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 αs ln2(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{}^{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-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 ∆Σ and ∆G, along with the g1 structure function,∆ΣxQ2∼∆GxQ2∼g1xQ2∼1x3.66αsNc2π\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\Delta \\Sigma \\left(x,{Q}^2\\right)\\sim \\Delta G\\left(x,{Q}^2\\right)\\sim {g}_1\\left(x,{Q}^2\\right)\\sim {\\left(\\frac{1}{x}\\right)}^{3.66\\sqrt{\\frac{\\alpha_s{N}_c}{2\\pi }}} $$\\end{document}in complete agreement with the earlier work by Bartels, Ermolaev and Ryskin [5].