Volume 78B, number 5 PItYSICS LETTERS 23 October 1978

TESTING WEAK INTERACTION MODELS AFTER THE NULL BISMUTH RESULT

David BAILIN and Norman DOMBEY

School of Mathematical and Physical Sciences, University of Sussex, Brighton BN1 9OH Sussex, England

Received 3 July 1978

The theories advanced to explain the absence of parity violation in bismuth vapour need further experiments to discrimi- nate between them. We show that any observed asymmetry in ep scattering with polarized electrons must have a specific y- dependence if it is to be consistent with the bismuth experiments. Different charge asymmetries in e+e - ~ ,,+g- and u-+p --+ ~t+-X are also predicted in the different types of theory.

Tile failure to detect parity violation in bisnmth vapour [1,2] at the level predicted ~ 1 by the standard renorlnalizable unified theory of weak and electromag- netic interactions [4] has stimulated tile construction of models which can accommodate this data. The least changed is the "mixed" model [5] which introduces a right-handed doublet including the electron (and a neutral heavy lepton). This results in the electronic part of the neutral current coupling to Z 0 being pure- ly vector, and this removes the coherent component of the parity-violating amplitude so there is no en- hancement factor in heavy atoms. No new gauge vec- tor mesons are introduced and the underlying group is still SU(2) X U(1); the mass spectrum of the gauge bosons is also unchanged from the simple Weinberg model [4].

We have written an alternative fommlation of SU(2) X U(1) [6] in which vector and axial currents are interchanged. In this model, which we call weak SU(2) X U(1) (as the photon is not one of the gauge bosons of the model), the electronic part of the neu- tral current coupling to Z 0 is purely vector for sin20 w

+1 A recent letter (3) has reported a parity violation in agree- ment with the predictions of standard unified SU(2) x U(1). We here discount this result as it is in disagreement not only with the two other published experiments from Oxford and Washington [ 1 ] but also with the first of the second generation experiments to give results [2]: the errors from a preliminary analysis of this data are over three times smaller than the errors quoted in ref. [3].

1 = a and thus there is again no coherent nuclear contri- bution to the parity-violating amplitude. There is now a new relatively light gauge boson X 0 (in place of the photon) which couples through a pure axial coupling to electrons and nucleons. If one insists on construct- ing a unified theory of weak and electromagnetic in- teractions, which therefore contains the photon as well as the X °, an SU(2) × U(1) X U(1) theory results [7]; weak SU(2) × U(I) is then an approximate version of SU(2) X U(1) × U(1) for small photon mixing angle.

Another class of models which is consistent with tile bismuth data are those in which in lowest order there is no parity violation in neutral currents at all. Examples of such models are the left-right symmetric

theories SU(2)L X SU(2)R × U(1) [8]. In these mod- els there are also extra gauge bosons, with axial vector couplings, which may be light [9].

In view of the new light bosons with axial couplings predicted in many of these theories in both class ! (the models in which there is no large coherent parity viola- tion) and class II (models in which there is no parity violation), weak interaction effects may show up sooner than was expected in the Weinberg model. In particu- lar although the clearest signal of the presence of weak interactions would be parity violation, for example an asymmetry in the scattering of polarized electrons by unpolarized nucleons, it is useful to look at the effect of new gauge bosons with axial couplings irrespective of whether they conserve parity or not. In this paper we consider tile results of both types of experiment in terms of the various theories.

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Volume 78B, number 5 PHYSICS LETTERS 23 October 1978

First let us look at deep inelastic scattering of po- larized electrons by an unpolarized proton: ep ~ eX. The presence of a parity-violating interaction (in class l theories) coupling electrons to hadrons will lead to different cross-sections o L and o R for the scattering of left and right polarized electrons, characterized by an asymmetry

do L do R A 1 = d o L + d o N 4 - 0 . (1)

In gauge theories the parity-violating interaction is generated by the exchange of a (neutral) gauge boson Z whose coupling we take to be

~° Z = k~k'Yc~(gv k - gkTs)kZC~, (2)

where the summation is over all leptons (e,/1 .... ) and quarks (u, d, ...). The dominant contribution to A 1 arises from interference between the electromagnetic and weak amplitudes. The cross sections are most easily evaluated using the quark-par ton model in the stand- ard way. Then we find

= 2t { EqgeAgqQquq(x) A1 (m 2 5 t)e ~ EqQ2qUq(X)

+ Eqg~gqQquq(X) y ( 2 - y ) t (3)

EqQ~Uq(X) 1 + (1 3,)2 J '

where t is the momentuln transfer to the hadrons and x andy are the usually defined scaling variables; u = E - E ' , where E, E ' is the energy of the initial and final electron and x = - t/2mpU, y = u/E. The summa- tion is over the constituents q of the proton and Qq is the electric charge of q (in units of e). Evidently A 1 is determined by a structure function depending on x which is unknown, even when the nrodel-dependent constants gxkr and g~ are specified. As an estimate we ignore non-valence quarks and assume that the parton distribution functions Uu(X ) and Ud(X ) for u and d quarks in a proton satisfy Uu(X ) = 2Ud(X). For Itl "~

2 the estimate for the Weinberg model [4] is m 2

A w Gt {4 s+~(4sin2Ow-1) Y ( 2 - Y ) } - X/~ 47ro~ sin20w - 3 1 + ( 1 _ ~ 2

Itl =(5.3 x 10 -s) m 2

P 594

(4)

1 for sin20w = z. Similarly weak SU(2) × U(1) 16] pre- dicts

A1 = 3 4rraK 2 (4 sin20w - I)

y(2 } -y) +(4 sin20w - s) 1 +(1 _ ~ ) 2 '

1 K2 where K = rnz cos Ow/rn w. Taking sin20w = z and = 1 this gives

s Itl y ( 2 y ) , =(5.3 x io- 1 + d Z T ) 2 (s)

P can easily be distinguished from A 1 w by its y- which

dependence, especially at low y-values. All class 1 mod- els will have this y-dependence; for example the mixed model [5] also yields an asymmetry

AM Gt { y ( 2 - y) ] = ~ 4 ~ s (4 s i n 2 0 w - 2) 1 +(1 - 7 ) 2)

= ( 1 3 . 2 × 1 0 - 5 ) 1 t ! y ( 2 - y ) (6) rn: 1 +(1 _ y ) 2

1 S ABD. for sin20w = a -- a factor g larger than We should stress again that these estimates assume that the patton distribution functions are identical, otherwise there is an unknown x-dependence which can easily modify these calculations by, say, 40%.

It is well known that all experiments looking for parity violation require polarization measurements, and that these are notoriously difficult. We thus turn next to tests not involving polarization effects.

As noted in the introduction, there is some interest in detecting a new, possibly light, gauge boson X cou- pled axially to electrons, muons and hadrons. The ef- fects of such a boson may be observed in charge asym- metries in deep inelastic e+-p or/a±p scattering. (Here perhaps muon beams are preferred on experimental grounds.) The electromagnetic amplitude, generated by photon exchange, interferes with the weak ampli- tude, generated by X exchange, resulting in a differ- ence between the ~÷p and ~ - p inelastic cross sections. This is because the vector bilinear, appearing in the elec- tromagnetic amplitude, and the axial bilinear, appearing in the weak amplitude, have opposite behaviour under charge conjugation. Thus a non-zero charge asymmetry

A2 _~ do(/-t+p) - d o ( u - p ) (7 t do(u+p) + d o ( u - p )

Volume 78B, number 5 PItYSICS LETTERS 23 October 1978

will ensue. Since the asymmetry is not in itself a parity- violating effect, it will arise so long as the X has some axial coupling to both muons and hadrons, irrespec- tive of whether these interactions are parity conserv- ing or not. In class II theories, for example, all neu- tral current interactions conserve parity, so the X will have a purely axial coupling to muons and quarks. In the Weinberg model, the only neutral gauge boson is Z which for general 0 w has a parity-violating (i.e. vec- tor and axial vector) coupling to muons and quarks. So long as X is the lightest (neutral) gauge particle in the theory the dominant contr ibution to A 2 arises from the interference of the amplitudes from T and X exchange. We assume that the axial part of the cou- pling of X is given by

(8)

where, as before, k = e,/~, ..., u, d ... Then the asym- metry calculated using the quark parton model is, for It[ <~ rex2,

-2 t G~ YqGqQquq(X) y ( 2 - y ) A 2 - (9)

m2x e2 yqQ2Uq (x) 1 + (1 3,)2 '

where t, x, y, Qq, Uq are the quantities defined earlier. Again the asymmetry involves an unknown structure function even when the model-dependent paranreters G q are specified. In the Weinberg model the only con- tribution is from Z-exchange, and again taking u u = 2u d we find (for any value of sin 0w)

AW Gt 5 y(2 - y) - X/247r~ 3 1 +(1 - y ) 2

= ( 1 3 . 2 × 10 -5 ) t y ( 2 y ) (10) m 2 1 + (1 _ y ) 2

P For the mixed model we have

A M = 0 , ( l l )

since the muonic neutral current is purely vector. But in other models, of both class I and class iI, which have a light X different from Z, the scale of the asymmetry is controlled by the ratio

where g is the usual charged current coupling constant

satisfying g2/8m2 = G/V/2. Now any gauge theory will specify G~/g, so that experimental information on A 2 may be translated into information on the ratio of the gauge boson masses mw/m x. For example in weak su(2) × u(1)

~/ sin20w 1 (G g)2 = ~ a ,

and since G q = -QqG~ in this model, we have

A~ D 4Gtsin2Ow(mw]2 y ( 2 - y )

= -5 i m w)2 y ( 2 - - y ) t (31.6 X 10 ) _ -

\ r e x - 1+(1 _ y ) Z m 2" (12)

Another place where this same quanti ty G~/m x = G~/m x occurs is in the hyperfine splitting of hydrogen. The additional axial exchange of X between the pro- ton and the electron causes a shift [10] &, in the split- ting u = ta T PS, which is given by

. e p 8v GAGA 12memp - - - - , (13)

v m2e 2 gp x

where gp = 5.58 is the proton's g-factor, and G p is the static coupling of the proton to the X-bosom In weak SU(2) × U(1) for example G p = 1 r e , 3 t, Atg A + g~) , where g i = 1.25 is the isovector axial renormalization and g~ is the isoscalar axial renormalization. Then using

1 (GeA/g) 2 = sin20w ~ a we find

(gu/u) BD = (1.9 2.3) X lO-7(mw/mx) 2 , (14)

depending on whether we take g~ = 383 [11] (as in- dicated by the quark model) or g~ = g ~ ( w h i c h is true if axial charge is conserved). In the mixed model, of c o u r s e ,

( ~ / v ) M = 0 , (15)

and in the Weinberg model Z-exchange gives

(St,/u) w = 5.8 × 10 - 8 (16)

(for any value of 0w), which is about an order of mag- nitude smaller than the theoretical uncertainty [12] (in this case the main uncertainty is theoretical, not experimental).

The opposite behaviour of vector and axial bilinears under charge conjugation will also be observed in e+e -

/a+/~ , causing a forward-backward asymmetry

A 3 = X f -Nb/Nf+Xb, (17)

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Volume 78B, number 5 PHYSlCS LETTERS 23 October 1978

where Nf, N b are the number o f / a - ' s travelling in the forward and backward direction with respect to the incident e - bealn. Now [13]

3 (cX12 A 3 . . . . . s (18)

2e 2 \ m x !

for s ~ rex2. As before, any determination ofA 3 will give information on GeA/mx in theories with new neu- tral gauge bosons.

For example it has been reported that A 3 < 3% for V~= 6.8 GeV [14]. It follows from (18) in weak

i l l SU(2) X U(1) for sin20 w = a that

m x / m w > 0.64 . (19)

In the Weinberg model, A 3 is completely specified and is independent of sin20w

w A 3 = - 3Gs/16x/2rro~ = 0.31% at x/) -= 6.8 GeV. (20)

The mixed model, of course, gives no asymmetry here. The bound (19) can be made stronger by combining

it with the measured ratio of neutral current to charged current neutrino cross sections. This ratio determines the quantity K = mzCOS 0w/m w (6) where K = 1 in the Weinberg model or mixed model, hr other models K :~ 1 and is determined by the appropriate mass formulae. For example in weak SU(2) × U(1)

,,,2cos Ow +., xsin20w : , ( 2 1 )

SO

g2 = 1 - (rex/row) 2 sin20w. (22)

Experimentally K 2 = 0.98 -+ 0.05 [15] and thus work- ing within two standard deviations for the value of K, we can say that

0.64 < m x / m w < 0.69 . (23)

Similarly bounds can be obtained in other nrodels. To summarize: the tests discussed here all explore

tire structure of the electronic (or muonic) -hadronic neutral current interaction. The observation of any polarization asymmetry A 1 immediately falsifies all class Ii theories since they have no parity violation. The null bismuth result would then indicate that only class I theories are acceptable and these typically have a polarization asymmetry with a characteristic y-de-

41 It is wrongly stated in ref. [6] that there is no forward- backward asymmetry for sin20w ~.

pendence which should be observable. There is, how- ever, a difficulty in fixing the overall scale for the pre- dicted asymmetry, since it generally involves an un- known and lnodel-dependent structure function in the

variable x. The bismuth data also indicate that any axial-component of the electron's neutral current can only be coupled to a purely axial hadronic current, and we have described tests which are sensitive to the existence of each of these currents. If a new light neu- tral boson X exists, its effects should be observable with these tests; furthermore the existence of X will cause the vahle of ~ determined in neutral current neu- trino experiments to deviate from unity. If indeed K does not deviate fl-Om unity, this would provide the best evidence yet that SU(2) X U(1) is really the under- lying theory of weak and electromagnetic interactions.

We thank Roger Phillips for pointing out an error in our previous paper [6] and for drawing our atten- tion to ref. [9] where similar calculations to these are done within the framework of left-right symmetric models (i.e. models of class 11).

[1] L.L. Lewis et al., Phys. Rev. Lett. 39 (1977) 795; P.E.G. Baird et al., Plays. Rev. Lett. 39 (1977) 798.

[2] F.N. Fortson, in: Proc. of Int. Conf. on Neutrino physics and neutrino astrophysics, Purdue 1978 (to be pub- lished).

[3] L.M. Barkov and Zolotoryov, JETP, L26 (1978) 369. [4] S. Weinberg, Phys. Rev. Lett. 19 (1967) 1264. [5] S. Weinberg, in: Proc. of 1977 Int. Symposium on Lep-

ton and photon interactions at high energies (DESY, 1977) p. 619.

[6] D. Bailin and N. Dombey, Nature 271 (1978) 20. [7] G.G. Ross and T. Weiler, Caltech Report 68--620 (1977),

to be published; D. Darby and G. Gramner, Illinois Report (TIt)-77-39 (1977), to be published.

[8] R.N. Mohapatra and D.P. Sidhu, Phys. Rev. Lett. 38 (1977) 667.

[91 V. Barger and R.J.N. Phillips, Wisconsin Report COO- 881-17 (1978) to be published.

[10] S. Drell and J. Sullivan, Phys. Lett. 19 (1965) 516. [11] S. Adler, Phys. Rev. Dll (1975) 3309. [12] H. Grotch and D.R. Yennie, Rev. Mod. Phys. 41 (1969)

350. [13] B. Kayser, S.P. Rosen and E. Fischbach, Phys. Rev. Dll

(1975) 2547. [14] V. Elias, J. Pati and A. Salam, Phys. Lett. 73B (1978)

451. [151 L. Sehgal, in: Proc. of lnt. Conf. on Neutrino physics

and neutrino astrophysics, Purdue 1978 (to be pub- lished).

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