Hypergeometric representation of the two-loop equal mass sunrise diagram

TL;DR

The paper addresses the analytic evaluation of the equal-mass two-loop sunrise diagram in arbitrary space-time dimension $d$ by applying Tarasov's dimensional-recurrence method. It derives a dimension-shifting difference equation and a mass-derivative differential equation, then solves the recurrence to obtain an explicit expression for $J_3^{(d)}$ in terms of $_2F_1$ and Appell's $F_2$ functions, including its imaginary part on the cut and the threshold value. The work provides integral representations for the hypergeometric functions and demonstrates consistency with known results and epsilon-expansions, illustrating the method's power for complex multi-loop integrals. This approach offers a path to analytic continuations and precise expansions, with potential applicability to a broader class of Feynman integrals and kinematic configurations.

Abstract

A recurrence relation between equal mass two-loop sunrise diagrams differing in dimensionality by 2 is derived and it's solution in terms of Gauss' 2F1 and Appell's F_2 hypergeometric functions is presented. For arbitrary space-time dimension d the imaginary part of the diagram on the cut is found to be the 2F1 hypergeometric function with argument proportional to the maximum of the Kibble cubic form. The analytic expression for the threshold value of the diagram in terms of the hypergeometric function 3F2 of argument -1/3 is given.