Dissecting the collinear structure of quark splitting at NNLL

We explore the collinear limit of final-state quark splittings at order alpha(2)(s). While at general NLL level, this limit is described simply by a product of leading-order 1 -> 2 DGLAP splitting functions, at the NNLL level we need to consider 1 -> 3 splitting functions. Here, by performing suitable integrals of the triple-collinear splitting functions, we demonstrate how one may extract B-2(q)(z), a differential version of the coefficient B-2(q) that enters the quark form factor at NNLL and governs the intensity of collinear radiation from a quark. The variable z corresponds to the quark energy fraction after an initial 1 -> 2 splitting, and our results yield effective higher-order splitting functions, which may be considered as a step towards the construction of NNLL parton showers. Further, while in the limit z -> 1 we recover the standard soft limit results involving the CMW coupling with scale kt, the z dependence we obtain also motivates the extension of the notion of a physical coupling beyond the soft limit.

Automated summary

This study examines the collinear limit of final-state quark splittings at order alpha(2)(s) and highlights the necessity to consider 1 -> 3 splitting functions at the NNLL level. By performing suitable integrals, the extraction of B-2(q)(z) from the quark form factor coefficient B-2(q) is demonstrated, providing insights into the intensity of collinear radiation from a quark. Additionally, the results yield effective higher-order splitting functions, laying the groundwork for the construction of NNLL parton showers.