Methods for deriving functional equations for Feynman integrals

TL;DR

This paper reviews two methods for deriving functional equations for Feynman integrals, demonstrating how they can simplify complex integrals in various kinematic scenarios by relating them to simpler ones.

Contribution

It introduces and compares two methods for obtaining functional equations for Feynman integrals, with detailed applications to one- and two-loop integrals.

Findings

Functional equations relate complex Feynman integrals to simpler ones.

Methods enable expressing integrals in general kinematics through simpler integrals.

Discussion of similarities with Abel integral addition theorems.

Abstract

We present short review of two methods for obtaining functional equations for Feynman integrals. Application of these methods for finding functional equations for one- and two- loop integrals is described in detail. It is shown that with the aid of functional equations Feynman integrals in general kinematics can be expressed in terms of simpler integrals. Similarities between functional equations for Feynman integrals and addition theorem for Abel integrals are shortly discussed.