Dissecting the collinear structure of quark splitting at NNLL

TL;DR

This paper analyzes the collinear structure of quark splittings at NNLL order, deriving effective splitting functions and exploring their implications for parton shower modeling and the extension of physical couplings.

Contribution

It introduces a method to extract the differential coefficient _2^q(z) from triple-collinear splitting functions, advancing NNLL parton shower development.

Findings

Derived _2^q(z) from triple-collinear splitting functions.

Connected the z-dependent results to the extension of physical couplings.

Recovered standard soft limit results involving the CMW coupling.

Abstract

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.