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

TL;DR

<3-5 sentence high-level summary> Introduces and rigorously defines linearly reducible elliptic Feynman integrals, showing that they admit an all-orders $\epsilon$-expansion expressed through a 1-fold integral over a polylogarithmic inner polylogarithmic part (IPP). When the IPP depends on a single elliptic curve, the class can be solved algorithmically in terms of elliptic multiple polylogarithms ($\text{eMPL}$) using integration-by-parts identities and canonical differential equations in $\epsilon$-form. The paper develops the Feynman trick to map the IPP to a generalized topology, derives canonical differential equations whose kernels align with $\text{eMPL}$ integration kernels, and demonstrates the approach on the unequal-mass sunrise and triangle-with-bubble, including analytic continuation to the physical region. It also discusses a non-planar Higgs+Jet triangle where the IPP involves multiple algebraic curves, highlighting directions for extending the framework to more intricate algebraic structures.

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 $ε$-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.

Figures (2)

• Figure 1: Plots of the first two epsilon orders of $S_{1,1,1}(s,m_1^2,m_2^2,m_3^2)$
• Figure 2: Plots of the first two epsilon orders of $T_{1,2,1,1}(s,m^2)$