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

Abstract

We evaluate the maxcut of the two loops sunrise amplitude with three different masses by direct use of the loop momenta in 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 4th order homogeneous equation valid for the maxcut, and show that for arbitrary dimension d the six functions do satisfy the equation separately. 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 considered.

Keywords

• Sunrise
• Differential equations
• Elliptic integrals
• Elliptic polylogarithms