All orders structure and efficient computation of linearly reducible elliptic Feynman integrals

TL;DR

This paper introduces a method to solve linearly reducible elliptic Feynman integrals using a combination of direct integration, differential equations, and elliptic multiple polylogarithms, enabling explicit solutions up to any order.

Contribution

It defines linearly reducible elliptic Feynman integrals and provides an algorithmic approach to solve them in terms of elliptic multiple polylogarithms (eMPLs).

Findings

Integrals can be solved as a 1D integral over a polylogarithmic integrand.

Differential equations can be put into ε-form and solved in terms of eMPLs.

Explicit solutions are obtained for two- and three-point integrals.

Abstract

We define linearly reducible elliptic Feynman integrals, and we show that they can be algorithmically solved up to arbitrary order of the dimensional regulator in terms of a 1-dimensional integral over a polylogarithmic integrand, which we call the inner polylogarithmic part (IPP). The solution is obtained by direct integration of the Feynman parametric representation. When the IPP depends on one elliptic curve (and no other algebraic functions), this class of Feynman integrals can be algorithmically solved in terms of elliptic multiple polylogarithms (eMPLs) by using integration by parts identities. We then elaborate on the differential equations method. Specifically, we show that the IPP can be mapped to a generalized integral topology satisfying a set of differential equations in $\epsilon$-form. In the examples we consider the canonical differential equations can be directly solved in terms of eMPLs up to arbitrary order of the dimensional regulator. The remaining 1-dimensional integral may be performed to express such integrals completely in terms of eMPLs. We apply these methods to solve two- and three-points integrals in terms of eMPLs. We analytically continue these integrals to the physical region by using their 1-dimensional integral representation.