# Numerical Multi-Loop Calculations via Finite Integrals and One-Mass EW-QCD Drell-Yan Master Integrals

## TL;DR

This paper demonstrates that using finite integrals as a basis significantly improves the efficiency of numerical multi-loop Feynman integral evaluations, enabling calculations that were previously challenging or infeasible.

## Contribution

It introduces a basis of finite integrals for numerical evaluation, leading to substantial performance improvements in multi-loop integral calculations.

## Key findings

- Performance improvements with SecDec3 for two-loop integrals.
- Order of magnitude speedup in three-loop form factor evaluations.
- Numerical accessibility of complex topologies previously difficult to compute.

## Abstract

We study a recently-proposed approach to the numerical evaluation of multi-loop Feynman integrals using available sector decomposition programs. As our main example, we consider the two-loop integrals for the $\alpha \alpha_s$ corrections to Drell-Yan lepton production with up to one massive vector boson in physical kinematics. As a reference, we evaluate these planar and non-planar integrals by the method of differential equations through to weight five. Choosing a basis of finite integrals for the numerical evaluation with SecDec3 leads to tremendous performance improvements and renders the otherwise problematic seven-line topologies numerically accessible. As another example, basis integrals for massless QCD three loop form factors are evaluated with FIESTA4. Here, employing a basis of finite integrals results in an overall speedup of more than an order of magnitude.