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][2][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 αs ln 2 (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 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][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, 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][2][3]. We also construct closed double-logarithmic evolution equations in the large-Nc and large-Nc&N f limits, with Nc and N f 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,3.66 αs Nc 2π , in complete agreement with the earlier work by Bartels, Ermolaev and Ryskin [5].

Single-Logarithmic Corrections to Small-$x$ Helicity Evolution

“…At small x in the double-logarithmic approximation, the quark polarized dipole amplitude 103.2 relates to the quark helicity distribution as follows [2,7].…”

“…With all these definitions, we now have a closed system of integral equations that can be solved to determine the high-energy asymptotic of G(x 2 10 , z min , z pol ), which in turn implies the small-x asymptotic of ∆Σ(x, Q 2 ) at large-N c . Note that, at large N c , the quark and gluon polarized dipole amplitudes satisfy Q(x 2 10 , z min , z pol ) = 1 4 G(x 2 10 , z min , z pol ).…”

“…Since the presentation of this work in April 2021, there have been updates on our small-x evolution. The updates are detailed in [2].…”