Dissecting the collinear structure of quark splitting at NNLL

We explore the collinear limit of final-state quark splittings at order $\alpha_s^2$. While at general NLL level, this limit is described simply by a product of leading-order $1\to 2$ DGLAP splitting functions, at the NNLL level we need to consider $1\to3$ splitting functions. Here, by performing suitable integrals of the triple-collinear splitting functions, we demonstrate how one may extract $\mathcal{B}^q_2(z)$, a differential version of the coefficient $B^q_2$ 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 $1 \to 2$ 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 $z \to 1$ we recover the standard soft limit results involving the CMW coupling with scale $k_t$, the $z$ dependence we obtain also motivates the extension of the notion of a physical coupling beyond the soft limit.