Quark and gluon helicity evolution at small x: revised and updated

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