Abstract
We consider two-loop two-, three-, and four-point diagrams with elliptic subgraphs involving two different masses, m and M. Such diagrams generally arise in matching procedures within nonrelativistic QCD and QED and are relevant, e.g., for top-quark pair production at threshold and parapositronium decay. We present the obtained results in several different representations: series solution with binomial coefficients, integral representation, and representation in terms of generalized hypergeometric functions. The results are valid up to terms of O(ε) in d=4−2ε space-time dimensions. In the limit of equal masses, m=M, the obtained results are written in terms of elliptic constants with explicit series representations.
Highlights
During the last two decades, great progress has been made in the calculation of multiloop Feynman diagrams
Calculations yielding polylogarithms are often related to massless problems, sometimes to massive ones where the unitary cuts of a Feynman diagram do not cross more than two massive lines
We considered a class of two-loop diagrams with two different masses, m and M, in the special kinematic regime defined by Eq (1), which play a crucial role in nonrelativistic effective field theories like NRQCD and NRQED
Summary
During the last two decades, great progress has been made in the calculation of multiloop Feynman diagrams. It should be noted that, in dimensional regularization with d = 4 − 2ε space-time dimensions, all one-loop diagrams with arbitrary masses and kinematics can, at least in principle, be expressed in terms of polylogarithms at any order in ε.2. We consider diagrams with only one or two independent parameters The former case is represented by the so-called single-scale integrals, which, by dimensional reasons, can be written as a product of a scale and a numerical factor. In the latter case, the considered diagram is expressed in terms of a function of one variable. In Appendix A, we explain the Frobenius solution for a system of differential equations
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