PHYSICAL REVIEW D VOLUME 17, NUMBER 3 1 FEBRUARY 1978

Transverse-momentum and angular distributions of hadroproduced muon pairs

E. L. Berger High Energy Physics Division. Argonne National Laboratory. Argonne. Illinois 60439

J. T. Donohue'" Fermi National Accelerator Laboratory. Batavia. Illinois 60510

S. Wolframt

High Energy Physics Division. Argonne National Laboratory. Argonne. Illinois 60439 (Received 23 September 1977)

We study the angular distribution of muons in the dimuon rest frame from pp ---> fkjiX at high energy and large dimuon mass. Including smearing due to quark transverse momenta, we show that the Drell-Yan model predicts a polar angular distribution in the t -channel helicity frame of 1 + acos28 with a ~ 0.8. Experimental deviations from this prediction would cast serious doubt on the Drell-Yan picture.

Datal on the cross section for pp- p.'ji.x as a function of the p.'ji. invariant mass suggest that the p.-pair spectrum is dominated by a few narrow resonances (I/!, I/!', T, ... ), superimposed on a sig­nificant continuum. The normalization, mass dependence, and scaling properties of the con­tinuum were predicted rather accurately on the basis of the Drell-Yan modeF illustrated in Fig. 1. The quark x distributions used in the model coincide with those deduced from studies of lepton­hadron interactions3; x is the fraction of proton longitudinal momentum carried by a quark. Fur­ther tests of the Drell-Yan model are imperative. In this article we provide predictions for the an­gular distribution of the muons in their center­of-mass system with respect to both t- and s­channel reference axes. For this it is necessary to make an explicit quantitative study of the ef­fects of allowing the quarks to carry finite mom­enta transverse to the direction of the incident hadron momentum_ Transverse momenta distort the angular distributions of the muons.

In computations with quark-parton models, it has often been assumed that only the quark mom­entum components parallel to the incident hadron momentum direction need be considered. How­ever, it is clear that the finite size of the proton

q

FIG. 1. Drell-Yan diagram for the production of massive dimuon pairs in pp collisions. q denotes a quark and q an antiquark.

17

implies that the quarks within have finite com ­ponents of momentum k in all directions. As is evident from Fig. 1, the dimuon system is pro­duced with transverse momentum P T = 0 unless the quarks themselves have finite k T • By con­trast, the data4 exhibit a (lp T 1>1JIl which increases with Q2 =M,.n2 up to MJJij. "'" 3 GeV2, and then levels off at ( IPT 1>1lii "'" 1.16 ± 0.12 GeV up to the largest masses measured (Mllii ~ 15 GeV). In the Drell­Yan model, (P T 2)1lii = 2(k T 2). for s-oo and MIlii-oo. Quark transverse momenta are deduced also from other processes,5 with ( IkT I> in the range 0.5 to 1 GeV.6

We suppose therefore that there exists a prob­ability distribution Gj/p(Xj , KTj ) for finding in a proton a quark of type i with momentum kj

= «(vs / 2)lx j , kTJ ). According to the Drell-Yan mod­el, the differential cross section for the process pp - p."jix is

:4~ = 2;: J dx ldx2d2i{Tld2i{T2 0 (4)(P - kl - k2) •

da =2 dMIJIl2dyd2P T . (1)

Here "7 denotes an antiquark. We take the quarks to be massless7 so that the muons from qq - p.p. have an angular distributionS aa/dcose*a: (1 + cos2 e *) in their center-of-mass system, with 9 * measured relative to the quark momentum direction.

As in the work of Feynman, Field, and Fox,6 for example, we assume that the quark distribu­tion functions may be factored into x-dependent and IkT I-dependent parts,

G(x,kT)=XR-lXG(x)f(lkTI), (2)

with XR2=[X2+4k//s ]. For xG(x), we employ the

858

17 TRANSVERSE·MOMENTUM AND ANGULAR DISTRIBUTIONS OF ... 859

TABLE 1- Possible choices for the quark transverse-momentum distributions. The value of A is determined from fits to ( I P T 1)"iL'

Model f(k T ) (k/ ). (k T ). A </l T2). (GeV2) (l kTI). (GeV)

Exponential e -AlkTI 6/ 71.2 2/71. . 71. = 2.7 GeV-1 0.8 0.7

Gaussian e -AkT2 1/ 71. t ( 7T/A)1 /2 71. = 1.2 GeV-2 0.8 0 .8

Inver se power (k T2 + Af" A/ (n -2) (n -lhtrB(~,n - ~) n = 4 71./ 2 "' 0.6-JI n = 8 71./ 6 "' 0.35-JI n = 4 for q;

n = 8 for q

Field-Feynman quark distributions,3 while we investigate three different forms for the kT de­pendence. Our choices are summarized in Table 1. The values for n in our function (k T 2 +Ar" are motivated by constituent-counting rules. 9 The only free parameter in each case is A. We de­termine it by requiring that Eq. (1) reproduce the result (PT>"ji=1.16±0.12 GeV for large M"ji. In Fig. 2, we show the transverse-momentum dis­tribution provided by our models, at y "" 0 and MJIli = 5 GeV. Similar results are obtained for other values of MJIli. Although all our models yield the same PT-integrated cross section and (lp T I>JIli' some differences are apparent in Fig. 2, par­ticularly the value of the cross section at P T = O. Changes in the quark x distributions do not ap­preciably alter any of our results. The curves in

.. I > .. '" .0

E

"-.. ."

-----b ."

N

10-10

10-11 -- EXPONENTIAL

- - GAUSSIAN

- --- I NV ERSE POWER

2

PT (GeV)

FIG. 2. Differential cross section da/ d4p= 2da/ dM ~lidydP T2 versus P T at y"' 0, M ~Ii= 5 GeV and Plab

= 400 GeV /e . y is the c .m. r apidity of the dimuon. Results obtained from our three models are s hown .

71. = 2 .5 GeV 2 1.25 (q) 0 .95 (q)

0.40 (q) 0.55 (7j)

Fig. 2 are absolutely normalized inasmuch as the Field-Feynman x distributions are themselves normalized. After integrating over P T, we obtain distributions d 2(J/ dM dy at y ~ 0 in good agreement with those of previous computations 2 and with recent data. 1

Examining the variation of (lp T I>"ji as MJIli changes, we find that in all cases (lpT I>"ji is nearly independent of MJIli for M,.-p.:<:: 3.5 GeV. Below 3.5 GeV, all but the Gaussian show a fall of (Ip T I>JIli from -1.1 GeV to -0.8 for MJIli-0.5 GeV, in qual­itative agreement with the data. 4 While pleasing, the agreement at low mass should perhaps be re­garded as fortuitous since there are theoretical and experimental reasons for lack of confidence in the model for M $ 4 GeV. We are mostly con-cerned here with large MJIli. ..

Having determined acceptable forms for !(kT), we turn to a discussion of the angular distribution of muons in the dimuon center-of-mass frame. With the advent of la.rge angular acceptance ex­periments on pp - IJ./iX, a measurement of this important distribution becomes possible. If the quarks have no momentum transverse to the mom­enta of their parent hadrons, then the muons should exhibit the same (approximately 1 + cos2 e *) distribution with respect to the incident hadron axis as they must with respect to the quark axis. In the IJ."jI rest fra~e, convenient polar axes for the discussion of this angular distribution are the beam direction (t-channel or Gottfried-Jackson frame) or the recoil (X) momentum direction (s­channel helicity or Jacob-Wick frame). We write the final angular distributions integrated over cf>

asB (1 + a cos 28), and we discuss our results in terms of a.

The most direct technical method for obtaining a is to compute angular moments t1 of the Drell­Yan cross section. For the moment t~, we replace a in Eq. (1) by D~o (CP', 8',0)8-, with D~ = f(3 cos 28' -1). Here (8' , CP ') are angles which define the quark direction relative to our chosen system of reference axes. Noting further that

860 E. L. BERGER, J. T. DONOHUE, AND S . WOLFRAM 17

j dxldx.pikTldZ({T204(P - kl - k 2) =t jdO'XR1XR2 ,

(3)

we derive

viOt~dald4p=t L: f dO'XIGI/P(Xl)XP1/p(X2) I

X/I ( IkTlI }f7( IkT21 )D~o a. (4)

In Eq. (4), the XI and kTi must be expressed in terms of 9' and CP'. Finally,

3vTOt~ a= 0

4-l1Ot2 (5 )

We also evaluated the moments t~ and t~, con­nected with cp dependence, and we shall discuss their values elsewhere. A summary of our re­sults for a is presented in Fig. 3. All our com­putations are done at P1ab = 400 GeV Ie, but little change is observed in the values of a for P1ab

= 1200 GeV Ie. Again taking MI.rlI = 5 GeV as a typ­ical mass in the Drell-Yan continuum region, we show in Fig. 3(a) the variation of a versus xF

for the dimuon. In the t-channel frame, a is nearly independent of X F, but a va:ries consider­ably if s-channel axes are used. The rapid vari­ation of a in the s-channel frame at small P T and small x F is due simply to the fact that the s-chan­nel axes are ill-defined in this kinematic regime. (The recoil system is nearly motionless.) In the small P T , small xF region which contributes most to the cross section, the t-channel frame is there­fore preferable.

In Fig. 3{b), we present the variation of at with MI.rlI at X F f'::j 0 for our three models. After a rapid rise at small M",-p" at becomes roughly constant, taking on values a t .<:0.8 forM",jl.<:5 GeV. Finally, we comment on the P T dependence of a, not shown here. For MiMi. and x F, our results exhibit a sys­tematic decrease of a as P T is increased. For large enough P T , at becomes negative, cor­responding to preferential muon emission trans­verse to the beam axis.

We conclude that if t-channel axes are used, the naive Drell-Yan model prediction dal dcos e

*Permanent address: Laboratoire de Physique Theorique, Universite de Bordeaux I, France.

tUsual address : St. John's College, Oxford, England. IS. W. Herb et al., Phys. Rev. Lett. 39, 252 (1977) and references ther ein ; D. Antreasyn elal., ibid. 39, 906 (1977) and r eferences therein. -

101 0.8

0.6

0.4

0

1.0 Ibl

0.8

0.6 <>

0.4 '/ /

0.2 ~/

,II 0

7

M~~ '5 G.v

- s -CHANNEL

--- I-CHANNEL

y"O

I-CHANNEL

-- EXPONENTIAL

-- GAUSSIAN

-----. I NVERSE POWER

M~~ IGoV I

1.0

FIG. 3. (a) O! versus X F = 2PL /1S at Mp jJ =5 GeV for our exponential model. P L is the C.m. longitudinal mo­mentum of the dimuon. The solid (dashed) curve shows values of O! determined with respect to the s-channel (t - channel) polar axis. (h) O! versus M p jJ at x F '" 0 for all three models. The t -channel axes are used. In (a) and (h), results are averaged over P T and are for Plab

=400 GeV/c .

a: (1 + a cos2 e), with a'" 1, is nearly preserved even after integration over quark transverse mom­enta. Measurement of large deviations of a from unity (Le., a ~ 0.8) at large Mij,(> 4 GeV) and mod­est P T would cast serious doubt on the validity of the naive Drell-Ya:n picture. 10

This work was performed under the auspices of the United States Energy Research and Develop­ment Administration.

2S. Drell and T.-M. Yan, Ann. Phys. (N.Y.) 66, 578 (1971). For r ecent discussions consult R. F. Peierls, T. L. Trueman, and L. L. Wang, Phys. Rev. D 16, 1397 (1977); C. QUigg, Rev. Mod. Phys. 49, 297 (1977).

3R. D. Field and R. P. F eynman, Phys. Re~ D 15, 2590 (1977).

17 TRANSVERSE·MOMENTUM AND ANGULAR DISTRIBUTIONS OF ... 861

4J. G. Branson et ai., Phys. Rev. Lett. 38, 1334 (1977); W. Innes, Columbia-FNAL-SUNY Collaboration, SLAC Topical Conference on Particle Physics, 1977 (unpub­lished).

5These are essentially [cf. F. Close, F. Halzen, and D. Scott, Phys. Lett. 68B, 447 (1977)1 the following: (i) the value of uL/ur in deep-inelastic scattering, and (ii) the transverse momenta of final-state hadrons in various processes, although this measures a con­volution of production and decay distributions, as dis­cussed in Ref. 6.

GR . Feynman, R. D. Field, and G. C. Fox, Nucl. Phys. B128, 1 (1977).

7Strange and charmed quarks do not contribute appreci­ably to the Drell-Yan process. If the u and d quarks have an effective mass m, the angular distribution be­comes [1+ (Q2-4m 2)/(Q2 + 4m 2)cos20*1; cf. K. V. Vasavada, Phys. Rev. D 16, 146 (1977). Note that the quarks must be very massive in order to obtain isotropy in cosO* for typical dimuon masses M V1/ <= Q 2).

8(;p -violating couplings could give rise to cosO terms (for pP rather than pp as the external hadrons), but spin-2 exchange is necessary for a cos40 term.

9M. Duong-Van, SLAC Report No. SLAC-Pub-1819, 1976 (unpublished); J. F. Gunion, Phys. Rev. D 14, 1400 (1976). '

loThe Drell-Yan model and our analysis ar.e not appli­cable for M ~ p in the region of obvious re!;onances (1jJ, T (9.5), ... ). For example, in one model of IjJ had­roproduction (Ref. 11), two gluons fuse to produce a X state, which then decays, X - ljJy. If the 1/;- x mass difference is ignored, then muons from ljJ's arising from spin-O X's will be isotropic (O!= 0). For higher spin X's, a set of X density matrix elements leading to O! = 1 can always be chosen. We shall make further statements about the resonance regions in another article.

11M. Einhorn and S. Ellis, Phys. Rev. D 12, 2007 (1975); C. E. Carlson and R. Suaya, ibid. 14, 3115 (1976).