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 resulted in an evolution equation for the gluon field-strength F12 and quark "axial current" ψ¯γ+γ5ψ operators (sandwiched between the appropriate light-cone Wilson lines) in the double-logarithmic approximation (DLA: summing powers of αsln2(1/x) with αs the strong coupling constant). In this work, we observe that an important mixing of the above operators with another gluon operator, D←iDi, 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, Di−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 (Di−D←i, F12, and ψ¯γ+γ5ψ), generalizing the previous results. We also construct closed DLA evolution equations in the large-Nc and large-Nc&Nf limits, with Nc and Nf the numbers of quark colors and flavors, respectively. Solving the large-Nc equations numerically we obtain the following small-x asymptotics of the quark and gluon helicity distributions ΔΣ and ΔG, along with the g1 structure function, ΔΣ(x,Q2)∼ΔG(x,Q2)∼g1(x,Q2)∼(1x)3.66αsNc2π√, in complete agreement with the earlier work by Bartels, Ermolaev and Ryskin.