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 $\alpha_s\,\ln^2(1/x)$ 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 betwee…