Special case of sunset: reduction and epsilon-expansion
A. Onishchenko, O. Veretin
TL;DR
The paper tackles two-loop sunset diagrams with two mass scales at threshold and pseudothreshold, where existing Tarasov-based reductions fail. It develops explicit IBP recurrence relations to reduce all integrals to a small master basis, then employs a differential-equation approach to obtain ε-expansions as series in the mass ratio r = m/M. Boundary data are furnished via asymptotic expansions, including Appell/F4 and Lauricella representations, allowing explicit expressions for the master integrals J111, J112, and J211 up to high orders in r and ε. These results provide practical, high-precision formulas for threshold kinematics, with significant utility for matching calculations in QCD/NRQCD contexts and related two-scale problems.
Abstract
We consider two loop sunset diagrams with two mass scales m and M at the threshold and pseudotreshold that cannot be treated by earlier published formula. The complete reduction to master integrals is given. The master integrals are evaluated as series in ratio m/M and in epsilon with the help of differential equation method. The rules of asymptotic expansion in the case when q^2 is at the (pseudo)threshold are given.