The planar double box integral for top pair production with a closed top loop to all orders in the dimensional regularisation parameter

TL;DR

This paper presents a systematic computation of the planar double box integral for top pair production with a closed top loop, expanding it in the dimensional regularisation parameter using differential equations transformed into an epsilon-linear form.

Contribution

The authors develop a method to solve elliptic multi-scale integrals by transforming differential equations into an epsilon-linear form, enabling all-order Laurent expansion in dimensional regularisation.

Findings

Successfully computed the Laurent expansion of the integral.

Demonstrated the method's applicability to elliptic multi-scale integrals.

Provided a systematic approach for similar Feynman integrals.

Abstract

We compute systematically for the planar double box Feynman integral relevant to top pair production with a closed top loop the Laurent expansion in the dimensional regularisation parameter $\varepsilon$. This is done by transforming the system of differential equations for this integral and all its sub-topologies to a form linear in $\varepsilon$, where the $\varepsilon^0$-part is strictly lower triangular. This system is easily solved order by order in the dimensional regularisation parameter $\varepsilon$. This is an example of an elliptic multi-scale integral involving several elliptic sub-topologies. Our methods are applicable to similar problems.