Tree-level splitting amplitudes for a quark into four collinear partons

We compute in conventional dimensional regularisation the tree-level splitting amplitudes for a quark parent in the limit where four partons become collinear to each other. This is part of the universal infrared behaviour of the QCD scattering amplitudes at next-to-next-to-next-to-leading order (N3LO) in the strong coupling constant. Further, we consider the iterated limit when m′ massless partons become collinear to each other within a bigger set of m collinear partons, as well as the limits when one gluon or a qq¯\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \overline{q} $$\end{document} pair or two gluons become soft within a set of m collinear partons.