Abstract
In this paper we study the calculation of multiloop Feynman integrals that cannot be expressed in terms of multiple polylogarithms. We show in detail how certain types of two- and three-point functions at two loops, which appear in the calculation of higher order corrections in QED, QCD and in the electroweak theory (EW), can naturally be expressed in terms of a recently introduced elliptic generalisation of multiple polylogarithms by direct integration over their Feynman parameter representation. Moreover, we show that in all examples that we considered a basis of pure Feynman integrals can be found.
Highlights
Polylogarithms (MPLs) [5,6,7] in high-energy physics [8, 9], and the study of their analytical, algebraic [10,11,12] and numerical [13] properties, have been crucial steps to systematise both strategies
We show in detail how certain types of two- and three-point functions at two loops, which appear in the calculation of higher order corrections in QED, QCD and in the electroweak theory (EW), can naturally be expressed in terms of a recently introduced elliptic generalisation of multiple polylogarithms by direct integration over their Feynman parameter representation
Our task is to find an order of integration of the Feynman parameters such that linear reducibility is achieved for all integrations except the last one, which will in turn require the introduction of elliptic multiple polylogarithms (eMPLs)
Summary
Our goal in this paper is to show explicitly how the notion of elliptic multiple polylogarithms (eMPLs) developed in refs. [25, 40, 44, 52] can be put into action for a wide range of Feynman integrals known not to be expressible in terms of ordinary MPLs. Since elliptic curves are isomorphic to complex tori, eMPLs can be described as iterated integrals over functions related to the torus, and were originally defined as such in refs. They are defined as iterated integrals of kernels that are rational functions on the elliptic curve with at most logarithmic singularities in all variables, x. We conclude this short exposition of pure eMPLs with a comment: much like with ordinary MPLs, one can associate a concept of length and of weight to eMPLs and to quantities which arise from evaluating eMPLs at special points, for example the periods and quasi-periods of the elliptic curve defined in eqs. Using the formalism revised in the rest of this paper we will show how certain Feynman integrals which evaluate to functions beyond MPLs can be brought to neat expressions in terms of combinations of pure eMPLs (2.24) of uniform weight by direct integration of their Feynman parametrisation
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