Threshold and pseudothreshold values of the sunset diagram

TL;DR

This paper derives analytic expressions for the threshold and pseudothreshold values of the two-loop sunset diagram with arbitrary masses, expressing the results primarily through dilogarithms of mass ratios. Using refined Feynman-parameter techniques, inverse-rescaling of alpha variables, and a decomposition into simpler master integrals, the authors obtain $\varepsilon$-expansions for the relevant integrals at both the threshold and the pseudothreshold and ensure full permutation symmetry. They introduce dilogarithmic functions $T^{\pm}$ and angle variables $\theta_i$ to capture the analytic structure, including logarithmic and $\pi$-dependent terms. The results enable precise near-threshold expansions, facilitate checks against known equal-mass limits, and offer a methodology extendable to higher-propagator diagrams and more complex cuts.

Abstract

Analytic results for the threshold and pseudothreshold values of the sunset diagram with arbitrary masses are obtained in terms of dilogarithms of ratios of the masses.