P H Y S I C A L R E V I E W D V O L U M E 2 1 , N U M B E R 9 1 M A Y 1 9 8 0

Parton-model relation without quantum-chromodynamic modifications in lepton pair production

C. S. Lam Stanford Linear Accelerator Center, Stanford, California 94305

and Department of Physics, McGill Uniuersity, Montreal, Canada

Wu-Ki Tung Department of Physics, Illinois Institute of Technology, Chicago, Illinois 60616

(Received 22 January 1980)

It is pointed out that, in contrast to the recently discovered large quantum-chromodynamics (QCD) correction to the Drell-Yan formula, a certain quark-parton-model relation between structure functions for lepton pair production is not subject to any first-order QCD modification. Both parallelism and contrast to the Callan-Gross relation in deep-inelastic scattering are spelled out. Implications on the lepton angular distribution for both low and high q, are discussed. The case is made that this relation provides a unique opportunity to test the "QCD-improved quark-parton model."

Recent quantum-chromodynamics (QCD) cal- culations' reveal a large (-100%) order-a, cor- rection to the Drell-Yan cross-section formula2 for lepton pair production (LPP) in hadron col- lisions. This makes the integrated cross section a dubious testing ground for the QCD-corrected parton model, and raises fundamental questions about the viability of the perturbative QCD ap- proach. It is , therefore, important to study other aspects of this approach.

There is much more to the quark-parton model and its QCD modifications in L P P than just the integrated Drell-Yan cross-section formula. The lepton angular distributions a r e controlled by structure functions which obey parton-model re- l a t i o n ~ ~ ' ~ similar to those between Fi and F2 in deep-inelastic scattering (DIS). How a r e these relations affected by perturbative QCD correc- tions? The answer to this question is quite sur- prising: At least one of these relations-the ex- act counterpart of the c a l l a n - ~ r o s s ~ relations- is not modified at all by first-order QCD correc- tions, although individual te rms in this relation may be subject to large corrections. In the r e s t of this note, we spell out explicitly the paral- lelism a s well a s the contrast between the DIS and L P P cases in the perturbative QCD approach and discuss the experimental implications of these results.

(hence the lepton pair). The trace of this tensor is related to the cross section (integrated over the lepton center-of-mass angles) via the form- u l a 2 9 4

wherein IVI represents the effective mass of the lepton pair and CY the fine structure constant.

The tensor W,, can be decomposed into four structure functions4 in much the same way as i n DIS :

~ h e r e g , , = g , , - ~ , ~ , / ~ ~ , P = P 1 + P 2 , p = P i - Pz, *il = g P , ~ ' / & , and 3, = g , , , p V / 6 . These structure functions can be determined experi- mentally from Eq. (2) together with lepton angu- l a r distribution measurements. Equation (3) closely resembles the corresponding formula defining Wi and W2 in DIS (where the las t two terms a r e absent).

A basic result of the quark-parton model in DIS is the Callan-Gross r e l a t i ~ n . ~ It is usually writ- ten a s

The L P P process is described by a tensor am- where W, is the helicity structure function for the plitude6 W, corresponding to the current corre- longitudinal virtual photon. We can recast this lation function result in the not-so-familiar form:

w,=2wi. (PIP2 I J~(Z)J,(O) (PIPz ) (I)

(5)

In Ref. 4 we showed that the same equation fol- similar to that of DIS. Here P I and P, a r e the four lows a s a general consequence of Ihe quark-par- momenta of the colliding hadrons, S = -(Pi + P~)', ton model in LPP. In terms of the invariant and q is the momentum of the virtual photon structure functions Wl to W4 of LPP, Eq. (5)

2 1 - 2712 @ 1980 The American Physical Society

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FIG. 1. Lowest-order diagram for LPP cross section.

takes the form

whereas in terms of the helicity structure func- tions

Here W,, is the "double-flip" structure func- t i ~ n . ~ These results a re to be compared with Eq. (4) for DIS. Of the three equivalent ver- sions, Eqs. (5), (6), and (7), only the f i r s t one is form-invariant when going over from L P P to DIS .

It is known that in perturbative QCD the struc- ture functions become functions of q 2 . In addi- tion:

(i) for DIS, the Callan-Gross relation is mod- ified by small first-order QCD terms7; and

(ii) for LPP, the Drell-Yan cross-section form- ula, which is essentially [(cf. Eq. (2)] is sub- ject to very substantial first-order correction^.^ It is natural, therefore, to ask how Eq. (5) i s affected by QCD effects in LPP.

If we represent the basic parton-parton amp- litude by Fig. 1, then first-order QCD correc- tions come in three forms: the "annihilation" diagrams, Fig. 2; the "Compton" diagrams, Fig. 3; and the "vertex correction" diagrams, Fig. 4. The annihilation and Compton diagrams have

( c ) ( d )

FIG. 2. Order-a, "annihilation diagrams" for LPP cross section.

( c ) ( d )

FIG. 3. Order-a, "Compton diagrams" for LPP cross section.

been studied by us8 and othersgfi0 in connection with high-q events for which they should repre- sent the dominant mechanism. I t has been noticed8 that the parton-model structure-function relation, Eq. (5), is also satisfied by these QCD diagrams. Since this result holds for all values of q,, i t means that these diagrams do not in- troduce any first-order QCD correction to the parton relation Eq. (5) even at low q,. We now point out that the remaining f i rs t order-diagrams Fig. 4, give r i se to an amplitude with the same tensor structure a s the zeroth-order amplitude of Fig. 1-as any anomalous magnetic moment term from the modified vertex drops out upon takingthe trace of the Dirac matrices. It follows then, even though the overall normalization may be corrected by some factor, the decomposition into invariant structure functions is not affected and Eq. '(5) remains intact.

We conclude, therefore, in L P P the parton- model relation WE = 2 Wi is not modified by first- order QCD corrections at all-in contrast to both the Callan-Gross relation (for DLS) and the Drell- Yan cross-section formula [essentially WE, cf. Eq. (2)]. This appears to be a rather remarkable result; we a r e not aware of any other parton- model result which is not affected by QCD cor- rections. For this reason, we sketch in the Appendix a derivation of Eq. (5) from the dia- grams Figs. 1-4 which is more direct than those given before. 8-10 I t will be interesting to find out to what extent this result can be extended to higher-order diagrams.

We now make a few additional remarks about Eq. (5) and i t s experimental implications:

FIG. 4. Order-a, "vertex-correction diagrams" for LPP cross section.

2714 C . S . L A M A N D W U - K I T U N G 21 -

(i) The relation = 2W1 is closely related to the sp in- i nature of the charged parton (i.e., quark). The fact that this relation is not mod- ified by f irst-order QCD effects when both sides themselves may be subject to corrections of order 100°/o certainly represents a unique signa- ture of the QCD-quark-parton picture.

(ii) In t e rms of helicity structure functions, this relation takes the form WL = 2 W,,, Eq. (7). Al- though for LPP, the helicity structure functions depend on the choice of coordinate axes4 (e. g., Gottfried-Jackson, C ollins-Soper , etc .). this relation remains frame independent-i.e., if the QCD-quark-parton model is correct , the two structure functions NTL and W,, must be related by Eq. (7), for any choice of axes in the lepton- pair center-of-mass frame. This strong result again demonstrates the significance of this r e - lation.

We know the angular distribution of the leptons in the r e s t f rame of the pair is given by4

Written in the form (8)

Eq. (7) implies

independent of the choice of axes. (iii) For q12 << Ad2, kinematic constraints re -

quire WAA to be small4 and of order q>/M2. The dynamical relation Eq. (7) res t r ic t s WL to stay the same a s 2WAA, hence i t must also be small. The angular distribution of the leptons should be close to 1 + cos29*. [w, i s required by kinemat- ics to be o(~,/M) and small also.] Now, make the transition to the large-q, region. There is no reason for WAA to stay small: the angular dis- tribution will be very different from 1 + cos28*, but Eq. (7) i s s t i l l in force and WL must keep pace with the change in W,, a s q, increases. Note, although the rate of high-q, events i s of order a,(M2), the predicted change in angular distribution from low to high qL is of order 1 and the relation between WL and W,, remains pre- cise throughout. In other words, the test of QCD furnished by this relation does not concern just smal l correction effects; it involves quan-

tities of order unity. For comparison, in DIS, the Callan-Gross r e -

lation reads WL = 0 in the parton limit and ac- quires a small a , correction when QCD i s taken into account. Since there is no q, variable in DIS, the transition in angular distribution dis- cussed above simply does not exist here.

(iv) So far , we have neglected all masses (m,) and intrinsic transverse momenta (Pi,) of the partons inside the hadron. It i s relatively easy to see that incorporating these effects in the zeroth-order parton model (Fig. 1) incurs order m,2/;1/12 and p , , 2 / ~ 2 corrections to the basic r e - lation Eq. (5).4 In current QCD jargon, these represent "higher-twist" effects. In DIS a s well a s elsewhere, the higher-twist effects occur alongside with calculable QCD (twist two) correc- tion terms, thereby complicating the phenomeno- logy considerably. Because of the unique situa- tion here in L P P where f irst-order QCD correc- tion is absent, deviations from Eqs. (5)-(7) can only come from the higher-twist effects. There- fore, we have a cleaner source of information on the size of ( p i L 2 ) , etc.

Incorporating parton intrinsic transverse mo- menta in f irst-order QCD diagrams, Figs. 2-4, is much more complicated. However, here the effects a r e of order a , p i 1 2 / ~ 2 and represent only a very small perturbation if M%S large.

(v) One may wonder why Eq. (5) is not subject to f irst-order QCD corrections in L P P while the same relation (i. e., Callan-Gross relation) in DJS is-after all, the Feynman diagrams in- volved in the two cases a r e identical except for line reversal. The technical explanation for this discrepancy lies in the fact that in DIS, correc- tions to the Callan-Gross relation come from Figs. 2 and 3 as a result of integration over the momenta of the two final-state partons; by con- t ras t , in LPP, there i s only one parton in the final state-the momentum of which is fixed by conservation, hence, there is no integration in- volved.

(vi) We mention, for completeness, that in ad- dition to Eq. (5), there a r e other parton-model relations4 which follow from the Drell-Yan mech- anism for low q, and from the QCD annihiliation diagrams, Fig. 2, for high q,. 8v '0 Those rela- tions a r e interesting in their own right and, in conjunction with Eq. (5), yield definite predic- tions on the angular distribution of the leptons. However, these predictions do depend on the choice of helicity f rames and a r e subject to QCD corrections, hence a r e not a s striking a s Eq. (5).

The work of C.S.L. was supported in part by the

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Department of Energy under Contract No. DE-ACO3- 76SP00515 and in part by the National Science and Engineering Research Council of Canada and the Quebec Department of Education. The research of W. -K.T. was supported in part by the National Science Foundation under Grant No. Phy. 78- 11576.

APPENDIX

In the literature, all structure-function rela- tions were derived by f i rs t calculating the full tensor amplitude. A more direct derivation of WE = 2 Wi is given in this appendix. The Feynman gauge is used and partons a r e assumed to be massless.

In Figs. 1 and 2(a), the cross-section tensor WUv are, respectively, proportional to

and

Using the identities

we immediately obtain the relation TE = 2Ti for both Fig. 1 and Fig. 2(a). Similarly we can drive the relation for Figs. 2(b), 3(a), and 3(b).

The derivation for Figs. 2(c), 2(d), 3(c), and 3(d) a r e more complicated in that the massless nature of the partons have to be used. We merely exhibit how this is done for Fig. 2(c). There

Using p3 =p2 +p5, p4 =pi -p5, p;=pi = p i =0, we get

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