**TMD splitting functions and the corresponding evolution equation**

*A. Kusina*

in XXV International Workshop on Deep-Inelastic Scattering and Related Subjects

Contribution: pdf

**Abstract**

We generalize the Catani Hautmann formalism to calculate the $P_{gq}$ and $P_{qq}$ transverse momentum dependent splitting functions. Then we use the obtained $P_{gq}$ 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 $P_{gq}$ 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.