Elliptic Feynman integrals and pure functions

Abstract

We propose a variant of elliptic multiple polylogarithms that have at most logarithmic singularities in all variables and satisfy a differential equation without homogeneous term. We investigate several non-trivial elliptic two-loop Feynman integrals with up to three external legs and express them in terms of our functions. We observe that in all cases they evaluate to pure combinations of elliptic multiple polylogarithms of uniform weight. This is the first time that a notion of uniform weight is observed in the context of Feynman integrals that evaluate to elliptic polylogarithms.

Three-loop contributions to the ρ parameter and iterated integrals of modular forms

“…This form matches precisely the structure of elliptic Feynman integrals conjectured in ref. [59]. In particular, we see that the functions f…”

“…We refer to ref. [59] for a detailed discussion. Let us make some comments about the iterated integrals defined in this section.…”

“…The period is multiplied by a linear combination of eMPLs of uniform weight two. If we assign weight one to the period Ψ 1 (t) [59], then f…”

Algorithms and tools for iterated Eisenstein integrals

“…Second, the functions U i are pure functions of uniform weight [63] in the sense of ref. [47]. We emphasise that this statement depends on our choice for the periods of the elliptic curve: in eq.…”

“…Their description as iterated integrals on a complex torus had been formalised by mathematicians already in 2011 [39,40] and had later found application in the calculation of one-loop scattering amplitudes in string theory [41][42][43][44]. More recently, an alternative formulation of eMPLs naturally defined on an elliptic curved defined by a polynomial equation instead of on a torus has been proposed [27,45,46], which has made it possible to naturally compute different previously out-of-reach multi-loop Feynman integrals [47][48][49]. While eMPLs are a rather large class of functions that can encompass different kinds of physical problems, for the particular case of the two-loop sunrise graph, 1 a special subclass of eMPLs has been shown to be sufficient, i.e.…”