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

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.

Comments: Minor edits to appear in SciPost. 7 pages, 3 figures