Finite Integrals from Feynman Polytopes

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.