Proper definition and evolution of generalized transverse momentum distributions

TL;DR

This work identifies a fundamental flaw in the standard GTMD definitions, showing that rapidity divergences render them ill-defined unless a soft radiation factor is included. By incorporating a soft function analogous to the TMD formalism, the authors obtain a RD-free, properly evolvable definition of GTMDs and derive their evolution kernel, which is spin independent and compatible with NNLL resummation. They compute the NLO structure for an unpolarized quark GTMD, illustrating the cancellation of rapidity divergences and establishing a consistent framework linking GTMDs to TMDs and GPDs. The results provide a practical path to lattice calculations and potential experimental access, and they extend naturally to gluon GTMDs through the same soft-factor structure.

Abstract

We consider one of the most fundamental sets of hadronic matrix elements, namely the generalized transverse momentum distributions (GTMDs), and argue that their existing definitions lack proper evolution properties. By exploiting the similarity of GTMDs with the much better understood transverse momentum distributions, we argue that the existing definitions of GTMDs have to include an additional dependence on soft gluon radiation in order to render them properly defined. With this, we manage to obtain the evolution kernel of all (un)polarized quark and gluon GTMDs, which turns out to be spin independent. As a byproduct, all large logarithms can be resummed up to next-to-next-to-leading-logarithmic accuracy with the currently known perturbative ingredients.

Figures (2)

• Figure 1: Kinematics for GTMDs in the symmetric frame.
• Figure 2: Evolved GTMD $F_{1,1}^q$ from initial scale $Q_0=\sqrt{2}~{\rm GeV}$.