Generalized transverse momentum distributions at small-$x$

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.