The maxcut of the sunrise with different masses in the continuous Minkoskean dimensional regularisation

TL;DR

This paper computes the maxcut of a two-loop sunrise amplitude with three different masses using Minkowski space regularisation, deriving related functions and analyzing their properties across dimensions.

Contribution

It introduces a Minkowski space regularisation approach to evaluate the maxcut of the sunrise amplitude, deriving new integral representations and analyzing their dimensional dependencies.

Findings

Six functions satisfy the homogeneous equation for arbitrary dimension d.

Only four functions are linearly independent in dimensions 2, 3, and 4.

The equal mass limit of the functions is briefly discussed.

Abstract

We evaluate the maxcut of the two loops sunrise amplitude with three different masses by using the Minkoskean (as opposed to the usual Euclidean) continuous dimension regularisation, obtaining in that way six related but different functions expressed in the form of one-dimensional finite integrals. We then consider the $4$th order homogeneous equation valid for the maxcut,and show that for arbitrary dimension $d$ the six functions do satisfy the equation. We further discuss the $d=2,3,4$ cases, verifying that only four of them are linearly independent. The equal mass limit is also shortly discussed.