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

Abstract

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 [1–3] resulted in an evolution equation for the gluon field-strength F¹² and quark “axial current” $\overline{\psi }\gamma$⁺γ⁵ψ operators (sandwiched between the appropriate light-cone Wilson lines) in the double-logarithmic approximation (summing powers of α_s ln²(1/x) with α_s the strong coupling constant). In this work, we observe that an important mixing of the above operators with another gluon operator, ${}_D{}^{\leftarrow i}$Dⁱ, also sandwiched between the light-cone Wilson lines (with the repeated transverse index i = 1, 2 summed over), was missing in the previous works [1–3]. 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, Dⁱ − ${}_D{}^{\leftarrow i}$ , which is related to the Jaffe-Manohar polarized gluon distribution [4]. In this work we include this new operator into small-x helicity evolution, and construct novel evolution equations mixing all three operators (Dⁱ − ${}_D{}^{\leftarrow i}$, F¹², and $\overline{\psi }\gamma$⁺γ⁵ψ), generalizing the results of [1–3]. We also construct closed double-logarithmic evolution equations in the large-N_c and large-N_c&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 ∆Σ and ∆G, along with the g₁ structure function,$$∆\mathrm{\Sigma }\left(x,Q^2\right)\sim ∆G\left(x,Q^2\right)\sim g_1\left(x,Q^2\right)\sim {\left(\frac{1}{x}\right)}^{3.66\sqrt{\frac{{\alpha }_s{N}_c}{2\pi }}}$$

in complete agreement with the earlier work by Bartels, Ermolaev and Ryskin [5].