On the finite basis topologies for multi-loop high-multiplicity Feynman integrals

TL;DR

This paper develops a systematic method for reducing high-multiplicity multi-loop Feynman integrals to a finite set of basis topologies, simplifying calculations in quantum field theory.

Contribution

It introduces an explicit reduction to finite basis topologies at any loop order, with detailed analysis at two loops, and demonstrates computational advantages over traditional schemes.

Findings

Finite basis topologies identified at two loops in four dimensions.

Reduction simplifies high-multiplicity integrals, reducing computational complexity.

Formal bounds on the complexity of special functions in perturbative calculations.

Abstract

In this work, we systematically analyse Feynman integrals in the `t Hooft-Veltman scheme. We write an explicit reduction resulting from partial fractioning the high-multiplicity integrands to a finite basis of topologies at any given loop order. We find all of these finite basis topologies at two loops in four external dimensions. Their maximal cut and the leading singularity are expressed in terms of the Gram determinant and Baikov polynomial. By performing an Integration-By-Parts reduction without any cut constraint on a numerical probe for one of these topologies, we show that the computational complexity drops significantly compared to the Conventional Dimensional Regularization scheme. Formally, our work implies an upper bound on the rigidity of special functions appearing in the iterated integral solutions at each loop order in perturbative Quantum Field Theory. Phenomenologically, the integrand-level reduction we present will substantially simplify the task of providing high-precision predictions for future high-multiplicity collider observables.