Loop Integrals, R Functions and their Analytic Continuation

TL;DR

This paper discusses a method for performing the analytic continuation of one-loop integrals using the R function notation, ensuring mathematical accuracy across all kinematic regions, with explicit examples for two- and three-point cases.

Contribution

It introduces a general, mathematically rigorous approach for the analytic continuation of one-loop integrals using R function notation, applicable to various kinematic regions.

Findings

Provides a systematic method for analytic continuation of loop integrals.

Explicitly demonstrates the approach for two- and three-point integrals.

Ensures accurate determination of resulting functions across all kinematic regions.

Abstract

To entirely determine the resulting functions of one-loop integrals it is necessary to find the correct analytic continuation to all relevant kinematical regions. We argue that this continuation procedure may be performed in a general and mathematical accurate way by using the ${\cal R}$ function notation of these integrals. The two- and three-point cases are discussed explicitly in this manner.