Precise numerical evaluation of the two loop sunrise graph Master Integrals in the equal mass case
S. Pozzorini, E. Remiddi
TL;DR
The paper addresses the precise numerical evaluation of the two equal-mass, two-loop sunrise Master Integrals for arbitrary momentum transfer in $d=2$ and $d=4$ by leveraging differential equations and accelerated power-series expansions around singular points. It develops a robust double-precision FORTRAN routine, SUNRISE, which computes $S(2,z)$ and related quantities with up to 22 terms per expansion, achieving a relative accuracy better than $10^{-15}$ across the entire real axis. The method expresses the second integral and the $d=4$ finite parts in terms of $S(2,z)$ and its derivatives, while using region-specific expansions and Bernoulli-variable transformations to ensure rapid convergence. The work enables fast, stable numerical radiative-correction calculations in cases where closed analytic forms are unknown, with practical runtime and precision suitable for high-precision phenomenology.
Abstract
We present a double precision routine in Fortran for the precise and fast numerical evaluation of the two Master Integrals (MIs) of the equal mass two-loop sunrise graph for arbitrary momentum transfer in d=2 and d=4 dimensions. The routine implements the accelerated power series expansions obtained by solving the corresponding differential equations for the MIs at their singular points. With a maximum of 22 terms for the worst case expansion a relative precision of better than a part in 10^{15} is achieved for arbitrary real values of the momentum transfer.