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 F 12 and quark “axial current” ψ ¯ γ $$ \overline{\psi}\gamma $$ + γ 5 ψ 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 ← i $$ {}_D{}^{\leftarrow i} $$ D i , 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, D i − D ← i $$ {}_D{}^{\leftarrow i} $$ , 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 (D i − D ← i $$ {}_D{}^{\leftarrow i} $$ , F 12, and ψ ¯ γ $$ \overline{\psi}\gamma $$ + γ 5 ψ), 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 g 1 structure function, ∆ Σ x Q 2 ∼ ∆ G x Q 2 ∼ g 1 x Q 2 ∼ 1 x 3.66 α s N c 2 π $$ \Delta \Sigma \left(x,{Q}^2\right)\sim \Delta G\left(x,{Q}^2\right)\sim {g}_1\left(x,{Q}^2\right)\sim {\left(\frac{1}{x}\right)}^{3.66\sqrt{\frac{\alpha_s{N}_c}{2\pi }}} $$ in complete agreement with the earlier work by Bartels, Ermolaev and Ryskin [5]