TMD splitting functions in kT factorization: the real contribution to the gluon-to-gluon splitting

TL;DR

This paper derives a new transverse momentum dependent gluon-to-gluon splitting function within $k_T$-factorization, unifying and extending existing kernels like BFKL, DGLAP, and CCFM, and confirming its consistency through direct QCD diagram calculations.

Contribution

The paper introduces a generalized $k_T$-dependent gluon splitting function that reduces to known kernels in various limits, unifying different approaches within a consistent formalism.

Findings

The new splitting function reduces to BFKL in the high energy limit.

It matches the DGLAP gluon splitting function in the collinear limit.

It converges to the CCFM kernel in the soft limit.

Abstract

We calculate the transverse momentum dependent gluon-to-gluon splitting function within $k_T$-factorization, generalizing the framework employed in the calculation of the quark splitting functions in [1-3] and demonstrate at the same time the consistency of the extended formalism with previous results. While existing versions of $k_T$ factorized evolution equations contain already a gluon-to-gluon splitting function i.e. the leading order Balitsky-Fadin-Kuraev-Lipatov (BFKL) kernel or the Ciafaloni-Catani-Fiore-Marchesini (CCFM) kernel, the obtained splitting function has the important property that it reduces both to the leading order BFKL kernel in the high energy limit, to the Dokshitzer-Gribov-Lipatov-Altarelli-Parisi (DGLAP) gluon-to-gluon splitting function in the collinear limit as well as to the CCFM kernel in the soft limit. At the same time we demonstrate that this splitting kernel can be obtained from a direct calculation of the QCD Feynman diagrams, based on a combined implementation of the Curci-Furmanski-Petronzio formalism for the calculation of the collinear splitting functions and the framework of high energy factorization.