Sunrise integrals with two internal masses and pseudo-threshold kinematics in terms of elliptic polylogarithms

Abstract

We consider a set of two-loop sunrise master integrals with two different internal masses at pseudo-threshold kinematics and we solve it in terms of elliptic polylogarithms to all orders of the dimensional regulator.

Feynman Integrals

“…Further examples of elliptic Feynman integrals can be found in [373][374][375][376][377][378][379][380][381][382].…”

The SAGEX Review on Scattering Amplitudes, Chapter 3: Mathematical structures in Feynman integrals

“…There is a natural class of iterated integrals that generalise MPLs to functions of elliptic type, called elliptic multiple polylogarithms (eMPLs) [195,196]. eMPLs have first appeared in physics in the context of one-loop scattering amplitudes in string theory [197][198][199][200], but they have also been used to express several multi-loop Feynman integrals that cannot be expressed in terms of eMPLs [64,[201][202][203][204][205] (see refs. [185-188, 206, 207] for alternative definitions of elliptic generalisations of polylogarithmic functions that are closely related to eMPLs).…”

On series and integral representations of some NRQCD master integrals

“…Then the problem of obtaining integral representation for this master integral is reduced to the problem of getting integral representations for corresponding hypergeometric functions. The latter problem can be solved using the techniques presented in [61,69]. First consider (d = 2 − 2ε)…”

“…In the present short note, we use an example a set of two-loop master integrals arising in the process of matching of QCD to NRQCD to introduce several new methods for obtaining their series and integral representations, which are either exact in the value of space-time dimension or expanded to any prescribed order for its fixed values. The mentioned techniques are analytical Frobenius method 1 as introduced in [62], the use of Feynman parameter trick 2 and differential equations with respect to the latter [27,48,61,68] and integral representations for hypergeometric functions [61,68,69].…”