Abstract
We calculate transverse momentum dependent quark splitting kernels $P_{gq}$ and $P_{qq}$ within $k_T$-factorization, completing earlier results which concentrated on gluon splitting functions $P_{gg}$ and $P_{qg}$. The complete set of splitting kernels is an essential requirement for the formulation of a complete set of evolution equations for transverse momentum dependent parton distribution functions and the development of corresponding parton shower algorithms.
Highlights
JHEP01(2016)181 the CCFM equation is the Pgg splitting function
Note that in [18] it has been observed that inclusion of non-singular pieces of the DGLAP gluon splitting function into CCFM evolution strongly affects the solution of the evolution equation
In this paper we extended the method developed by Catani and Hautmann for the determination of transverse-momentum-dependent parton splitting functions to splittings of initial kT -dependent quarks, based on factorization of cross-sections in the high energy limit
Summary
We start our presentation with a short review of the results of [12, 25] which allowed for the definition of the TMD Pqg splitting function and eventually of the sea-quark density. To ensure gauge invariance in the presence of off-shell splitting kernels, factorization of the process qg∗ → qZ in the high-energy limit as realized by the reggeized quark formalism [31, 41] has been employed. While [12] concentrates on factorization of a particular process, namely qg∗ → qZ, one can show that the resulting matrix elements and TMD splitting functions are process-independent To this end we recall the details of the high-energy factorization of the qg∗ → qZ matrix element: within the reggeized quark formalism, the entire process is described using a single diagram, figure 1, with the qq∗ → Z and g∗q∗ → q sub-amplitudes connected by reggeized. The second term can be neglected and one remains with the projector p/α1α2n/β1β2 which only contracts the Dirac indices of the qq∗ → Z and g∗q∗ → q sub-amplitudes respectively and leads to a complete factorization of both processes
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