# Multiple polylogarithms with algebraic arguments and the two-loop EW-QCD Drell-Yan master integrals

## TL;DR

This paper demonstrates that certain complex Feynman integrals with algebraic singularities can be expressed using multiple polylogarithms, enabling efficient numerical evaluation of two-loop corrections in Drell-Yan processes.

## Contribution

It introduces a method to evaluate integrals with unrationalizable roots using multiple polylogarithms, applied to two-loop Drell-Yan master integrals.

## Key findings

- Successfully evaluated two-loop master integrals for Drell-Yan production.
- Optimized functional basis for fast numerical evaluation.
- Extended the class of integrals expressible with multiple polylogarithms.

## Abstract

We consider Feynman integrals with algebraic leading singularities and total differentials in $\epsilon\,\mathrm{d}\ln$ form. We show for the first time that it is possible to evaluate integrals with singularities involving unrationalizable roots in terms of conventional multiple polylogarithms, by either parametric integration or matching the symbol. As our main application, we evaluate the two-loop master integrals relevant to the $\alpha \alpha_s$ corrections to Drell-Yan lepton pair production at hadron colliders. We optimize our functional basis to allow for fast and stable numerical evaluations in the physical region of phase space.