Constraints on spin-dependent parton distributions at large x from global QCD analysis

We investigate the behavior of spin-dependent parton distribution functions (PDFs) at large parton momentum fractions x in the context of global QCD analysis. We explore the constraints from existing deep-inelastic scattering data, and from theoretical expectations for the leading x ->1 behavior based on hard gluon exchange in perturbative QCD. Systematic uncertainties from the dependence of the PDFs on the choice of parametrization are studied by considering functional forms motivated by orbital angular momentum arguments. Finally, we quantify the reduction in the PDF uncertainties that may be expected from future high-x data from Jefferson Lab at 12 GeV.


I. INTRODUCTION
Recently a new global next-to-leading order (NLO) analysis [1] of spin-dependent parton distribution functions (PDFs) was performed by the JAM (Jefferson Lab Angular Momentum) Collaboration [2], in which particular attention was paid to the valence quarkdominated region at high parton momentum fractions x and low four-momentum transfers Q 2 . This region requires careful treatment of the potentially important 1/Q 2 power corrections associated with target mass and higher twist contributions to the inelastic cross sections, as well as nuclear smearing effects when using deuterium and 3 He data. The analysis [1] indeed found significant effects on the polarized leading twist PDFs when twist-3 and twist-4 power corrections in both the spin-dependent g 1 and g 2 structure functions were taken into account. In particular, the ∆d + distribution (defined as ∆d + ≡ ∆d + ∆d) was found to have a significantly larger magnitude at x 0.2 than in previous global analyses, driven partly by a large and positive twist-4 correction to the neutron g 1 structure function.
Analyses such as those in Ref. [1] that systematically incorporate subleading effects in an effort to accommodate data over a broad range of kinematics can therefore provide a more solid basis for extracting reliable information on PDFs and their uncertainties, especially in regions such as at large x where data are relatively scarce [3]. In fact, the absence of high-precision polarization data at high x, particularly for the neutron (or 3 He), has meant that spin-dependent PDFs are essentially unconstrained for x 0.5 − 0.6. This is rather unfortunate, given that polarized PDFs, and ratios of polarized to unpolarized PDFs, are quite sensitive to the details of nonperturbative quark-gluon dynamics in the nucleon at high x, with theoretical predictions differing in some cases even in sign [4].
While most global PDF analyses do not include conditions such as in Eq. (1) in order to avoid introducing theoretical bias into the PDF extraction, Brodsky, Burkardt and Schmidt [14] proposed a simple parametrization of PDFs in which the large-x constraints of Eq. (1) were built in. Only a limited set of data was analysed in Ref. [14], however, although a subsequent global analysis utilizing the pQCD expectations was performed by Leader, Sidorov and Stamenov (LSS) [15]. This found that a reasonably good fit to the available data was indeed possible, with the feature of a steep rise in the ∆d/d ratio at intermediate values of x. Later high-precision data from the E99-117 experimental at Jefferson Lab [16] observed the first evidence of a turn over in the A n 1 neutron polarization asymmetry from negative to positive values, although at somewhat larger values of x (x ∼ 0.5 − 0.6) than those in the fit of Ref. [15].
In a more recent analysis, Avakian et al. [17] showed that inclusion of L z = 1 components in the lowest three-quark Fock state of the nucleon, in addition to the usual L z = 0 configurations, can generate additional terms that behave as (1 − x) 5 log 2 (1 − x) at large x, which can play an important role numerically. Generalizing the pQCD-inspired parametrization from Ref. [14] to include the subleading log 2 (1 − x) terms, Avakian et al. showed that the large-x asymmetry data could be well fitted while preserving the asymptotic constraints of Eq. (1). In particular, the new ∆d/d ratio was found to remain negative until x ≈ 0.75, as suggested by the E99-117 data [16], before rising towards unity at higher x values.
While the analysis of Ref. [17] showed the potential of high-x data to reveal information about the orbital angular momentum (OAM) of quarks in the nucleon, it was not based on a comprehensive global analysis of all available data. The goal of the present work is to examine the behavior of spin-dependent PDFs in the x ∼ 1 region in the context of a global QCD analysis, including the effects of the x → 1 constraints in Eq. (1) and of the log 2 (1 − x) terms inspired by pQCD.
We begin our discussion in Sec. II by summarizing the recent global analysis [1] from the JAM Collobaration, which we use as the baseline fit for our large-x studies. To explore the dependence on the choice of parametrization and allow for more direct connection with quark orbital angular momentum, we also consider a simplified functional form which uses a smaller number of parameters. The effects on the fits of additional terms in the PDF parametrizations induced by nonzero orbital angular momentum are investigated, together with the impact on the ∆u and ∆d PDFs from imposing constraints for the x → 1 behavior from perturbative QCD. In Sec. III we repeat the global analysis using in addition pseudodata generated at the kinematics of future data from several experiments planned at the 12 GeV energy upgraded Jefferson Lab, and quantify the resulting reduction in the PDF errors at high x. Finally in Sec. IV we draw some conclusions of the present analysis and outline steps for future work.

II. PARTON DISTRIBUTIONS AT LARGE x
For our exploration of the large-x region we use as a baseline the PDFs from the JAM analysis in Ref. [1]. The JAM PDFs were obtained from a global NLO fit to all available data on inclusive polarized deep-inelastic scattering asymmetries for Q 2 > Q 2 0 = 1 GeV 2 and W > 1.87 GeV. Inclusion of low-Q 2 and low-W data necessitated a careful treatment of the subleading 1/Q 2 contributions, to both the g 1 and g 2 structure functions, from target mass and higher twist corrections, as well as nuclear smearing effects for deuterium and 3 He data. By fitting directly the longitudinal and transverse polarization asymmetries, where available, one avoids introducing biases that would otherwise arise in fits to the spindependent structure functions, which are often extracted from the experimental asymmetries under different assumptions for the spin-averaged structure functions.
A standard parametrization was used for the polarized quark, antiquark and gluon distributions in terms of four parameters plus an overall normalization, where ∆q + ≡ ∆q + ∆q, at the input scale Q 2 0 . At large values of x the antiquark and gluon PDFs play a negligible role, so that the inclusive DIS data alone are sufficient to determine the ∆u + and ∆d + distributions individually. A total of over 1,000 data points were used in the analysis, giving an overall χ 2 per data point of 0.98 (see Table I). The resulting ∆u + and ∆d + distributions are illustrated in Fig. 1, together with the ratios to the spin-averaged PDFs, ∆u + /u + and ∆d + /d + , at Q 2 = 1 GeV 2 .
Unlike the Q 2 dependence of PDFs, which is determined by the Q 2 evolution equations to a given order in the strong coupling α s [18], the x dependence of PDFs is generally not accessible directly from pQCD calculations. An exception is the kinematic region at large x, where hard gluon exchange between the quarks in the leading three-quark Fock state including the JAM, SIMP and OAM fits, with or without the x → 1 constraint in Eq. (1), and including leading twist (LT) contributions only. For the JAM fit we also considered the cases where the ∆d + PDF was forced to cross zero at x = 0.5 and x = 0.75.
component of the nucleon can be used to determine the dominant contributions to the x dependence of the PDFs in the x → 1 limit [9]. Typically one finds that the quark PDFs in the nucleon behave as ∼ (1 − x) 2ns−1 , where n s is the minimum number of partons that are spectators to the hard collision [9,14], so that for n s = 2 the leading exponent is equal to 3.
More generally, the exponent on (1 − x) also depends logarithmically on Q 2 [19], although the scale from which this should evolve is a priori unknown. Nevertheless, global PDF fits do find parameters β in Eq.
(2) that are close to the pQCD (or quark "counting rule") predictions; for the JAM fit, for instance, one has β u = 3.3 and β d = 4.0 for ∆u + and ∆d + , respectively, at the input scale Q 2 0 . Of course, the additional polynomial terms in (2) with coefficients and η obscure the direct connection between the x dependence of the fitted distributions and the predicted pQCD behavior. To make the connection more explicit, we consider a fit based on a simplified functional form, with parameters and η in Eq. (2) set to zero. The resulting fit, labeled "SIMP" in Fig. 1, gives similar ∆u + and ∆d + distributions to those from the full JAM analysis, albeit with a slightly larger overall χ 2 value. The leading (1 − x) exponents in this case are reduced slightly to β u = 2.5 and β d = 3.4.
As well as specifying the leading x → 1 behavior of the PDFs, the pQCD counting rules also predict that the dominant contribution to the cross section, in a reference frame where the nucleon is moving fast along the z-axis, is from scattering off quarks with the same helicity as that of the nucleon. This implies that asymptotically the helicity-aligned distributions dominate both the unpolarized and polarized PDFs, so that the ratio ∆q + /q + → 1 as x → 1 for all quark flavors q, as in Eq. (1). In this scenario the A 1 polarization asymmetries in DIS are therefore expected to approach unity for both the proton and neutron.
Unfortunately, current data cannot definitively confirm the pQCD expectations. While the proton A p 1 asymmetries, which have been measured to x ≈ 0.7 in the DIS region, are consistent with an approach towards unity in the x → 1 limit, the neutron (or 3 He) data extend only to x ≈ 0.6 and are generally consistent with a zero or negative asymmetry. The dearth of high-x data means that the spin-dependent PDFs, and particularly the ∆d + distributions, are essentially unconstrained above this region. Consequently the spin-dependent PDFs obtained from global analyses often violate the positivity condition |∆q(x)| ≤ q(x) at large x (although strictly speaking these need not be satisfied beyond leading order). This can be seen in Fig. 1 for both the JAM and SIMP (and other) fits, and in general will be a feature of any global fit which does not a priori impose the positivity constraint (or else fits at Q 2 = 3 GeV 2 with proton (left) and 3 He (right) A 1 polarization asymmetry data for similar Q 2 values. The proton data are from the SLAC E143 [22] and HERMES [23] experiments, while the 3 He data are from the SLAC E142 [24], HERMES [25] and Jefferson Lab E99-117 [16] experiments.
fit the individual helicity-aligned and helicity-antialigned distributions separately). Note that in the present work the unpolarized fits are based on the recent NLO global analysis from Ref. [20]. Ideally one should perform a global fit of polarized and unpolarized data simultaneously and extract the helicity-aligned and antialigned distributions. This will be particularly important in future analyses that include semi-inclusive data sets, where transverse momentum distributions, which are expected to be different for different helicity states [21], will play a more important role.
To explore the effect of the x → 1 constraints (1) on PDFs in the context of the JAM global analysis, we consider a modified fit in which the ∆u + /u + and ∆d + /d + ratios are both forced to unity at x = 1. The resulting fit, denoted by "JAM+" in Fig. 1 and Table I, shows that one can indeed obtain a reasonable description of data, consistent with the pQCD x → 1 limit, with an overall χ 2 /N dat = 1.01 that is only slightly larger than for the unconstrained fit. This is confirmed also in Fig. 2, where the JAM and JAM+ fits are compared with data on the A 1 asymmetries for the proton and 3 He from SLAC, HERMES, and Jefferson Lab.
The increase in χ 2 is associated with the reduced magnitude of ∆d + in the intermediate-x region, x 0.3, which in order to maintain the normalizations required by the triplet and octet axial vector charges [1], becomes slightly more negative (with larger magnitude) at smaller x, x 0.2, where considerably more data exist. The ∆u + distribution, on the other hand, which was already large and positive in the JAM fit, undergoes relatively little change with the x = 1 constraint. Note that the constraint (1) cannot be accommodated by the SIMP parametrization, as without nonzero or η terms in Eq.
(2) the ∆d + distribution cannot change sign at any x.
Interestingly, the turn-over in ∆d + from negative to positive values occurs at relatively large values of x, x ≈ 0.95, which would be challenging to observe experimentally. This is significantly higher than the turn-over found in the earlier LSS analysis [15] at x ∼ 0.5, which was subsequently found to be in conflict with the neutron asymmetry data from the E99-117 experiment at Jefferson Lab [16]. Indeed, the existing data tend to disfavor fits with positive d quark polarization over the measured x range. We studied this by forcing a zero crossing in ∆d + at x = x 0 , with the distribution becoming positive for x > x 0 . For the JAM fit with x 0 = 0.75 the χ 2 /N dat increased slightly compared to the JAM+ fit, but the increase was significantly greater, to χ 2 /N dat = 1.06, when the crossing was set at a lower value, x 0 = 0.5.
Of course, the behavior of leading twist PDFs at large x is also influenced to some extent by the effect of higher twist corrections, which become more important as x → 1.
Using either the JAM or SIMP parametrizations without including the 1/Q 2 corrections generally results in a significantly worse fit, with χ 2 /N dat values increasing form ≈ 1 to ≈ 1.1 in Table I,  Although formally subleading in the x → 1 limit compared with the dominant ∼ (1 − x) 3 contributions expected from the L z = 0 component, numerically the log terms can play an important role. In particular, Avakian et al. found that by using the pQCD-inspired parametrization from Ref. [14] supplemented by the subleading log 2 (1−x) terms, the large-x asymmetry data could be well fitted while preserving the asymptotic constraints of Eq. (1).
Furthermore, the resulting ∆d + /d + ratio remained negative until x ≈ 0.75, as suggested by the E99-117 data [16], before rising towards unity at higher x values.
To explore the importance of the additional log 2 (1 − x) terms in the context of a global QCD analysis of all data, at small and high x, we use as a basis the simplified parametrization with = 0 = η in Eq.
(2), together with the log term inspired by the OAM arguments, with arbitrary relative normalization N . It is reasonable as a first approximation to assume that the x → 0 behavior of the OAM-inspired term is the same as the standard term, To reduce the number of parameters that can be reliably determined from the existing data, we also fix β = 5 in accordance with the pQCD derivation [17], even though the corresponding power of (1 − x) in the first term of Eq. (3) remains a free parameter.
The resulting fit, denoted by "OAM" in Fig. 1, is of comparable quality to the JAM fit the quark orbital angular momentum [26]. Its appearance in the present analysis serves more to illustrate the possible role played by OAM in understanding PDFs at large x, and to explore the systematic uncertainties that may arise from different assumptions about the functional forms used for the PDF parametrizations. Fits including only terms with L z = 0 and L z = 1 [17], which can be interpreted in terms of relative contributions from different orbital states, will be reported elsewhere [27].
A scenario in which one finds qualitatively different fits with comparable χ 2 values, or fits which differ by amounts that are larger than the uncertainties from the propagation of experimental errors, indicates a lack of information at large x, and an underestimate of the systematic errors in this region. In the absence of clearer theoretical constraints at x 1, the problem can be best addressed of course by the availability of new data at higher x values than are currently available, which we discuss in the next section.

III. CONSTRAINTS FROM FUTURE DATA
Constraining the behavior of the polarization asymmetries A 1 , and consequently of the spin-dependent PDFs, in the limit as x → 1 is one of the featured goals of the experimental physics program planned for the 12 GeV energy upgraded CEBAF accelerator at Jefferson Lab. Data from several experiments are expected to be collected for values of x as high as ≈ 0.8 for DIS kinematics [28], and even higher x in the nucleon resonance region. This should significantly reduce the PDF uncertainties for x 0.5, especially for the ∆d + distribution, which will be more strongly constrained by new data on the 3 He asymmetry.
To estimate the possible impact of the new Jefferson Lab data we use the projected statistical and systematic uncertainties for the proposed experiments at the x and Q 2 values where the asymmetries will be measured [28]. The pseudodata are generated by randomly distributing the central values of the points around the JAM fit in Fig. 2 for hydrogen, deuterium and 3 He targets (distributing them around any of the other fits considered in this analysis would be equally suitable). The reduction in the PDF uncertainties, illustrated in Fig. 3, is significant, with the relative error on ∆u + and ∆d + decreasing by ∼ 70% for x = 0.6 − 0.8 at the input scale Q 2 = 1 GeV 2 .
Reductions in the spin-dependent PDF errors such as these, combined with similarly dramatic reductions expected for the uncertainty on the unpolarized d quark distribution (or the d/u ratio) [29], should at the very least allow one to discriminate between a ∆d/d ratio that remains negative, as in simple quark models, and one that approaches ∼ 1 in the x → 1 limit, as predicted by pQCD arguments. Beyond this there may be additional constraints on the x → 1 behavior of spin-dependent PDFs from an electron-ion collider [30][31][32], particularly if the spectator tagging technique [33] in semi-inclusive DIS from the deuteron or 3 He can be extended to polarized beams and targets.

IV. CONCLUSIONS
The aim of this analysis was to investigate whether existing data from polarized leptonnucleon DIS is able to provide any constraints on the x → 1 behavior of spin-dependent PDFs in the context of a global QCD analysis. Using the recent JAM fit as a baseline, we showed that demanding the polarized to unpolarized PDF ratios ∆q + /q + to approach unity at x = 1 results in equally good fits to the available data, even though the resulting changes to the ∆d + PDF are significant in the intermediate-x region. With dramatically different behaviors for the ∆d + /d + ratio allowed for x 0.3, this highlights the critical need for precise data sensitive to the d quark polarization at large x values.
We have investigated the recent suggestion that inclusion of Fock states in the nucleon wave function with nonzero orbital angular momentum gives rise to additional contributions to PDFs proportional to (1 − x) 5 log 2 (1 − x) [17] which could play an important role numerically. Employing an extension of the typical functional form used in standard PDF analyses which allows for the log dependence, we find that the generalized parametrization is also able to provide a good description of the existing DIS data, with or without the x = 1 constraint. While there has been a first indication of a rise above unity of the neutron ( 3 He) polarization asymmetry for x 0.6 [16], the data still generally prefer a negative ∆d + distribution at large x even with the x = 1 limit built in, although the cross over to positive values depends on the parametrization chosen (at ≈ 0.95 for the JAM+ and ≈ 0.65 for OAM+).
Further progress on this problem is expected soon with new data expected from several experiments at the 12 GeV energy upgraded Jefferson Lab, which aim to measure polarization asymmetries of protons, deuterons and 3 He up to x ∼ 0.8 in DIS kinematics [28].
Using the projected statistical and systematic errors from these experiments, we explored the possible impact on the PDFs and their uncertainties in this region. We find reductions in both the ∆u + and ∆d + PDFs of up to ≈ 70% for x ≈ 0.6 − 0.8 in the JAM fit, with significant reductions also at smaller x values. This should considerably narrow the range of possible asymptotic x → 1 behaviors of the ∆q + /q + ratios, and for the first time provide critical tests of the various theoretical scenarios that have been proposed to describe PDFs in the large-x region [4][5][6][7][8][9]14].