Unveiling Regions in multi-scale Feynman Integrals using Singularities and Power Geometry

Abstract

We introduce a novel approach for solving the problem of identifying regions in the framework of Method of Regions by considering singularities and the associated Landau equations given a multi-scale Feynman diagram. These equations are then analyzed by an expansion in a small threshold parameter via the Power Geometry technique. This effectively leads to the analysis of Newton Polytopes which are evaluated using a Mathematica based convex hull program. Furthermore, the elements of the Gröbner Basis of the Landau Equations give a family of transformations, which when applied, reveal regions like potential and Glauber. Several one-loop and two-loop examples are studied and benchmarked using our algorithm which we call ASPIRE.

v2,29 pages, plain latex, 12 figures, 3 tables, mathematica programs given as ancillary files,corresponds to version to appear in European Physical Journal C; compared to v1, sign of U polynomial corrected everywhere and in the new notebooks. The sign did not alter any of the results. Extensive discussions added, and clarifications inserted. Reference added