Generalized transverse momentum distributions at small-$x$

TL;DR

This paper calculates the complete set of leading-twist gluon and sea-quark GTMDs at small-$x$, establishing universal relations and connecting them to TMDs and GPDs, with implications for phenomenological modeling.

Contribution

It provides the first comprehensive calculation of small-$x$ gluon and sea-quark GTMDs, revealing universal relations and linking them to known distributions.

Findings

All gluon GTMDs expressed via gluon dipole operators.

Universal relations between different GTMDs at small-$x$.

Sea-quark GTMDs related to gluon dipoles with new helicity-flip results.

Abstract

We compute the complete set of the leading-twist gluon and sea-quark generalized transverse momentum distributions (GTMDs) in the small-$x$, or eikonal, approximation at vanishing skewness $\xi = 0$. All the gluon GTMDs become expressed in terms of the basic gluon dipole operator featuring also proton helicity-flips. Consequently, we establish universal relations between otherwise distinct GTMDs that hold in the small-$x$ limit. The obtained results are systematically projected onto the transverse momentum dependent distributions (TMD) and the generalized parton distribution (GPD) cases, recovering known results where available. In case of sea-quarks all the GTMDs are given in terms of the gluon dipole convoluted with a hard kernel. We generalize the unpolarized sea-quark GTMD to non-zero momentum transfers and find new results for the proton helicity-flip distributions. We pay special attention to their perturbative, high transverse momentum, tails to confirm that it becomes governed by the small-$x$ gluon GPDs. The obtained results provide guidance to the phenomenological modeling of the GTMDs but also allow for their explicit computations.