A practical criterion of irreducibility of multi--loop Feynman integrals

TL;DR

This work presents a practical irreducibility criterion for multi-loop Feynman integrals with respect to IBP identities by tying irreducibility to the existence of zero-gradient (stable) points of a Gram-determinant-type polynomial derived from loop momenta under on-shell constraints. It replaces the earlier hand-crafted approach with a computational framework that uses integral representations and a $1/D$ expansion to construct special IBP solutions $s(\underline{n})$ that cannot be expressed as linear combinations of related integrals, thereby proving irreducibility. The method is demonstrated on four-loop massless propagator topologies, identifying irreducible integrals for each topology and outlining an algorithm to enumerate all such irreducibles. Together, these results provide a practical tool to reduce large families of Feynman integrals and enhance the efficiency of high-order quantum correction computations.

Abstract

A practical criterion for the irreducibility (with respect to integration by part identities) of a particular Feynman integral to a given set of integrals is presented. The irreducibility is shown to be related to the existence of stable (with zero gradient) points of a specially constructed polynomial.