Expansion by regions: revealing potential and Glauber regions automatically

TL;DR

The paper advances automatic detection of all regions in Minkowski-space asymptotic expansions by extending the existing asy.m to asy2.m, capable of uncovering both potential and Glauber regions. It leverages parametric representations with Symanzik polynomials and introduces tools like AlphaRepExpand[] and WilsonExpand[] to decompose integrals into region-specific contributions, including handling of generic propagator powers via analytic regulators. Through threshold and threshold-like threshold/threshold-like Glauber examples, the authors show how domain decompositions and variable changes reveal previously missed regions and how these regions map consistently between parametric and loop-momentum space. The work provides a robust, automated pipeline for complete asymptotic expansions in Minkowski-space limits, with practical guidance and demonstrable consistency checks across multiple integral topologies.

Abstract

When performing asymptotic expansions using the strategy of expansion by regions, it is a non-trivial task to find the relevant regions. The recently published Mathematica code asy.m automates this task, but it has not been able to detect potential regions in threshold expansions or Glauber regions. In this work we present an algorithm and its implementation in the update asy2.m which also reveals potential and Glauber regions automatically.

Figures (1)

• Figure 1: One-loop five-point integral exhibiting a Glauber contribution