Two-Loop Quark and Gluon Form Factors in Dimensional Regularisation

TL;DR

This work computes the two-loop corrections to massless quark and gluon form factors in dimensional regularisation to all orders in $\epsilon$, reducing the loop integrals to a small set of master integrals. A key result is the exact expression for the crossed triangle master integral $A_6$ in terms of generalised hypergeometric functions, expandable to arbitrary $\epsilon$-order with the HypExp package. The authors provide high-order $\epsilon$-expansions for the form factors (up to $\mathcal{O}(\epsilon^2)$ at two loops and $\mathcal{O}(\epsilon^4)$ for lower-order terms), enabling precise ultraviolet renormalisation and infrared factorisation relevant for three-loop analyses. This work facilitates the extraction of the complete infrared pole structure at three loops and demonstrates the utility of HypExp for multi-loop QFT calculations, with potential extensions to higher-loop two-point functions.

Abstract

We compute the two-loop corrections to the massless quark form factor $γ^* \to q\bar q$ and gluon form factor $H\to gg$ to all orders in the dimensional regularisation parameter $ε=(4-d)/2$. The two-loop contributions to the form factors are reduced to linear combinations of master integrals, which are computed in a closed form, expressed as $Γ$-functions and generalised hypergeometric functions of unit argument. Using the newly developed HypExp-package, these can be expanded to any desired order, yielding Laurent expansions in $ε$. We provide expansions of the form factors to order $ε^2$, as required for ultraviolet renormalisation and infrared factorisation of the three-loop form factors.