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QCD Corrections to the Polarized Drell-Yan Process

The results of a recent calculation of the QCD corrections to the longitudinally polarized Drell-Yan process are summarized. In addition to the parity conserving double spin asymmetry which is already accessible in the Drell-Yan production of lepton pairs at fixed target energies, we consider parity violating single spin asymmetries appearing only in the production of massive vector bosons at polarized hadron colliders. The prospects for extracting polarized quark distributions from asymmetries in vector boson production at RHIC are briefly discussed.


Introduction
The production of lepton pairs in hadron collisions, the Drell-Yan process [1], is one of the most powerful tools to probe the structure of hadrons. Its parton model interpretation is straightforward -the process is induced by the annihilation of a quark-antiquark pair into a virtual photon which subsequently decays into a lepton pair. The Drell-Yan process in protonproton or proton-nucleus collisions therefore provides a direct probe of the antiquark densities in protons and nuclei. Experimental data on the Drell-Yan process in unpolarized collisions are crucial to constrain the behaviour of the sea quark distributions at large x and to determine the flavour structure of the light quark sea. It is therefore natural to expect that a measurement of the Drell-Yan cross section in polarized hadron-hadron collisions will yield vital information on the polarization of the quark sea in the nucleon, which is presently only poorly constrained [2,3,4] from deep inelastic scattering data [5].
Apart from the invariant mass M distribution, one usually studies the distribution of the lepton pairs as function of the Feynman parameter x F or of the hadron-hadron centre-of-mass rapidity y. The resulting distributions at fixed invariant mass can be directly related to the x-dependence of the parton distributions in beam and target. Moreover, most fixed target experiments have only a limited kinematic coverage in x F or y, such that only these distributions can be measured without extrapolation into experimentally inaccessible regions.
All calculations for Drell-Yan cross sections can be immediately applied to the production of vector bosons at hadron colliders, which is mediated by exactly the same partonic subprocesses. In the case of vector boson production with polarized beams, one finds however one crucial difference to the Drell-Yan process, which can be viewed as the production of an offshell photon of invariant mass M. The parity violating couplings of the vector bosons give rise to non-vanishing single spin asymmetries, which are absent in the 'classical' Drell-Yan process. These single spin asymmetries, measured in vector boson production with one polarized and one unpolarized hadron beam, allow for a study of the various polarized quark distributions. The single spin asymmetries can at lowest order be directly related [6] to the ratio of polarized and unpolarized quark distributions, and they are largely dominated by one particular quark (or antiquark) flavour in most regions of y.
The double and single spin asymmetries in massive vector production will soon become experimentally accessible with the start of the spin physics programme at RHIC [7]. It will on the other hand be very difficult to study the Drell-Yan process at RHIC, since its cross section is already vanishingly small at RHIC energies ( √ s = 500 GeV). The ideal place to study this process would be e.g. the proposed HERA-N experiment [8] operating a fixed polarized nucleon target in the polarized HERA proton beam [9].
The rather large QCD corrections to the unpolarized Drell-Yan cross section [10] suggest that a reliable interpretation of the Drell-Yan process in terms of partonic distribution functions is only possible if higher order corrections are taken into account. Following closely the method of the unpolarized calculation [10], we have calculated [11] the next-to-leading order corrections to the x F -and y-distributions of lepton pairs produced in collisions of longitudinally polarized hadrons. This calculation has recently been extended [12] to single spin asymmetries in vector boson production at hadron colliders.
A fully consistent numerical study of spin asymmetries in the Drell-Yan process and in vector boson production at next-to-leading order was until now not possible, as the polarized parton distributions could only be determined at leading accuracy. With the recently calculated polarized two-loop splitting functions [13], the polarized distributions can now be determined to next-to-leading order from fits [2,3,4] to polarized structure function data. We will present some consistent next-to-leading order results for these asymmetries below.
2 Double and single spin asymmetries at next-to-leading order Truncated up to O(α s ), the cross section for the double polarized Drell-Yan process receives contributions from the qq-annihilation process at leading and next-to-leading order and the quark-gluon Compton scattering process. It can be expressed as: where analytic expressions for the parton level cross sections in the MS-scheme are listed in [11]. The expression for the y-distribution of the Drell-Yan pairs takes a similar form. Integration  over x F yields the Drell-Yan mass distribution which agrees with earlier results [14]. The numerical importance of these different contributions is illustrated in Figure 1, which shows the ratio between polarized and unpolarized Drell-Yan cross sections for the collision of a proton beam (E p = 820 GeV) on a fixed proton target [8]. All curves are obtained with the polarized GS(A) parton densities [3] and are shown for M = 8 GeV. The polarized subprocess contributions are normalized to the full unpolarized cross section at next-to-leading order. The relative magnitude of the individual contributions is similar to the unpolarized Drell-Yan process [10]. The O(α s ) correction to the qq-annihilation process enhances significantly the lowest order prediction while the quark-gluon Compton process contributes with a sign opposite to the annihilation process. However, the relative magnitude of annihilation and Compton process depends on the magnitude of the gluon distribution at large x, which is completely undetermined at present [3,4]. This uncertainty prevents the sensible prediction of a K-factor between the cross sections at leading and next-to-leading order.
The theoretical uncertainty inherent to an extraction of the polarized quark distribution functions from future data on the polarized Drell-Yan process can be estimated by varying the mass factorization scale µ F which is used in the evaluation of the asymmetry. It is found [11] that the absolute value of the asymmetry changes by less than 0.01 if µ F is varied between 0.5 M and 2 M. This uncertainty has to be compared to the discrepancy between predictions for the asymmetry obtained with different parameterizations of the polarized parton distributions [3,4], which lie in the band A = −0.05 . . . + 0.05.
In the QCD corrections to single spin asymmetries [12,15], one has to discriminate two different contributions from the quark-gluon Compton scattering process: a gluon from the polarized hadron scattering of a quark from the unpolarized hadron (denoted by gq) and a gluon from the unpolarized hadron scattering off a quark from the polarized hadron (denoted by qg). The calculation of these QCD corrections is very similar to the corresponding calculation for the double spin asymmetries and will be presented in [12]. Figure 2 illustrates the magnitude of the individual partonic subprocesses to the single polarized W + production cross section in proton-proton collisions at RHIC ( √ s = 500 GeV). As above, the contributions are normalized to the full unpolarized cross section at next-to-leading order. The polarized proton is moving qg, (1) gq,(1) Figure 2: Contributions of the individual parton level subprocesses to the single polarized W + production cross section at RHIC (see text).
in the +y direction, such that A L (y > 0) is dominated by ∆u(x, Q 2 ) in the range x > ∼ 0.2, while A L (y < 0) mainly reflects ∆d(x, Q 2 ) in the x < ∼ 0.2 [6]. Figure 2, obtained with the polarized GS(A) parton distributions for W + production at RHIC, illustrates the sizable impact of the next-to-leading order corrections on the single polarized cross section. Like in the case of the double polarized cross section, one finds an enhancement of the cross section due to corrections to the annihilation process, which is partly compensated by the Compton scattering process contributing with opposite sign. It is found that the contribution from the gq Compton scattering process becomes relevant only for A L (y < 0), while being completely negligible in A L (y > 0). Consequently, an extraction of the large x behaviour of the polarized quark distributions at next-to-leading order from a measurement of A L (y > 0) at RHIC will not suffer from the uncertainty on the polarized gluon distribution, since only the qg Compton scattering process (with the known unpolarized gluon distribution) contributes. The determination of the polarized antiquark distributions from A LL (y) or A L (y < 0) on the other hand requires always the knowledge of the polarized gluon distribution for a sensible prediction of the next-to-leading order corrections.
Variation of the mass factorization scale µ F allows again for an estimate of the theoretical uncertainty on the single spin asymmetries due to unknown higher order corrections. It is found that this uncertainty is below 0.01 for the whole range of y, thus indicating the perturbative stability of the next-to-leading order calculation for this asymmetry. Even smaller values are obtained for the double spin asymmetry A LL (y) in vector boson production at RHIC.
The double and single spin asymmetries in vector boson production at RHIC are evaluated for different up-to-date parameterizations [3,4] of the polarized parton distribution functions in Figure 3. It can clearly be seen that these parameterizations yield significantly different predictions for all asymmetries, thus reflecting the range of quark polarizations in the nucleon still allowed by present polarized deep inelastic scattering data.
The single spin asymmetry A L (y > 0) is dominated by contributions from valence quarks at large x, which are already reasonably well constrained. However, in the case of W − production, related to ∆d(x, Q 2 ), a discrepancy of about a factor of 2 between different parameterizations can still be seen in this region. A measurement of this asymmetry at RHIC will thus provide valuable supplementary information on the polarized valence quarks, although they are believed to be already well constrained from deep inelastic scattering data. The situation is even more drastic for A L (y < 0) and A LL (y). These asymmetries are dominated by contributions from polarized sea quarks, which are only weakly constrained by present deep inelastic scattering data. This is clearly reflected in Figure 3: on the basis of the present parameterizations, it is not even possible to predict the sign of these asymmetries, neither their magnitude. Measuring these asymmetries at RHIC will clearly provide the first direct information on the shape and the flavour structure of the polarized light quark sea. The discrepancies between different parameterizations are far larger than the theoretical error on all asymmetries estimated above. It is therefore apparent that a measurement at RHIC will enable an unambiguous discrimination between the parameterizations, which is not affected by theoretical uncertainties. However, a proper extraction of the polarized sea quark distributions at next-to-leading order from A L (y < 0) and A LL (y) will require some knowledge on the behaviour of the polarized gluon distribution at large x, which enters in the determination of the K-factor.

Summary
In summary, the complete O(α s ) corrections to the x F -and y-dependence of the longitudinally polarized Drell-Yan cross section have been derived [11]. These results have been recently extended [12] to double and single spin asymmetries in vector boson production in collisions of longitudinally polarized hadrons. These corrections are quantitatively similar to the corrections in the unpolarized case and hence sizable even at collider energies. They enable a consistent next-to-leading order determination of the polarization of quarks in the nucleon from future measurements of lepton pair production at fixed target energies or from massive vector boson production at RHIC.
A reliable quantification of the next-to-leading order effects in the polarized Drell-Yan process requires at least an approximate knowledge of the polarized gluon distribution ∆G(x, Q 2 ). The lack of information on this quantity affects in particular the asymmetries A LL (y) and A L (y < 0), which both probe the sea quark polarization in the nucleon. The asymmetry A L (y > 0), probing mainly the quark polarizations at large x, is on the other hand completely insensitive to the polarized gluon distribution.
Finally, we have illustrated the discriminative power of a measurement of asymmetries in vector boson production at RHIC between different up-to-date parameterizations [3,4] of the polarized parton distribution functions. It was especially shown that the scale uncertainty on these asymmetries at next-to-leading order is far smaller than the deviation between the different parameterizations.