Polarized Parton Distribution Functions in the Nucleon

TL;DR

This work analyzes world polarized DIS data to extract longitudinally polarized parton distributions by fitting the spin asymmetry $A_1$ under positivity and counting-rule constraints at a low scale $Q_0^2=1$ GeV$^2$. It develops LO and NLO DGLAP evolution within the $\,\overline{MS}$ scheme and a flexible parametrization $\Delta f_i(x)=h_i(x)f_i(x)$, yielding three AAC distributions (one LO, two NLO). The results show a small quark spin content $\Delta\Sigma$ in NLO and a substantial, but poorly constrained, gluon polarization $\Delta g$, with the small-$x$ sea remaining a major uncertainty. The analysis highlights the need for future high-energy polarized experiments to fix the sea-quark polarization and to pin down the spin contributions of quarks and gluons more precisely.

Abstract

Polarized parton distribution functions are determined by using world data from the longitudinally polarized deep inelastic scattering experiments. A new parametrization of the parton distribution functions is adopted by taking into account the positivity and the counting rule. From the fit to the asymmetry data A_1, the polarized distribution functions of u and d valence quarks, sea quarks, and gluon are obtained. The results indicate that the quark spin content is ΔΣ=0.20 and 0.05 in the leading order (LO) and the next-to-leading-order (NLO) MS-bar scheme, respectively. However, if x dependence of the sea-quark distribution is fixed at small x by "perturbative QCD" and Regge theory, it becomes ΔΣ=0.24 ~ 0.28 in the NLO. The small-x behavior cannot be uniquely determined by the existing data, which indicates the importance of future experiments. From our analysis, we propose one set of LO distributions and two sets of NLO ones as the longitudinally-polarized parton distribution functions.

Figures (9)

• Figure 1: $Q^2$ evolution results for the proton structure function $g_1^{\, p}$. The initial LO and NLO-1 $g_1$ structure functions are evolved to those at 60 GeV$^2$ by the LO and NLO DGLAP evolution equations.
• Figure 2: Calculated LO and NLO spin asymmetries $A_1$ for the proton are compared with the experimental results by the SMC, SLAC-E143, and HERMES collaborations at $x \approx 0.117$. The theoretical curves are obtained by using our LO and NLO-1 fitting results at $Q^2$=1 GeV$^2$.
• Figure 3: Comparison of our calculations with the experimental asymmetry $A_1(x, Q^2)$ data for the (a) proton, (b) neutron, and (c) deuteron. Our results are obtained at $Q^2$=5 GeV$^2$ with the optimum parameters in Tables \ref{['T:LO']} (LO) and \ref{['T:NLO']} (NLO-1). The NLO and LO results are shown by the solid and dotted lines, respectively.
• Figure 4: Experimental spin-dependent structure functions $xg_1(x, Q^2)$ are compared with our LO results for the (a) proton, (b) neutron, and (c) deuteron. Our fitting results are calculated at $Q^2$=1, 5, 20 GeV$^2$ by using the LO evolution equations with the optimum parameters in Table \ref{['T:LO']}, and they are shown by the dashed, solid, and dotted curves, respectively. The experimental data are obtained from the $A_1(x, Q^2)$ data and the $F_2(x, Q^2)$ calculated with the unpolarized GRV distributions and $R_{1990}(x, Q^2)$.
• Figure 5: Experimental data of $xg_1(x, Q^2)$ are compared with our NLO-1 results for the (a) proton, (b) neutron, and (c) deuteron. The notations are the same as those in Fig. \ref{['fig:g1lo']}.