PHYSICAL REVIEW D VOLUME 39, NUMBER 9 1 MAY 1989 Scattering in nuclei and QCD Keith Kastella and George Sterman Institute for Theoretical Physics, State University of New York at Stony Brook, Stony Brook, New York 11 794-3840 Joseph Milana Physics Department, Oregon State University, Corvallis, Oregon 97331 (Received 2 December 1988) We argue that double scattering in nuclei can be treated perturbatively in QCD, and derive ex- pressions for double-scattering contributions to short-distance cross sections. These cross sections are sensitive to the distribution of gluons at low x. I. INTRODUCTION Important atomic-number (A) dependence has long been observed in high-P, hadron-nucleus reactions, and more recently in lepton pair production in nuclei.' One plausible explanation for A dependence has been found in multiple scattering of a single parton of the projectile hadr~n.~ When all these scatterings involve large momentum transfer, they may be treated as localized QCD subprocesses.3 In the limit where one scattering is soft, however, these models require an infrared ~ u t o f f , ~ and are sensitive to nonperturbative effects. The importance of soft scatterings over nuclear scales was pointed out in Ref. 5. Although these soft effects turn out to be higher they may still play a role because they grow with nuclear dimensions. In the following, we study sequential multiple scatter- ing in the context of perturbative QCD. We argue that double scattering, at least, may be treated perturbatively as a factorizing, first-nonleading-twist effect, and that to leading power in the nuclear radius (or equivalently in our approximation, A 'I3), the double-scattering cross section is determined by the distribution of gluons in a bound nucleon. In particular, we find that the leading A dependence is sensitive to the small-x behavior of the gluon distrib~tion.~ This is in contrast to models in which multiple scattering is treated in terms of the quark-nucleon cross s e ~ t i o n . ~ Finally, triple or higher scatterings should be suppressed by extra factors of A 'I3 /R ;e2 (R is the internucleon separation), which we take to be small. When both scatterings are hard, the model of Ref. 3 (which we refer to as the "hard-scattering" model) should be adequate, and we review it in Sec. 11. In Sec. I11 we present arguments which suggest that double hard scattering is a perturbative higher-twist effect, and that the leading noncancelling contribution involves the phys- ical degrees of freedom of the soft gluon. Sections IV-VII describe the process of verifying this result at lowest order. Section IV describes the diagrammatic ap- proach, in which the projectile parton is treated perturba- tively, and the target nonperturbatively. We develop our approximations for the target structure in Sec. V, and an- alyze the projectile in Sec. VI, showing how short- and long-distance effects factorize and the leading power can- cels. In Sec. VII we show that the remaining contribu- tion involves matrix elements of the gluonic field strength. Finally, in Sec. VIII we summarize our results on soft scattering, combine them with the hard-scattering model, and discuss the interplay of nuclear structure with the gluon distribution in determining A dependence. 11. HARD-SCATTERING MODEL In order to set the context, we briefly review the hard- scattering model3 specialized to double scattering, and neglecting "intrinsic" transverse momentum. Consider the single-particle inclusive cross section for projectile h with momentum Ph to produce hadron h' with momen- tum Phv on nucleus A, with momentum transfer Q 2= -2Ph-Ph,. For simplicity at high energy, we drop projectile masses in the kinematics. The active parton, of type i from hadron h, has momentum where vp=6p+. This parton scatters elastically into the final momentum I'=l +Q after two collisions, and then fragments into the observed hadron h' with momentum fraction z, so that The parton sequentially absorbs momenta (Q -q)p and qp from the target. Our main interest will be in the "soft regions" for which q, << Q, or / Q, -9, I << Q,. For its part, the nucleus is modeled in its rest frame as a sphere of radius R =Ro A'/3 centered at the origin, with uniform number density p=3/4aR i. The parton- model cross sections h 'J( 1 ;I' ) r E 'd ui'j/d 31' for projec- tile partons to make the transition i+j in flavor and I-tl' in momentum on a single-nucleon target with momentum P, are 2586 01989 The American Physical Society