Domains of Definition for Feynman Integrals over Real Feynman Parameters

Abstract

Given a Feynman diagram, the corresponding integral over real Feynman parameters is meaningful and analytic in a certain domain in the space of the Lorentz invariants formed from the external momenta, each of which is on the mass shell. In the case where all the masses are equal, the intersection of these domains for all proper convergent diagrams is studied. For the cases of four and five external lines, the real intersections are explicitly found; for the case of six, seven, and eight external lines, procedures for finding the real intersections are given. A knowledge of the real intersection makes it possible to construct geometrically a subset of the complex intersection. Generalization to unequal external masses is briefly considered.