We state the weak and strong factorization theorems for the Drell-Yan cross section and outline the ingredients involved in their proof. We also discuss the physical picture implied by the factorization results and its phenomenological consequences

Bodwin, G. T.

Brodsky, S. J.

Lepage, G. P.

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Ths submitted manuscript has been authoredby * wntractor of the U. S. Governmentunder ''contr&ct No. W-31-109-ENG-38-Accordingly, the U. S- Government retains anonexclusive, royalty-free license to publishor reprtttJuce the published form of thiscontribution, or allow others to do so, forU. S. Government purposes.JUi2oBiiANL-HEP-CP-85-32Apr i l , 1985FACTORIZATION OF THE DRELL-YAN CROSS SECTIONGeoffrey T. BodwinHigh Energy Physics DivisionArgonne National LaboratoryArgonne, I l l i n o i s 60439Stanley J . BrodskyStanford Linear Accelerator CenterStanford UniversityStanford, California 94305G. Peter LepageLaboratory of Nuclear StudiesCornell UniversityI thaca, New York 14853ANL/HEP-CP—85-32DE85 014978AbstractWe state the weak and strong factorization theorems for the Drell-Yancross section and outline the ingredients involved in their proof. We. alsodiscuss the physical picture implied by the factorization results and itsphenomenological consequences.DISCLAIMERThis report was prepared as an account of work sponsored by an agency of the United StatesGovernment. Neither the United States Government nor any agency thereof, nor any of theiremployees, makes any warranty, express or implied, or assumes any legal liability or responsi-bility for the accuracy, completeness, cr usefulness of any information, apparatus, product, orprocess disclosed, or represents that its use would not infringe privately owned rights. Refer-ence herein to any specific commercial product, process, or service by trade name, trademark,manufacturer, or otherwise does not necessarily constitute or imply its endorsement, recom-mendation, or favoring by the United States Government or any agency thereof. The viewsand opinions of authors expressed herein do not necessarily state or reflect those of theUnited States Government or any agency thereof.*A talk presented by GTB at the XX Recontre de Moriond, Les Arcs, France,March 10-17, 1985.mm. of THIS ooamarr« mtmimQIntroductionThe basic Drell-Yan process for lepton pair production is shown inFig.l. A quark from one hadron annihilates with an anti-quark from anotherhadron to produce a virtual photon, which in turn produces a lepton pair. Wefind it convenient to work in the hadron center of mass frame, with one hadronmoving in the plus-z direction and the other hadron moving in the minus-zdirection. In terms of the light-cone variables (p = p + pz, p~ = p - pz,p.), the upper hadron has momentum P^=(0,P,0 .) and the lower hadron hasmomentum ?2 = (P»0,0 . ) . The invariant momentum-squared of the virtual photonisQ2 = x x_P2 , (1)where x and x~ are the longitudinal momentum fractions of the quark andantiquark respectively.In QCD we would like to maintain the simple parton model picture ofFig. 1 as much as possible. Of course the basic Drell—Yan process can be"dressed" with any number of QCD corrections, such as real and virtual gluonradiation, real and virtual pair production, quark-gluon interactions, andgluon-gluon interactions. Thus, the naive parton picture cannot survive.However, we would still like to be able to associate a factor in the crosssection with each hadron (the structure function) and a remaining factor witha perturbatively calculable QCD correction to the basic parton model process(the hard process).The QCD corrections to the cross section can be characterized in terms ofregions of momentum space of the gluonic momenta. The regions that are ofimportance for the discussion of factorization are(1) Collinear(a) to + :(b) to - : ,(2) Central U+~*r-*l)(a) U.V. : |*U|»P(b) Hard : *pHP(c) Soft : m<|*u|«P(d) Very Soft : |iy|(3) Glauber :Here m is a typical hadronic mass and we think of P as a large momentum scale(P»m). The very soft region is the one discussed by Landshoff and Stirling ̂in connection with the on-shell—off-shell ambiguity in the K-factor. TheGlauber region corresponds to the kinematically allowed region of gluonmomentum for the case of nearly on-shell to nearly on-shell quark-quarkscattering( semi-classical scattering) by gluon exchange.Let us consider the role played by each of these regions in a diagram-matic calculation of the cross section. The U.V. region is removed by theusual renormalization counter-terms. The contributions from the hard regionare perturbatively calculable because of asymptotic freedom. However, thecollinear, soft, very soft, and Glauber regions potentially destroy thecalculability of radiative corrections, and thus factorization, by driving theloop integrals into regions of small invariant momentum-squared.A further obstacle to factorization was pointed out by Bodwin, Brodsky,and Lepage '„ They showed that spectator interactions like the one shown inFig.2 give a leading twist (leading in 1/P ) contribution to the crosssection. Since such interactions can involve both the upper and lower hadron,they cannot be associated with either hadron or with the basic process in anyobvious way. What is worse, they can become entangled with the QCD cor-rections to the basic process or with each other.The Factorization ProgramA number of authors have discussed programs for putting the Drell-Yancross section into a factored fornr'. However, the factorization scheme towhich we refer here is the one due to Collins, Soper, and Sterman '»-**.Their proposal consists of two parts:(1) Weak Factorization:At large Q , da/dQ dQ is a convolution of a hard process with structurefunctions S^h an<* ̂ q/2» w n e r e the subscripts indicate that these are thestructure functions for finding a quark or antiquark in a hadron 1 or 2.Them's contain all the collinear and all spectator contributions.(2) Strong Factorization:The Drell-Yan and deep-inelastic structure functions are simply related:In the context of nucleus collisions strong factorization appears para-doxical. For deep-inelastic scattering we know that, away from kinematicboundaries, cross sections are approximately proportional to nuclear mass:Then strong factorization implies that, for pion-nucleus collisions, theDrell-Yan cross section is also proportional to A :It seems that nuclear matter is transparent to quarks. That is, a quark onthe back face of the nucleus has the same probability as a quark on the frontface to annihilate with an antiquark from the pion. Of course we know thatthis picture must break down for a macroscopic target: pions beams aredepleted when they pass through a macroscopic length of material. Thus theremust be a constraint on the region of validity of any factorization theoremthat depends on the length of the target.Let us now outline a procedure by which leading twist contributions tothe Drell-Yan cross section can be put into the weak factored form'» '.(1) Or>e deforms loop—integration contours into their complex planes in orderto eliminate contributions from the Glauber region. This step makes useof the analyticity properties of the Feynman amplitudes and the unitaritycancellation of final state interactions. It is at this stage that atarget length condition emerges.(2) For each set of graphs in nth order, G^n', one constructs a set of col-linear subtractions, S^n\ such that G^n'-s'n^ contains only centralcontributions. It turns out that G^n'-s'n' contains only interactionsinvolving the active quark and antiquark.(3) One uses Ward identities to disentangle the S^n' into a topologicalform that is consistent with factorization.(4) Soft divergences cancel by a generalized Bloch-Nordsieck mechanism°'.(5) Contributions from the very soft region have the net effect of puttingthe active quark and antiquark on-shell.The result of this procedure is shown schematically in Fig.3. The crosssection is now topologically factored into structure functions (associatedwith each hadron) and a perturbatively calculable part involving only theactive quark and antiquark (acea£Ta\) • The net effect of the collinear andspectator interactions is to append an "eikonal" line—indicated by a doubleline in the figure—on to uhe hadron wave functions. The eikonal lineactually represents a path-ordered exponential of the line integral (along thecollision axis) of the hadron's chromodynamic field. It is the effect of thelong-range part of the hadronic vector potential, which is a pure gauge. Thehadronic wave functions alone are, of course, gauge-dependent, but the eikonallines combined with the wave functions are a gauge-invariant structure.Strong factorization is easily proven by making use of the analyticityproperties of the Feynman amplitudes and a cancellation between real, andvirtual emission . The cancellation requires that one integrate over chelepton pair transverse momentum.Given these results, we can explain the resolution of the A "paradox."The nucleus is not transparent to nuclear matter. There are spectatorinteractions in the Drell-Yan process, but to leading twist they have the sameeffect as the spectator interactions in deep-inelastic scattering. That is,they result in gauge-artifact eikonal lines. Their effect vanishes if onewrites the Drall-Yan cross section in terms of the deep-inelastic structurefunctions. Their effect is manifest for a long target, where the weak7 2factorization procedure breaks down, and for da/dQ dQ., where there is nostrong factorization connection between structure functions.Implications for PhenomenologyNow let us discuss the phenomenological consequences of the factorizationresults.First, there is a target length condition that must be satisfied iffactorisation is to hold. Namely,Q2»x_LMk2, <* A2/3 . (5)qHere, L is the target length, M is the target mass and k . is a typical softgluon momentum transfer ~200 MeV. For uranium, then, we expect the A*behavior that is characteristic of the factorized cross section to change tothe A ' behavior that is characteristic of interactions on the nuclearsurface at Q2~10 GeV2.If Q satisfies the target length condition, then the Drell-Yan and deep-inelastic structure functions are related by Eq.(2). That is, the normal-ization of the Drell-Yan cross section is completely fixed by the deep-inelastic results. This means that the K-factor is not an arbitrarynormalization factor but, rather, a specific consequence of higher-order QCDcorrections to the parton model. Eq.(2) also implies that if the EMCeffect ' is present in deep-inelastic scattering, it should appear in leptonpair-production as well.8Finally, the presence of spectator interactions implies a smearingof the lepton pair transverse momentum distribution that increases withnuclear size. If we attempt to estimate this effect by assuming the spectatorinteractions to be semi-classical, then we arrive at the result 'a + bA1/3 , (6)where b~(.l Gev) . The Chicago-Princeton pion-nucleus Dreli-Yan data ' for2Q as a function of target nucleus mass are shown in Fig.4. As can be seen,2the data show no indication of the predicted growth of Q. with nuclear size.However, they do not quite rule it out. Furthermore, we know from the EMCeffect that there is a softening of the quark longitudinal momentumdistribution with increasing nuclear size. Lorentz invariance implies acorresponding softening of the transverse momentum distribution. Bodwin,12}Brodsky, and Lepage have estimated the size of this effect and find it tobe —10% to -20% for Fe—that is roughly equal in magnitude and opposite insign to the spectator interaction smearing. If the softening effect saturates2at an A of about 27, then one might find that the nuclear dependence of Q, issomething like the dashed curve in Fig.5. Since the total effect is small atlarge A, it would be more difficult to observe than a simple linear dependenceon A1/3.AcknowledgementsWe wish to thank J. Collins, D. Soper, and G. Stennan for manyilluminating discussions on techniques for implementing the factorizationprogram. We also wish to acknowledge the contribution of J. Ralston, whofirst suggested to us the possiblility of nuclear softening of transversemomentum distributions.References1) P. V. Landshoff and W. J. Stirling, Zeit. Phys. C14, 251 (1982).2) G. T. Bodwin, S. J. Brodsky, and G. P. Lepage, Phys. Rev. Lett. 47, 1799(1981); and in Particles and Fields 2 (Proceedings of the 1981 BanffSummer School), edited by A. N. Kamal and A. Capri (Plenum Press, 1982).3) S. B. Libby and G. Sterman, Phys. Rev. D18, 3252 (1978) and Phys. Lett.78B, 618 (1978). R. K. Ellis, H. Georgi, M. Machacek, H. D. Politzer, andG. C. Ross, Nucl. Phys. B152, 285 (1979); S. Gupta and A. H. Mueller,Phys. Rev. D20 118 (1979); J. C. Collins and G. Sterman, Nuc. Phys. B185,172 (1981).4) J. C. Collins, D. E. Soper, and G. Sterman, Phys. Lett. 109B, 388 (1982).5) J. C. Collins, D. E. Soper, and G. Sterman, Phys. Lett. 126B. 275 (1983).6) G. T. Bodwin ANL-HEP-PR-8 -64 (Argonne National Lab.), September 1984.7) J. C. Collins, D. E. Soper, and G. Sterman, Phys. Lett. 134B, 263 (1984)and OITS 287 (University of Oregon, Eugene), January 1985.8) See also J. Frenkel, J. G. M. Gatheral, and J. C. Taylor, Nuc. Phys. B223,307 (1983).r£ • •r.!• 1 0b9) See Ref.6. Strong factorization is also discussed in Ref.5.10) J. J. Aubart, .et_al., Phys. Lett. 123B, 275 (1983).11) M. L. Swartz, _et̂ _al., Phys. Rev. Lett. 53 (1984) 32.12) G. T. Bodwin, S. J. Brodsky, and G. P. Lepage, unpublished.Figure CaptionsFig. 1 The basic Drell-Yan process.Fig. 2 An example of a spectator interaction.Fig. 3 The Weak Factored Form for the Drell-Yan Cross Section.Fig. A Q. Dependence on Nuclear Mass. The solid and broken lines are ofthe form given in Eqn.(3). The dashed line is indicative of theeffect of nuclear softening as explained in the text.MESON i MESON 1MESON 2 MESON 2Fig. 1Fig. 2CM><£AV0Be Cu Sn Wb=(.2GeV)2 —Ib = (.l GeV)2WITH NUCLEARSOFTENINGA l / 3Fig. 4

Factorization of the Drell-Yan cross section

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Factorization of the Drell-Yan cross section

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