Unregulated Divergences of Feynman Integrals

TL;DR

This paper investigates unregulated divergences in Feynman integrals, providing criteria to identify them and demonstrating their connection to degeneracies of regions through parametric and Mellin-Barnes representations.

Contribution

It introduces a criterion for identifying unregulated divergences in Feynman integrals and proves their link to degeneracies of regions using multiple representations.

Findings

Unregulated divergences indicate degeneracies of regions.

Parametric and Mellin-Barnes representations yield consistent analysis.

Presence of unregulated divergences affects asymptotic expansions.

Abstract

Feynman integrals can be expanded asymptotically with respect to some small parameters at the integrand level, a technique known as the expansion by regions. A naive expansion by regions may break down due to divergences not regulated by the spacetime dimension, exemplified by the rapidity divergences. A criterion to identify unregulated divergences is provided in this article. The analysis is conducted using both parametric and Mellin-Barnes representations, leading to a consistent conclusion. Based on this analysis, it is proven that the presence of unregulated divergences implies the degeneracies of regions.