Quark and Gluon Helicity Evolution at Small $x$: Revised and Updated

We revisit the problem of small Bjorken-$x$ evolution of the gluon and flavor-singlet quark helicity distributions in the shock wave ($s$-channel) formalism. Earlier works on the subject in the same framework resulted in an evolution equation for the gluon field-strength $F^{12}$ and quark “axial current” $\bar{\psi }{\gamma }^{+}{\gamma }^5\psi$ operators (sandwiched between the appropriate light-cone Wilson lines) in the double-logarithmic approximation (DLA: summing powers of ${\alpha }_s{\ln }^2(1\mathrm{/}x)$ with ${\alpha }_s$ the strong coupling constant). In this work, we observe that an important mixing of the above operators with another gluon operator, ${\overleftarrow{D}}^i{D}^i$, also sandwiched between the light-cone Wilson lines (with the repeated index $i=1,2$ summed over), was missing in the previous works. This operator has the physical meaning of the sub-eikonal (covariant) phase: its contribution to helicity evolution is shown to be proportional to another sub-eikonal operator, $D^i-{\overleftarrow{D}}^i$, which is related to the Jaffe-Manohar polarized gluon distribution. In this work we include this operator into small-$x$ helicity evolution, and construct a novel evolution mixing all three operators ($D^i-{\overleftarrow{D}}^i$, $F^{12}$, and $\bar{\psi }{\gamma }^{+}{\gamma }^5\psi$), generalizing the previous results. We also construct closed DLA evolution equations in the large-$N_c$ and large-$N_c\mathrm{\&}{N}_f$ limits, with $N_c$ and $N_f$ the numbers of quark colors and flavors, respectively. Solving the large-$N_c$ equations numerically we obtain the following small-$x$ asymptotics of the quark and gluon helicity distributions $\mathrm{\Delta }\mathrm{\Sigma }$ and $\mathrm{\Delta }G$, along with the $g_1$ structure function, $$\mathrm{\Delta }\mathrm{\Sigma }(x,Q^2)\sim \mathrm{\Delta }G(x,Q^2)\sim g_1(x,Q^2)\sim {\left(\frac{1}{x}\right)}^{3.66\sqrt{\frac{{\alpha }_s{N}_c}{2\pi }}},$$ in complete agreement with the earlier work by Bartels, Ermolaev and Ryskin.