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

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 single-logarithmic corrections to equations, order $\alpha_s\ln(1/x)$ of kernel. new include effects running coupling and unpolarized evolution, both which are parametrically significant order. large-$N_c$ large-$N_c\& N_f$ approximations equation computed. (Here, $N_c$ $N_f$ numbers colors flavors, respectively.) Their solutions will provide more precise estimates distribution small $x$, contributing resolution proton spin puzzle.