Helicity Evolution at Small $x$: Quark to Gluon and Gluon to Quark Transition Operators

We include the quark to gluon and gluon to quark shock-wave transition operators into the small Bjorken-$x$ evolution equations for helicity in the flavor-singlet channel derived earlier. While such transitions do not affect the large-$N_c$ version of the evolution equations for helicity, the large-$N_c \& N_f$ equations are affected. ($N_c$ and $N_f$ are the numbers of quark colors and flavors, respectively.) We derive the corresponding corrected large-$N_c \& N_f$ equations for the polarized dipole amplitudes contributing to the flavor-singlet quark and gluon helicity distributions in the double-logarithmic approximation (DLA), resumming powers of $\alpha_s \, \ln^2 (1/x)$ with $\alpha_s$ the strong coupling constant. We solve these equations iteratively and extract the polarized splitting functions up to four loops. We show that our splitting functions agree with the fixed-order perturbative calculations up to and including the existing three-loops results. Similar to the large-$N_c$ helicity evolution in the shock-wave approach, our large-$N_c \& N_f$ small-$x$ splitting functions agree with those obtained in the infrared evolution equations framework up to three loops, but appear to slightly disagree at four loops.