Canonical differential equations for the elliptic two-loop five-point integral family relevant to $t\bar t +$jet production at leading colour

Abstract

We present differential equations (DEs) in canonical form for a family of two-loop five-point Feynman integrals containing elliptic functions and nested square roots. This is the only family for which canonical DEs were not yet available among those required to compute the two-loop leading-colour amplitude for top-pair production in association with a jet at hadron colliders. We write the DEs in terms of one-forms having (locally) simple poles at all singular points, and highlight the `duplet' structure that generalises the even/odd charge of square roots to nested roots. All transcendental functions in the DEs are expressed in closed form using complete elliptic integrals, while one-forms free of elliptic functions are given in terms of logarithms, including those with the nested roots. In addition to marking a significant step towards next-to-next-to-leading order QCD predictions for an important LHC process, this work represents the first time that a canonical basis of integrals involving elliptic functions has been obtained for a five-particle process.