Closing in on the Spin-Dependent Gluon Distribution
By Daniel de Florian and Werner Vogelsang
Two decades have passed since the European Muon Collaboration
(EMC) at CERN discovered that the spins of the quarks and
anti-quarks in the proton provide only a small fraction of the
proton's spin [1]. This finding, which became famously known as
the "proton spin crisis," initiated much theoretical activity
and motivated further experimental studies of nucleon spin
structure. The spin of the proton being 1/2, the smallness of
the quark and anti-quark spin contribution implies that the
spins of the gluons or orbital angular momenta of the partons
must contribute significantly, or both. The determination of the
gluon spin contribution to the proton spin, ∆G, is a major goal
of the RHIC spin program. Key to this is to measure the
spin-dependent gluon distribution ∆g(x), with its dependence on
the momentum fraction carried by the gluon. ∆G is obtained from this by
∆G=∫01 dx ∆g(x).
Theorists have presented various estimates of ∆G. A recent model calculation [2], for example, obtains a value of ∆G ~ 0.3, which is of the order of the "missing part" of the proton spin, at a scale of 1 GeV. One would regard values of this size as quite "natural". In the wake of the EMC result, it was proposed that an explanation for the smallness of the quark spin contribution should be sought in a shielding" of the quark spins by a perturbative gluonic contribution to DIS associated with the axial anomaly of QCD [3]. In essence, positively polarized gluons in the nucleon can produce pairs of sea quarks and anti-quarks with net negative polarization, which may be seen by the virtual photon in DIS, hence suppressing the value found for the total quark and anti-quark spin contribution. Since this process is formally only a higher-order contribution, ∆G must be very large in order to be able to generate a sufficient amount of negative sea quark polarization. Typical values needed for this mechanism to be of phenomenological relevance are ∆G ~1.5 (again at a scale of 1 GeV), three times larger than the proton spin itself! If realized, such a large polarization of the confining fields inside a nucleon would certainly be a phenomenon as puzzling as the proton spin crisis itself.
Prior to RHIC, information on the spin structure of the nucleon came from polarized deep-inelastic lepton scattering (DIS) off fixed targets. As DIS is an electromagnetic probe and gluons carry no electric charge, sensitivity to the gluon distribution arises only through the scaling violations of the DIS structure function, described by the DGLAP equations, or through higher-order diagrams. Since the DIS experiments were all fixed-target, the lever arm for studying scaling violations was quite limited. As a result, information on the spin-dependent gluon distribution remained elusive.
In polarized proton-proton collisions at RHIC, however, one can study processes for which gluons enter directly at the lowest order of perturbation theory and in fact even dominate the physics in certain kinematical regions. Examples are pion and jet production at transverse momenta, pT, of a few GeV or more, whose spin-averaged cross sections receive substantial contributions from gluon-gluon and quark-gluon scattering. Neutral-pion and jet production have become "flagship" observables for PHENIX and STAR, respectively. Earlier results [4,5] already indicated that the gluons in the proton are relatively little polarized, likely much less than expected within the "anomaly scenario" we discussed above. This finding was in line with results from dedicated studies in lepton scattering made by the Hermes (DESY) and Compass (CERN) experiments [6]. But it is the most recent precise (still preliminary) RHIC data collected during Run 6 [7,8] that allow to close in on the spin-dependent gluon distribution ∆g(x).
Previously, information about gluon polarization from RHIC data was obtained by confronting theoretical next-to-leading order (NLO) perturbative calculations [9] of the spin asymmetries with the data, essentially choosing a few different ∆g(x) distributions and seeing which of them described the data best. The new preliminary Run 6 RHIC data, however, justify a more serious theoretical analysis. The goal is to simultaneously analyze all the available data from (inclusive and semi-inclusive) DIS and RHIC, to really extract the best set of spin-dependent parton distribution functions. In all this the - often important - next-to-leading order corrections should be included. Such an analysis is referred to as a "global" NLO QCD analysis. One reason why it is important to analyze all data sets simultaneously is that the theoretical description links them all: for example, the polarized quark and anti-quark distributions play a role not only in DIS, but also for the spin asymmetries at RHIC. Likewise, better knowledge of ∆g(x) from RHIC will also lead to better constraints on the sea quark distributions, since the latter are generated in part by gluon splitting.
The precise method of extracting the parton distributions from data is based on a χ2 minimization procedure: the parton distributions are parameterized at an initial scale and evolved to the scale relevant for each data point, where then the spin asymmetry and the resulting χ2 contribution for the data point is computed. The process is repeated, typically a few thousand times, varying parameters of the initial distributions until a minimum in χ2 is found. NLO global analyses of the DIS data became available about a decade ago. Many such analyses have subsequently been presented, typically showing large differences in the sea quark distributions and, of course, the almost undetermined gluon density. It turns out that including data from pp collisions at RHIC in the NLO global analysis is technically a very challenging task. This is related to the facts that two parton distributions enter in the theoretical calculation of the pp cross section (as opposed to only one in DIS), that the kinematics in pp scattering is in general more complex than in DIS, and that the NLO terms in the partonic cross sections are harder to evaluate numerically.
In collaboration with Rodolfo Sassot (Buenos Aires U.) and Marco Stratmann (RIKEN) we have recently succeeded in performing the first global NLO analysis of the DIS and RHIC data [10]. We have overcome the technical obstacles just described by adopting a combination of various techniques, involving a re-organization of the calculation, the use of Mellin moments, and Monte-Carlo sampling. The analysis yields very interesting results, shown in Fig.1. For the first time, a strong constraint on ∆g(x) is found, thanks in large part to the RHIC data. The gluon distribution turns out to be small in the region of momentum fraction x accessible at RHIC, quite possibly having a node. As a result of the advent of more precise semi-inclusive DIS data [12] and of a new set of fragmentation functions [13] that describes the observables well in the unpolarized case, also the various polarized u,d,s sea quark distributions are constrained better now. We find that they are far from SU(3) symmetric: the ubar distribution is mainly positive, while the dbar anti-quarks carry opposite polarization. The strange sea quark density shows a sign change, which is due to a certain tension between the inclusive DIS data, which demand a negative integral of ∆s, and the semi-inclusive DIS data, which prefer a positive ∆s at medium x. We have also extracted estimates of the uncertainties of the various parton distributions. The shaded bands in Fig.1 show which distributions are allowed if one permits an increase of ∆ χ2=1 (green) or ∆ χ2/χ2=2% (yellow). We note that future improvements of our study will include a more detailed account of the experimental errors and theoretical uncertainties. The constraints on ∆g(x) will further improve when higher statistics data will become available from RHIC, hopefully in the next run.
Fig.1: Polarized sea quark and gluon densities found in the global NLO analysis [10], compared to those of previous fits [11]. The shaded bands correspond to alternative fits with ∆ χ2 = 1 (green) and ∆ χ2/χ2 = 2% (yellow).
As we have seen, the RHIC data so far show that ∆g(x) is small in the measured x region, implying that also its integral over that region is constrained to be small. The χ2 profile shown in Fig.2 as a function of the integral ∫0.050.2 dx ∆g(x) at Q2=10 GeV2 confirms this. If we integrate our fitted gluon distribution over all 0<x<1, we find a slightly negative value, ∆G=-0.084, for the gluon spin contribution to the proton spin. It is, however, not yet possible to make a reliable statement about ∆G. This will require to constrain ∆g(x) at lower x than so far accessible. It is not ruled out that there are significant contributions to ∆G from x<0.02 or so. It is useful to recall in this context that the spin crisis was only discovered because the EMC experiment was able to go to lower x than its predecessors! Measurements at RHIC's higher pp center-of-mass energy of 500 GeV, and/or at more forward angles, will likely be important for going to lower x, and a high-energy electron-proton polarized collider (EIC) could become the ultimate tool to explore the small x region.
Fig.2: The χ2 profile (left) for variations of the integral of ∆g(x) over the region 0.05<x<0.2, at Q2=10 GeV2. The plot on the right shows the individual contribution from each experiment to the total χ2.
More information can be found in http://arXiv.org/abs/0804.0422.
References:
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[7] K. Boyle [PHENIX Collaboration], talk at the 2007 RHIC AGS Users' Meeting, BNL, June 2007
[8] M. Sarsour [STAR Collaboration], talk at the 2007 APS DNP meeting, Newport News, Virginia, Oct. 2007
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[13] D. de Florian, R. Sassot and M. Stratmann, Phys. Rev. D75, 114010 (2007)
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