On the maximal cut of Feynman integrals and the solution of their differential equations

TL;DR

This paper demonstrates that the maximal cut of Feynman integrals offers a straightforward way to obtain homogeneous solutions, significantly simplifying the process of solving their differential equations in multi-loop calculations.

Contribution

It introduces a novel approach using maximal cuts to find homogeneous solutions, enhancing the efficiency of solving differential equations for Feynman integrals.

Findings

Maximal cuts provide explicit homogeneous solutions.

Simplifies solving coupled differential equations.

Improves computational efficiency in multi-loop integrals.

Abstract

The standard procedure for computing scalar multi-loop Feynman integrals consists in reducing them to a basis of so-called master integrals, derive differential equations in the external invariants satisfied by the latter and, finally, try to solve them as a Laurent series in $\epsilon = (4-d)/2$, where $d$ are the space-time dimensions. The differential equations are, in general, coupled and can be solved using Euler's variation of constants, provided that a set of homogeneous solutions is known. Given an arbitrary differential equation of order higher than one, there exist no general method for finding its homogeneous solutions. In this paper we show that the maximal cut of the integrals under consideration provides one set of homogeneous solutions, simplifying substantially the solution of the differential equations.