survive. It

w xwas shown by Neuberger 7 , on very generalgrounds, that a cancellation takes place indeed, withthe disastrous result of leaving physical observables

0in the embarrassing indeterminate form . The situa-0

tion becomes even more confused when, as it hap-pens for chiral gauge theories, a gauge invariantdiscretization is missing. One possible scheme for

w xtheir quantization, known as the Rome approach 8 ,requires gauge-fixing and BRST quantization as fun-damental ingredients. While the validity of this ap-proach can be checked within perturbation theory,

w xmost of the objections raised against it 9 concernthe lack of control of the gauge fixing procedure at


( )M. TestarPhysics Letters B 429 1998 349–353350

the non-perturbative level, although Neuberger’s ar-gument is not directly relevant in this case.

In this paper we will show how the paradoxw xpresented in Ref. 7 can be avoided, through a slight

modification of the BRST scheme, in the case oftheories admitting a gauge invariant discretization.

In Section 2 we will review the Neuberger’sargument. In Section 3 we will clarify its Gribov-likenature through a very simplified one dimensionalintegral. In Section 4 we will discuss the modifica-tion required in order to obtain a consistent BRSTquantization.

2. The paradox

w xFollowing Ref. 7 we consider a discretizedgauge-invariant theory in a finite volume, so that thefunctional integral reduces to a finite-dimensionalone. The expectation value of any gauge invariant

Ž .operator OO U is given by:

1yS ŽU . ySŽU .² :OO s DDUe OO U , ZZ' DDUeŽ .H H

ZZ

1Ž .

where DDU denotes the group-invariant integrationŽ .measure over links and S U is the gauge invariant

euclidean action.The process of gauge-fixing requires, first of all,

the choice of a gauge condition, i.e. a gauge nonŽ .invariant function, f U, x , and the introduction of

auxiliary degrees of freedom: the grassmannian ghostand anti-ghost variables, c and c, and the Lagrangemultipliers l, to be integrated over the whole realaxis.

After gauge-fixing, the expectation value in Eq.Ž .1 , is replaced by:

a1 2yS ŽU . y lH² :OO s DDm e e exp d fc OO UŽ .2H HX ž /ZZ

a2X ySŽU . y lHZZ ' DDm e e exp d fc 2Ž .2H Hž /

where

DDm'DDU dl dc dc 3Ž .

Ž 2 .and d denotes the nilpotent d s0 BRST transfor-mation which acts on the links U as an infinitesimalgauge transformation with parameter c and:

1dcs il , dcs cc , dls0 4Ž .

2

Ž .In Eq. 2 both the measure and the integrand are² :BRST invariant. The expectation value OO com-

Ž .puted through Eq. 2 is formally equal to the oneŽ .computed through Eq. 1 .

However let us examine the quantity:

a2yS ŽU . y lHF t ' DDm e e exp t d fc OO UŽ . Ž .2H HOO ž /

5Ž .

Ž .which coincides with the numerator in Eq. 2 , whenthe real parameter t is equal to 1. We have:

dF tŽ .OO

dt

a2yS ŽU . y lHs DDm d fc e e exp t d fc OO UŽ .2H H Hž /

a2yS ŽU . y lHs DDm d fc e e exp t d fc OO UŽ .2H H Hž /

s0 6Ž .

because the integral of a total BRST variation van-ishes identically. On the other hand we also have:

a2yS ŽU . y lHF 0 s DDm e e OO U s0 7Ž . Ž . Ž .2HOO

according to Berezin rules of grassmannian integra-tion, because there are no ghost or anti-ghost vari-

Ž . Ž . Ž .ables in the integrand of Eq. 7 . Eqs. 6 and 7imply:

F 1 s0 8Ž . Ž .OO

Similar considerations show that:

ZZX s0 9Ž .

0Ž .thus reducing Eq. 2 to an indeterminate form .

0We stress that these manipulations are completely

justified by the finite dimensionality of the regular-ized functional integral.

( )M. TestarPhysics Letters B 429 1998 349–353 351

3. The nature of the paradox

In order to visualize where the problem comesŽ .from, we consider a toy abelian model with only

one degree of freedom, one ‘‘link’’ variable U,which we choose to parametrize through its phase,as:

Useia A 10Ž .Ž .In Eq. 11 , A is a variable with values ranging

pbetween " and a is a parameter, reminiscent ofa

the lattice spacing, in more realistic situations, whoselimit a™0 will be used to connect the periodic,compact case to the non compact one. The integralwe consider is the trivial one:

p

aNN' dA 11Ž .H py

a

‘‘Gauge fixing’’ proceeds exactly as in Section 2: weŽ .choose a periodic function f A and consider:

paq` 2X a y lNN s dA dl dc dc e exp d cf AŽ .Ž .2H H Hp

y yà

12Ž .Ž 2 .where d is the BRST variation d s0 , defined by:

d Asc , dcs il , dcs0 , dls0 13Ž .dcs0 is of course peculiar to the present abeliansituation.

Ž . Ž .Eqs. 12 and 13 give:

paq` X2X a y l i l f Ž A. yc f Ž A.cNN s dA dl dc dc e e e2H H Hp

y yà

paq` 2 Xa y l i l f Ž A.s dA dl e e f AŽ .2H Hp

y yà

p2Ž .f A2p

a ys df A e s0 14Ž . Ž .2 a( H pa y

a

Ž .for a periodic f A .Ž .Why must we choose a periodic f A ?

Because we want the expectation value of a BRSTvariation to vanish identically, as a consequence ofBRST invariance.

Let us choose, for instance:X

G'd cF A s ilF A ycF A c 15Ž . Ž . Ž . Ž .A simple computation gives:

² :G

paq` X2a y l i l f Ž A. yc f Ž A.c' dA dl dc dc e e e G2H H Hp

y yà

p2Ž .f A2p d

a ys dA F A e 16Ž . Ž .2 a( H p ž /a dAy

a

Ž . ² : Ž .Eq. 16 shows that G is zero, only if both f AŽ .and F A are periodic functions.

Ž .Choosing as0 in Eq. 14 , we get:

p

X XaNN s2p dA f A d f A 17Ž . Ž . Ž .Ž .H py

a

which discloses the Gribov-like nature of the zeroŽ . Ž . Xobtained in Eq. 14 : Eq. 17 tells us that NN gets

Ž .contributions only from the zeros of f A . Each zerocontributes alternatively a factor "1 and the period-

Ž .icity of f A implies an even number of zeros.

4. The solution

There is, however, a way out which exploits thecompact nature of the variable A. The basic idea issimple: it amounts to change the BRST formulation,

Ž .so that in Eq. 17 the periodic d-function appears.As we will show in a moment, this allows the choice

Ž .of a function f A periodic up to a shift, i.e.:

2p 2p kf Aq s f A q 18Ž . Ž .ž /a a

with an integer k, still leaving a periodic integrand inŽ .Eq. 17 . In this way the number of zeroes is odd and

the Gribov problem is overcome.To be more precise, let us start recalling some

Ž . Ž .properties of the periodic d-function, d x . d xP Pw xis defined as 10 :

` 2pd x ' d xyn 19Ž . Ž .ÝP ž /ansy`

( )M. TestarPhysics Letters B 429 1998 349–353352

and may also be represented as:`a

ina xd x s e 20Ž . Ž .ÝP 2p nsy`

The modification I propose, is to allow the vari-Ž .able l in Eq. 12 to assume discrete values only:

l 'an 21Ž .n

Ž .with integer n, and replace Eq. 12 by:

pq` a2XX a y lnNN sa dA dc dc e exp d cf AŽ .Ž .2Ý H Hp

ynsy` a

22Ž .

The BRST transformations have the same form asŽ .those defined in Eq. 13 , only with l replaced by

Ž .l . The same steps leading to Eq. 14 give now:n

pq` a2XX Xa y l i l f Ž A.n nNN sa dA e e f A 23Ž . Ž .2Ý H p

ynsy` a

Ž .We can now choose a function f A obeying theŽ . Ž .condition stated in Eq. 18 : while f A itself cannot

be expressed globally in terms of the link U, definedŽ . i ln f Ž A. XŽ .in Eq. 10 , both e and f A can, being

strictly periodic functions of A. We can easily checkthat the consequences expected from BRST invari-

Ž .ance are satisfied by the new definition, Eq. 23 . Inparticular the expectation value of G , defined in Eq.Ž . Ž .15 , still vanishes for a periodic F A . However, in

XX w xgeneral, NN /0: the argument of Ref. 7 , reviewedin Section 2, fails to apply in this case because,while ei ln f Ž A. is a periodic function, eitln f Ž A. is notperiodic, except for integer values of t.

It is worth noticing that, in the ‘‘continuum limit’’,Ž .a™0, Eq. 23 converges to the usual BRST expres-

sion:

aq` q` 2XX Xy l i l f Ž A.lim NN s dA dl e e f AŽ . .2H Ha™0 y` y`

24Ž .

thus recovering a continuous valued l variable.The above arguments seem to translate directly

into the much more complicated case of lattice gaugetheories where we have a set of Lagrangian multipli-

Ž .ers, l x , for each space-time point x. The basicmove is the discretization of the range of the La-

Ž .grange multipliers, l x , which allows the introduc-n

tion of ‘‘twisting’’ gauge fixings, as exemplified inŽ .Eq. 18 . We stress again that this amounts to a non

Ž .conventional choice of the gauge condition, f A, x ,which should not be expressible in terms of the

Ž . i a A Ž x .lin k v a riab le s , U x s e , w h ileexp iHd4 xl x f A , x should.Ž . Ž .Ž .n

In realistic cases, of course, it is not easy toidentify gauge-fixing conditions apt to avoid the

Ž . Ž .vanishing results implied by Eqs. 8 and 9 : it is,however, most encouraging that, in presence of dis-

Ž . Ž .cretized l ’s, Eqs. 8 and 9 cannot be shown to ben

valid any more.

5. Conclusions

We have proposed a possible way out to an oldparadox concerning non-perturbative BRST symme-try in lattice regularized gauge theories. The solutionconsists in changing the BRST formulation by allow-ing the Lagrange multipliers, which enforce the gaugecondition, to range over discrete values only. Thismodification gives more freedom in the choice of thegauge condition, allowing, in simple examples, tocircumvent the paradox 3. Even more convincing isthe fact that, within this modified BRST formulation,the steps leading to the paradox cannot be imple-mented.

In the continuum limit, the modified BRST for-mulation naturally converges towards the conven-tional continuum one, in the sense that the dis-cretized Lagrange multipliers become continuousvariables as a™0.

It is hard to think that the modified BRST willhave any impact on actual numerical simulations,because its implementation seems rather involved.To this we must add that Neuberger’s argument,although relevant to gauge-fixings used in practice,was always ignored by the practitioners in the field.

Are there any relevant consequences for the Romeapproach? I don’t think so. In the Rome approachone starts from a discretized theory which is not

3 The possibility of a multivalued gauge-fixing condition isw xhinted in the concluding remarks of Ref. 7 , but it is not pursued

any further.

( )M. TestarPhysics Letters B 429 1998 349–353 353

gauge invariant and tries to merge it, as a™0, intoŽa continuum gauge fixed, gauge invariant i.e. satis-

.fying the continuum BRST identities theory. Asstressed several times, within the Rome approachone does not deal with regularized, exactly gauge-in-variant systems.

Acknowledgements

I thank the CERN Theory Division for the kindhospitality. I also thank G. C. Rossi for discussions.

References

w x Ž .1 K. Wilson, Phys. Rev. D 14 1974 2455.w x Ž .2 L.D. Faddeev, V.N. Popov, Phys. Lett. B 25 1967 29.

w x Ž .3 C. Becchi, A. Rouet, R. Stora, Phys. Lett. B 52 1974 344;I.V. Tyupkin, Gauge invariance in field theory and statisticalphysics in operatorial formulation, preprint of LebedevPhysics Institute n. 39, 1975.

w x Ž .4 V.N. Gribov, Nucl. Phys. B 139 1978 1.w x Ž .5 I.M. Singer, Comm. Math. Phys. 60 1978 7.w x Ž .6 P. Hirschfeld, Nucl. Phys. B 157 1979 37; B. Sharpe, J.

Ž .Math. Phys. 25 1984 3324.w x Ž .7 H. Neuberger, Phys. Lett. B 183 1987 337.w x8 A. Borrelli, L. Maiani, G.C. Rossi, R. Sisto, M. Testa, Nucl.

Ž .Phys. B 333 1990 335; A. Borrelli, L. Maiani, G.C. Rossi,Ž .R. Sisto, M. Testa, Phys. Lett. B 221 1989 360; L. Maiani,

Ž .G.C. Rossi, M. Testa, Phys. Lett. B 261 1991 479; B 292Ž .1992 397; M. Testa, The Rome Approach to Chirality,invited talk at APCTP - ICTP Joint International ConferenceŽ .AIJIC 97 on Recent Developments in NonperturbativeQuantum Field Theory, Seoul, Korea, 26-30 May 1997,hep-latr9707007.

w x9 H. Neuberger, Remarks on Lattice BRST Invariance, hep-latr9801029.

w x10 M.J. Lighthill, Introduction to Fourier analysis and gener-alised functions, Cambridge Univ. Press, 1958.

18 June 1998


q yQCD corrections to e e ™u d s cat Lep 2 and the Next Linear Collider:

ž / 1CC11 at OO a s

Ezio Maina a,2, Roberto Pittau b,3, Marco Pizzio a,4

a Dip. di Fisica Teorica, UniÕersita di Torino and INFN, Sezione di Torino, Õ. Giuria 1, 10125 Torino, Italy`b Theoretical Physics DiÕision, CERN CH-1211 GeneÕa 23, Switzerland


Abstract

QCD one-loop corrections to the full gauge invariant set of electroweak diagrams describing the hadronic processq ye e ™u d s c are computed. Four-jet shape variables for WW events are studied at next-to-leading order and the effects of

QCD corrections on the determination of the W-mass in the hadronic channel at Lep 2 and NLC is discussed. We comparethe exact calculation with a ‘‘naive’’approach to strong radiative corrections which has been widely used in the literature.q 1998 Elsevier Science B.V. All rights reserved.

1. Introduction

The measurement of W-mass to high precision isone of the main goals of Lep 2 and will provide a

Ž . w xstringent test of the Standard Model SM 1,2 . Infact, the mass of the W boson in the SM is tightlyconstrained and an indirect determination of M canW

be obtained from a global fit of all electroweak data.w x q0.009The fit gives 3 M s80.338"0.040 GeVW y0.018

where the central value corresponds to M s300H

1 Work supported in part by Ministero dell’ Universita e della`Ricerca Scientifica.

2 E-mail: [email protected] E-mail: [email protected] E-mail: [email protected].

GeV and the second error reflects the change of MW

when the Higgs mass is varied between 60 and 1000GeV. A disagreement between the value of MW

derived from the global fit and the value extractedfrom direct measurement would represent a majorfailure of the SM. Alternatively, an improvement inthe value of the W-mass can significantly tightenpresent bounds on the Higgs mass. These studies willbe continued and improved at the Next Linear Col-

Ž .lider NLC which is expected to reduce the error onthe measurement of W-mass down to about 15 MeVw x4 .

In order to extract the desired information fromWW production data, theoretical predictions withuncertainties smaller than those which are foreseenin the experiments are necessary. This requires a


( )E. Maina et al.rPhysics Letters B 429 1998 354–362 355

careful study of all radiative corrections which haveto be brought under control. In this paper we will beconcerned with QCD corrections which are knownw x5,6 to modify the shape of the W-mass peak and thedistributions of others kinematical variables. Sinceevent shape variables are often used to extract WWproduction from the background it is highly desirable

Ž .to have a complete next-to-leading order NLOstudy of these distributions for WW events. Further-

Ž 2 2 .more, calculations of QCD corrections to OO a asw xfour-jet production have recently appeared 7,8 .

Combining these results with a complete calculationŽ 4.of QCD corrections to all OO a four-fermion pro-

cesses it would be possible to obtain NLO predic-tions for any four-jet shape variable at Lep 2 andNLC providing new means of testing perturbativeQCD.

W-pair production can result in a variety of four-fermion hadronic final states. The simplest one is

q yu d s c. A gauge invariant description of e e ™

u d s c requires in the unitary gauge the eleven dia-grams shown in Fig. 1. This amplitude is known asCC11 in the literature. The subset of three diagrams

Ž . Ž .labeled e and f in Fig. 1, in which both intermedi-ate W ’s can go on mass-shell, is known as CC03 andis often used for quick estimates of WW production.Numerically the total cross sections obtained fromCC03 and those obtained from CC11 at Lep 2 and

w xNLC energies differ by a few per mil 9 . Other finalŽ .states to which W-pairs can decay are u u d d c c s s

Žand, through CKM mixing, u u s s u u b b, c c d d,.c c b b . The additional diagrams which appear in

these latter sets contribute at most at the per-millevel to the total cross section and tend to produceevents which resemble very little WW or even sin-gle-W events since only electroweak neutral virtualvector bosons appear in them. There are also four-quark final states where only neutral intermediatevector boson, including gluons, play a role like

Ž . Ž .u u c c, d d s s d d b b, s s b b , u u u u c c c c andŽ .d d d d s s s s, b b b b . An additional important con-

tribution to four-jet production, indistinguishablefrom four-quark events, originates from eqey™

q q g g.In this letter we present the complete calculation

of QCD corrections to CC11. While this is only aŽ 4.first step in the calculation of all OO a four-ferm-

ion processes at NLO, this reaction includes the mostimportant source of background, namely single Wproduction, providing a natural setting for a firststudy of the role of QCD corrections to gauge invari-ant sets of four-quark production diagrams.

In most instances QCD corrections have beenincluded ‘‘naively’’ with the substitution G ™W

Ž .G 1q2r3 a rp and multiplying the hadronicW sŽ .branching ratio by 1qa rp . This prescription iss

q yFig. 1. Tree level diagrams for e e ™u d s c. The dashed lines are W ’s.

( )E. Maina et al.rPhysics Letters B 429 1998 354–362356

exact for CC03 when fully inclusive quantities arecomputed. However, it can only be taken as an orderof magnitude estimate even for CC03 in the presenceof cuts on the jet directions and properties, as dis-

w xcussed in 5 . It is well known that differentialdistributions can be more sensitive to higher ordercorrections than total cross-sections in which virtualand real contributions tend to cancel to a largedegree. It is therefore necessary to include higherorder QCD effects into the predictions for WW pro-duction and decay in a way which allows to imposerealistic cuts on the structure of the observed events.The impact of QCD corrections on the angular distri-bution of the decay products of a W and theirapplication to on-shell W-pair production is dis-

w xcussed in Ref. 10 .

2. Calculation

One-loop virtual QCD corrections to eqey™

u d s c are obtained by dressing all diagrams in Fig. 1with gluon loops. Defining suitable combinations ofdiagrams one can organize all contributions in a very

w xmodular way 6 . All QCD virtual corrections toŽ 4.OO a four-fermion processes can be computed us-

ing the resulting set of loop diagrams, see Fig. 1.q yThe real emission contribution for e e ™u d s c

can be obtained attaching a gluon to the quark linesof the diagrams shown in Fig. 1 in all possiblepositions. This results in fifty-two diagrams. Therequired matrix element has been computed using

w xthe formalism presented in Ref. 11 with the help ofŽ . w xa set of routines PHACT 12 which generate the

building blocks of the helicity amplitudes semi-auto-matically.

The calculation of the virtual corrections has beenperformed in two different ways, with identical re-sults. In the first case we have used the standard

w xPassarino–Veltman 13 reduction procedure, whilein the second we have used the new techniques

w xpresented in 14 . The matching between real andw xvirtual corrections has been implemented as in 6 ,

w xusing the dipole formulæ of Ref. 15 . All integra-tions have been carried out using the Monte Carlo

w xroutine VEGAS 16 .An important ingredient for accurate predictions

of W-pair production is the effect of initial state

Ž .radiation ISR . In the absence of a calculation of allŽ .OO a corrections to four-fermion processes, these

effects can only be included partially. In contrastwith Lep 1 physics a gauge invariant separation ofinitial and final state radiation is not possible. Onlythe leading logarithmic part of ISR is gauge invariantand universal. These contributions can be includedusing structure functions. Part of the non-logarithmicterms have been computed for CC03 and some other

w xfinal states in 17 using the current splitting tech-w xnique 18 . This corresponds to splitting the electri-

cally neutral t-channel neutrino flow into two oppo-sitely flowing charges, assigning the y1 charge tothe initial state and the q1 charge to the final state.In this way a gauge-invariant definition of ISR canbe given. However, there are clearly cancellationsbetween initial and final state radiation whose rele-vance is difficult to estimate in the absence of acomplete calculation. Quite recently the full calcula-

q y qŽ .tion of all OO a corrections to e e ™W mn hasm

w xbeen published 20 . Non factorizable QED correc-w xtions to CC03 have been studied in 21 . We have

only included the leading logarithmic part of ISRusing the b prescription in the structure functions,

Ž 2 .where b s ln srm y 1. Beamstrahlung effectshave been ignored. We have not included Coulombcorrections to CC03, which are known to have asizable effect, particularly at threshold. They couldhowever be introduced with minimal effort.

3. Results

In this section we present a number of crossq ysections and distributions for e e ™u d s c. We

have used a s .123 at all energies. ISR is includeds

in all results. The width of the W-boson is kept fixedŽ .and includes OO a corrections.s

w xFor the Lep 2 workshop 1 the so calledADLOrTH set of cuts have been agreed on:Ø the energy of a jet must be greater than 3 GeV;Ø two jets are resolved if their invariant mass is

larger than 5 GeV;Ø jets can be detected in the whole solid angle.

For the NLC a slightly different set calledNLCrTH has been chosen. The NLCrTH set ofcuts differs from the ADLOrTH set in that a mini-mum angle of 58 is required between a jet and either


Žbeam as appropriate for the larger bunch size at theNLC and the corresponding larger bunch disruption

.at crossing and that two jets are resolved if theirinvariant mass is larger than 10 GeV. Both set ofcuts will also be referred to as ‘‘canonical’’ in thefollowing. We have preferred a different criterion fordefining jets which is closer to the actual practice ofthe experimental collaborations. For mass reconstruc-

w xtion studies we have used the Durham scheme 19 ,with y s1.=10y2 at Lep 2 energies. At the NLCD

we have adopted a smaller cut y s1.=10y3 inD

order to have an adequate fraction of events with atleast four jets. The four-momenta of the particleswhich have to be recombined have been simplysummed. If any surviving jet had an energy smallerthan 3 GeV it was merged with the jet closest in theDurham metric.

w xPrevious studies 9 have shown that the differ-ences between the total cross sections obtained fromCC11 and those obtained with CC03 are at the permil level. Much larger effects have been found inobservables like the average shift of the mass recon-structed from the decay products from the true W-mass.

In Fig. 2 we compare the NLO spectrum of theaverage reconstructed W-mass with the naive-QCDŽ .nQCD result at Lep 2 energies. All events with at

least four observed jets have been retained in Fig. 2.The two candidate masses are obtained forcing firstall remaining five-jet events to four jets, merging thetwo partons which are closest in the Durham scheme,and then selecting the two pairs which minimize

2 2XD s M yM q M yM . 1Ž . Ž . Ž .M R1 W R2 W

where M and M are the two candidate recon-R1 R2

structed masses and M is the input W-mass.W

In Fig. 2 the dashed line refers to the nQCDresults while those of the full NLO calculation aregiven by the solid line. The corresponding crosssection are given in Table 1. Fig. 2a is obtainedusing only the basic set of cuts described above.Since all experiments restrict their analysis to aregion around the expected W-mass, we have studiedthe effect of requiring that both reconstructed masseslie within 10 GeV of the input mass. The result isshown in Fig. 2b. Finally we have tried to take intoaccount some form of experimental smearing, inorder to determine whether the distortion of the massdistribution we observe does survive in a more real-istic setting. To this aim we have smeared the recon-

Ž .structed masses entering Eq. 1 , using a gaussiandistribution with a 2 GeV width. This proceduregives Fig. 2c. It is worth mentioning that simulatingexperimental smearing at NLO is far more compli-

'Fig. 2. Average mass distribution at s s175 GeV. All ADLOrTH cuts are applied. The continuous histogram is the exact NLO resultwhile the dashed histogram refers to nQCD. The corresponding cross sections can be found in Table 1.


Table 1'Cross sections in pb at s s175 GeV

E )3 GeV, y s0.01 E )3 GeV, y s0.01, E )3 GeV, y s0.01,j D j D j D

NM yM N-10 GeV NM yM N-10 GeV,R i W R i W

smeared

Ž . Ž . Ž .NLO 1.1493 4 0.7895 5 0.7758 9Ž . Ž . Ž .nQCD 1.1069 3 1.0545 3 1.0479 3

cated than at tree level. In order to preserve thedelicate cancellations between the real emission crosssection and the subtraction terms, both contributionsto four-jet quantities must be smeared, event byevent, by the same amount. Fig. 2 shows that atNLO the mass distribution is shifted towards lowermasses and a long tail for rather small averagemasses is generated, with a corresponding reductionof the high-mass part of the histogram. This tail is

eliminated when only reconstructed masses in thevicinity of the expected W-mass are retained. Evenwith this additional cut, however, the NLO distribu-tion is clearly different from the nQCD one. The

Ž .reduction in cross section see Table 1 shows that alarge number of soft gluons is exchanged, at theperturbative level, between decay products of differ-ent W ’s. In our simplified treatment of experimentaluncertainties these differences are still visible though

'Fig. 3. Average mass distribution at s s500 GeV. All NLCrTH cuts are applied. The continuous histogram is the exact NLO result whilethe dashed histogram refers to nQCD.


somewhat reduced. If we try to quantify the massshift using the standard quantity:

1 M qM y2 MR1 R2 W² :DM s ds . 2Ž .H ž /s 2

² : Ž .we obtain D M s y0.229 1 GeV andNLO

² : Ž .D M sy0.0635 4 GeV when both recon-nQCD

structed masses are required to lie within 10 GeV ofthe input mass M s80.23 GeV and no smearing isW

applied.In Fig. 3 we compare the NLO spectrum of the

average reconstructed W-mass with the naive-QCDŽ .nQCD result at the NLC, using the procedurealready described for the Lep 2 case but for a smallerminimum Durham cut y s1.=10y3. Because ofD

the larger relative momentum of the two W ’s it isless likely that partons from the decay of one W endup close to the decay products of the other W-boson,

therefore the difference between the two distribu-tions is smaller than at Lep 2 energies.

'In Fig. 4 we present the distributions at s s175w xGeV of the following four-jet shape variables 22 :

wŽØ the Bengtsson-Zerwas angle: x s / p =BZ 1. Ž .x Ž .p , p =p Fig. 4a ;2 3 4

Ø the Korner–Schierholz–Willrodt angle: F s¨ KSW� wŽ . Ž .x wŽ .1r2 / p =p , p =p q/ p =p ,1 4 2 3 1 3

Ž .x4 Ž .p =p Fig. 4b ;2 4

Ø the angle between the two least energetic jets;w x Ž .a s/ p , p Fig. 4c ;34 3 4

Ž . )Ø the modified Nachtmann–Reiter angle: u sNRwŽ . Ž .x Ž ./ p yp , p yp Fig. 4d .1 2 3 4

The numbering is1 . . . ,4 of the jet momenta pi

corresponds to energy-ordered four-jet configura-Ž .tions E )E )E )E . We compare the exact1 2 3 4

NLO results with the distributions obtained in nQCDand with the results obtained from the standard

Ž . Ž . Ž . Ž .Fig. 4. a , b The full NLO results solid line is compared with the nQCD prediction dashed line and with the tree level background'Ž . Ž . Ž . Ž .distributions from q q g g chain-dotted line and q q q q dotted line . c , d Four-jet shape variables at s s175 GeV.1 1 2 2


q ybackground reactions e e ™q q g g, which is theq ydominant contribution, and e e ™q q q q .1 1 2 2

It should be explicitly mentioned that the NLOŽ 2 3.OO a a QCD corrections for the above backgrounds

distributions are known and shown to be relevant inw xthe measurement of the QCD color charge 8 . How-

ever, for the sake of simplicity, we decided to com-pare our results with the background at the tree level.

In all subplots of Fig. 4 the full NLO results aregiven by the solid line and the nQCD prediction isgiven by the dashed line. The q q g g and q q q q1 1 2 2

tree level background distributions are given by thechain-dotted and the dotted line respectively. Theshape variables are computed following the proce-

w xdure outlined in Ref. 23 where the Durham clusteralgorithm is complemented by the E0 recombinationscheme, namely if the two particles i and j are

merged the pseudo-particle which takes their placeremains massless, with four-momentum:

E qEi jE sE qE , p s p qp . 3Ž . Ž .new i j new i j< <p qpi j

Using this clustering procedure all five-jet events areconverted into four-jet events, then each event isused in the analysis if min y )y with yi, js1,4 i j cut cut

s0.008.w xIn 23 it has been shown that standard parton

shower Monte Carlo programs like JETSET do notreproduce well the observed distribution of four-jetshape variables at Lep, which are instead well de-scribed by HERWIG 5.9A which includes four par-ton matrix elements. The discrepancy can be par-tially explained by the different distributions for the

'Ž . Ž . Ž . ŽFig. 5. a , b Four-jet shape variables at s s500 GeV. The full NLO results solid line is compared with the nQCD prediction dashed. Ž . Ž .line and with the tree level background distributions from q q g g chain-dotted line and q q q q dotted line .1 1 2 2


q q g g and the q q q q final states, which can be1 1 2 2

clearly seen in Fig. 4, since the four quark final stateis included in the parton shower programs onlypartially. It is precisely in correspondence with thepeaks of the q q q q distributions that the dis-1 1 2 2

crepancy between data and simulations is larger.From Fig. 4 it is apparent that at NLO four-jet

shape variables distributions are significantly modi-fied with respect to leading order results, which areindistinguishable from the nQCD distributions. It isalso evident that four-jet distributions in WW eventsare markedly different from the background distribu-tions and can be useful in separating the two sam-ples. At Lep 2 energies the Korner–Schierholz–Wil-¨lrodt angle F seems to be the most effectiveKSW

variable for this purpose, while the angle betweenthe two least energetic jets a is of little use, being34

almost flat over the whole range for all samples.From our results, namely if we assume that the treelevel background distributions closely resemble theactual behaviour of the background 5, it appears thatthe differences between CC11 distributions and thedominant q q g g background decrease when NLOcorrections to CC11 are included.

It should be stressed that four-jet shape variablesin WW events measure the correlations between thehadronic decays of the two W ’s and therefore itshould be explicitly checked whether existing codes,NLO calculations or parton shower Monte Carloprograms, successfully reproduce the experimentalcurves.

'The shape-variable distributions at s s500 GeVare given in Fig. 5. The only difference with respectto the Lep 2 analysis is a smaller value for the jetseparation parameter y s0.001. In particular nocut

minimum angle between jets and either beam isrequired. A separation based on shape-variables ofWW events from the background seems, at firstsight, to be more difficult at the NLC than at Lep 2.Only for the angle between the two least energeticjets a the signal and background distribution are34

significantly different. The former peaks in the back-ward direction the latter is almost flat. On the con-trary, the Bengtsson-Zerwas angle and the Korner–¨

5 w xNagy and Trocsanyi 8 , however, find large corrections, of´ ´the order of 100% for the D parameter and for the acoplanarity.

Schierholz–Willrodt angle distribution from CC11are almost indistinguishable from those generatedfrom the q q g g background. The sensitivity of theNachtmann–Reiter angle is similar at the two ener-gies. The signal distribution peaks at small angles,particularly at higher energies, while the backgroundis flatter, with a large tail which extends to 1808. Thedistributions obtained in nQCD are closer to the fullNLO results than at lower energies.

4. Conclusions

We have described the complete calculation ofQCD radiative corrections to the process eqey™

u d s c which are essential in order to obtain theoreti-cal predictions for W-pair production with per milaccuracy. The amplitudes we have derived are com-pletely differential, and realistic cuts can be imposedon the parton level structure of the observed events.We have presented the distribution of the averagereconstructed W-mass and the distribution of severalfour-jet shape variables at Lep 2 and NLC energies.The so called naive-QCD implementation of NLOcorrections fails in both instances.

References

w x1 The most complete review of W-pair production at Lep 2 canbe found in the Proceedings of the Workshop on Physics at

Ž .Lep 2, G. Altarelli, T. Sjostrand, F. Zwirner Eds. , Cern¨96-01.

w x2 A. Ballestrero et al., Determination of the mass of the Ww xboson, in Ref. 1 , vol. 1, p. 141.

w x3 The LEP Electroweak Working Group and the SLD HeavyFlavour Group, LEPEWWGr96-02.

w x4 E. Accomando et al., DESY 97-100, hep-phr9705442.w x Ž .5 E. Maina, M. Pizzio, Phys. Lett. B 369 1996 341.w x Ž .6 E. Maina, R. Pittau, M. Pizzio, Phys. Lett. B 393 1997 445.w x7 J.M. Campbell, E.W.N. Glover, D.J. Miller, Phys. Lett. B

Ž .409 1997 503; E.W.N. Glover, D.J. Miller, Phys. Lett. BŽ .396 1997 257; Z. Bern, L. Dixon, D.A. Kosower, S.

Ž .Weinzierl, Nucl. Phys. B 489 1997 3; A. Signer, L. Dixon,Ž . Ž .Phys. Rev. Lett. 78 1997 811; Phys. Rev. D 56 1997

4031; Z. Bern, L. Dixon, D.A. Kosower, hep-phr9708239.w x Ž .8 Z. Nagy, Z. Trocsanyi, Phys. Rev. Lett. 79 1997 3604,´ ´

hep-phr9708344 and hep-phr9712385.w x9 E. Accomando et al., Event generators for WW physics, in

w xRef. 1 , vol. 2, p. 3.w x Ž .10 K.J. Abraham, B. Lampe, Nucl. Phys. B 478 1996 507.


w x Ž .11 A. Ballestrero, E. Maina, Phys. Lett. B 350 1995 225.w x12 A. Ballestrero, in preparation.w x Ž .13 G. Passarino, M. Veltman, Nucl. Phys. B 160 1979 151.w x Ž .14 R. Pittau, Comm. Phys. Comm. 104 1997 23, hep-

phr9712418.w x Ž .15 S. Catani, M.H. Seymour, Nucl. Phys. B 485 1997 291.w x Ž .16 G.P. Lepage, Jour. Comp. Phys. 27 1978 192.w x Ž .17 D. Bardin, D. Lehner, T. Riemann, Nucl. Phys. B 477 1996

27.w x18 D. Bardin, M. Bilenky, A. Olchevski, T. Riemann, Phys.

Ž . w Ž . xLett. B 308 1993 403 Erratum: B 357 1995 725 .w x19 S. Catani, Yu.L. Dokshitser, M. Olsson, G. Turnock, B.R.

Ž .Webber, Phys. Lett. B 269 1991 432; N. Brown, W.J.Ž .Stirling, Z. Phys. C 53 1992 629.

w x20 J. Fujimoto, T. Ishikawa, Y. Kurihara, Y. Shimizu, K. Kato,N. Nakazawa, T. Kaneko, hep-phr9707260.

w x21 W. Beenakker, A.P. Chapovskii, F.A. Berends, Phys. Lett. BŽ . Ž .411 1997 203; Nucl. Phys. B 508 1997 17; A. Denner, S.

Dittmaier, M. Roth, hep-phr9710521 and hep-phr9803306.w x22 J.G. Korner, G. Schierholz, J. Willrodt, Nucl. Phys. B 185¨

Ž . Ž .1981 365; O. Nachtmann, A. Reiter, Z. Phys. C 16 1982Ž .45; M. Bengtsson, P.M. Zerwas, Phys. Lett. B 208 1988

Ž .306; M. Bengtsson, Z. Phys. C 42 1989 75; S. Betke, A.Ž .Richter, P.M. Zerwas, Z. Phys. C 49 1991 59.

w x23 R. Barate et al., ALEPH Collaboration, CERN–PPEr97-002.

18 June 1998


Energy scale and coherence effects in small-x equations

Marcello CiafaloniDipartimento di Fisica dell’UniÕersita, Firenze and INFN, Sezione di Firenze, Italy`

Received 22 January 1998Editor: R. Gatto

Abstract

I consider the next-to-leading high-energy cluster expansion of large-k double jet production in QCD, and I determine thecorresponding one-loop quark and gluon impact factors for a self-consistent energy scale. The result shows that coherentangular ordering of emitted gluons holds for hard emission also, and singles out a scale which in essentially the largestvirtuality in the process. Both remarks are relevant for the precise determination of the BFKL kernel at the next-to-leadinglevel. q 1998 Elsevier Science B.V. All rights reserved.

PACS: 12.38.Cy

Various attempts to understand the small-x HERAw xdata on structure functions and final states 1 have

stimulated the analysis of high-energy QCD beyondw xthe leading-logarithmic approximation 2 .

After several years of investigation, the high-en-Ž .ergy vertices needed for the next-to-leading NL

w xapproximation have been computed 3–8 , and theirreducible part of the NL kernel of the BFKL

w x w xequation 2 has been written out explicitly 9–11 . ItŽ .turns out that the NL terms provide sizeable correc-

tions to the hard Pomeron intercept and to the singletw xanomalous dimensions 10 .

However, the definition of the NL kernel is notfree of ambiguities which have prevented, so far, acomplete quantitative analysis of the NL results. Themost important ambiguity is due to the dependenceof some NL features on the determination of the

physical scale of the squared energy s, in the log sdependence of the cross sections.

In a hard process like DIS, the scale of s is takento be Q2, the virtuality of the photon or electroweakboson involved. Thus the structure functions arebasically dependent on the Bjorken scaling variable`

2 Ž 2 .xsQ rs, with scaling violations induced by a Q .s

Similar considerations can be made for double DISw x12 or quarkonium production, where two hard scalesare present.

On the other hand, the high-energy cluster expan-sion needed for the definition of the NL kernel, hasbeen mostly investigated in the case of parton-partonscattering, in which no physical hard scale is present.

Ž .In order to define such scale s , we have to deal witha partial cross section, by fixing the virtualitiesk ,k of the off-shell gluon Green’s function by1 2


( )M. CiafalonirPhysics Letters B 429 1998 363–368364

Ž .Fig. 1. a k-factorized decomposition of double jet productionŽ . ) Ž .and b qg k ™ qg squared matrix element in the fragmenta-2

tion region of A. Wavy lines denote gluon exchange.

Ž .high-energy factorization Fig. 1a . Experimentally,w xthis corresponds 13 to the cross section associated

with the production of two jets with given transversemomenta k ,k and large relative rapidity, in the1 2

fragmentation regions of the incoming partons A, B.Ž .We can then define the scale s k ,k as the one0 1 2

occurring in the logarithmic behaviour of the gluonŽ .Green’s function G k ,k ;s , whose loop expansion1 2

reads

sGs1q K log qconst.0ž /s k ,kŽ .0 1 2

2s s1 2q K log qK log qconst. q . . . ,0 NL2 ž /ž /s s0 0

1Ž .

where we have introduced the leading and next-to-leading operator kernels K , K . It becomes then0 NL

clear that the determination of K on the basis ofNL

an explicit two-loop calculation is dependent on thechoice of s : a change in the latter is a NL effect0

which induces a change in K , for the expansionNLŽ .1 to be left unchanged.

More precisely, I will assume in the following ak-factorized form for the differential cross section

vA Bds dv s psH ž /w x w xd k d k 2p i s k ,k vŽ .1 2 0 1 2

=h A k G k ,k hB k , 2Ž . Ž . Ž . Ž .v 1 v 1 2 v 2

w x 2Ž1qe . 1qewhere d k sd krp in Ds4q2e di-mensions,

y11G s 1y K qK 3Ž . Ž .v 0 NL

v

is the Mellin transform of the gluon Green’s func-1 A, B Ž .tion and h are process dependent impact fac-

tors, to be determined also, which may carry collinearŽ .singularities due to the initial massless partons

A,B.In this note I will present arguments for a choice

of s which is consistent with the collinear proper-0

ties of the partonic process and I will give thecorresponding one-loop determination of the impactfactors. These results can then be used to yield a

w xprecise determination of K at two-loop level 10 .NL

It turns out that the basic physical issue is theso-called ‘‘coherent effect’’ in the space-like jets,

w xnoticed by the author long ago 14 . This effect canbe stated by saying that the leading log squaredmatrix element for gluon emission holds in a dynam-ically restricted phase space, in which angular order-ing of the emitted gluons is required. The angularrestriction determines, among other features, the en-ergy scale also.

) Ž .Let me start noting that the one-loop Ag k ™2) Ž . w xAg k impact factor involves the known 5 parti-2

X ) Ž .cle-particle reggeon vertex AA g k at virtual2) Ž . X Ž .level, and the Ag k ™A g q squared matrix2

Ž .element Fig. 1b at real emission level. The latterquantity can be extracted from the exact squared

w xmatrix elements of Ellis and Sexton 15 , and hasbeen investigated by various authors at matrix ele-

w xment level 3,8,16 but has not yet been provided inexplicit form in D dimensions, to the author’s knowl-edge. I thus derive it directly for the quark case.

By using the notation of Fig. 1 and the Sudakovparametrization

k 21m m m m m mp yp sk ,z p y p qk ,1 3 1 1 1 2 11yz sŽ .1

k 22m m m mk sz p y p qk , 4Ž .2 2 2 1 21yz sŽ .2

1 An alternative formulation, not adopted here, could include anŽ .v-independent NL factor in Eq. 3 .

( )M. CiafalonirPhysics Letters B 429 1998 363–368 365

we want to compute the q q ™q q g squaredA B A B

matrix element in the limit of large subenergy s22qyz kŽ .1 2

s sz s4s s qsk qk ,Ž .2 1 1 1 2z 1yzŽ .1 1

5Ž .and fixed subenergy s , so as to cover the parton A1

fragmentation region. Since s is large, we can use2w xk-factorization 4 to write

2< <)Mq g ™ q g

2ab m l< <s u p A u p ´ 2 pŽ . Ž .Ý 3 ml 1 2spins colours

=g 6Cs F

, 6Ž .22kŽ .2

w xwhere, by current conservation, we can use 4 theFeynman gauge

g pu qqu g g pu yqu gŽ . Ž .m 3 l l 1 mab a b b aA s t t yt tml 2 p q 2 p q3 1

1w xq t ,t k yk g q ku qqu gŽ . Ž .Ž ma b 1 2 l 2 ml2< <k1

y qqk g . 7Ž . Ž ..1 ml

By using physical polarizations with ePqsePp2

s0 and the high-energy kinematics we can write thecolour decomposition

ab m l ab a b Žq. w a b x Žy.� 4A e 2 p sA s t ,t A q t ,t A ,ml 2

8Ž .with

1" "A sC pu p Peyeu pu quŽ .1 2 1 22

1Ž" .qC pu p Peq euqupu ,Ž .3 2 3 22

4 1 1 4y yC s q , C s y ,1 32 2< < < <p q p qk k1 31 1

1 1q qC sy , C s . 9Ž .1 3p q p q1 3

By performing the straightforward gamma func-tion algebra and the polarization sum in Ds4q2e

dimensions, we obtain, for each colour structure, theresult

22 2 2< <A s 2 s P e , z C 2 p qqC 1yz 2 p qŽ . Ž . Ž .Žq 1 3 3 1 1 1

qC C 2 p q 1yz q2 p qyz k 2 ,Ž .Ž . .1 3 3 1 1 1 2

10Ž .

where the splitting function of q™g type

1 1 2 2P sP e , z s 1q 1yz qe zŽ . Ž .Ž .g q q2C 2 zF

11Ž .

appears to factor out in front of the transversemomentum dependence. Finally, by introducingphase space and coupling constant

2 w x w x1 dz d k d k1 1 2dfsp ,2qe 2ž / s z 1yzŽ .4pŽ . 1 1

g 2G 1yeŽ .sa s , 12Ž .s 1qe4pŽ .

and performing the colour algebra, we obtain

2Ž1.ds p 2a C NŽ .s F es 2 2 2/w x w xd k d k dz N y1 k k1 2 1 C 1 2quark

=P e , z N a 1yz qP qyz kŽ . Ž . Ž .q 1 C s 1 1 2

P 22G 1ye pŽ . q qyz kŽ .1 2

2 2C a z kF s 1 1q 22p q qyz kŽ .1 2

=

er24pŽ .N ' . 13Ž .ež /G 1yeŽ .

This result agrees with the one extracted from Ref.Ž .15 in the high-energy limit.

Ž .The cross section in Eq. 13 contains two colourfactors, which have a simple interpretation, depend-ing on the collinear singularities involved. The C 2

F2 ŽŽ .2 .term, with singularities at q s0 qyz k s0 ,1 2

comes from the Sudakov jet region, in which theŽ .emitted gluon is collinear to the incoming outgoing

quark.On the other hand, the C C term is not reallyF A

2 Ž .2singular at either q s0 or qyz k s0, except1 2

for z s0, which corresponds to the central region.1

It comes from the ‘‘coherent’’ region in which thegluon is emitted at angles which are large withrespect to the q q X scattering angle, and is thusA A

sensitive to the total q q X charge C . This is theA A A

region we are interested in, which is relevant for the


energy scale, because it tells us how the leadingmatrix element, valid in the central region, is cut-offin the fragmentation region.

Ž .Since the C C term in Eq. 13 acquires theF A2 < <Ž . < <1rq singularity for q 1yz 4 k z , we can1 1 1

write roughly, in the quark case,

ds Ž1.Ž0. Ž0.sp h k h kŽ . Ž .q 1 q 2/w x w xd k d k dz1 2 1 coherent

=as

P e , z Q q 1yz yk z ,Ž . Ž .Ž .q 1 1 1 12q

2a C N N as F e C sŽ0.h k ' , a ' . 14Ž . Ž .sq 1r22 2 pN y1 kŽ .C

The latter is the coherence effect prescription of Ref.Ž .14 , in which angular ordering

q k1X) , u )u 15Ž .q p1z 1yz1 1

replaces the smooth behaviour of the differentialŽ . Ž .cross-section in Eq. 13 . The expression 14 is

actually exact upon azimuthal averaging in k of Eq.1Ž . < <13 , at fixed q .

Ž .Let me stress the point that in Eq. 14 the q™gŽ .splitting function ;P e , z factors out in front ofq 1

the leading matrix element for any Õalue of z , not1w xonly for z <1 as assumed originally 14 . There-1

Ž . Ž .fore, Eqs 13 and 14 form the basis for the interpo-Ž 2 2 .lation between Regge region z <1, any k rq1 1

Ž 2 2 .and collinear region k rq <1, any z , in small-x1 1w xequations of CCFM type 14,17 .

Ž .We are now in a position to start checking Eqs 1Ž .and 2 and determining scale and impact factors. Let

me start with a qualitative argument based on Eq.Ž . Ž .14 . By introducing the gluon quark rapidity ys

' 'Ž .Ž Ž .. Ž .log z s rq Y s log s rk , the restriction 151 1 1

on half the phase space becomes

q '0-y-Y y log Q s log s rMax q ,k .Ž .Ž .1 qk 11k1

16Ž .

'In other words, the pure phase space qr s -z1'Ž Ž ..-1, or 0-y- log s rq is cut-off by coherence'if q-k , yielding qr s -z -qrk .1 1 1

This means that, by adding the remaining phasespace related to the fragmentation region of B, thedifferential cross section is roughly

ds Ž1.Ž0. Ž0.,p h k h kŽ . Ž .1 2/w x w xd k d k1 2 coherent

' 'a s ss= log q log . 17Ž .2 ž /Max q ,k Max q ,kq Ž . Ž .1 2

Therefore, the physical scale for the energy isŽ . Ž .Max q,k P Max q,k , which can be roughly1 2

replaced by the expression

s k ,k sMax k 2 ,k 2 , 18Ž . Ž .Ž .0 1 2 1 2

which has the same behaviour in the regions q<k1

,k ,q,k 4k and q,k 4k .2 2 1 1 2

We have thus understood the mechanism by whichthe physical scale is provided by the largest virtual-ity, in qualitative agreement with what we knowabout deep inelastic scattering.

The coherence argument above can be made moreprecise by using the exact squared matrix element in

Ž .Eq. 13 , in the phase space region connected withthe fragmentation of A. In fact it is easy to check therepresentation

ds hŽ0. k hŽ0. kŽ . Ž .1 q 1 q 2dz spH 1 /w x w xd k d k dz G 1yeŽ .qr s' 1 2 1 C CF A

='a ss

log qh q ,k , 19Ž . Ž .q 12 ž /Max q ,kq Ž .1

where the appropriate scale has been subtracted, and

1h q ,k s dz P e , zŽ . Ž .Hq 1 1 q 1

0

=1yz qP qyz k 1 kŽ . Ž .1 1 2 1

y q log Qk q2 1z qqyz kŽ . 11 2

20Ž .

Ž .is a finite constant which, by Eq. 14 , Õanishes inthe q™0 limit, thus eliminating the qs0 singular-ity in front.

Ž .The result 19 , upon symmetrization, coincidesŽ .with Eq. 17 for the logarithmic piece. Furthermore,

the constant piece h can be integrated over k atq 1

( )M. CiafalonirPhysics Letters B 429 1998 363–368 367

fixed k and interpreted as one-loop contribution to2Ž .the quark A impact factor in Eq. 2 :

Ž1. 2 w xh k a k d kŽ . sq 2 2 1s h q ,kŽ .H q 1Ž0. 2 2/ G 1yeh k k qŽ .Ž .q 2 1real

3eX 12sa k y y2c 1 y .Ž .Ž .s 2 4ž /4e

21Ž .Ž .The 1re pole in Eq. 21 comes as no surprise,

because it corresponds to the collinear divergence atk 2 s0 due to the q™g transition of the initial1

massless quark11y12C g vs0 s dz P e , z yŽ . Ž . Ž .ˆ HF g q qž /z0

3 1sy q e , 22Ž .4 41the term being included in the leading kernel K .0z

Ž .Eq. 19 yields, upon symmetrization, the scaleŽ . Ž .Max q,k P Max q,k mentioned before. The1 2

Ž 2 2 .latter can be translated into the scale Max k ,k of1 2Ž .Eq. 18 by the change in the constant piece

k q2d h q ,k s log Q y log Q , 23Ž . Ž .q 1 k k qk2 1 1k k1 1

which vanishes at qs0 also and, upon k integra-1

tion, corresponds to the change in impact factorX1Ž1. Ž0.d h rh sa c 1 . 24Ž . Ž .sq q 2

Ž . Ž . Ž . Ž .Eqs. 18 , 19 , 21 and 24 summarize theone-loop results for scale and quark impact factor atreal emission level. The virtual corrections are knownw x 25 for both C and C C colour factors. The Su-F F A

dakov C 2 term turns out to cancel completely withFŽthe corresponding z - integrated real emission vs01

.moment and is thus of little concern here.w xThe C C term, on the other hand, yields 5F A

dsŽ0. Ž0. 2sp h k h k 2v kŽ . Ž . Ž .q 1 q 2 1/w x w xd k d k1 2 virtual

=s NF112Ž1qe . 1qed q p log q yŽ . 62 ž /3Nk C1

5 NFX 85y 3c 1 q y e ,Ž . 18ž /9 NC

2a G 1qeŽ .s2 2 ev k sy k . 25Ž . Ž .2e G 1q2eŽ .

Here the logarithmic term provides the gluon trajec-

tory renormalization, which regularizes the leadingBFKL kernel, the second term in square brackets is abeta-function coefficient which yields the runningcoupling for the overall a 2 factor, and half the lasts

Ž .term adds up to Eq. 21 to yield the total one-loopcorrection to the quark impact factor, for the scale

Ž . Ž .Max q,k P Max q,k :1 2

3e 1Ž1. Ž0. 2h k rh k sa k y qŽ . Ž . Ž .sq q 4ž /4e

1 5 NFX67q yc 1 y .Ž .18ž /2 9 NC

26Ž .Ž .In the final result, Eq. 26 , we have singled out

Ž .the collinear singularity due to Eq. 22 , so that itsfinite part is the last term in square brackets.

The above calculation can be repeated for initialgluons. If Ds4 the squared gg )™gg matrixelement can be extracted from the helicity vertices of

Ž .Ref. 8 to yield2Ž1.ds p 2a NŽ .s C

s P zŽ .g 12 2 2/w x w xd k d k dz N y1 k kŽ .1 2 1 C 1 2gluon

=N a 1yz q qyz k qz 2 k 2Ž . Ž .C s 1 1 2 1 1

, 27Ž .22p q qyz kŽ .1 2

where441qz q 1yzŽ .

P z s 28Ž . Ž .g 2 z 1yzŽ .is related to the g™g splitting function. The Su-dakov and coherent regions have in this case thesame colour factor, due to the final gluons’ identity.

The main difference with the quark case comesfrom the non singular part of the splitting function,

Ž .which yields a different constant piece in Eq. 19and thus a different real emission contribution to thegluon impact factor. The latter has the expected

Ž .collinear divergence related to g vs0 , and aˆg g

finite term, whose precise determination requiresŽ .generalizing Eq. 27 to 4q2e dimensions, and

adding the gg )™qq contribution. The scale deter-mination remains, of course, unchanged.

I thus conclude that the NL one-loop calculationdetermines the partonic impact factors with consis-tent collinear behaviour, depending on the choice of

Ž 2 2 .scale s s Max k ,k . With this choice the LL0 1 2


logarithmic term has no spurious qs0 singularitiesŽ .as pure phase space would imply and the NLconstant piece has no qs0 singularity at all. Fur-thermore, no spurious k 2 s0 singularities are pre-1

Žsent as, for instance, the scale s sk k would0 1 2.imply .

This means that the left-over ambiguity comesfrom a change of scale keeping the behaviour of Eq.Ž .18 in all the regions k <k ,k <k ,q<k ,k1 2 2 1 1 2

Ž . Ž .as, for instance, s s Max k ,q P Max k ,q0 1 2

does. In this case the change of scale can be reab-sorbed in a change of impact factor without spoiling

Ž .the collinear properties, as already done in Eq. 23for the example just mentioned.

Once a self-consistent scale is chosen and thecorresponding impact factors are found at one-looplevel, then the two-loop calculation will determinethe NL kernel unambiguously. The correspondingresults and physical implications will be reportedelsewhere.

Note added. During the refereeing process of thepresent paper, a preprint by Fadin and Lipatov ap-

Ž .peared hep-phr9802290 where the NL correctionsto the impact factors are not discussed, but a form ofthe NL kernel is presented which is claimed to be in‘‘disagreement’’ with the ‘‘irreducible part’’ ob-

w xtained in 10 .I just stress the point that such a comparison is

misleading at the present stage, and it will only makesense after the energy-scale dependent contributionsare included, and the NL impact factors are deter-mined along the lines of the present work.

Acknowledgements

I wish to thank Gianni Camici for a number ofdiscussions and suggestions, Dimitri Colferai for

helpful remarks, Keith Ellis for quite helpful corre-spondence on his work, and Victor Fadin, GiuseppeMarchesini and Gavin Salam for stimulating conver-sations. This work is supported in part by M.U.R.S.T.Ž .Italy .

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w x Ž .13 A.H. Mueller, H. Navelet, Nucl. Phys. B 282 1987 727.w x Ž .14 M. Ciafaloni, Nucl. Phys. B 296 1988 49.w x Ž .15 K. Ellis, J.C. Sexton, Nucl. Phys. B 269 1986 445.w x16 V. Del Duca, C.R. Schmidt, hep-phr9711309.w x17 S. Catani, F. Fiorani, G. Marchesini, Phys. Lett. B 234

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18 June 1998


Isolated photons in perturbative QCD

Stefano Frixione 1

Theoretical Physics, ETH, Zurich, Switzerland

Received 4 February 1998Editor: R. Gatto

Abstract

I present a definition of the cross section for the production of an isolated photon plus n jets which only depends upondirect photon production, and it is independent of the parton-to-photon fragmentation contribution. This prescription, basedon a modified cone approach which implements the isolation condition in a smooth way, treats in the same way quarks andgluons and can be directly applied to experimental data in hadron-hadron, photon-hadron and eqey collisions. The case ofseveral, isolated photons in the final state can also be dealt with in the very same way. q 1998 Elsevier Science B.V. Allrights reserved.

1. Introduction

Photons are produced in scattering phenomena bytwo different mechanisms. In the direct process, thephoton enters the partonic hard collision, character-ized by a large energy scale. In the fragmentation

Ž .process, a QCD parton quark or gluon fragmentsnon-perturbatively into a photon, at a scale of theorder of the typical hadronic mass. The former pro-cess is computable in perturbative QCD, while thelatter is not; all the unknowns of the fragmentation

Žmechanism are collected into two functions thequark-to-photon and gluon-to-photon fragmentation

.functions which, although universal, must be deter-mined by comparison with the data. Direct photonsare usually well isolated from the final state hadrons,while photons produced via fragmentation usually lieinside hadronic jets. From the experimental point ofview, it is not difficult to select a data sample in

1 Work supported by the Swiss National Foundation.

which the direct mechanism is dominant over thefragmentation mechanism, by rejecting all thoseevents where the photon is not isolated from hadronictracks. A clean sample of well-isolated photons isextremely useful for a variety of topics, like a de-tailed understanding of the underlying parton picture,to constrain the gluon density of the proton in anintermediate x range, and to obtain an efficientbackground rejection in Higgs searches at futurecolliders.

In perturbative QCD, the problem is more in-volved. It is not possible to separate sharply thephoton from the partons; in fact, this would constrainthe phase space of soft gluons, thus spoiling thecancellation of infrared divergences which is crucialin order to get a sensible cross section. Two methodshave been devised to tackle this problem. In the cone

w xapproach 1 , a cone is drawn around the photonŽaxis; if only a small hadronic energy compared to

.the photon energy is found inside the cone, thepartons accompanying the photon are clustered with


( )S. FrixionerPhysics Letters B 429 1998 369–374370

a given jet-finding algorithm. In the democratic ap-w xproach 2 the photon is treated as a parton as far as

the jet-finding algorithm is concerned. At the end ofthe clustering procedure, the configuration corre-sponds to an isolated photon event only if the ratioof the hadronic energy found inside the jet contain-ing the photon over the total energy of the jet itself issmaller than a fixed amount, usually less than 10%.The democratic approach is more suited than thecone approach to extract the non-perturbative parton-to-photon fragmentation functions from the data. This

w xpoint of view was adopted in Ref. 3 .In this paper, I will deal with the problem of

defining an isolated-photon cross section which min-imizes the contribution of the fragmentation mecha-nism. In particular, I will show that it is possible tomodify the cone approach in order to get a crosssection which only depends upon the direct process.I argue that this prescription is infrared safe at anyorder in perturbative QCD. The paper is organized asfollows: in Section 2 I sketch the main ideas in asimple way. In Section 3 I give a precise definitionof the isolation conditions. Finally, Section 4 con-tains my conclusions.

2. A simple formulation of the problem

In the cone approach, the first naive procedure isto draw a cone around the photon axis, and toimpose that no quark or gluon is found inside thecone. With this definition, the configurations where aparton is collinear to the photon are rejected, andtherefore the contribution of the fragmentation pro-cess is exactly zero. Unfortunately, this prescriptionis not infrared safe: soft gluons can not be emittedinside the cone, thus spoiling the cancellation ofinfrared singularities. One may relax the definition,by allowing a small amount of hadronic energyinside the cone. This restores the correct infraredbehaviour, but at the same time introduces a depen-dence upon the fragmentation functions, sincecollinear configurations are not forbidden any longer.Another possibility, which only works at next-to-leading order, is to allow soft gluons inside the cone,but to exclude the quarks. This prescription, how-ever, is non-physical and its predictions can not be

straightforwardly compared with experimental re-sults.

Therefore, to define an infrared-safe cross section,there should be no region of forbidden radiation inthe phase space, while to eliminate the dependenceupon the fragmentation functions such a region mustexist; these two requirements are seemingly incom-patible. I will argue in the following that actually thisis not the case. In fact, the fragmentation mechanismin QCD is a purely collinear phenomenon; therefore,to eliminate its contribution to the cross section, it issufficient to veto the collinear configurations only. In

Žpractice, this can be achieved in the following way Irestrict for the moment to the case of eqey colli-

. Ž .sions . A cone of fixed half-angle d is drawn0

around the photon axis. Then, for all dFd , the0Ž .total amount of hadronic energy E d found insidetot

the cone of half-angle d drawn around the photonaxis is required to fulfill the following condition

E d F KKd 2 , 2.1Ž . Ž .tot

Ž 2where KK is some energy scale the form KKd ischosen for illustrative purposes; it will be general-

. Ž .ized in the following . According to Eq. 2.1 , a softgluon can be arbitrarily close to the photon. On the

Ž .other hand, Eq. 2.1 implies that the energy of aparton emitted exactly collinear to the photon mustvanish. Therefore, the contribution of the fragmenta-tion process is restricted to the zero-measure setzs1.

In the following section, I will give a precisedefinition of the isolation condition, refining Eq.Ž . Ž .2.1 . Here, I stress that Eq. 2.1 does not spoil thecancellation of soft gluon effects, and achieves theisolation of the photon in a smooth way, which canbe easily implemented at the experimental level.

3. Isolated-photon plus jets cross section

I start from the class of scattering events whosefinal state contains a set of hadrons, labelled by theindex i, with four-momenta k , and a hard photoni

with four-momentum k . I assume to be in a kine-g

matic regime where the masses of the hadrons areŽ .small compared to their transverse energies. Also,

in a real experimental situation, we may think of the

( )S. FrixionerPhysics Letters B 429 1998 369–374 371

k as the four momenta deposited in the i th calori-i

metric cell, instead of the four momentum of the i th

hadron. Fix the parameter d , which defines the0

so-called isolation cone, and apply to each event theŽ .following procedure isolation cuts .

1. For each i, evaluate the angular distance Rig

between i and the photon. The angular distance isdefined, in the case of eqey collisions, to be

R sd , 3.1Ž .ig ig

where d is the angle between the three-momentaig

of i and g . In the case of hadronic collisions Idefine instead

2 2R s h yh q w yw , 3.2Ž . Ž . Ž .(ig i g i g

where h and w are the pseudorapidity and az-imuthal angle respectively.

2. Reject the event unless the following condition isfulfilled

E u dyR F XX d for all dFd ,Ž .Ž .Ý i ig 0i

3.3Ž .

where E is the energy of hadron i and, due toiŽ .u dyR , the sum gets contribution only fromig

those hadrons whose angular distance from thephoton is smaller than or equal to d . The function

2 Ž .XX , which plays the role of KKd in Eq. 2.1 , isˆfixed and will be given in the following. Thefunction XX must vanish when its argument tends

Ž .to zero, XX d ™0 for d™0. At hadron collid-ers, the transverse energy E must be used in-i T

stead of E .i3. Apply a jet-finding algorithm to the hadrons of

Ž .the event therefore, the photon is excluded . Thiswill result in a set of mqmX bunches of well-col-limated hadrons, which I denote as candidate jets.

Ž X.m m is the number of candidate jets which lieŽ .outside inside the isolation cone, in the sense of

Ž .the angular distance defined by Eqs. 3.1 orŽ .3.2 .

4. Apply any other additional cuts to the photon andto the m candidate jets which lie outside the coneŽfor example, the cut over the minimum observ-

Ž .able transverse energy of the jets must be ap-.plied here .

An event which is not rejected when the isolationcuts are applied is by definition an isolated-photonplus m-jet event. The key point in the above proce-dure is step 2: hadrons are allowed inside the isola-

Ž .tion cone, provided that Eq. 3.3 is fulfilled. This inturn implies the possibility for a candidate jet to beinside the isolation cone. It would not make muchsense to define a cross section exclusive in thevariables of such a jet, which can not be too hard.For this reason, in the physical observable that Idefine here, the jets which accompany the photon arethe candidate jets outside the isolation cone whichalso pass the cuts of step 4. The resulting crosssection is therefore totally exclusive in the variablesof these jets and of the photon, and inclusive in thevariables of the hadrons found inside the isolationcone. Notice that this is not equivalent to applying ajet-finding algorithm only to the hadrons lying out-side the isolation cone; in fact, such a procedure isnot infrared-safe.

I define

n1ycosd

XX d sE e , 3.4Ž . Ž .g g ž /1ycosd0

Žwhere E is the photon energy in the case of hadrong

collisions, E must be replaced by the transverseg

.energy of the photon, E . I will useg T

e s1, ns1. 3.5Ž .g

The reason for this choice will be discussed in thefollowing. Here, I stress that this choice is arbitraryto a large extent. The main feature of the function XX

is that

lim XX d s0. 3.6Ž . Ž .d™0

In QCD, any jet cross section is easily written inw xterms of measurement functions 4 . Given a N-par-

� 4Nton configuration k , the application of a jet-i is1

finding algorithm results in a set of M jets with� 4Mmomenta q . This can be formally expressed bya as1

the measurement function

M N� 4 � 4SS q ; k , 3.7Ž .Ž .N a ias1 is1

which embeds the definition of the jet four-momentain terms of the parton four-momenta. It has been


shown that, at next-to-leading order and for an arbi-trary type of collisions, the infrared-safeness require-ment on the cross section can be formulated in termsof conditions relating the measurement functions SSN

Ž w x.for different N see for example Refs. 5–7 . Theseconditions can be extended without any difficultiesto higher perturbative orders. Here, I stress that the

Ž .measurement function in Eq. 3.7 implements aninfrared-safe jet cross section definition, which I willapply to the partons accompanying the photon in acandidate isolated-photon event. By labeling the par-tons in such a way that

R G R if i ) j, 3.8Ž .ig jg

I define

M N� 4 � 4SS k , q ; kž /g , N g a ias1 is1

NM N� 4 � 4sSS q ; k = II , 3.9Ž .ŁŽ .N a i ias1 is1

is1

i

II su XX min R ,d y E u d yR .Ž . Ž .Ž . Ýi ig 0 j 0 jgž /js1

3.10Ž .

Ž .It is easy to understand that Eq. 3.9 is equivalent tothe isolation cuts described above. In particular, thequantity Ł N II is equivalent to step 2. Therefore,is1 i

SS is the measurement function relevant for theg , N

isolated-photon plus jets cross section: it vanisheswhen applied to those parton configurations where

Ž . Ž .the photon is non-isolated. Eqs. 3.9 and 3.10 canbe straightforwardly used to construct a Monte Carlo

w xprogram as described in Ref. 8 , which evaluatesisolated-photon plus jets cross section at next-to-leading order.

I now turn to the discussion of the main featuresof the definition of isolated photon given in thispaper. Clearly, it is sufficient here to investigate thebehaviour of the cross section when one or morepartons are inside the isolation cone. First of all, I

Ž .observe that for a soft parton E™0 the isolationŽ Ž ..cone does not exist at all see Eq. 3.10 . This

ensures that the cancellation of soft gluon effectswill take place as in ordinary infrared-safe jet cross

Ž . Ž .sections. Secondly, Eqs. 3.3 and 3.6 imply that aparton is softer the closer to the photon axis: a partonexactly collinear to the photon is necessarily soft.

Thus, when a quark gets collinear to the photon, thedamping associated with the quark vanishing energysuppresses the collinear divergence. Therefore, thereis no need for a final-state collinear counterterm.

To be more quantitative, I start by considering thecase of a quark inside the isolation cone. I restrict forthe moment to the case of eqey collisions. Theleading behaviour of the partonic amplitude squared

Ž .is 1r 1yy , where y is the cosine of the anglebetween the quark and the photon. The contributionto the isolated-photon cross section from the regioninside the isolation cone 2 is therefore

u XX d y yEŽ .Ž .Ž .1s ; dy dE E , 3.11Ž .H Hcone 1yycosd 00

where the factor E originates from the phase space,Ž .and the condition of Eq. 3.3 has been enforced with

a theta function. The upper integration bound in E isirrelevant in what follows, and will be neglected.

Ž .Using Eq. 3.4 we get

2 n2 2E e 1 1yy1g gs ; dyHcone ž /2 1yy 1ycosdcosd 00

E2e 2g g

s , 3.12Ž .4n

provided that nG1r2. As previously anticipated,the damping associated with the energy of the quarkwhich gets soft cancels the effects of the collineardivergence, for reasonable choices of XX . The factthat there is no need for a final-state collinear coun-terterm is consistent with the fact that the contribu-tion from the fragmentation function is also vanish-ing; in QCD, fragmentation is a rigorously collinear

Ž .phenomenon, and therefore Eq. 3.3 would implyzs1. Thus, the contribution of the fragmentationfunction is restricted to a zero-measure set in thephase space.

I now turn to the case of a gluon inside theisolation cone. The leading behaviour of the partonicamplitude squared is 1rE2. Using the subtraction

2 Ž .Notice that, by integrating 1r 1y y outside the isolationŽ .cone, one gets a term proportional to log 1ycosd , as expected0

in isolated-photon production.

( )S. FrixionerPhysics Letters B 429 1998 369–374 373

method, 3 the contribution to the finite part of theŽcross section again, from the region inside the isola-

.tion cone will read

u XX d y yE y1Ž .Ž .Ž .1s ; dy dE .H Hcone Ecosd 00

3.13Ž .Ž .Using Eq. 3.4 we get

n1yy1

s ; dylog E eHcone g g ž /ž /1ycosdcosd 00

s 1ycosd log E e yn . 3.14Ž . Ž .Ž .Ž .0 g g

The case of hadronic collisions is only slightlyŽ .more complicated. The u function in Eqs. 3.11 and

Ž .3.13 is now

u XX R y yE . 3.15Ž . Ž .Ž .Ž .T

This function clearly constrains the energy of theŽ .parton, since EsE coshh y , where I have explic-T

itly indicated that the pseudorapidity of the partondepends upon y. In order not to spoil the conclusions

Ž . Ž .of Eqs. 3.12 and 3.14 , one must have E™0Ž .when y™1, or, which is equivalent, coshh y must

Žtend to a finite constant when y™1 notice thatŽ .1ycos R y tends to zero at the same rate of 1yy.for y™1 . This is indeed the case, since for the

Ž .very definition of y one gets coshh y ™coshh . Ing

isolated-photon production, the photon is observed inthe central region of the detector, and coshh is ofg

order one.By increasing the number of partons inside the

isolation cone the situation rapidly becomes ratherinvolved, preventing us to perform analytical calcula-tions. However, from the definition of the isolationcuts, it should be clear that, given a configurationwhich fulfills the isolation criteria, it is always possi-ble that a parton emits a soft gluon, or splits into twocollinear partons, without changing abruptly thephysical observables. On the other hand, if a config-

3 With the subtraction method, the divergent part of the crosssection is evaluated strictly in the soft limit, Es0. As previouslystressed, the isolation cone does not constrain at all the phasespace of soft partons, and therefore this divergent part will becancelled by the corresponding virtual contribution as customaryin perturbative QCD.

uration does not fulfill the isolation criteria, if aparton emits a soft gluon or splits into two collinearpartons, we get a configuration which still does notfulfill the isolation cuts. Therefore, any QCD in-

Žfrared configuration soft gluons, quarks and gluons. Žcollinear to each other can be locally that is, prior

.to any phase-space integration subtracted, and thecorresponding singularities are cancelled by the vir-tual contributions. The only singularities which are

Žleft unsubtracted are the QED ones quarks collinear.to the photon . However, in this case the singularity

Ž .is damped by the mechanism described in Eq. 3.12 .The fact that soft gluon emission and collinear

splitting do not modify the physical observables isformally equivalent to the infrared safeness of thecross section to all perturbative orders. However, ithas to be stressed that the isolation cuts have animpact on the local subtraction of singularities. This

Ž . Ž .can be clearly seen from Eqs. 3.13 and 3.14 ,where the integration over the gluon energy E, con-strained by the isolation cuts, results in a singularŽ . Žbut integrable function of y without any isolationcondition, the integral over E would simply give

.zero . It seems reasonable to assume that, at higherorders, the isolation cuts always define a functionwhich, although singular in some regions of thephase space, has a finite integral: however, no formalproof will be given here. This fact has been explic-itly verified in the case where a quark and a gluonare inside the isolation cone, which is relevant forthe next-to-leading order cross section of isolated-photon production in eqey collisions.

We can understand the impact of the isolationcondition on the radiative corrections by looking at

Ž .Eq. 3.14 . The definitions of isolated photon in thecone approach which are usually adopted in theliterature can be recovered by setting ns0 ande se . e is the maximum amount of hadronicg c c

Ž .energy normalized to the photon energy allowedinside the cone, and to minimize the contribution ofthe fragmentation mechanism it must be a small

Ž .number. Therefore, we see from Eq. 3.14 that,comparing to the case where the isolation condition

Ž .is extremely loose ns0, e s1 , there is a sizeableg

negative correction loge . On the other hand, withc

the definition given in this paper, e does not need tog

be small. The isolation condition is obtained in asmooth way, controlled by the function XX . From


Ž .Eq. 3.14 , the smaller is n, the more moderate willŽ .be the negative corrections. This is the reason for

Ž .the choice of e and n made in Eq. 3.5 . Finally,g

notice that in standard approaches d is usually0

taken to be of the order of 208. With the currentdefinition, d can be chosen to be larger. This0

Ž .implies that the logarithms log 1ycosd , which0

are present in the isolated-photon cross section, donot get large.

A final remark is still in order. When d,0, theŽ .value of XX d is below the energy threshold E ofth

hadronic calorimeters, and therefore the condition ofŽ .Eq. 3.3 is not experimentally meaningful any

longer. The data can therefore get a contributionfrom the fragmentation process, with z)z s1yth

E rE . However, from the experimental point ofth g

view there is nothing special in the region d,0Žobviously, there are no singularities in particle de-

.tectors ; since E is typically of the order of fewth

hundred MeV, and the photon is required to be hard,z ,1, and the contribution of the fragmentationth

mechanism will be small. In other words, one couldŽ .relax Eq. 3.3 by allowing quasi-soft partons to be

collinear with the photon; this would imply thepresence of a collinear singularity in the direct crosssection, which would be cancelled by a proper frag-mentation cross section. After the cancellation, onewould be left with a very small finite contribution.

Ž .By strictly imposing Eq. 3.3 , this small contribu-tion is zero from the beginning, and the fragmenta-tion cross section is simply not there.

4. Conclusions

I presented a definition of isolated-photon plusjets cross section which is based upon an isolationcone and a jet-finding algorithm which excludes thephoton. It has been argued that this prescription,which treats identically quarks and gluons, is in-frared safe to all orders. Hadrons are allowed insidethe isolation cone, at the border of which they can beas energetic as the photon itself, but are required tobe softer the closer they are to the photon axis,

eventually becoming soft in the exact collinear limit.This fact implies that the contribution of the photonscoming from the fragmentation of quarks and gluonscan be neglected in QCD. Therefore, for hadron-hadron and photon-hadron collisions, the only non-calculable parts which enter the theoretical predic-tion are the parton densities of the incoming hadrons.In the case of eqey collisions, the isolated photonplus jets cross section is fully calculable in perturba-tion theory. The prescription given here is also appli-cable without any modification to the case of several,isolated photons in the final state.

Acknowledgements

I would like to thank C. Grab, Z. Kunszt, M.Mangano, H. Niggli, S. Passaggio, G. Ridolfi and A.Signer for many discussions and useful comments. Iam especially indebted to S. Catani, who found amistake in a preliminary version of this paper, and toP. Nason, whose suggestions influenced the finalform of the paper.

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18 June 1998


Local multiplicity fluctuations in hadronic Z decay

L3 Collaboration

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a I. Physikalisches Institut, RWTH, D-52056 Aachen, FRG 4

III. Physikalisches Institut, RWTH, D-52056 Aachen, FRG 4

b National Institute for High Energy Physics, NIKHEF, and UniÕersity of Amsterdam, NL-1009 DB Amsterdam, The Netherlandsc UniÕersity of Michigan, Ann Arbor, MI 48109, USA

d Laboratoire d’Annecy-le-Vieux de Physique des Particules, LAPP,IN2P3-CNRS, BP 110, F-74941 Annecy-le-Vieux CEDEX, Francee Johns Hopkins UniÕersity, Baltimore, MD 21218, USA

f Institute of Physics, UniÕersity of Basel, CH-4056 Basel, Switzerlandg Institute of High Energy Physics, IHEP, 100039 Beijing, China 5

h Humboldt UniÕersity, D-10099 Berlin, FRG 4

i UniÕersity of Bologna and INFN-Sezione di Bologna, I-40126 Bologna, Italyj Tata Institute of Fundamental Research, Bombay 400 005, India

k Boston UniÕersity, Boston, MA 02215, USAl Northeastern UniÕersity, Boston, MA 02115, USA

m Institute of Atomic Physics and UniÕersity of Bucharest, R-76900 Bucharest, Romanian Central Research Institute for Physics of the Hungarian Academy of Sciences, H-1525 Budapest 114, Hungary 6

o Massachusetts Institute of Technology, Cambridge, MA 02139, USAp INFN Sezione di Firenze and UniÕersity of Florence, I-50125 Florence, Italy

q European Laboratory for Particle Physics, CERN, CH-1211 GeneÕa 23, Switzerlandr World Laboratory, FBLJA Project, CH-1211 GeneÕa 23, Switzerland

s UniÕersity of GeneÕa, CH-1211 GeneÕa 4, Switzerlandt Chinese UniÕersity of Science and Technology, USTC, Hefei, Anhui 230 029, China 5

u SEFT, Research Institute for High Energy Physics, P.O. Box 9, SF-00014 Helsinki, Finlandv UniÕersity of Lausanne, CH-1015 Lausanne, Switzerland

w INFN-Sezione di Lecce and UniÕersita Degli Studi di Lecce, I-73100 Lecce, Italy´x Los Alamos National Laboratory, Los Alamos, NM 87544, USA

y Institut de Physique Nucleaire de Lyon, IN2P3-CNRS,UniÕersite Claude Bernard, F-69622 Villeurbanne, France´ ´z Centro de InÕestigaciones Energeticas, Medioambientales y Tecnologicas, CIEMAT, E-28040 Madrid, Spain 7

aa INFN-Sezione di Milano, I-20133 Milan, Italyab Institute of Theoretical and Experimental Physics, ITEP, Moscow, Russiaac INFN-Sezione di Napoli and UniÕersity of Naples, I-80125 Naples, Italyad Department of Natural Sciences, UniÕersity of Cyprus, Nicosia, Cyprus

ae UniÕersity of Nijmegen and NIKHEF, NL-6525 ED Nijmegen, The Netherlandsaf Oak Ridge National Laboratory, Oak Ridge, TN 37831, USA


ag California Institute of Technology, Pasadena, CA 91125, USAah INFN-Sezione di Perugia and UniÕersita Degli Studi di Perugia, I-06100 Perugia, Italy´

ai Carnegie Mellon UniÕersity, Pittsburgh, PA 15213, USAaj Princeton UniÕersity, Princeton, NJ 08544, USA

ak INFN-Sezione di Roma and UniÕersity of Rome, ‘‘La Sapienza’’, I-00185 Rome, Italyal Nuclear Physics Institute, St. Petersburg, Russia

am UniÕersity and INFN, Salerno, I-84100 Salerno, Italyan UniÕersity of California, San Diego, CA 92093, USA

ao Dept. de Fisica de Particulas Elementales, UniÕ. de Santiago, E-15706 Santiago de Compostela, Spainap Bulgarian Academy of Sciences, Central Lab. of Mechatronics and Instrumentation, BU-1113 Sofia, Bulgariaaq Center for High Energy Physics, Korea AdÕ. Inst. of Sciences and Technology, 305-701 Taejon, South Korea

ar UniÕersity of Alabama, Tuscaloosa, AL 35486, USAas Utrecht UniÕersity and NIKHEF, NL-3584 CB Utrecht, The Netherlands

at Purdue UniÕersity, West Lafayette, IN 47907, USAau Paul Scherrer Institut, PSI, CH-5232 Villigen, Switzerland

av DESY-Institut fur Hochenergiephysik, D-15738 Zeuthen, FRG¨aw Eidgenossische Technische Hochschule, ETH Zurich, CH-8093 Zurich, Switzerland¨ ¨ ¨

ax UniÕersity of Hamburg, D-22761 Hamburg, FRGay High Energy Physics Group, Taiwan, ROC

Received 18 December 1997Editor: K. Winter

Abstract

Local multiplicity fluctuations in hadronic Z decays are studied using the L3 detector at LEP. Bunching parameters areused for the first time in addition to the normalised factorial moment method. The bunching parameters directly demonstratethat the fluctuations in rapidity are multifractal. Monte Carlo models show agreement with the data, reproducing the trend,although not always the magnitude, of the factorial moments and bunching parameters. q 1998 Elsevier Science B.V. Allrights reserved.

1. Introduction

Hadronic final states of eqey collisions provide afavourable environment for QCD studies. Initially,the hadronic system is simply a quark-antiquark pair.

1 Also supported by CONICET and Universidad Nacional deLa Plata, CC 67, 1900 La Plata, Argentina.

2 Also supported by Panjab University, Chandigarh-160014,India.

3 Supported by Deutscher Akademischer Austauschdienst.4 Supported by the German Bundesministerium fur Bildung,¨

Wissenschaft, Forschung und Technologie.5 Supported by the National Natural Science Foundation of

China.6 Supported by the Hungarian OTKA fund under contract num-

bers T14459, T19181 and T24011.7 Supported also by the Comision Interministerial de Ciencia y´

´Technologia.

The energy of this pair is dissipated during a com-plex, non-linear QCD parton shower and non-per-

Ž .turbative hadronisation process. Monte Carlo MCprograms incorporating the QCD parton shower andphenomenological models of hadronisation and reso-nance decay have been successful in describing globalfeatures.

In this paper we use high statistics data from theL3 experiment at LEP to study fluctuations in thecharged particle multiplicity distribution in smallregions of phase space, as a function of the size ofthe region. If particles were independently produced,the local multiplicity distribution would be a Poisso-nian. A deviation of this distribution from a Poisso-nian measures dynamical local multiplicity fluctua-tions, which are a consequence of short-range corre-lations between final-state particles. Parton showers,fragmentation, resonance decays and Bose-Einsteininterference can all contribute to these correlations.


Fluctuations are often studied using the nor-Ž .malised factorial moments NFMs of the local mul-

Ž .tiplicity distribution, P d , which is the probabilityn

to find n particles inside a phase space bin of size d

w x Ž .1 . The NFM of order q, F d , is defined byq

² w q x:nF d s , 1Ž . Ž .qq ² :n

`

w q x w q x² :n s n P d ,Ž .Ý nnsq

nw q x sn ny1 . . . nyqq1 . 2Ž . Ž . Ž .Ž . Ž .If P d is a Poisson distribution, F d s1 for alln q

Ž .q. If there are fluctuations, F d deviates fromq

unity. Further, if the fluctuations are self-similar, i.e.,Ž . yf q Ž .F ld sl F d , then a power-like increaseq q

Ž . yf qemerges with decreasing d , F d Ad , whereqŽ .the intermittancy indices f s qy1 d , and d areq q q

the anomalous fractal dimensions. Local fluctuationsŽare classified as monofractal d is independent ofq

. Ž . Žq or multifractal d is a function of q . See recentqw x .reviews 2 .

Local fluctuations in eqey processes have beenw xstudied in several experiments 3–5,7,8,6,9,11,10 .

The data do indeed exhibit an approximate power-likerise of the NFMs with decreasing d , especially whenevaluated in two- and three-dimensional phase spacevariables. All four LEP experiments have found thatcurrent MC models can, in general, describe theNFMs, even without additional tuning. Exceptionshave, however, been found in rapidity defined with

w xrespect to the sphericity axis by OPAL 7 and byw x ŽDELPHI 9 for restricted charge-multiplicity and.p regions .T

Recently, it has been realized that the factorialmoment method poorly reflects the information onlocal fluctuations, since the NFM of order q containsa contamination from lower-order correlation func-

w xtions 2 . As a result, the nature of the fluctuations isdifficult to determine from the behaviour of theNFMs. Factorial moments also suffer from a statisti-cal bias due to the finite size of the event sample.This is because measurements of the NFMs are

Ž .dominated by the first few terms of expression 2 .In most cases this leads to a significant underesti-mate of the measured NFMs with respect to their

w xtrue values 12 .

An alternative to the NFMs is provided by bunch-Ž . Ž . w xing parameters BPs , h d , 13,14 which are de-q

fined by

q P d P dŽ . Ž .q qy2h d s , qs2,3, . . . .Ž .q 2qy1 P dŽ .qy1

3Ž .

They are more sensitive than the NFMs to the varia-Ž .tion in the shape of P d with decreasing d . In then

Ž .case of self-similar fluctuations, one expects h d2yd 2 Ž .Ad . For multifractal local fluctuations, the h dq

are d-dependent functions for all qG3, while forŽ .monofractal behaviour h d is independent of d forq

w xqG3 13 . For independent particle production, theŽBPs are d-independent constants for a Poisson dis-

Ž . .tribution, h d s1 , for all q.q

From an experimental point of view, the BPs havew x Ž .several advantages 14 : 1 They are less severely

affected by statistical bias than the NFMs, since theBP of order q depends only on the behaviour of themultiplicity distribution near multiplicity nsqy1;Ž .2 for the calculation of the BP of order q, one onlyneeds to be able to resolve q particles in a bin, rather

Ž .than all particles as for the NFMs; and 3 manysystematic errors cancel in the ratio of probabilities.

When defined in this way, the NFMs and BPsboth require dividing phase space into a number ofbins. This has the disadvantage of losing information

Ž .on local fluctuations in particle density ‘‘spikes’’that are divided by bin boundaries. To remedy thisproblem with the NFMs, and in addition to increasethe effective statistics, normalized density strip inte-

w xgrals have been proposed 15 . Analogously, a neww xtype of bunching parameter has been suggested 14 ,

which can be used to study the fluctuation of theŽ .number of spikes defined in Section 2 per event.

Ž .Generalized integral bunching parameters GBPsare defined by

q P e P eŽ . Ž .q qy2x e s , 4Ž . Ž .q 2qy1 P eŽ .qy1

Ž .where P e is the probability of an event to have qq

spikes of size less than e , irrespective of how manyparticles are inside each spike. For purely indepen-dent particle production, with the multiplicity distri-

Ž .bution characterised by a Poissonian, x e s1 forq

all q.


In this paper we study fluctuations in rapidity,defined with respect to the thrust axis, using factorialmoments and, for the first time, bunching parame-ters. We also study fluctuations in the four-momen-tum difference using generalized bunching parame-ters.

2. Methods

In order to improve the accuracy we use thew xbin-averaged ‘‘horizontal’’ NFMs 1 and BPs

w x13,14 : The NFM of order q is calculated using thestandard definition:

M w q x² :1 nmF M s ,Ž . Ý qq ² :M nms1

nw q x sn n y1 . . . n yqq1 , 5Ž . Ž . Ž .m m m m

where n is the number of particles in bin m,m² :n sNrM, N is the average multiplicity for fullphase space, MsDrd is the total number of bins,and D represents the full phase space volume. The‘‘horizontal’’ BP of order q is calculated using

q N M N MŽ . Ž .q qy2h M s ,Ž .q 2qy1 N MŽ .qy1

M1N M s N m ,d , 6Ž . Ž . Ž .Ýq qM ms1

Ž .where N m,d is the number of events having qq

particles in bin m and M has the same meaning asfor the NFMs.

Note that bin-averaging, as used in the abovedefinitions, is only justified for a flat single-particledensity distribution. To be able to study non-flatdistributions, we transform the original phase spacevariable to one in which the underlying density is

w xuniform 16 .For the generalized bunching parameters we need

to define the number of spikes of size less than e . Todo so we need both a measure of the size of a spikeand a method of assigning particles to spikes. For thespike size, s , we use the so-called Grassberger-

w xHentschel-Procaccia counting topology 17 for whichs is the maximum of all pairwise distances betweenparticles in the spike. As distance between particles i

and j we use the squared four-momentum difference2 Ž .2Q sy p yp . Spikes are then defined in thei j i j

following way: Consider all possible combinationsof two or more particles. For each combination s isdetermined. If s is larger than some maximum, e ,the combination is discarded. Of the remaining com-binations, those completely contained in anotherŽ .larger combination are also discarded. Combina-

Žtions left after this procedure are called spikes of.size less than e . Note that while the assignment of

Žparticles to spikes is not unique the same particle.can be in more than one spike , the number of spikes

is unambiguously defined.The GBPs are then given by

q S Q2 S Q2Ž . Ž .q qy22x Q s , 7Ž .Ž .q 2 2qy1 S QŽ .qy1

Ž 2 .where S Q is the number of events having qq

spikes of size less than Q2.

3. Data samples and analysis procedures

We use data, corresponding to an integrated lumi-nosity of 52 pby1, collected at a centre of mass

'energy of s ,91.2 GeV during the 1994 LEP run-Ž .ning period. Hadronic events are selected using 1

energy deposits in the electromagnetic and hadronicŽ .calorimeters, and 2 momenta measured in the Time

Ž .Expansion Chamber TEC and the Silicon Mi-Ž .crovertex Detector SMD . The L3 detector is de-

w xscribed in detail in Ref. 18 .First, a loose calorimeter-based selection is per-

formed in order to reject non-hadronic background.Using clusters with energy larger than 100 MeV, werequire

EC E EH I0.6- -1.4, -0.4, -0.4,C C' E Es

13-N -75,cl

where EC is the total energy observed in theŽ .calorimeters, E E is the energy imbalance inH I

Ž .the plane perpendicular parallel to the beam direc-tion, and N is the number of calorimeter clusters.cl

To ensure that the event is contained in the barrel< <region of the calorimeters we require cosu -0.74,thr

where u is the polar angle of the event thrust axis.thr


To obtain a sample with well-measured chargedtracks, a further selection is performed using onlytracks which have passed certain quality cuts. The

Ždistance of closest approach projected onto the.transverse plane of a track to the nominal interac-

tion vertex is required to be less than 5 mm and itstransverse momentum must be larger than 100 MeV.To ensure that the event lies within the full accep-tance of the TEC and SMD, the direction of thethrust axis, as determined from the charged tracks,

< <must satisfy cosu -0.7. Events are then selectedthr

using criteria similar to the above calorimeter-basedselection, but using tracks:

p pÝ Ýi I ii i

)0.15, -0.75,' ps Ý ii

pÝ H ii

-0.75, N )4,chpÝ ii

where p is the momentum of particle i and the sumi

runs over all tracks of an event, and where N is thech

number of charged tracks. The resulting sample con-tains about 1 million events.

The experimental distributions are corrected forselection and acceptance losses using two samples,‘‘generator level’’ and ‘‘detector level’’ of eqey™

hadrons MC events generated with JETSET 7.4 PSw x19 including initial-state photon radiation. At thegenerator level particles with lifetime ct)1 cm areassumed stable. The detector level sample has passed

w xa full detector simulation 20 including time-depen-dent variations of the detector response based oncontinuous detector monitoring and calibration. Ithas been reconstructed with the same program as thedata and passed through the same selection proce-dure.

In this paper we study fluctuations in the rapiditywith respect to the thrust axis, y, and the square ofthe pairwise four-momentum difference, Q2. Bothfor the calculation of Q2 and for the grouping oftracks into small bins of rapidity, the resolution ofthe angle between pairs of tracks is of crucial impor-tance. For this reason we impose additional stringentquality cuts on track reconstruction, which results in

rejection of 39% of the tracks. With this selection weachieve very good agreement between data and sim-ulation for the distributions of the difference in anglebetween pairs of tracks for both the azimuthal angleabout, and the polar angle with respect to, the beam,as is shown in Fig. 1a and 1b, respectively.

The uncorrected distributions of y, and Q2, arecompared to the detector level MC distributions inFigs. 1c and 1d, respectively. There is reasonableagreement, which indicates the quality of both thedetector simulation and the JETSET predictions. Itshould be noted that these distributions have notbeen used in the tuning of JETSET’s parameters.

The distributions of NFMs and BPs are correctedbin-by-bin for detector effects. The corrected value

Ž .Fig. 1. Distributions of a the difference in polar angle of pairs ofŽ .tracks, du , b the difference in azimuthal angle of pairs of tracks,

Ž .df, c the single-particle rapidity with respect to the thrust axis,Ž . 2y, and d the inclusive four-momentum difference squared, Q ,

Ž .for uncorrected data points compared to the predictions ofŽ .JETSET after detector simulation histogram .


in a bin is found by multiplying the value calculateddirectly from the data by a correction factor given bythe ratio C sM genrM det, where M gen and M det areq q q q q

the corresponding NFM or BP calculated from thegenerator- and detector-level MC samples, respec-tively. These corrections, which tend to be larger forsmaller bin size, are in no case larger than about 5%.

To reduce possible systematic bias, the minimumbin size is chosen comparable to the experimental

w xresolution, which was estimated 21,22 by MC sim-ulation. In the case of Q2, the smallest bin size islarge enough that the measurements are not strongly

Ž 0 q y .affected by Dalitz decays p ™e e g or by pho-ton conversions.

The error bars on the results include contributionsfrom both statistical and systematic errors on the rawquantities and on the correction factors. The statisti-

Ž . Ž .cal errors on the F M and h M are derived fromq q

the covariance matrix of the NFMs and BPs. Thestatistical errors for the GBPs are derived according

w xto the expression obtained in Ref. 14 . Systematicerrors on the raw quantities have been estimated byvarying the event- and track-selection criteria. Assystematic error on C , we take half of the differenceq

between the correction factors determined usingJETSET and those using HERWIG.

w xThe predictions of the JETSET 7.4 PS 19 , ARI-w x w xADNE 4.08 23 and HERWIG 5.9 24 models, all

of which have been tuned to reproduce globalevent-shape and single-particle inclusive distribu-

w xtions 10,25 , are compared to the data. The Bose-Einstein effect is a potential source of particle corre-lations. JETSET and ARIADNE include the samemodelling of this effect. 8 HERWIG does not incor-porate a Bose-Einstein model. We also compare thedata with predictions of JETSET without Bose-Ein-stein interference. 9 The errors on the JETSET pre-dictions include both statistical and systematic errors.These systematic errors were estimated by varying,by one standard deviation, the following JETSET

w xparameters tuned in Ref. 10,25 : the Lund fragmen-

8 The Bose-Einstein model used is the luboei model of JET-SET.

9 The parameters of JETSET were retuned with Bose-EinsteinŽ .interference disabled. This resulted in changes of PARJ 21 ,

Ž . Ž .PARJ 42 , and PARJ 81 from 0.411, 0.886, and 0.311 to 0.343,1.1, and 0.312, respectively.

tation function parameter b, the width of the Gauss-ian p and p hadronic transverse momentum distri-x y

bution, and the value of L used for a in partons

showers. Systematic errors on the other MC predic-tions are similarly determined. The errors on theARIADNE predictions are comparable to those onJETSET, while those of HERWIG are about 50%larger. The errors on the MC results are dominatedby the systematic errors.

4. Results

4.1. Fluctuations in rapidity, y

To study fluctuations inside jets, we first deter-mine the thrust axis and analyse the NFMs and BPs

< <in the full rapidity range y F5. Since the single-particle rapidity distribution is non-uniform, we first

w xtransform y to a uniformly distributed variable 16 .The horizontally normalised NFMs are shown in

Fig. 2 as a function of the number of bins, M, in the

Fig. 2. NFMs as a function of the number of bins, M, in thetransformed rapidity defined with respect to the thrust axis. In thisand the following figures the shaded areas represent the errors onthe JETSET predictions. The errors on the ARIADNE predictionsare comparable whereas those on the HERWIG predictions areabout 50% larger.


Fig. 3. BPs as a function of the number of bins, M, in thetransformed rapidity defined with respect to the thrust axis.

transformed rapidity. They rise with increasing Mand then saturate. All of the MCs are in reasonableagreement with the data. It is also seen that theBose-Einstein effect in JETSET raises the values ofthe NFMs.

Fig. 3 shows the results for the horizontally nor-Ž .malised BPs. All higher-order q)2 BPs show an

approximate power-law increase with increasing M.The MCs also show a power-law behaviour, al-though some differences in slope are apparent. Thatthese BPs vary with M is a direct indication that the

10 w xfluctuations in y are multifractal 13 , as is ex-w xpected in QCD 26–28 .

The second-order BP decreases with increasingM up to Mf20, which is found to correspond tothe value of M at which the maximum of the

Ž .multiplicity distribution P d first occurs at ns0.n

The large errors on the data for MQ30 arise mainly

10 This conclusion is possible without measuring the intermit-tency indices f . In contrast, to reveal multifractality with NFMsq

one must first fit the NFMs by a power law. Because of theŽ .saturation of the F M observed in Fig. 2, this procedure isq

fraught with ambiguity.

from the systematic error assigned from the differ-ence between correction factors determined usingJETSET and HERWIG. The data are shown usingthe JETSET correction factors.

The second-order BP is related to the width of thew xmultiplicity distribution 13,14 . Hence, HERWIG’s

overestimation of h means that HERWIG’s local2

multiplicity distributions are too broad. This agreeswith ALEPH’s conclusion from a direct measure-

Ž .ment of the dispersion for various large intervals ofw xrapidity 29 , but emphasizes the contribution of the

low values of n in this discrepancy.To study the second-order BP in more detail, we

split h into two components:2

h M sh Ž"" . M qh Žqy. M . 8Ž . Ž . Ž . Ž .2 2 2

Ž"" .Ž . Ž .The definition of h M is as in Eq. 6 , but2Ž . Ž"" .Ž .with N m,d replaced by N m,d , the number2 2

of events having two like-charged particles insideŽqy.Ž .bin m of size d . Analogously, h M is con-2

structed from the number of events having two oppo-sitely charged particles in the bin. Note that for

Ž"" .Ž . Žqy .Ž .combinatorial reasons, h M - h M .2 2

However, both would be independent of M in thecase of independent production.

Ž"" .Ž . Žqy.Ž .Fig. 4 shows that h M and h M be-2 2Ž"" .Ž .have differently. While h M shows the ex-2

Ž .pected rise and saturation of the data at large M ,Žqy.Ž .h M shows a decrease at low M.2

ŽThe anti-bunching tendency decrease of h with2.increasing M seen for unlike-charged particles for

MQ20 is also seen in all MCs. Resonance decays

Fig. 4. The second-order BP as a function of the number of bins,M, in the transformed rapidity defined with respect to the thrustaxis for like-charged and unlike-charged particle combinations.


are a likely explanation of this effect. Their decayparticles tend to be of opposite charge and aretypically separated in rapidity by d y;0.5–1.0. As aresult, a rapidity separation of this order of magni-tude is more frequent between unlike-charged parti-

Žqy.Ž .cles than between like-charged and h M is2Ž"" .Ž . Ž .much larger than h M at small M large d y .2

However, this difference decreases rapidly with de-creasing d y until about Ms20, which correspondsto d ys0.5.

For like-signed combinations the MCs all showgood agreement with the data. However, for unlike-charged combinations HERWIG overestimates thedata, while the other MCs agree reasonably well.The errors on the JETSET predictions are domi-nantly systematic. They are mainly due to the uncer-tainty on the parameter responsible for the width ofthe Gaussian hadronic transverse momentum distri-bution in the Lund model. This shows the impor-tance of fragmentation for the bunching parameters.For h Žqy., as for h , the error at small M comes2 2

mainly from differences in the correction factors;other errors are comparable to the errors on thecorresponding points for h Ž"" .. The effect of Bose-2

Einstein interference in JETSET is to increase thevalue of h Ž"" . and to decrease that of h Žqy..2 2

4.2. Fluctuations in four-momentum difference, Q2

The behaviour of the GBPs for the invarianttwo-particle squared four-momentum difference,Ž 2 . 2x Q , is shown in Fig. 5 as a function of ylnQ .q

All GBPs, for both data and Monte Carlo, rise2 Ž 2 .similarly with increasing ylnQ decreasing Q .

This corresponds to a bunching effect for all orders,similar to the behaviour of fluctuations in y. TheMC models show a similar trend with ylnQ2 but

Ž 2 .tend to underestimate the values of x Q ; HER-q

WIG agrees best with the data.Given the difference in behaviour observed in the

previous section between h Žqy. and h Ž"" ., we now2 2

define second-order GBPs for multiparticle spikesconsisting entirely of particles of the same charge,x Žsc ., and for spikes containing particles of different2

Ždc . Ž .charge, x . These GBPs are defined as in Eqs. 42Ž .and 7 except that q now refers to the number of sc

and dc spikes, respectively. We plot in Fig. 6 thebehaviour of x sc and x dc. A difference is observed2 2

Fig. 5. GBPs as a function of the squared four-momentum differ-ence Q2 in GeV2 between two charged particles.

between these two quantities. For sc spikes a strongŽ Žsc .Ž 2 . .bunching effect x Q )1 is seen at large2

ylnQ2. This is well reproduced by HERWIG. In thecase of dc spikes, the bunching is smaller and tendsto saturate at large ylnQ2.

In contrast to the BPs of the previous section,resonances have little effect on the GBPs. This is

Fig. 6. The second-order GBP for spikes of same-charged parti-Ž . Ž .cles sc and spikes of differently charged particles dc as a

function of the squared four-momentum difference Q2 in GeV 2

between two charged particles.


because the most copiously produced resonances de-cay to two particles with a Q2 so large that theparticles are necessarily in different spikes for the

Ž 2 .spike sizes which we consider ylnQ R2 . Thespike multiplicity distribution is therefore not stronglyaffected.

It is interesting to observe that JETSET’s treat-ment of BE correlations decreases the values of x2

for sc spikes, the opposite of the behaviour forh Ž"" .. This comes about because JETSET’s mod-2

elling of Bose-Einstein interference increases thenumber of sc spikes. This leads to higher values forh Ž"" .. However, x is mainly influenced by the2 2

shape of the spike-multiplicity distribution ratherthan by the average spike multiplicity.

5. Conclusions

Local charged particle multiplicity fluctuations inrapidity with respect to the thrust axis have beenstudied using factorial moments and, for the firsttime, bunching parameters, which are sensitive tofurther details. Also, fluctuations of spikes have beenstudied using generalized bunching parameters de-fined in terms of the four-momentum difference, Q2.Bunching parameters directly demonstrate a multi-fractal behaviour of the fluctuations in rapidity, as isexpected from QCD.

Monte Carlo models, which have been tuned toreproduce global event-shape distributions and sin-gle-particle inclusive distributions provide a reason-able description of fluctuations in these variables.They reproduce the trend, although not always themagnitude, of the normalized factorial moments,bunching parameters and generalized integral bunch-ing parameters. It thus appears that the ingredients of

Žthese MC models coherent parton shower, string or.cluster fragmentation, and resonance decays are suf-

ficient to explain the fluctuations observed in thedata.

Acknowledgements

We wish to express our gratitude to the CERNaccelerator divisions for the excellent performance of

the LEP machine. We acknowledge the effort of allengineers and technicians who have participated inthe construction and maintenance of this experiment.We thank V. I. Kuvshinov, J.-L. Meunier and R.Peschanski for useful discussions and comments.

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a I. Physikalisches Institut, RWTH, D-52056 Aachen, FRG 4

III. Physikalisches Institut, RWTH, D-52056 Aachen, FRG 4

b National Institute for High Energy Physics, NIKHEF, and UniÕersity of Amsterdam, NL-1009 DB Amsterdam, The Netherlandsc UniÕersity of Michigan, Ann Arbor, MI 48109, USA

d Laboratoire d’Annecy-le-Vieux de Physique des Particules, LAPP,IN2P3-CNRS, BP 110, F-74941 Annecy-le-Vieux CEDEX, Francee Johns Hopkins UniÕersity, Baltimore, MD 21218, USA

f Institute of Physics, UniÕersity of Basel, CH-4056 Basel, Switzerlandg Institute of High Energy Physics, IHEP, 100039 Beijing, China 5

h Humboldt UniÕersity, D-10099 Berlin, FRG 4

i UniÕersity of Bologna and INFN-Sezione di Bologna, I-40126 Bologna, Italyj Tata Institute of Fundamental Research, Bombay 400 005, India

k Boston UniÕersity, Boston, MA 02215, USAl Northeastern UniÕersity, Boston, MA 02115, USA

m Institute of Atomic Physics and UniÕersity of Bucharest, R-76900 Bucharest, Romanian Central Research Institute for Physics of the Hungarian Academy of Sciences, H-1525 Budapest 114, Hungary 6

o Massachusetts Institute of Technology, Cambridge, MA 02139, USAp INFN Sezione di Firenze and UniÕersity of Florence, I-50125 Florence, Italy

q European Laboratory for Particle Physics, CERN, CH-1211 GeneÕa 23, Switzerlandr World Laboratory, FBLJA Project, CH-1211 GeneÕa 23, Switzerland

s UniÕersity of GeneÕa, CH-1211 GeneÕa 4, Switzerlandt Chinese UniÕersity of Science and Technology, USTC, Hefei, Anhui 230 029, China 5

u SEFT, Research Institute for High Energy Physics, P.O. Box 9, SF-00014 Helsinki, Finlandv UniÕersity of Lausanne, CH-1015 Lausanne, Switzerland

w INFN-Sezione di Lecce and UniÕersita Degli Studi di Lecce, I-73100 Lecce, Italy´x Los Alamos National Laboratory, Los Alamos, NM 87544, USA

y Institut de Physique Nucleaire de Lyon, IN2P3-CNRS,UniÕersite Claude Bernard, F-69622 Villeurbanne, France´ ´z Centro de InÕestigaciones Energeticas, Medioambientales y Tecnologicas, CIEMAT, E-28040 Madrid, Spain 7

aa INFN-Sezione di Milano, I-20133 Milan, Italyab Institute of Theoretical and Experimental Physics, ITEP, Moscow, Russiaac INFN-Sezione di Napoli and UniÕersity of Naples, I-80125 Naples, Italy


ad Department of Natural Sciences, UniÕersity of Cyprus, Nicosia, Cyprusae UniÕersity of Nijmegen and NIKHEF, NL-6525 ED Nijmegen, The Netherlands

af Oak Ridge National Laboratory, Oak Ridge, TN 37831, USAag California Institute of Technology, Pasadena, CA 91125, USA

ah INFN-Sezione di Perugia and UniÕersita Degli Studi di Perugia, I-06100 Perugia, Italy´ai Carnegie Mellon UniÕersity, Pittsburgh, PA 15213, USA

aj Princeton UniÕersity, Princeton, NJ 08544, USAak INFN-Sezione di Roma and UniÕersity of Rome, ‘‘La Sapienza’’, I-00185 Rome, Italy

al Nuclear Physics Institute, St. Petersburg, Russiaam UniÕersity and INFN, Salerno, I-84100 Salerno, Italyan UniÕersity of California, San Diego, CA 92093, USA

ao Dept. de Fisica de Particulas Elementales, UniÕ. de Santiago, E-15706 Santiago de Compostela, Spainap Bulgarian Academy of Sciences, Central Lab. of Mechatronics and Instrumentation, BU-1113 Sofia, Bulgariaaq Center for High Energy Physics, Korea AdÕ. Inst. of Sciences and Technology, 305-701 Taejon, South Korea

ar UniÕersity of Alabama, Tuscaloosa, AL 35486, USAas Utrecht UniÕersity and NIKHEF, NL-3584 CB Utrecht, The Netherlands

at Purdue UniÕersity, West Lafayette, IN 47907, USAau Paul Scherrer Institut, PSI, CH-5232 Villigen, Switzerland

av DESY-Institut fur Hochenergiephysik, D-15738 Zeuthen, FRG¨aw Eidgenossische Technische Hochschule, ETH Zurich, CH-8093 Zurich, Switzerland¨ ¨ ¨

ax UniÕersity of Hamburg, D-22761 Hamburg, FRGay National Central UniÕersity, Chung-Li, Taiwan, ROC

az Department of Physics, National Tsing Hua UniÕersity, Taiwan, ROC

Received 18 February 1998Editor: K. Winter

Abstract

Using the data collected with the L3 detector at LEP between 1990 and 1995, corresponding to an integrated luminosityof 149 pby1, the t longitudinal polarisation has been measured as a function of the production polar angle using the t

yy y yŽ . Ž .decays t ™h n hsp,r,a and t ™ ll n n llse,m . From this measurement the quantities AA and AA , whicht 1 ll t e t

depend on the couplings of the electron and the t to the Z, are determined to be AA s0.1678"0.0127"0.0030 ande

AA s0.1476"0.0088"0.0062, consistent with the hypothesis of e–t universality. Under this assumption a value oft2AA s0.1540"0.0074"0.0044 is obtained, yielding the value of the effective weak mixing angle sin u s0.2306"0.0011.ll W

q 1998 Elsevier Science B.V. All rights reserved.

1 Also supported by CONICET and Universidad Nacional deLa Plata, CC 67, 1900 La Plata, Argentina.

2 Also supported by Panjab University, Chandigarh-160014,India.

3 Supported by Deutscher Akademischer Austauschdienst.4 Supported by the German Bundesministerium fur Bildung,¨

Wissenschaft, Forschung und Technologie.5 Supported by the National Natural Science Foundation of

China.6 Supported by the Hungarian OTKA fund under contract num-

bers T14459, T19181 and T24011.7 Supported also by the Comision Interministerial de Ciencia y´

´Technologia.

1. Introduction

q yw xParity violation 1 in the process e e ™Z™ ffgives rise to non-zero polarisation of the Z and thefermions even in the case of unpolarised electron andpositron beams. The t leptons decay inside thedetector allowing their polarisation to be measured.The t longitudinal polarisation, PP , is defined as:t

s ysR LPP ' , 1Ž .t

s qsR L

where s and s are the cross sections for theR L

production of ty with positive and negative helicity,


respectively 8. The dependence of PP on the scatter-t

ing angle in the improved Born approximation at thew xZ pole is given by 2 :

AA 1qcos2u q2 AA cosuŽ .t ePP cosu sy , 2Ž . Ž .t 21qcos u q2 AA AA cosuŽ . e t

where u is the angle between the ty and the incom-ing electron beam directions. The quantities AA ande

AA depend on the neutral current couplings of thet

electron and the t to the Z:

2 g gA Vll llAA ' ; llse,t , 3Ž .ll 2 2g qgA Vll ll

where g and g are the effective axial–vectorA Vll ll

w xand vector couplings 3 respectively. According toŽ .Eq. 2 , AA and AA can be determined simultane-e t

Ž .ously from a measurement of PP cosu . This mea-t

surement is done by exploiting the kinematics of thew xt decay products 2,4–6 under the assumption of

Ž .pure V–A structure in the t decay. The latter isw xsupported by experimental results 7 . The measure-

ment of AA and AA tests lepton universality in thee t

neutral current and, together with the measurementsof forward–backward asymmetries and cross sec-tions, improves significantly the precision of elec-troweak parameters. It also gives the relative sign ofg and g .A Vll ll

w x Ž .In the Standard Model 8 , Eq. 3 provides aprecise determination of the effective weak mixing

2w xangle 3 , sin u :W

22 1y4sin uŽ .WAA s ; llse,t . 4Ž .ll 221q 1y4sin uŽ .W

In this paper, semileptonic t decays ty™hynt

Ž . 9 yhsp, r, a and purely leptonic t decays t ™1y Ž .ll n n llse, m are analysed using the data col-ll t

lected in 1994 and 1995. The results are combinedwith our previous published ones, based on 1990 to

w x1993 data 9 . Other measurements of PP have beent

8 Formulae are given for the decay of the ty. In the analysis thecharge conjugate decays are also used.

9 No distinction between charged pions and kaons is made inty™hy

nt decay. Both decay modes are combined in the analy-sis and referred to as ty™pyn .t

w x q yperformed at LEP 10,11 and at lower energy e ew xcolliders 12 .

2. Data analysis

The data used in this analysis were collected atcentre-of-mass energies around the Z mass, corre-sponding to a total integrated luminosity of 80 pby1,where 80% is on the Z peak and 10% at energies' 's sm y2 GeV and s sm q2 GeV each. TheyZ Z

w xare taken with the L3 detector 13 upgraded with aw xSilicon Microvertex Detector 14 , which improves

the measurement of track parameters and vertexdetermination. We exploit this new detector particu-larly in the analysis of ty™ayn ™pqpypyn1 t t

decays.q y q yŽ .The e e ™t t g events are selected by re-

quiring low track multiplicity. The identification ofthe different t decay products is performed sepa-rately in each hemisphere of the event, which isdefined by the plane perpendicular to the event thrustaxis. Electrons, muons and a mesons are identified1

in the central part of the detector, which covers< <cosu - 0.72. Pions and r mesons are also identi-fied in the end–cap regions, extending the angular

< <coverage to cosu - 0.94.For efficiency and background estimates, Monte

Carlo events are generated using the programs KO-w x q y q yŽ . q yRALZ 15 for e e ™m m g and e e ™

q yŽ . w x q y q yŽ .t t g , BHAGENE 16 for e e ™e e g ,q y q y q yw xDIAG36 17 for e e ™e e ff, where ff is e e ,

q y q y q yw xm m , t t or qq, and JETSET 18 for e e ™Ž .qq g . The Monte Carlo events are passed through a

full detector simulation based on the GEANT pro-w xgram 19 , which takes into account the effects of

energy loss, multiple scattering, showering and timedependent detector inefficiencies. These events arereconstructed with the standard L3 reconstructionprogram. The number of Monte Carlo events in eachprocess is about ten times larger than the data sam-ple.

2.1. Particle identification

Electrons are identified by an energy deposit inŽ .the electromagnetic calorimeter BGO which is


electromagnetic in shape and consistent in positionand energy with a track in the central tracker. Muonsare identified by a track in the muon spectrometeroriginating from the interaction point with a mini-mum-ionising particle response in the BGO and thehadron calorimeter. The electron and muon identifi-cation efficiencies, which are estimated from Monte

Ž .Carlo, are shown in Fig. 1 a as functions of thenormalised particle energy, E rE , where E isll beam ll

the lepton energy and E is the beam energy. Thebeam

average identification efficiencies are 82% and 70%for electrons and muons, respectively.

The identification of ty™pyn , ty™ryn andt t

ty™ayn decays is based on the central tracker and1 t

w xcalorimeter information. An algorithm 9 is appliedto disentangle overlapping neutral electromagneticclusters in the vicinity of the impact point of thecharged hadron in the BGO. Around the impactpoint, which is predicted by the central tracker, ahadronic shower, whose shape is nearly energy inde-pendent, is subtracted from the energy deposit. Elec-tromagnetic neutral cluster identification criteria are

Fig. 1. Identification efficiencies for the different t decay chan-nels as a function of the corresponding polarisation-sensitive

y y y yŽ .variables described in the text: a t ™e n n and t ™m n n ,e t m t

Ž . y y Ž . y y Ž . y y Ž .b t ™p n , c t ™ r n and d t ™a n 3-prong .t t 1 t

applied to the remaining local maxima of energydeposit. The energies and angles are determined forthe accepted electromagnetic neutral clusters. Twodistinct neutral clusters form a p 0 if their invariantmass is within 40 MeV of the p 0 mass. A singleneutral cluster is considered a p 0 if its energyexceeds 1 GeV. Its transverse energy profile must beconsistent with either a single electromagnetic showeror a two–photon hypothesis for which the invariantmass is within 50 MeV of the p 0 mass. The ty™

pyn identification requires that there are no p 0andt

no neutral clusters with energy greater than 0.5 GeVin the vicinity of the py.

To identify ty™ryn decays, one p 0 is requiredt

in the hemisphere. The invariant mass of the pyp 0

system must be in the range 0.45 to 1.20 GeV and itsenergy must be larger than 5 GeV. The efficiency forty™pyn and ty™ryn identification is esti-t t

mated from Monte Carlo. The average efficienciesŽ . Ž . Žare 66% 64% and 71% 61% in the barrel end–

.caps for p and r, respectively.The a meson decays via the channels ay™1 1

y 0 0 Ž . y y q y Ž .p p p 1-prong and a ™p p p 3-prong1Ž .with equal probability. The a 1-prong is identified1

as a charged track accompanied by two p 0. Theinvariant mass of the three pion system is required tobe greater than 0.45 GeV. To identify the a in the1

3-prong decay mode three charged tracks and no p 0

are required. The ty™ayn identification effi-1 t

ciency, estimated from Monte Carlo, is 36% for the1-prong and 45% for the 3-prong mode.

Ž . Ž .Figs. 1 b to 1 d show the dependence of thehadron identification efficiencies on the kinematicvariables used to measure PP ; these are the nor-t

malised pion energy, E rE , in the ty™pynp beam t) ) w x y ydecay, the angles u and c 5 for t ™r n ,r r t

where:

4 m2 E yqE 0 m2 qm2t p p t r

)cosu s y ;r 2 2 2 2'm ym m ymst r t r

m E yyE 0r p p)cosc s 5Ž .r 2 2 y 0p qpm y4m p p( r p

w x y yand the v variable 20 for the t ™a n decay,a 1 t1

which is defined in terms of the energies and angles


of the three pions coming from the a and contains1

all the information about the polarisation of the t .

2.2. EÕent selection and background rejection

Events with at least one identified t decay areretained for the measurement of PP . Additional cri-t

teria are applied to reduce background from Bhabha,dimuon and two–photon events and from cosmic ray

w xmuons 9 .The shape of residual Bhabha and dimuon back-

ground is estimated from data samples selected byrelaxing the requirement on the energy of electronsand muons, respectively, in the hemisphere oppositeto the identified one. The distribution of the remain-ing two–photon background is estimated from MonteCarlo. Its normalisation is determined from datausing events with large acollinearity. The back-ground from cosmic muons is estimated from acontrol sample selected in data by loosening the cuton the distance of the muons to the eqey interactionvertex. The background from hadronic Z decays isnegligible. The fraction of misidentified t decays ineach channel is determined using simulated eqey™

q yŽ .t t g events.The number of selected decays in each exclusive

final state, the selection efficiency and the back-ground fractions are given in Table 1 for each t

decay mode, for the 1994–1995 data. The totalnumber of t decays analysed including the 1990–1993 data is 137 092, corresponding to an integratedluminosity of 149 pby1.

Table 1Ž .Number of selected decays, selection efficiencies ´ and frac-

tions from t and non-t backgrounds for the t decay channelsused to measure PPt

Ž . Ž .Channel Decays ´ % Bkg. %

in 4p t non-ty yt ™e n n 16300 45 1.6 3.6e ty yt ™m n n 13920 40 0.9 4.7m ty yt ™p n 12104 42 9.8 3.7ty yt ™ r n 22634 38 11.0 1.0ty y Ž .t ™a n 1 prong 4 172 22 32.7 0.01 ty y Ž .t ™a n 3 prong 4 159 27 16.7 0.01 t

3. Measurement methods

The t longitudinal polarisation, PP , is measuredt

in nine cosu intervals ranging from y0.94 to 0.94.The angle u is approximated by the polar angle ofthe event thrust axis in the ty jet direction. Thefollowing methods are used to measure PP : thet

analysis of the exclusive single t decays, the analy-sis of inclusive hadronic t decays, the analysis ofacollinearity in p X final states, where X is any1-prong t decay, and an analytical fit to the energyspectra of the exclusive single t decays.

The individual measurements of PP for each tt

decay channel in a given cosu bin are correctedaccounting for the statistical correlation when both t

decays in an event are selected for the polarisationmeasurement. The resulting values are combinedwith those from the inclusive analysis taking intoaccount the statistical correlation arising from theoverlap between the samples. Finally the combina-tion with the results from the acollinearity method isperformed. The correlation matrices used in the com-bination procedure are obtained with a fast MonteCarlo simulation which provides very high statistics

w xevent samples 21 .Ž .In order to obtain AA and AA as given in Eq. 2 ,e t

the measured values are corrected for QEDBremsstrahlung, g exchange and g-Z interferencecontributions, in bins of cosu using the program

w xZFITTER 22 . These corrections are small at the Zpeak and reach the level of a few per cent foroff-peak data.

Decays with wrongly assigned charge migratebetween bins of opposite sign of cosu . The fractionof such events on average is 1.4% in the barrel and7.5% in the end–caps. A correction is applied to themeasured value of PP in each cosu interval tot

account for this effect.

3.1. Analysis of exclusiÕe single t decays

This method uses spectra of charged and neutralparticles in each t decay channel generated by the

w xMonte Carlo program KORALZ 15 for positive andnegative ty helicity. The spectra for the two helicitystates obtained after detector simulation and recon-struction are fitted to the data distributions, using a


Fig. 2. The measured spectra for the polarisation–sensitive vari-Ž . Ž .ables described in the text for the t decaying to a electron, b

Ž . Ž .muon, c p and d a mesons. The data are compared to the1

results of the fits. The two helicity components and the back-ground are shown separately.

binned maximum likelihood method that accountsw xfor finite statistics both in data and Monte Carlo 23 .

For the electron and muon spectra, the normalisa-tion of the Bhabha and dimuon background is left

Fig. 3. The data spectrum of the cosc ) distribution in four slicesr

of cosu ) for the ty™ ryn decays.r t

free in the fit, whereas the two–photon and cosmicbackgrounds are fixed to the estimated amounts. Thenon-t background is fixed in the fit of hadronspectra. The background from other t decays isincluded in the distributions of each helicity andvaried in the fit simultaneously with the signal ones.

The spectra of the charged decay products for they y y y y ychannels t ™e n n , t ™m n n , t ™p ne t m t t

and ty™ayn , using 1994 and 1995 data, are shown1 t

in Fig. 2. The angular distributions for the ty™rynt

decays are shown in Fig. 3 for the same data set.Good agreement between data and Monte Carlo dis-tributions is observed. Compatible distributions forthe 1990 to 1993 data have been already publishedw x9 .

3.2. Analysis of inclusiÕe hadronic t decays

The measurement of the t polarisation from ex-clusive hadronic t decays, in spite of the highsensitivity, suffers from losses induced by the identi-fication of p 0 in ty™ryn and ty™ayn decays.t 1 t

Most of these events are recovered by selecting aninclusive sample of 1-prong hadronic t decays where

0 w xno explicit identification of p is attempted 10 .Hadronic t decays are identified by one chargedtrack matching an energy deposit in the calorimetersnot consistent with an electromagnetic shower or aminimum–ionising particle. The invariant mass mh

Fig. 4. The distribution Q of the semileptonic t decays used inthe inclusive analysis for the first mass bins: m -0.3 GeV.h


of the charged hadron and all the neutral deposits ina cone of 308 around the track is calculated and theresulting spectrum is divided in three bins: m -0.3h

GeV, 0.3-m -0.9 GeV, 0.9-m -m GeV.h h t

The polarisation–sensitive variables Q and C aredefined as a generalisation of cosu ) and cosc ) inr r

Ž . 0Eq. 5 , by replacing m with m and E with ther h p

sum of the energies of all the neutral particles mea-sured in the BGO in a 308 cone around the chargedhadron track. The distribution of these variables isshown in Figs. 4 and 5, for data and Monte Carlo,for the three mass bins specified. In the first bin,which is mainly populated by ty™pyn decays, thet

polarisation is extracted by fitting the spectrum of Q

only. In the second and third mass bins, which arepopulated mainly by ty™ryn and ty™ayn de-t 1 t

cays respectively, the polarisation is obtained from atwo-dimensional fit to Q and C .

The statistical correlation between the results fromthis analysis and those from the exclusive analysis ofsingle t decays is 38% for ty™pyn , 50% fort

ty™ryn and 6% for ty™ayn .t 1 t

3.3. Analysis of acollinearity in p X final states

Using the acollinearity between the t decay prod-w xucts to measure PP 24 , additional detector informa-t

Fig. 5. The distributions Q and C of the semileptonic t decaysused in the inclusive analysis for the second and third mass bins:Ž . Ž .a,b 0.3- m -0.9 GeV and c,d 0.9- m - m GeV.h h t

Fig. 6. Acollinearity spectrum obtained in 1994 for ty™pynt

decays recoiling against a 1-prong t decay.

tion is exploited giving, in particular, almost inde-pendent systematic errors. The selected sample con-sists of 6763 events, collected in 1993 and 1994 10,with a ty™pyn decay recoiling against a 1-prongt

t decay. For these events, the acollinearity is definedas j'pya where a is the angle between the12 12

charged pion track and the track in the oppositehemisphere.

The tracks are required to be within the region of< <polar angle cosu -0.72. As an example, the

acollinearity spectrum for the 1994 data is shown inFig. 6.

The correlation in the PP measurement betweent

this method and the exclusive analysis of single t

decays is 30%.

3.4. Analytical fit to exclusiÕe t decays energy spec-tra

We have measured the t longitudinal polarisationusing a completely independent analysis method,based on the comparison of the t decay energyspectra with analytical functions. These functions arecalculated for all the 1-prong t decays, for t leptons

w xwith positive or negative helicity 25 . Mass effects

10 w xIn our previous publication 9 only 1991 and 1992 data wereused in the acollinearity method.


and initial and final state radiation are included.Selection and detector effects are taken into accountby a convolution of the theoretical expressions withthe resolution function and correcting for acceptance.The resolution function is extracted from test beamdata and from calibration samples selected in datafor this purpose.

This method is used to measure PP using thety y y y ydecay channels t ™e n n , t ™m n n , t ™e t m t

pyn and ty™ryn . The results obtained with thist t

method are used as a cross–check, especially of thesystematics concerning the resolution and calibrationof the detectors.

4. Systematic errors

The main sources of systematic errors are theuncertainties in the energy scales of the differentsubdetectors, uncertainties in the background estima-tions and possible biases due to the event selection.

The scale uncertainties of the BGO and muonspectrometer are estimated by comparing the detec-tor responses for Bhabha and dimuon data with thebeam energy. The central tracker momentum scaleuncertainty is determined by comparing its momen-tum measurement with that of the muon spectrome-ter. At low energy, a cross calibration of the centraltracker with the BGO and the muon spectrometer isperformed using electrons and muons from t decaysand two–photon events. The p 0 peak position in theinvariant mass of photon pairs is used as an addi-tional constraint to the BGO calibration. The BGOand the muon spectrometer scale uncertainties areestimated to be 0.5% at low energy and 0.05 % athigh energy. The scale uncertainties are interpolatedlinearly for intermediate energies. This procedure ischecked using radiative Bhabha and dimuon events.The momentum scale of the central tracker is veri-

Table 2Systematic error on AA and AA from different sourcest e

energy scale background selection theory total

AA 0.0039 0.0012 0.0046 0.0010 0.0062t

AA 0.0002 0.0019 0.0023 0.0001 0.0030e

AA 0.0027 0.0010 0.0033 0.0007 0.0044ll

Table 3The results on AA and AA from 1990 to 1995 data, for thet e

different methods and channels. The errors are statistical only.Note that these values are subject to correlations, which are takeninto account for the determination of the final result

method channel AA AAt e

y yExclusive t ™e n n 0.121"0.031 0.257"0.046e ty yt ™m n n 0.144"0.033 0.206"0.047m ty yt ™p n 0.142"0.015 0.146"0.023ty yt ™ r n 0.155"0.012 0.147"0.019ty yt ™a n 0.191"0.056 0.214"0.0841 t

y yInclusive t ™h nt 0.152"0.015 0.185"0.033

y yAcollinearity t ™p n 0.111"0.041 0.128"0.058t

y yAnalytical t ™e n n 0.139"0.032e ty yt ™m n n 0.130"0.035m ty yt ™p n 0.152"0.017ty yt ™ r n 0.161"0.023t

fied to 0.5% from 1 to 45 GeV. The hadroncalorimeter scale is estimated at low energy by acomparison with the central tracker momentum mea-surement and at high energy by the peak position ofthe r resonance in the p "p 0 invariant mass distri-bution. The scale uncertainty is estimated to be 1%in the barrel and 3% in the end–caps, independent ofthe hadron energy. For the evaluation of the system-atic errors, the responses of the individual subdetec-tors are varied according to their scale uncertaintiesusing a fast Monte Carlo simulation.

The normalisation of the non-t background isvaried within its statistical error. The correspondingchange of PP is assigned as the systematic error. Int

y y y ythe case of t ™e n n and t ™m n n , the nor-e t m t

malisation of Bhabha and dimuon background is afree parameter in the fit and its uncertainty is alreadyincluded in the statistical error. The systematic errordue to background from other t decays is estimatedby varying the branching fractions of the contribut-

w xing decay channels within their errors 26 .Particle identification and background rejection

are designed to be nearly independent of the energyof the t decay products in order to keep polarisationbiases to a minimum. This is checked by comparingMonte Carlo energy and momentum distributions forelectrons and muons from Bhabha, dimuon andtwo–photon control data samples. Good agreement


Fig. 7. The longitudinal polarisation of the t lepton PP as at

function of the cosine of the polar angle of the ty. Included arethe data from 1990 to 1995. The solid line corresponds to the twoparameter fit of AA and AA and the dashed line to the onee t

parameter fit of AA under the assumption of lepton universality.ll

between data and Monte Carlo is found. Finally theimportant selection cuts are varied. The change inthe value of PP is assigned as a systematic error.t

The systematic error due to the uncertainty in thecharge confusion correction is negligible.

The theoretical error accounts for uncertainties iny y Ž . ythe decay radiation for t ™p n g and t ™t

y Ž . y yr n g and in the modelling of the a in t ™a nt 1 1 ty y Ž .decays. Structure dependent effects in t ™p n gt

w x y y27 amount to 0.002 in PP for the t ™p n finalt ty y Ž .state. The effect of decay radiation in t ™r n gt

is estimated to be 0.001. The a resonance and the1

a ™rp decay are analysed with different theoreti-1w xcal approaches 28 . The differences in the a mass1

and width are propagated to PP leading to a system-t

atic error of 0.010 on the ty™ayn result.1 t

The systematic errors on AA and AA are obtainedt e

by combining the individual ones in quadrature,allowing for correlations and polar angle depen-dences. They are summarised in Table 2.

5. Results

The values of AA and AA obtained with thet e

different analysis methods and decay channels areshown in Table 3. The results obtained using differ-ent analysis methods are in good agreement. Further-more, the results for the different t decay modesagree with each other.

The PP results from 1990 to 1995 data, given int

each cosu interval after combining the differentmethods, are shown in Fig. 7 and listed in Table 4together with the statistical and systematic errors.

Ž .From a fit to the PP distribution using Eq. 2 thet

final result on AA and AA is determined to be:t e

AA s0.1476"0.0088 stat . "0.0062 sys.Ž . Ž .t

AA s0.1678"0.0127 stat . "0.0030 sys. .Ž . Ž .e

These are consistent with lepton universality. Underthis assumption, these results are combined to yield:

AA s0.1540"0.0074 stat . "0.0044 sys. .Ž . Ž .ll

The ratio of the vector to the axial-vector effectivecouplings is g rg s0.0775"0.0044 and theV All ll

Table 4Values of PP from 1990 to 1995 data, given as a function of cosu , corrected for QED Bremsstrahlung, g exchange and g-Z interference.t

The correction applied to the measured PP values is shown in the last columnt

cosu PP Dstat. Systematic errors D QEDt

Energy scale Selection Background

y0.94 , y0.83 0.008 0.048 0.017 0.010 0.010 0.000y0.72 , y0.55 y0.009 0.025 0.008 0.004 0.003 0.000y0.55 , y0.35 0.011 0.023 0.008 0.004 0.003 y0.001y0.35 , y0.12 y0.080 0.025 0.008 0.005 0.003 y0.002y0.12 , 0.12 y0.171 0.023 0.008 0.005 0.003 y0.004

0.12 , 0.35 y0.212 0.025 0.008 0.005 0.003 y0.0050.35 , 0.55 y0.283 0.023 0.008 0.004 0.003 y0.0050.55 , 0.72 y0.283 0.024 0.008 0.004 0.003 y0.0050.83 , 0.94 y0.273 0.045 0.018 0.010 0.013 y0.005


corresponding value of the effective weak mixingangle is:

2sin u s0.2306"0.0011 .W

Acknowledgements

We wish to express our gratitude to the CERNaccelerator division for the excellent performance ofthe LEP machine. We acknowledge the contributionsof all the engineers and technicians who have partici-pated in the construction and maintenance of thisexperiment.

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18 June 1998


Search for an excess in the production of four-jet eventsq y (from e e collisions at s s130–184 GeV

OPAL Collaboration

K. Ackerstaff h, G. Alexander v, J. Allison p, N. Altekamp e, K.J. Anderson i,S. Anderson l, S. Arcelli b, S. Asai w, S.F. Ashby a, D. Axen ab, G. Azuelos r,1,

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R. Burgin j, P. Capiluppi b, R.K. Carnegie f, A.A. Carter m, J.R. Carter e,¨C.Y. Chang q, D.G. Charlton a,2, D. Chrisman d, P.E.L. Clarke o, I. Cohen v,

J.E. Conboy o, O.C. Cooke h, C. Couyoumtzelis m, R.L. Coxe i, M. Cuffiani b,S. Dado u, C. Dallapiccola q, G.M. Dallavalle b, R. Davis ac, S. De Jong l,

L.A. del Pozo d, A. de Roeck h, K. Desch h, B. Dienes af,3, M.S. Dixit g, M. Doucet r,E. Duchovni y, G. Duckeck ag, I.P. Duerdoth p, D. Eatough p, P.G. Estabrooks f,

E. Etzion v, H.G. Evans i, M. Evans m, F. Fabbri b, A. Fanfani b, M. Fanti b,A.A. Faust ac, L. Feld h, F. Fiedler z, M. Fierro b, H.M. Fischer c, I. Fleck h,

R. Folman y, D.G. Fong q, M. Foucher q, A. Furtjes h, D.I. Futyan p,¨P. Gagnon g, J.W. Gary d, J. Gascon r, S.M. Gascon-Shotkin q, N.I. Geddes t,

C. Geich-Gimbel c, T. Geralis t, G. Giacomelli b, P. Giacomelli d, R. Giacomelli b,V. Gibson e, W.R. Gibson m, D.M. Gingrich ac,1, D. Glenzinski i, J. Goldberg u,

M.J. Goodrick e, W. Gorn d, C. Grandi b, E. Gross y, J. Grunhaus v, M. Gruwe z,´C. Hajdu ae, G.G. Hanson l, M. Hansroul h, M. Hapke m, C.K. Hargrove g,

P.A. Hart i, C. Hartmann c, M. Hauschild h, C.M. Hawkes e, R. Hawkings z,R.J. Hemingway f, M. Herndon q, G. Herten j, R.D. Heuer h, M.D. Hildreth h,

J.C. Hill e, S.J. Hillier a, P.R. Hobson x, A. Hocker i, R.J. Homer a, A.K. Honma aa,1,D. Horvath ae,4, K.R. Hossain ac, R. Howard ab, P. Huntemeyer z, D.E. Hutchcroft e,´ ¨

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( )K. Ackerstaff et al.rPhysics Letters B 429 1998 399–413400

P. Jovanovic a, T.R. Junk h, J. Kanzaki w, D. Karlen f, V. Kartvelishvili p,K. Kawagoe w, T. Kawamoto w, P.I. Kayal ac, R.K. Keeler aa, R.G. Kellogg q,

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M. Thiergen j, M.A. Thomson h, E. von Torne c, E. Torrence h, S. Towers f,¨I. Trigger r, Z. Trocsanyi af, E. Tsur v, A.S. Turcot i, M.F. Turner-Watson h,´ ´

I. Ueda w, P. Utzat k, R. Van Kooten l, P. Vannerem j, M. Verzocchi j, P. Vikas r,E.H. Vokurka p, H. Voss c, F. Wackerle j, A. Wagner z, C.P. Ward e, D.R. Ward e,¨

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G. Yekutieli y, V. Zacek r, D. Zer-Zion h

( )K. Ackerstaff et al.rPhysics Letters B 429 1998 399–413 401

a School of Physics and Astronomy, UniÕersity of Birmingham, Birmingham B15 2TT, UKb Dipartimento di Fisica dell’ UniÕersita di Bologna and INFN, I-40126 Bologna, Italy`

c Physikalisches Institut, UniÕersitat Bonn, D-53115 Bonn, Germany¨d Department of Physics, UniÕersity of California, RiÕerside, CA 92521, USA

e CaÕendish Laboratory, Cambridge CB3 0HE, UKf Ottawa-Carleton Institute for Physics, Department of Physics, Carleton UniÕersity, Ottawa, Ont., K1S 5B6, Canada

g Centre for Research in Particle Physics, Carleton UniÕersity, Ottawa, Ont., K1S 5B6, Canadah CERN, European Organisation for Particle Physics, CH-1211 GeneÕa 23, Switzerland

i Enrico Fermi Institute and Department of Physics, UniÕersity of Chicago, Chicago, IL 60637, USAj Fakultat fur Physik, Albert Ludwigs UniÕersitat, D-79104 Freiburg, Germany¨ ¨ ¨

k Physikalisches Institut, UniÕersitat Heidelberg, D-69120 Heidelberg, Germany¨l Indiana UniÕersity, Department of Physics, Swain Hall West 117, Bloomington, IN 47405, USA

m Queen Mary and Westfield College, UniÕersity of London, London E1 4NS, UKn Technische Hochschule Aachen, III Physikalisches Institut, Sommerfeldstrasse 26-28, D-52056 Aachen, Germany

o UniÕersity College London, London WC1E 6BT, UKp Department of Physics, Schuster Laboratory, The UniÕersity, Manchester M13 9PL, UK

q Department of Physics, UniÕersity of Maryland, College Park, MD 20742, USAr Laboratoire de Physique Nucleaire, UniÕersite de Montreal, Montreal, Que., H3C 3J7, Canada´ ´ ´ ´

s UniÕersity of Oregon, Department of Physics, Eugene, OR 97403, USAt Rutherford Appleton Laboratory, Chilton, Didcot, Oxfordshire OX11 0QX, UK

u Department of Physics, Technion-Israel Institute of Technology, Haifa 32000, Israelv Department of Physics and Astronomy, Tel AÕiÕ UniÕersity, Tel AÕiÕ 69978, Israel

w International Centre for Elementary Particle Physics and Department of Physics, UniÕersity of Tokyo, Tokyo 113, Japanand Kobe UniÕersity, Kobe 657, Japan

x Institute of Physical and EnÕironmental Sciences, Brunel UniÕersity, Uxbridge, Middlesex UB8 3PH, UKy Particle Physics Department, Weizmann Institute of Science, RehoÕot 76100, Israel

z UniÕersitat HamburgrDESY, II Institut fur Experimental Physik, Notkestrasse 85, D-22607 Hamburg, Germany¨ ¨aa UniÕersity of Victoria, Department of Physics, P.O. Box 3055, Victoria, BC V8W 3P6, Canada

ab UniÕersity of British Columbia, Department of Physics, VancouÕer, BC V6T 1Z1, Canadaac UniÕersity of Alberta, Department of Physics, Edmonton, AB T6G 2J1, Canada

ad Duke UniÕersity, Department of Physics, Durham, NC 27708-0305, USAae Research Institute for Particle and Nuclear Physics, P.O. Box 49, H-1525 Budapest, Hungary

af Institute of Nuclear Research, P.O. Box 51, H-4001 Debrecen, Hungaryag Ludwigs-Maximilians-UniÕersitat Munchen, Sektion Physik, Am Coulombwall 1, D-85748 Garching, Germany¨ ¨

Received 29 January 1998Editor: K. Winter

Abstract

Events with four distinct jets from eqey collisions, collected by the OPAL detector at centre-of-mass energies between130 and 184 GeV, are analysed for a peak in the sum of dijet masses. This search is motivated by the ALEPHCollaboration’s observation of a clear excess of events with dijet mass sums close to 105 GeV in data taken atcentre-of-mass energies of 130 and 136 GeV in 1995. We have observed no significant excess of four-jet events compared tothe Standard Model expectation for any dijet mass sum at any energy. Our observation is inconsistent with the excessobserved by ALEPH in 1995. Upper limits are determined on the production cross-section as a function of the dijet masssum. q 1998 Elsevier Science B.V. All rights reserved.

1 And at TRIUMF, Vancouver, Canada V6T 2A32 And Royal Society University Research Fellow3 And Department of Experimental Physics, Lajos Kossuth University, Debrecen, Hungary4 And Institute of Nuclear Research, Debrecen, Hungary5 And Department of Physics, New York University, NY 1003, USA


1. Introduction

In a run of LEP in 1995 at centre-of-mass ener-'gies of s s130 and 136 GeV, the ALEPH Collabo-

w xration observed 1 an excess of events with fourdistinct jets compared with the Standard Model ex-pectation. Such an excess could be due to the pro-duction of new particles X and Y, each decaying intotwo hadronic jets in the process eqey™XY™ fourjets. The two particles could have equal or unequalmasses. Grouping the jets into pairs, calculating theirpair invariant masses M and M , and selecting thei j k l

combination yielding the smallest mass difference< <DMs M yM , ALEPH observed a clustering ofi j k l

nine events in a mass window 6.3 GeV wide centredaround MsM qM f105 GeV, with a Standardi j k l

Model expectation of 0.8 events in this mass win-dow. The choice of the combination with the mini-mum DM would tend to favour the selection ofparticles of equal mass or with a small mass differ-ence.

In response to this observation, the OPAL Collab-oration performed an analysis that closely followed

w xthe selection of Ref. 1 . In its 130 and 136 GeV datafrom 1995, OPAL observed seven events with Mbetween 60 and 130 GeV, with an expected StandardModel background of 6.4"0.6 events. In the signalregion indicated by ALEPH, OPAL observed oneevent, consistent with the expected Standard Model

w xbackground of 0.8"0.2 events 2 . The estimatedefficiency of the OPAL analysis and the resolutionon the dijet mass sum are similar to those obtainedby ALEPH. Consequently, the OPAL detector wouldbe expected to have a sensitivity comparable to theALEPH detector for a four-jet signal should oneexist. The ALEPH Collaboration also reported aslight excess at the higher centre-of-mass energies of

w x161 and 172 GeV 3 . The DELPHI Collaborationobserved no significant peak at 105 GeV in a similaranalysis using their 1995 data at 130 and 136 GeVw x4 . The L3 Collaboration likewise reported no ex-cess of events in the indicated mass window for' w xs s130–172 GeV 5 . Nonetheless, there has beena great deal of theoretical speculation on the cause of

w xthe excess observed by ALEPH 6–8 .'In 1997, LEP made short runs at s s130 and

136 GeV with an integrated luminosity similar tothat of 1995 at these centre-of-mass energies to test

again the signal hypothesis. We add these data to ourw xsample described in Ref. 2 , and also include data

collected at 161, 172 and 183 GeV. To search for theclass of events observed by ALEPH in a modelindependent fashion, we have performed analyses onthe OPAL data as close as possible to the ALEPH

w xanalyses at these energies 3 . However, above thekinematic threshold for W-pair production, a veto isimposed to suppress this new source of backgroundand results are presented with and without this re-quirement.

The comparison of results of the OPAL emulationof the ALEPH selection to the ALEPH observationof an excess does not depend upon the underlyingmodel of possible new physics if only the number ofobserved events is compared. In the context of amodel of the process eqey™XY™ four jets, aseparate analysis is also presented that is intended toimprove the sensitivity for values up to 30 GeV ofthe difference in mass between the two producedparticles. This broader search is motivated by thefact that an analysis performed by the ALEPH Col-laboration, using a kinematic fit which constrains the

w xmasses of the two dijet systems to be equal 9 ,indicates that the excess events are not consistentwith the hypothesis that the produced particles haveequal mass. Furthermore, compared with the ALEPHemulation analysis, this OPAL-specific analysis isestimated to be more sensitive to a four-jet signal ofequal-mass particle production at higher energies,and its efficiency is less dependent on the flavour ofthe final-state quarks. It is used in addition to theemulation of the ALEPH analysis to set cross-sectionlimits as function of the dijet mass sum, and also toprovide limits in the case of nonzero mass differ-ence.

2. The OPAL detector

A detailed description of the OPAL detector canw xbe found elsewhere 10 . OPAL’s nearly complete

solid angle coverage and excellent hermeticity en-able it to detect the four-jet final state with highefficiency. The central tracking detector consists of a

w xtwo-layer silicon microstrip detector 11 with polar


6 < <angle coverage cosu - 0.9, immediately sur-rounding the beam-pipe, followed by a high-preci-sion vertex drift chamber, a large-volume jet cham-ber and z-chambers, all in a uniform 0.435 T axialmagnetic field. A lead-glass electromagneticcalorimeter is located outside the magnet coil, which,in combination with the forward calorimeter, gamma

w xcatcher and silicon-tungsten luminometer 12 , com-plete the geometrical acceptance down to 24 mradfrom the beam direction. The silicon-tungsten lumi-nometer serves to measure the integrated luminosity

w xusing small-angle Bhabha scattering events 13 . Themagnet return yoke is instrumented with streamertubes for hadron calorimetry and is surrounded byseveral layers of muon chambers.

3. Data and Monte Carlo simulations

The data used in these analyses were collected inw xfive separate running periods. The energies 14 and

w xintegrated luminosities 13 for the five data samplesare given in Table 1. The 130 and 136 GeV data of1995 and 1997 are collectively referred to in thisletter as the 133 GeV data. The other three samplesare referred to as the 161 GeV data, the 172 GeVdata, and the 183 GeV data, and are analysed sepa-rately.

The main backgrounds for the selection of anoma-0 )lous four-jet events are Z rg ™qq production and

Standard Model four-fermion production processes.Monte Carlo samples modelling the backgrounds

w xhave been prepared using PYTHIA 5.7 15 for the0 ) w xZ rg ™qq process and EXCALIBUR 16 and

w xgrc4f 17 for the Standard Model four-fermion pro-cesses, all using JETSET 7.4’s parton shower and

w xhadronization models 15 . For the generation ofStandard Model four-fermion processes, the W massis taken to be 80.33 GeV. Two-photon processes

w xgenerated by PYTHIA, HERWIG 18 , and PHOJETw x19 were used to estimate the contribution of these

6 OPAL uses a right-handed coordinate system where the q zdirection is along the electron beam and where q x points to thecentre of the LEP ring. The polar angle, u , is defined with respectto the q z direction and the azimuthal angle, f, with respect tothe q x direction.

Table 1Summary of the data samples, luminosity-weighted centre-of-massenergies, year of collection, and integrated luminosities used inthese analyses

y1' Ž . Ž .s GeV Year HLL d t pb

130.3 1995 2.7136.2 1995 2.6

130.0 1997 2.6136.0 1997 3.4

161.3 1996 10.0

172.1 1996 10.3

182.7 1997 57.1

processes to the Standard Model background in theearly stages of the analysis.

The signal detection efficiencies were estimatedw xusing the HZHA generator 20 to simulate the pro-

duction of supersymmetric Higgs bosons eqey™0 0 q yh A ™bbbb as a model for the signal process e e

™XY™4 jets. Samples with decays into otherquark flavours were also used to check for flavourdependence. All Monte Carlo samples were pro-cessed through a full simulation of the OPAL detec-

w xtor 21 .

4. Analysis and results

The main features of the signal process are fourwell-defined, energetic, hadronic jets and a totalvisible event energy close to the centre-of-mass en-ergy. The Standard Model background expectationchanges considerably in size and composition be-

'tween s s133 GeV and 183 GeV. At 133 GeV, the0 )main background comes from Z rg ™qq both

with or without initial-state radiation and accompa-nied by hard gluon emission. Above the threshold for

q y q y 'e e ™W W at s s161 GeV, the backgroundfrom Standard Model four-fermion processes is im-

'portant and becomes larger with increasing s . Aq yprocedure to reject W W ™qqqq is implemented

for centre-of-mass energies of 161 GeV and above.Events are reconstructed from charged particle

Ž .tracks and energy deposits ‘‘clusters’’ in the elec-


tromagnetic and hadronic calorimeters. Tracks arerequired to originate from close to the interactionpoint, to have more than a minimum number of hitsin the jet chamber, and to have a transverse momen-tum greater than 0.1 GeV and a total momentum less

w xthan 100 GeV 22 . Energy clusters in the electro-magnetic and hadron calorimeters are required toexceed minimum energy thresholds. Tracks and clus-ters passing these quality requirements are then pro-cessed to reduce double-counting of energy and mo-mentum in the event by matching charged trackswith calorimeter clusters. The energy-momentum

w xflow obtained with this algorithm 23 is usedthroughout the analysis. Energy measured in the

< <forward detectors, covering cosu )0.985, has notbeen included in the analyses presented here.

Selection criteria emulating the ALEPH analysisw x1 will first be described followed by the descriptionof another set of requirements for event selection inan OPAL-specific analysis. Efficiencies and back-grounds for the two analyses are given followed byestimates of systematic errors on these quantities.

4.1. OPAL emulation of the ALEPH selection

The procedure for selecting four-jet hadronicevents and reconstructing the dijet masses is de-scribed below, emulating the ALEPH selection as

w xdescribed in Ref. 1 and subsequent modificationsw xand developments as described in Ref. 3 . The num-

ber of events retained after each cut in sequence isgiven in Table 2, together with the expectation from

0 )Z rg ™qq, Standard Model four-fermion and'two-photon background processes, for s s133 GeV

and for the sum over all centre-of-mass energies.Table 2 also lists the efficiencies for the reference

0 0 'h A Monte Carlo at s s133 GeV after each step.1. Events are required to have at least five charged

tracks and seven electromagnetic calorimeter clus-ters. The sum of the electromagnetic calorimetercluster energies should be at least 10% of thecentre-of-mass energy, and the electromagneticcalorimeter energy is required to be roughly bal-anced along the beam direction: ÝE cosu Fi i

0.65ÝE , where the sums run over measured elec-i

tromagnetic calorimeter clusters. The propertiesw xof this selection are detailed in Ref. 22 .

2. To remove events with a real Z0 and largeŽ .initial-state radiation radiative return events ,

< vis < Ž Ž ..events must satisfy p FK M y90 GeV ,z vis

where pvis is the momentum sum along the beamz2 2(direction and M s E yp is the total ob-vis vis vis

'served mass. K is a coefficient depending on s .For the 133 GeV sample, Ks0.75; for the 161GeV sample, Ks1.50; and for the 172 GeV and183 GeV samples, Ks1.65.

Table 20 )Event counts observed by OPAL at the various selection stages, with backgrounds estimated using PYTHIA for Z rg ™qq,

EXCALIBUR and grc4f for Standard Model four-fermion processes, and PYTHIA and PHOJET for two-photon processes. Signal0 0 Ž . 0 0efficiencies at 133 GeV for h A see text for M sM s55 GeV are also listedh A

Ž . Ž . Ž . Ž . Ž . Ž . Ž . Ž . Ž . Ž .Cut 1 2 3 4 5 6 7 8 9 10

0 )'s s133 GeV Z rg ™qq 3151. 1185. 153.2 55.1 50.8 45.7 22.5 18.0 14.6 14.64-Fermion 24.5 16.1 4.2 3.6 3.1 3.0 1.6 1.1 0.7 0.7gg 161.0 8.9 1.7 1.7 1.7 1.7 1.1 0.0 0.0 0.0

Total SM backg. 3337. 1211. 159.2 60.4 55.6 50.4 25.2 19.1 15.3 15.3Observed 3372 1165 147 51 47 42 19 16 13 13

Ž .Sig. effic., e % 99.8 90.2 69.0 68.6 67.4 67.2 65.4 48.6 45.4 45.4hA

0 )'All s Z rg ™qq 10456. 4450. 481.1 188.7 176.7 155.0 68.2 59.7 49.2 40.2130–184 GeV 4-Fermion 1098. 913.0 476.0 428.0 400.5 364.8 258.1 236.3 191.7 74.2

gg 450.7 26.3 2.7 2.2 2.1 1.8 1.0 0.0 0.0 0.0

Total SM backg. 12004. 5389. 959.9 618.9 579.4 521.6 327.4 295.9 240.8 114.3Observed 12617 5461 984 635 592 527 328 299 240 92


3. Jets are formed with the Durham jet-finding algo-w xrithm 24 with its resolution parameter y set tocut

0.008. Selected events are required to have fouror more jets. For events with five or more jets,the jet pair with the smallest invariant mass iscombined into a single jet and this procedure isrepeated until four jets are left.

4. The contribution of radiative return events is fur-ther reduced by requiring for each jet that theenergy observed in the electromagnetic calorime-ter, after subtracting the energy expected to havebeen deposited by the jet’s charged hadrons, isless than 80% of the jet energy. The expectedhadronic energy in the electromagnetic calorime-ter is calculated using a track-cluster matching

w xalgorithm 23 .5. All jet masses are required to exceed 1.0 GeV to

suppress further the contribution from radiativereturn events.

6. The energies and momenta of each jet are rescaledimposing conservation of energy and momentumfor the event using the beam energy constraint.The jet velocities b sp rE are held fixed in thei i i

scaling. If one or more scaling factors is negative,the event is rejected. The rescaled jet energies andmomenta are used in the following stages of theselection.

7. To suppress events involving gluon radiation, alltwo-jet combinations are required to have an in-variant mass of more than 19.2% of the centre-of-mass energy. This initially corresponded to 25GeV for the 133 GeV sample.

8. All combinations of jet pairs must have a sum ofŽ .the individual jet masses M qM )10 GeV.i j

9. All combinations of jet pairs must have a totalcharged multiplicity of at least 10.

10.W-pair veto: For the 161 GeV data sample, arequirement is placed on the dijet mass sum foreach of the three possible pairings of jets: MF150 GeV for the pairing with the smallest DM,MF152 GeV for the pairing with intermediateDM, and MF156 GeV for the pairing with thelargest DM. For the 172 and 183 GeV data, it is

< <required that My160 G10 GeV for the pairingwith the smallest mass difference if DM is lessthan 15 GeV, and the same condition is applied tothe pairing with the second-smallest mass differ-ence if the second smallest DM is less than 30

GeV. No W-pair veto is applied to the 133 GeVsample.The dijet mass sum for the combination with the

smallest dijet mass difference is shown in Fig. 1,separately for the data samples at the four different

'values of s , after the W-pair veto. The expectedStandard Model background distribution is shownwith the data for each case. No significant excess isobserved at any of the centre-of-mass energies.

The sensitivity of the analysis to a peak at aparticular dijet mass sum depends on the resolution

Fig. 1. The dijet mass sum for the combination with the smallestDM after the W-pair veto in OPAL’s emulation of the ALEPHanalysis shown separately for the different centre-of-mass ener-gies. Data are shown by the points and Standard Model back-grounds by the histograms. The hatched component of the back-ground histograms denotes Standard Model four-fermion pro-

0 )cesses, while the unhatched component denotes Z rg ™qq.The mass windows containing the region of interest are indicated

Ž .by the arrows. The dashed histogram in a illustrates a signalŽ . 0 0plus background that could be expected due to h A with bothdecaying to pairs of b-quark jets, and M 0 s M 0 s52.5 GeV,h A

normalized to the excess observed by ALEPH at E f133 GeVcmw x1 .


and may be affected by energy scale biases. Theresolution was investigated using the HZHA event

w x q y 0 0generator 20 to model the process e e ™h A .The masses of both the h0 and the A0 are taken to beequal to 55 GeV with negligible width. The resolu-tions found for the reconstructed dijet mass sum Mfor the combination with the smallest DM are s sM'2.0, 2.8, 3.0, and 3.0 GeV for s s133 GeV, 161GeV, 172 GeV, and 183 GeV, respectively. Theseresolutions do not include effects of significant non-Gaussian tails which arise from wrong jet-pair com-binations, where the correct jet-pair combination iswhen each of the two jets comes from the decay of

'the same particle. For example, at s s133 GeV,38% of the events fall into these tails. This definitionof the resolution and tails is the same as that used in

w xthe ALEPH publication 1 and the resolution valuesfound are similar to those of ALEPH. The degrada-tion of mass resolution with increasing energy arisesfrom the scaling of the jet energies to the beamenergy and also from the energy dependence of thedetector resolution.

Studies of the h0A0 signal Monte Carlo withsamples generated with input masses adding to 110GeV at each centre-of-mass energy show that thereconstructed M distributions have peaks at massesconsistent with this input value within their errors ofapproximately 0.5 GeV. Studies of events from ra-

0diative returns to the Z , qqg, were also used tocheck that the Z0 peak is well simulated in positionand shape, further indicating that there is no signifi-cant bias in M or degradation in resolution inherentto the selection or the mass reconstruction procedure.

For the 133 GeV signal Monte Carlo, if thereconstructed dijet mass sum for the jet-pair combi-nation having the smallest DM is required to be

Ž .within 2s 4.0 GeV of the generated mass sum,M

the efficiency obtained is 26.6%, which is 60% ofthe efficiency obtained without the requirement onM. The efficiencies and expected backgrounds bothbefore and after the mass window cut are similar to

w xthose obtained by ALEPH 1 so that for the sameintegrated luminosity, the observed number of eventscan be directly compared to the number observed in

w xRef. 1 . At higher centre-of-mass energies, accept-ing events only in a mass window of width "2sM

results in efficiencies of 18–21% which is 66–69%of the efficiency before the mass window require-ment.

To search for an excess of four-jet events withdijet mass sums near 105 GeV as motivated by Ref.w x < <1 , events satisfying My105 GeV -2s for theM

combination with the smallest DM were countedand the Standard Model backgrounds were esti-mated. These mass windows are shown in Fig. 1 andthe results of the searches within these mass regionsare given in Table 3 both before and after the W-pairveto, when applicable. No significant excess is seenin any sample. Combining data from all centre-of-mass energies, nine events are observed while 11.5"0.4 are expected from Standard Model processes.

Fig. 2 shows the distribution of the dijet masssum for the jet pairing with the smallest DM for allrunning periods combined, with and without theW-pair veto. The data agree well with the StandardModel background simulation and no excess is ob-

Table 3Observed event count and expected Standard Model background for selected events close to 105 GeV, for the combination with the smallestDM, before and after the W-pair veto. No W-pair veto is applied for the 133 GeV data. The mass window is chosen to allow events that arewithin "2s of 105 GeV to be included, where s is the expected experimental resolution on M as given in the text. Signal efficienciesM M

0 0 Ž .0 0apply to h A production with M sM s55 GeV in which case efficiencies are found for a mass window within "2s of 110 GeVh A M

Data sample Without W-pair veto With W-pair veto

Observed Expected Sig. eff., e Observed Expected Sig. eff., ehA hA

133 GeV 1 1.7"0.2 26.6% 1 1.7"0.2 26.6%161 GeV 1 1.5"0.1 26.2% 0 1.0"0.1 18.4%172 GeV 3 2.9"0.1 28.3% 0 1.8"0.1 20.7%183 GeV 13 11.5"0.4 26.0% 8 7.0"0.3 18.3%

Total 18 17.7"0.5 –- 9 11.5"0.4 –-


Fig. 2. The dijet mass sum in OPAL’s emulation of the ALEPHŽ .analysis for the combined 130–184 GeV samples. Plots a and

Ž .b show the distribution of the dijet mass sum before and afterthe W-pair veto, respectively. Data are shown by the points andStandard Model backgrounds by the histograms. The hatchedcomponent of the background histograms denotes Standard Modelfour-fermion processes and the unhatched component denotes

0 )Z rg ™qq. The mass window around 105 GeV whose width'accommodates the resolution at s s183 GeV, is shown with the

Ž .arrows. Plot c shows the sliding mass window scan for the sameanalysis after the W-pair veto. The hatched histograms show thetotal number of data events, and the solid line shows the StandardModel expectation; the line width indicates the Monte Carlostatistical error. An arrow is drawn at 105 GeV.

served in the region 99.0-M-111.0 GeV, wherethe width has been chosen to accommodate theresolution at the highest energy. A clear peak may beseen at twice the W mass in the sample before theW-pair veto.

To test for a peak in the dijet mass sum distribu-tion for arbitrary mass M and independent of his-togram binning, the positions of the mass windowswere scanned over the full range of M. The results

Ž .are shown in Fig. 2 c for the combined data sam-ples. The figure displays the event counts withinwindows of fixed width but whose centres are ad-justed in steps of 50 MeV. The width of the mass

window is "4.0, "5.6, "6.0 GeV, and "6.0 GeVfor the 133, 161, 172, and 183 GeV data samples,respectively, to reflect the resolution. The contents ofnearby bins in these scans have high statistical corre-lations. No significant excess is observed in the masswindow scan at any value of the dijet mass sum. Inparticular, no choice of binning produces a peak near105 GeV.

To check for a possible signal in the fraction ofevents with wrong jet-pair combinations, the dijetmass sum for the jet pairing with the second-smallestDM was also considered. If the mass difference of apair of objects produced together were 20 GeV, thecorrect jet pairing would yield the smallest DM forroughly half of the signal, and the second-smallestDM for most of the remainder. In the ALEPH

w xanalysis 1 , including the second combination to thedijet mass sum distribution resulted in three addi-tional events within the mass window with an addi-tional 1.2 events expected from Standard Modelprocesses. Fig. 3 shows the effect of adding the dijetmass sum distributions for the smallest and second-smallest DM combinations for different centre-of-

Fig. 3. The dijet mass sum in OPAL’s emulation of the ALEPHanalysis for both the combination with the smallest DM and the

Ž .combination with the second-smallest DM for a the 133 GeVŽ .data sample and b the combined 130–184 GeV samples. The

points and histograms are as in Fig. 1.


mass energies. The distributions agree well with theStandard Model prediction and no peak arises whenthe second combination is included.

4.2. OPAL-Specific analysis

In the above analysis, an emulation of the ALEPHselection criteria was applied to the OPAL data totest for the presence of events of the type observedby ALEPH. The selection described below is anOPAL-specific analysis in which the sensitivity hasbeen maximised for detecting a possible signal forthe process eqey™XY in the form of an excess ofevents with similar mass sums MsM qM in theX Y

four-jet topology. The analysis is designed to retainsensitivity even when the mass difference DMs< <M yM is as large as 30 GeV. Efficiencies andX Y

backgrounds are estimated for different values of Mand DM.

The cuts are designed to be as insensitive aspossible to the flavours of the final state quarks.Although the methods employed at each of the cen-tre-of-mass energies are similar, the optimal cut val-

'ues in most cases depend on s .1. The events must pass the hadronic final state

requirement of cut 1 in Section 4.1.2. The effective centre-of-mass energy after initial-

X'state radiation, s , calculated using the method'w xdescribed in Ref. 25 , has to be at least 0.87 s .

The measured visible mass, M , is required tovis' 'be between s y40 GeV and s q30 GeV at'133 GeV, between 100 and 200 GeV at s s161'GeV, between 110 and 210 GeV at s s172'GeV and between 120 and 220 GeV at s s183

GeV.3. The charged particles and calorimeter clusters are

grouped into four jets using the Durham algo-w xrithm 24 . The jet resolution parameter, y , at34

which the number of jets changes from three to'four, is required to be larger than 0.007 at s s

'133 GeV, and larger than 0.005 at s s161–183GeV. To discriminate against poorly recon-structed events, a kinematic fit imposing energyand momentum conservation is required to yield ax 2 probability larger than 0.01. Each of the fourjets is required to contain at least two tracks at133 GeV and at least one track at higher energies.

These kinematically constrained jets are used inthe subsequent calculation of dijet masses.

4. In the case of the 161–183 GeV data, the back-ground from eqey™Z0g is further reduced byeliminating those events where one of the fourjets has properties compatible with those of aradiative photon, namely that it has exactly oneelectromagnetic cluster, not more than two tracksŽ .possibly from a photon conversion , and energy

'between 45 and 65 GeV at s s161 GeV, be-'tween 52 and 72 GeV at s s172 GeV and'between 60 and 80 GeV at s s183 GeV.

5. The polar angle of the thrust axis, u , is requiredthr< < < <to satisfy cosu -0.9 at 133 GeV and cosuthr thr

-0.8 at 161–183 GeV.6. To reduce background from qq events, the event

w xshape parameter C 26 , which ranges between 0and 1 and is 0 for a perfect 2-jet event, is required

'to be larger than 0.7 at s s133 GeV and largerthan 0.6 at higher energies.

7. To ensure well-separated jets for better kinematicfits, the angle between any two jets is required toexceed 0.8 radians for 161–183 GeV data.

8. Above the WqWy threshold, explicit vetoesagainst the process eqey™WqWy are applied.

"'At s s161 GeV, the two W bosons are pro-duced with only a small boost. The two jetshaving the largest opening angle are assigned toone of the W " bosons and the two remaining jetsto the other. An event is rejected if both jet pairshave an invariant mass between 75 GeV and 90GeV.

'At s s172 and 183 GeV, a more sophisticatedveto is applied. The four jets are combined intopairs, and for all three combinations the event is

'refitted constraining the total energy to s andthe total momentum to zero, and also constraining

Žthe masses of the two jet pairs to be equal five.constraints . From the three combinations, the one

yielding the largest x 2 fit probability is consid-ered. If the jet pair mass from the fit exceeds 75GeV and the fit probability is at least 0.01, theevent is rejected.

9. To achieve good sensitivity for all DM less than30 GeV, we use two separate mass selections, onerelevant for unequal masses and one for equalmasses. In both selections, when searching for asignal with a hypothetical sum of masses, M , the0


Table 4Number of observed and expected Standard Model background events before and after the W-pair veto and after adding the mass selectionŽ .cut 9 centred at 105 GeV for the smallest DM combination. The quoted errors are statistical. No W-Pair veto has been applied to the 133GeV data

Data sample Without W-pair veto With W-pair veto After mass selection

Observed Expected Observed Expected Observed Expected

133 GeV 18 17.0"0.6 18 17.0"0.6 4 3.1"0.3161 GeV 11 15.8"0.3 8 13.6"0.3 2 2.7"0.1172 GeV 36 33.8"0.3 21 16.2"0.2 4 2.9"0.1183 GeV 190 210.1"1.2 70 81.6"0.8 6 8.9"0.3

Total 255 276.7"1.4 117 128.4"1.1 16 17.6"0.4

range M "2s is used, where s is 2.0 GeV0 M M'at s s133 GeV and 3.0 GeV for higher ener-Ž .gies. For unequal masses DM )5 GeV , the

event is selected if either the jet association withthe smallest mass difference or the one with thesecond smallest mass difference has a mass sumM in the range M "2s . For nearly equal0 M

Ž .masses DM -5 GeV , better sensitivity is ob-tained when considering only the jet associationwith the smallest mass difference. The resolutions varies only slowly with M and DM.M

Table 4 presents the number of observed eventsand the Standard Model expectations before and

q y Ž .after the W W veto cut 8 . The numbers of

observed events are consistent with the StandardModel expectations at all centre-of-mass energiesboth before and after the mass selection.

Table 5 shows the signal efficiencies for variousŽ .combinations of M , M together with the pre-X Y

dicted background and the numbers of observedevents after all cuts.

Fig. 4 shows the distributions of M for the jetassociations with the smallest DM, and for the jetassociation with the smallest and second-smallestDM, summed over all centre-of-mass energies. Glob-ally, the distributions show consistency between thedata and the Standard Model background prediction.In particular, there is no excess in the vicinity of M

Table 5Signal detection efficiencies, numbers of expected background events and number of observed data events, for various mass combinations inthe OPAL-specific analysis, after the mass selection, cut 9 of Section 4.2. The quoted errors are statistical

Ž .M , M 133 GeV 161 GeV 172 GeVfs 183 GeVX Y

Ž . Ž . Ž . Ž . Ž .GeV Effic. % Backgd. Data Effic. % Backgd. Data Effic. % Backgd. Data Effic. % Backgd. Data

Ž .50,50 30.6"1.5 3.2"0.3 3 38.0"2.2 2.4"0.1 3 31.0"1.5 2.5"0.1 4 19.4"1.8 4.8"0.2 4Ž .40,60 29.6"2.0 5.6"0.4 6 35.8"2.1 4.2"0.2 3 24.6"1.9 3.2"0.1 5 16.3"1.6 5.6"0.2 6

Ž .55,55 29.0"2.0 3.1"0.3 4 37.4"1.0 3.0"0.1 1 34.4"1.0 3.6"0.2 5 31.6"2.1 12.4"0.3 9Ž .50,60 38.4"1.5 5.6"0.4 7 42.0"2.2 5.0"0.2 1 32.6"1.5 5.1"0.2 6 22.2"1.9 16.2"0.3 12Ž .40,70 23.0"1.9 5.6"0.4 7 34.8"2.1 5.0"0.2 1 23.0"1.3 5.1"0.2 6 18.0"1.7 16.2"0.3 12

Ž .60,60 26.2"1.4 3.2"0.3 3 39.0"2.2 2.8"0.1 0 33.0"1.5 3.7"0.2 9 32.6"2.1 17.1"0.4 20Ž .50,70 30.2"2.1 5.6"0.4 4 41.2"2.2 4.8"0.2 1 34.4"2.1 5.5"0.2 11 23.9"1.9 23.5"0.4 30

Ž .60,70 24.2"1.9 2.2"0.2 4 34.2"2.1 4.7"0.2 2 33.6"1.5 5.5"0.2 6 32.2"2.1 28.2"0.4 24Ž .50,80 13.1"1.1 2.2"0.2 4 32.8"2.1 4.7"0.2 2 28.8"1.4 5.5"0.2 6 23.2"1.9 28.2"0.4 24

Ž .70,70 – – – 26.6"2.0 2.0"0.1 2 29.9"1.4 3.3"0.1 1 31.2"2.1 20.7"0.4 20Ž .60,80 – – – 33.8"2.1 4.4"0.2 3 31.0"2.1 6.0"0.2 4 27.7"2.0 32.3"0.5 28


Fig. 4. Distributions of M in the OPAL-specific analysis for thecombined 130–184 GeV samples after all selection requirements

Ž . Ž .except the mass selection cut 9 , a for the jet combination withŽ .the smallest DM and b for the jet combinations with the

smallest and second-smallest DM. The points and histograms areas in Fig. 1.

f105 GeV. The overlap of the OPAL-specific anal-ysis and the OPAL emulation of the ALEPH selec-tion has been evaluated in a typical Monte Carlo

' 0four-jet signal sample at s s133 GeV with M sh

M 0 s55 GeV. In this sample, 59% of the eventsA

selected by the OPAL emulation of the ALEPHanalysis are also selected by the OPAL-specific anal-ysis.

4.3. Systematic errors

'At s s133 GeV, since the efficiencies and ex-pected backgrounds for the OPAL emulation of theALEPH signal are similar to those obtained by

w xALEPH 1 , it is not necessary to consider systematiceffects in detail if only numbers of observed eventsare compared. However, to calculate limits oncross-sections, systematic errors on efficiencies andbackgrounds are estimated.

To emulate the ALEPH analysis, the chargedŽ .multiplicity requirement cut 9, Section 4.1 on all

combinations of two jets was necessary. Since theaim of the emulation analysis is to compare directly

the OPAL result with the ALEPH observation, andALEPH’s cross-section estimate was made assuminga model of four b-jets, we also assume this model.Varying the mean charged multiplicity in b-hadron

w xdecays by its measurement uncertainty 27 results inan estimated systematic error of 12% on signal de-tection efficiencies due to this effect. Including addi-tional uncertainties in the modelling of the cut vari-ables, energy scales, mass resolutions and limitedMonte Carlo statistics results in an estimated totalsystematic error of 13%.

In the OPAL-specific analysis, the signal detec-tion efficiencies are subject to a systematic error of9%, which includes an allowance for the final stateto contain any composition of quark flavours anduncertainties in modelling heavy hadron decays, theuncertainty on the simulation of the decay withregards to fragmentation and hadronization, the mod-elling of the cut variables, and the limited MonteCarlo statistics.

The total relative uncertainty on the residual back-ground is 20% for the OPAL emulation of theALEPH analysis, and 13% for the OPAL-specificanalysis. These errors include the uncertainty on themodelling of the hadronization process, on the pre-diction of the four-jet rate, W-pair cross-section, andthe modelling of the cut variables. The error due tothe limited Monte Carlo statistics is added in quadra-ture to this uncertainty. The systematic errors on theluminosity measurements range from 0.5% to 1.6%.

5. Cross-section upper limits

In the OPAL emulation of the ALEPH analysis,the number of observed events can be compared

w xdirectly to the ALEPH observation 1 because boththe observed background rate and the estimated effi-ciency are nearly identical to those obtained byALEPH. From the number of observed and expectedevents in the dijet mass sum window of 105"4

'GeV at s f133 GeV, we set a 95% confidenceŽ .level CL upper limit of 2.1 events that could be

attributed to additional cross-section from newphysics when scaled to the integrated luminosity ofthe 1995 ALEPH result. This can be compared toALEPH’s observation in 1995 of nine events with a


Standard Model expectation of 0.8 events. To calcu-late the probability that the OPAL observation isconsistent with the ALEPH observation in the pres-ence of a possible signal, the product is formed ofthe Poisson probability p that at least nine events1

were observed in ALEPH and p that no more than2

one event was observed in OPAL, given the Stan-dard Model backgrounds and assuming the presenceof a signal scaled by the integrated luminosity. Theprobability of an outcome no more likely than thatobserved in the data, i.e., the sum of Poisson proba-bilities of possible outcomes less than or equal top p , is found to be 2.6=10y4 , where the hypothe-1 2

sized signal cross-section has been chosen to maxi-mize this probability.

Assuming production of new particles X and Ysubsequently decaying to a final state of four b-jetsto determine efficiencies, cross-section upper limitsare set using the OPAL emulation of the ALEPHanalysis. From the number of observed and expectedevents in the dijet mass sum window as above at' Ž .s f133 GeV, a 95% confidence level CL upperlimit of 1.4 pb is determined for the productioncross-section at the dijet mass sum of 105 GeV.Systematic uncertainties of efficiency, backgroundand luminosity are taken into account using the

w xprocedure outlined in Ref. 28 .To combine data from different centre-of-mass

energies, we consider two different functions for theenergy dependence of the cross-section of a hypo-thetical signal. It is first assumed that the cross-sec-

Ž 2 .tion varies as b 3yb rs, typical for pair-produc-tion of spin-1r2 particles, where b is taken as theaverage velocity of the particles in the laboratory

w xframe 29 . Taking from Table 3 the total number ofobserved and expected events in resolution-depen-dent mass windows around 105 GeV, an upper limiton the production cross-section at 133 GeV of 0.58pb at 95% CL is found. Secondly, under the hypoth-esis that the signal cross-section varies as b 3rs,

w xtypical for the production of scalar particles 29 , theupper limit on the cross-section at 133 GeV iscomputed to be 0.31 pb at 95% CL. These limits canbe compared to ALEPH’s estimated cross-section of

w x3.1"1.7 pb 1 from their total number of excessevents.

The OPAL-specific analysis described in Section4.2 is used to obtain upper limits for the cross-sec-

tion of a possible signal process eqey™XY™ fourjets, in the presence of background from StandardModel processes, using Poisson statistics and incor-porating systematic uncertainties as described in Ref.w x q y 0 028 . The process e e ™h A was used to modelthe signal detection efficiencies. The resulting 95%

ŽCL upper limits, as function of the mass sum M '.M qM , are shown in Fig. 5, for DM close toX Y

zero and DMs30 GeV. A mass window of M"

2s is scanned across the distribution of the dijetM

mass sum in small steps. To account for a possiblediscrepancy between the mass scale of the data andthe Monte Carlo in a conservative manner, the masswindow is displaced by "0.5 GeV at each scanpoint. The largest data count in any of the threewindows including the nominal one and the smallestbackground estimation in any of the three windowsare used to compute the limit, with backgroundsubtraction as described previously.

When results at different centre-of-mass energiesare combined, the hypothetical production cross-sec-

Fig. 5. The 95% CL upper limits obtained with the OPAL-specificanalysis on the production cross-section of a possible signal as a

Ž .function of M for DM close to 0 solid lines and for DM -30Ž . Ž .GeV dashed lines . Plot a shows the limits computed using the

' Ž .data collected at s f133 GeV; plot b shows the limits using'the combined data from s s130–184 GeV assuming a cross-sec-

3 ' Žtion that varies as b r s scaled to s s133 GeV lines that end'. Žnear Ms130 GeV and s s183 GeV lines that extend to larger

.M .


tion is assumed to vary as b 3rs. The cross-sectionlimits are presented separately for the 133 GeV data,

Ž .and for all data 130–184 GeV combined. Limits onthe cross-section from the combined data sample are

' 'computed both at s s133 GeV and at s s183GeV. These results are independent of the flavour ofthe quarks from the decay of the hypothesized parti-cles and are valid for X and Y being scalars pro-duced predominantly by an s-channel process.

6. Conclusions

Following the ALEPH observation of a largeexcess of four-jet events with dijet mass sums around

' w x105 GeV at s f133 GeV 1 , a careful emulationof the ALEPH analysis has been performed usingOPAL data collected from eqey collisions at centre-of-mass energies between 130 and 184 GeV. Theprocess eqey™h0A0 was used to estimate the signaldetection efficiencies. The estimated sensitivity, massresolution, efficiency, and estimated backgrounds inthis analysis were similar to that of the ALEPHanalysis. No significant excess of four-jet eventswith dijet mass sums in the region close to 105 GeV,or any other region between 60 and 160 GeV, hasbeen observed in any of the data samples separatelyor combined, and our observations are consistentwith Standard Model predictions. The same conclu-sions are reached when an OPAL-specific analysis isemployed. Limits for the cross-section of a hypo-thetical process eqey™XY™ four jets are given asa function of the dijet mass sum M and the massdifference DM. The 95% confidence upper limitsobtained in both analyses for dijet mass sums near105 GeV are below the excess reported in 1995 by

w xALEPH 1 to a high degree of confidence. ALEPHw xhas recently analysed 30 new data at centre-of-mass

energies between 130 and 184 GeV and do notconfirm the previously reported excess.

Acknowledgements

We particularly wish to thank the SL Division forthe efficient operation of the LEP accelerator at allenergies and for their continuing close cooperation

with our experimental group. We thank our col-leagues from CEA, DAPNIArSPP, CE-Saclay fortheir efforts over the years on the time-of-flight andtrigger systems which we continue to use. In additionto the support staff at our own institutions we arepleased to acknowledge the Department of Energy,USA, National Science Foundation, USA, ParticlePhysics and Astronomy Research Council, UK, Nat-ural Sciences and Engineering Research Council,Canada, Israel Science Foundation, administered bythe Israel Academy of Science and Humanities, Min-erva Gesellschaft, Benoziyo Center for High EnergyPhysics, Japanese Ministry of Education, Science

Ž .and Culture the Monbusho and a grant under theMonbusho International Science Research Program,

Ž .German Israeli Bi-national Science Foundation GIF ,Bundesministerium fur Bildung, Wissenschaft,¨Forschung und Technologie, Germany, National Re-search Council of Canada, Research Corporation,USA, Hungarian Foundation for Scientific Research,OTKA T-016660, T023793 and OTKA F-023259.

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Ž .On page 350 in Eq. 12 the 2 in front of the large bracket should be omitted.

1 Ž .PII of the original article: S0370-2693 97 01152-0.


information entropy and number of principal components in shell...

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