Helicity Evolution at Small-x

Abstract

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 ${\alpha }_s{\ln }^2(1/x)$ in the polarization-dependent evolution along with the powers of ${\alpha }_s\ln (1/x)$ 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 $g_1$ structure function derived previously by Bartels, Ermolaev and Ryskin.

Comments: 33 pages, 20 figures; v2: typos corrected, a reference added; v3: more corrections included