Note on Closed-Form Expressions for Irreducible Loop Integrals

TL;DR

This paper introduces a novel approach to analyze irreducible loop integrals using probabilistic convergence ideas, deriving conditions for well-defined limits and providing closed-form expressions with incomplete Gamma-functions.

Contribution

It offers new conditions for regulator masses ensuring well-defined limits and derives explicit closed-form expressions for regularized integrals.

Findings

Conditions on regulator masses for convergence

Closed-form expressions using incomplete Gamma-functions

Enhanced understanding of loop integral regularization

Abstract

We provide a new analysis of the irreducible loop integrals first considered in a 2003 paper of Wu. Using convergence ideas from probability, we produce conditions on the regulator masses so that the integrals have well-defined limits in the limit required by the regularisation technique; we then derive closed-form expressions for the regularised forms of these integrals in terms of incomplete Gamma-functions.