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

TL;DR

This work tackles the numerical evaluation of multi-loop Feynman integrals for mixed EW-QCD corrections to Drell-Yan production with up to one massive vector boson. Using differential equations in a normal-form basis, the authors obtain analytic $\epsilon$-expanded solutions up to weight five and construct a real-valued, finite integral basis that dramatically improves numerical stability and speed when evaluated with sector decomposition (SecDec3) and FIESTA4 for massless three-loop form factors. They demonstrate substantial performance gains—accessing previously intractable seven-line topologies and achieving order-of-magnitude speedups—by combining finite integrals with automated basis construction via IBP reductions (Reduze 2.x). The results emphasize the viability of numerical approaches as primary tools for complex phenomenology and provide publicly available tooling to enable broader adoption.

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 $αα_s$ 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.

Figures (1)

• Figure 1: Selected planar and non-planar two-loop Feynman diagrams for Drell-Yan lepton production. The heavy wavy lines denote a $Z$ boson, the light wavy lines denote a photon, and the curly lines denote a gluon.