Small-$x$ Asymptotics of the Leading-Twist Flavor Singlet Quark TMDs

TL;DR

This paper derives the small-$x$ asymptotics of all leading-twist flavor-singlet quark TMDs using the Light-Cone Operator formalism, providing a comprehensive understanding of their behavior at high energies.

Contribution

It introduces a systematic approach to analyze small-$x$ TMDs via polarized dipole amplitudes and solves their evolution equations in the large-$N_c$ and DLA approximations.

Findings

Derived small-$x$ asymptotics for various TMDs including Sivers and Boer-Mulders.

Provided a unified framework for understanding TMD behavior at small-$x$.

Discussed implications for phenomenology and future research directions.

Abstract

In this paper, we investigate the small-$x$ behavior of the flavor-singlet, leading-twist quark Transverse Momentum Dependent parton distribution functions (TMDs) using the Light-Cone Operator Treatment. This formalism allows us to express TMD operators at small $x$ in terms of polarized dipole amplitudes, enabling a systematic approach to their small-$x$ evolution. We derive the evolution equations for these TMDs and solve them within the large-$N_c$ approximation under the linearized, Double-Logarithmic Approximation (DLA), where $N_c$ represents the number of quark colors. Expanding on previous work on unpolarized and helicity TMDs, we present the small-$x$ asymptotics for a comprehensive set of TMDs, including the Sivers function, helicity worm-gear, transversity, pretzelosity, Boer-Mulders, and transversity worm-gear distributions. Our results provide a complete picture of the small-$x$ asymptotic behavior for all leading-twist flavor-singlet quark TMDs. We also discuss the implications of our findings for phenomenological applications and outline potential avenues for further research, particularly in understanding non-linear effects and extending beyond the DLA and large-$N_c$ approximations.