TMD splitting functions and the corresponding evolution equation

TL;DR

This paper extends the formalism for transverse momentum dependent splitting functions, constructs a low-x gluon evolution equation including quark-to-gluon effects, and discusses its regularization and potential nonlinear generalizations.

Contribution

It generalizes the Catani Hautmann formalism to include Pgq and Pqq splitting functions and develops a low-x gluon evolution equation with non-diagonal quark-to-gluon splittings.

Findings

Derived the Pgq and Pqq TMD splitting functions.

Constructed a low-x gluon evolution equation with quark effects.

Demonstrated regularization of the Pgq singularity.

Abstract

We generalize the Catani Hautmann formalism to calculate the Pgq and Pqq transverse momentum dependent splitting functions. Then we use the obtained Pgq kernel to construct a low-x evolution equation for gluons that takes into account the effect of non-diagonal quark-to-gluon splittings. In order to write down a consistent equation we resum virtual corrections coming from the gluon channel and demonstrate that this implies a suitable regularization of the Pgq singularity, corresponding to a soft emitted quark. We also note that the obtained equation is in a straightforward manner generalized to a nonlinear evolution equation which takes into account effects due to the presence of high gluon densities.