Dissecting the collinear structure of quark splitting at NNLL
Abstract We explore the collinear limit of final-state quark splittings at order $$ {\alpha}_s^2 $$ α s 2 . 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 $$ {\mathrm{\mathcal{B}}}_2^q(z) $$ ℬ 2 q z , a differential version of the coefficient $$ {\mathrm{\mathcal{B}}}_2^q $$ 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.
