Finite Integrals from Feynman Polytopes

TL;DR

This paper introduces a geometric method using Newton polytopes to identify finite Feynman integrals, providing an algorithm that aligns with existing Landau-analysis results and advancing the understanding of integral convergence.

Contribution

It presents a novel geometric approach based on Newton polytopes and an algorithm for finding finite numerators in Feynman integrals, supported by a convergence theorem.

Findings

Algorithm successfully finds finite numerators in various examples.

Results agree with Landau-analysis approach.

Supports conjecture on parameter-space monomials and polytope interior.

Abstract

We investigate a geometric approach to determining the complete set of numerators giving rise to finite Feynman integrals. Our approach proceeds graph by graph, and makes use of the Newton polytope associated to the integral's Symanzik polynomials. It relies on a theorem by Berkesch, Forsg{\aa}rd, and Passare on the convergence of Euler--Mellin integrals, which include Feynman integrals. We conjecture that a necessary in addition to a sufficient condition is that all parameter-space monomials lie in the interior of the polytope. We present an algorithm for finding all finite numerators based on this conjecture. In a variety of examples, we find agreement between the results obtained using the geometric approach, and a Landau-analysis approach developed by Gambuti, Tancredi, and two of the authors.