Single-Logarithmic Corrections to Small-$x$ Helicity Evolution

TL;DR

This paper derives single-logarithmic corrections to small-$x$ helicity evolution equations, incorporating effects like running coupling and unpolarized evolution, to improve estimates of quark helicity distributions relevant to the proton spin puzzle.

Contribution

It introduces the first derivation of single-logarithmic corrections to small-$x$ helicity evolution equations, including effects of running coupling and unpolarized evolution.

Findings

Derived single-logarithmic corrections to helicity evolution equations.

Included effects of running coupling and unpolarized evolution.

Provided solutions in large-$N_c$ and large-$N_c$&$N_f$ limits.

Abstract

The small-$x$ quark helicity evolution equations at double-logarithmic order, with the kernel $\sim\alpha_s\ln^2(1/x)$, have been derived previously. In this work, we derive the single-logarithmic corrections to the equations, to order $\alpha_s\ln(1/x)$ of the evolution kernel. The new equations include the effects of the running coupling and the unpolarized small-$x$ evolution, both of which are parametrically significant at single-logarithmic order. The large-$N_c$ and large-$N_c\& N_f$ approximations to the equation are computed. (Here, $N_c$ and $N_f$ are the numbers of quark colors and flavors, respectively.) Their solutions will provide more precise estimates of the quark helicity distribution at small $x$, contributing to the resolution of the proton spin puzzle.