The quark and gluon form factors to three loops in QCD through to $\mathcal{O}\left( {{\epsilon^2}} \right)$

Abstract

We give explicit formulae for the $\mathcal{O}(\epsilon)$ and $\mathcal{O}\left( {{\epsilon^2}} \right)$ contributions to the unrenormalised three loop QCD corrections to quark and gluon form factors. These contributions have at most transcendentality weight eight. The $\mathcal{O}(\epsilon)$ terms of the three-loop form factors are required for the extraction of the four-loop quark and gluon collinear anomalous dimensions. The $\mathcal{O}\left( {{\epsilon^2}} \right)$ terms represent an irreducible contribution to the finite part of the form factors at four-loops. For the sake of completeness, we also give the contributions to the one and two loop form factors to the same transcendentality weight eight.