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

TL;DR

This paper presents a new method called ASPIRE for identifying regions in multi-scale Feynman integrals by analyzing singularities, Landau equations, and Newton Polytopes, improving the understanding of complex quantum diagrams.

Contribution

The paper introduces a novel algorithm that combines singularity analysis, Power Geometry, and Gr"{o}bner Bases to systematically identify regions in multi-scale Feynman integrals.

Findings

Successfully applied to one-loop and two-loop examples

Revealed regions like potential and Glauber in Feynman diagrams

Benchmarking shows effectiveness of the ASPIRE algorithm

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 Gr\"{o}bner 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.