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 [1,2,3] 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 [1,2,3]. 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 [1,2,3]. 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,
in complete agreement with the earlier work by Bartels, Ermolaev and Ryskin [5].
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Cougoulic, F., Kovchegov, Y.V., Tarasov, A. et al. Quark and gluon helicity evolution at small x: revised and updated. J. High Energ. Phys. 2022, 95 (2022). https://doi.org/10.1007/JHEP07(2022)095
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DOI: https://doi.org/10.1007/JHEP07(2022)095