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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 aproduct of leading-order $1\to 2$ DGLAP splitting functions, at the NNLL levelwe need to consider $1\to3$ splitting functions. Here, by performing suitableintegrals of the triple-collinear splitting functions, we demonstrate how onemay 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 ofcollinear radiation from a quark. The variable $z$ corresponds to the quarkenergy fraction after an initial $1 \to 2$ splitting, and our results yieldeffective higher-order splitting functions, which may be considered as a steptowards the construction of NNLL parton showers. Further, while in the limit $z\to 1$ we recover the standard soft limit results involving the CMW couplingwith scale $k_t$, the $z$ dependence we obtain also motivates the extension ofthe notion of a physical coupling beyond the soft limit.

Dissecting the collinear structure of quark splitting at NNLL