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

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” ψγ + γ 5 ψ operators (sandwiched between the appropriate light-cone Wilson lines) in the double-logarithmic approximation (summing powers of α s ln 2 (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 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 , 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 , F 12 , and ψγ + 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, ΔΣ(x,Q^2)∼Δ G(x,Q^2)∼g_1(x,Q^2)∼(1/x)^3.66√(α_sN_c/2π) in complete agreement with the earlier work by Bartels, Ermolaev and Ryskin [ 5 ].