We construct small-x evolution equations which can be used to calculate quark and anti-quark helicity TMDs and PDFs, along with the g 1 structure function. These evolution equations resum powers of α s ln 2 (1/x) in the polarization-dependent evolution along with the powers of α s ln(1/x) in the unpolarized evolution which includes saturation effects. The equations are written in an operator form in terms of polarization-dependent Wilson line-like operators. While the equations do not close in general, they become closed and self-contained systems of non-linear equations in the large-N c and large-N c & N f limits. As a cross-check, in the ladder approximation, our equations map onto the same ladder limit of the infrared evolution equations for the g 1 structure function derived previously by Bartels, Ermolaev and Ryskin [1].

We rederive the small-x evolution equations governing quark helicity distribution in a proton using solely an operator-based approach. In our previous works on the subject, the evolution equations were derived using a mix of diagrammatic and operator-based methods. In this work, we re-derive the double-logarithmic small-x evolution equations for quark helicity in terms of the "polarized Wilson lines", the operators consisting of light-cone Wilson lines with one or two non-eikonal local operator insertions which bring in helicity dependence. For the first time we give explicit and complete expressions for the quark and gluon polarized Wilson line operators, including insertions of both the gluon and quark sub-eikonal operators. We show that the double-logarithmic small-x evolution of the "polarized dipole amplitude" operators, made out of regular light-cone Wilson lines along with the polarized ones constructed here, reproduces the equations derived in our earlier works. The method we present here can be used as a template for determining the small-x asymptotics of any transverse momentum-dependent (TMD) quark (or gluon) parton distribution functions (PDFs), and is not limited to helicity.

Abstract:We determine the small-x asymptotics of the gluon helicity distribution in a proton at leading order in perturbative QCD at large N c . To achieve this, we begin by evaluating the dipole gluon helicity TMD at small x. In the process we obtain an interesting new result: in contrast to the unpolarized dipole gluon TMD case, the operator governing the small-x behavior of the dipole gluon helicity TMD is different from the operator corresponding to the polarized dipole scattering amplitude (used in our previous work to determine the small-x asymptotics of the quark helicity distribution). We then construct and solve novel small-x large-N c evolution equations for the operator related to the dipole gluon helicity TMD. Our main result is the small-x asymptotics for the gluon helicity distribution: ∆G ∼

We extend our earlier results for the quark helicity evolution at small x [1] to derive the small-x asymptotics of the flavor singlet and flavor non-singlet quark helicity TMDs and PDFs and of the g1 structure function. In the flavor singlet case we re-derive the evolution equations obtained in our previous paper on the subject [1], performing additional cross-checks of our results. In the flavor non-singlet case we construct new small-x evolution equations by employing the large-Nc limit. All evolution equations resum double-logarithmic powers of αs ln 2 (1/x) in the polarization-dependent evolution along with the single-logarithmic powers of αs ln(1/x) in the unpolarized evolution which includes saturation effects. We solve the linearized flavor non-singlet equation analytically, obtaining an intercept which agrees with the one calculated earlier by Bartels, Ermolaev and Ryskin [2] using the infra-red evolution equations. Our numerical solution of the linearized large-Nc evolution equations for the flavor singlet case is presented in the accompanying Letter [3] and is further discussed here.