Valence Quark Transversity at Small $x$

TL;DR

This paper develops a formalism to determine the small-$x$ behavior of valence quark transversity TMDs, deriving their asymptotics and relating them to polarized dipole amplitudes, with results consistent with previous theoretical predictions.

Contribution

It introduces a new formalism for small-$x$ asymptotics of transversity TMDs and derives their explicit small-$x$ behavior in the large-$N_c$ limit.

Findings

Derived the evolution equation for transversely polarized dipole amplitude.

Obtained the small-$x$ asymptotics of valence quark transversity TMDs.

Confirmed consistency with previous theoretical asymptotics.

Abstract

In our previous work we established a formalism which allows one to determine the small-$x$ asymptotics of any transverse momentum-dependent parton distribution function (TMD PDF) of the proton at small values of strong coupling. In this paper we apply this formalism to the valence quark transversity TMD. We relate the valence quark transversity to the transversely polarized dipole scattering amplitude, written in terms of the fundamental transversely-polarized "Wilson line" operator, an expression for which we derive explicitly as well. We then write down the evolution equation for the transversely polarized dipole amplitude. Solving that equation we arrive at the following small-$x$ asymptotics of the valence quark transversity in the large-$N_c$ limit: \begin{align} h_{1T}^v (x, k_T^2) \sim h_{1T}^{\perp \, v} (x, k_T^2) \sim \left( \frac{1}{x} \right)^{-1 + 2 \, \sqrt{\frac{\alpha_s \, N_c}{2 \, \pi}} } . \notag \end{align} This result is in agreement with one of the two possible small-$x$ asymptotics for the transverse structure function found previously by Kirschner, Mankiewicz, Sch\"{a}fer, and Szymanowski.