Main Text

Data (1) on the cross section for as a function of the invariant mass suggest that the -pair spectrum is dominated by a few narrow resonances , 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 model (2) illustrated in Fig. 1. The quark distributions used in the model coincide with those deduced from studies of lepton-hadron interactions (3); 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 center-of-mass system with respect to both - and -channel 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.

[ Figure 1 ] Drell-Yan diagram for the production of massive dimuon pairs in collisions. denotes a quark and an antiquark.

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 implies that the quarks within have finite components of momentum in all directions. As is evident from Fig. 1, the dimuon system is produced with transverse momentum unless the quarks themselves have finite . By contrast, the data (4) exhibit a which increases with up to , and then levels off at up to the largest masses measured (). In the Drell-Yan model, for and . Quark transverse momenta are deduced also from other processes, (5) with in the range 0.5 to 1 GeV. (6)

We suppose therefore that there exists a probability distribution for finding in a proton a quark of type with momentum . According to the Drell-Yan model, the differential cross section for the process is

Here denotes an antiquark. We take the quarks to be massless (7) so that the muons from have an angular distribution (8) in their center-of-mass system, with 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 -dependent and -dependent parts,

with . For , we employ the Field-Feynman quark distributions, (3) while we investigate three different forms for the dependence. Our choices are summarized in Table I. The values for in our function are motivated by constituent-counting rules. (9) The only free parameter in each case is . We determine it by requiring that Eq. (1) reproduce the result for large . In Fig. 2, we show the transverse-momentum distribution provided by our models, at and . Similar results are obtained for other values of . Although all our models yield the same -integrated cross section and , some differences are apparent in Fig. 2, particularly the value of the cross section at . Changes in the quark distributions do not appreciably alter any of our results. The curves in Fig. 2 are absolutely normalized inasmuch as the Field-Feynman distributions are themselves normalized. After integrating over , we obtain distributions at in good agreement with those of previous computations (2) and with recent data. (1)

[ Table 1 ] Possible choices for the quark transverse-momentum distributions. The value of is determined from fits to .

[ Figure 2 ] Differential cross section versus at and . is the c.m. rapidity of the dimuon. Results obtained from our three models are shown.

Examining the variation of as changes, we find that in all cases is nearly independent of for . Below 3.5 GeV, all but the Gaussian show a fall of from to for , 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 . We are mostly concerned here with large

Having determined acceptable forms for , we turn to a discussion of the angular distribution of muons in the dimuon center-of-mass frame. With the advent of large angular acceptance experiments on , 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 distribution with respect to the incident hadron axis as they must with respect to the quark axis. In the rest frame, convenient polar axes for the discussion of this angular distribution are the beam direction (-channel or Gottfried-Jackson frame) or the recoil momentum direction (-channel helicity or Jacob-Wick frame). We write the final angular distributions integrated over as (8) , and we discuss our results in terms of .

The most direct technical method for obtaining is to compute angular moments of the Drell-Yan cross section. For the moment , we replace in Eq. (1) by , as with . Here are angles which define the quark direction relative to our chosen system of reference axes. Noting further that

we derive

In Eq. (4), the and must be expressed in terms of and . Finally,

We also evaluated the moments and , connected with dependence, and we shall discuss their values elsewhere. A summary of our results for is presented in Fig. 3. All our computations are done at , but little change is observed in the values of for . Again taking as a typical mass in the Drell-Yan continuum region, we show in Fig. 3(a) the variation of versus for the dimuon. In the -channel frame, is nearly independent of but varies considerably if -channel axes are used. The rapid variation of in the -channel frame at small and small is due simply to the fact that the -channel axes are ill-defined in this kinematic regime. (The recoil system is nearly motionless.) In the small , small region which contributes most to the cross section, the -channel frame is therefore preferable.

In Fig. 3(b), we present the variation of with at for our three models. After a rapid rise at small , becomes roughly constant, taking on values for . Finally, we comment on the dependence of , not shown here. For and , our results exhibit a systematic decrease of as is increased. For large enough , becomes negative, corresponding to preferential muon emission transverse to the beam axis.

[ Figure 3 ] (a) versus at for our exponential model. is the c.m. longitudinal momentum of the dimuon. The solid (dashed) curve shows values of determined with respect to the -channel (-channel) polar axis. (b) versus at for all three models. The -channel axes are used. In (a) and (b), results are averaged over and are for .

We conclude that if -channel axes are used, the naive Drell-Yan model prediction , with , is nearly preserved even after integration over quark transverse momenta. Measurement of large deviations of from unity (i.e., ) at large () and modest would cast serious doubt on the validity of the naive Drell-Yan picture. (10)

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

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