Toric geometry and regularization of Feynman integrals

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.