Two-loop sunset diagrams with three massive lines

TL;DR

This work addresses the analytic evaluation of the two-loop sunset diagram with three massive lines in the kinematics $q^2=-m^2$ by introducing a master-formula approach that replaces a two-mass loop with a parameterized one-loop integral carrying an effective mass $M^2/[s(1-s)]$, enabling reconstruction of the two-loop result from one-loop building blocks. It derives explicit $x=m^2/M^2$ expansions for the general unequal-mass case and, in the equal-mass limit, obtains an $O(\varepsilon)$ analytic structure that involves elliptic-integral constants, with PSLQ relations connecting sums to $\ln 2$, $\zeta(2)$, and elliptic terms. The equal-mass case reduces to two master integrals whose series exhibit rapid convergence and elliptic behavior, while the general case provides closed-form series with harmonic sums and polylogarithmic-like structures. These results enable precise numerical evaluation (e.g., via Padé approximants) and have potential applications to hard Wilson coefficients in NRQED/NRQCD and near-threshold heavy-quarkonium production.

Abstract

In this paper, we consider the two-loop sunset diagram with two different masses, m and M, at spacelike virtuality q^2 = -m^2. We find explicit representations for the master integrals and an analytic result through O(epsilon) in d=4-2epsilon space-time dimensions for the case of equal masses, m = M.

Figures (1)

• Figure 1: (a) Two-loop sunset diagram $J_{abc}^{mMM}$ and (b) one-loop self-energy diagram $I_{ab}^{mM}$ with different masses, $m$ and $M$. In both cases, the external four-momentum $q$ satisfies the condition $q^2=-m^2$. The label attached to a given line indicates the mass square appearing in the respective propagator and the power to which the latter is raised.