Toric geometry and regularization of Feynman integrals

TL;DR

This paper explores the use of toric geometry to analyze and improve the regularization of Feynman integrals, providing new geometric insights and unifying various sector decomposition methods.

Contribution

It introduces a geometric framework using toric compactifications for Feynman integrals, connecting sector decomposition techniques with Newton polytopes and generalized permutahedra.

Findings

Refined convergence domain description for Mellin transforms of Laurent polynomials.

Unified geometric perspective on sector decompositions of Feynman integrals.

Demonstrated applications to dimensional regularization methods.

Abstract

We study multivariate Mellin transforms of Laurent polynomials by considering special toric compactifications which make their singular structure apparent. This gives a precise description of their convergence domain, refining results of Nilsson, Passare, Berkesch and Forsg\aa rd. We also reformulate the geometric sector decomposition approach of Kaneko and Ueda in terms of these compactifications. Specializing to the case of Feynman integrals in the parametric representation, we construct multiple such compactifications given by certain systems of subgraphs. As particular cases, we recover the sector decompositions of Hepp, Speer and Smirnov, as well as the iterated blow-up constructions of Brown and Bloch-Esnault-Kreimer. A fundamental role is played by the Newton polytope of the product of the Symanzik polynomials, which we show to be a generalized permutahedron for generic kinematics. As an application, we review two approaches to dimensional regularization of Feynman integrals, based on sector decomposition and on analytic continuation.