Feynman integrals as A-hypergeometric functions

Abstract

Abstract We show that the Lee-Pomeransky parametric representation of Feynman integrals can be understood as a solution of a certain Gel’fand-Kapranov-Zelevinsky (GKZ) system. In order to define such GKZ system, we consider the polynomial obtained from the Symanzik polynomials g = $$ \mathcal{U} $$ U + $$ \mathcal{F} $$ F as having indeterminate coefficients. Noncompact integration cycles can be determined from the coamoeba — the argument mapping — of the algebraic variety associated with g. In general, we add a deformation to g in order to define integrals of generic graphs as linear combinations of their canonical series. We evaluate several Feynman integrals with arbitrary non-integer powers in the propagators using the canonical series algorithm.