Helicity Evolution at Small $x$: the Single-Logarithmic Contribution

We calculate single-logarithmic corrections to the small-$x$ flavor-singlet helicity evolution equations derived recently in the double-logarithmic approximation. The new single-logarithmic part of the evolution kernel sums up powers of $\alpha_s \, \ln (1/x)$, which are an important correction to the dominant powers of $\alpha_s \, \ln^2 (1/x)$ summed up by the double-logarithmic kernel at small values of Bjorken $x$ and with $\alpha_s$ the strong coupling constant. The single-logarithmic terms arise separately from either the longitudinal or transverse momentum integrals. Consequently, the evolution equations we derive simultaneously include the small-$x$ evolution kernel and the leading-order polarized DGLAP splitting functions. We further enhance the equations by calculating the running coupling corrections to the kernel.