Spin-Spin Coupling at Small $x$: Worm-Gear and Pretzelosity TMDs

Abstract

We study the small-$x$ asymptotics of the leading-twist quark transverse momentum dependent parton distribution functions (TMDs) which encode couplings between the polarization of the quarks and that of their parent hadron, with at least one of the two polarizations in the transverse direction: the two worm-gear TMDs $g_{1T}$ and $h_{1L}^{\perp}$, and the pretzelosity $h_{1T}^{\perp}$. We apply the recently developed Light Cone Operator Treatment, finding that in the flavor non-singlet sector all three TMDs reduce in the small-$x$ limit to previously known polarized dipole amplitudes, and thus the large-$N_c$, linearized, Double Logarithmic Approximation (DLA) asymptotics have all been solved for previously. For the worm-gear TMD $g_{1T}$ we find \begin{align} g_{1T}^{\textrm{NS}} (x \ll 1, k_T^2) \sim \left( \frac{1}{x} \right)^0 , \end{align} while for the worm-gear TMD $h_{1L}^{\perp}$ we find \begin{align} h_{1L}^{\perp \textrm{NS}} (x \ll 1, k_T^2) \sim \left( \frac{1}{x} \right)^{-1}. \end{align} Finally, for the pretzelosity TMD $h_{1T}^{\perp}$ we find \begin{align} h_{1 T}^{\perp \textrm{NS}} (x \ll 1, k_T^2) \sim \left( \frac{1}{x} \right)^{-1 + 2 \sqrt{\frac{\alpha_s N_c}{2\pi}}}. \end{align} We have compiled these asymptotics together with those of the other five flavor non-singlet, leading-twist quark TMDs into a single unified table.