The $\varepsilon$-form of the differential equations for Feynman integrals in the elliptic case

TL;DR

This work shows that the differential-equation method, and in particular the $\varepsilon$-form, extends to Feynman integrals with elliptic characteristics by allowing transcendental basis transformations. Using the kite integral and its elliptic sunrise sub-sector, the authors define an eight- master-integral basis in which the system satisfies $(1/(2\pi i))\,d/d\tau\,\vec{I}=\varepsilon\,A\,\vec{I}$ with $A$ independent of $\varepsilon$ and consisting of modular forms for $\Gamma_1(6)$; kernels become modular forms after the base change $x\to\tau$. The $\tau$-variable reveals a rich modular structure, enabling the integrals to be expressed as iterated integrals of modular forms, and the approach relies on a transcendental, elliptic-period–based normalization (e.g., dividing by the maximal-cut term). This broadens the applicability of the $\varepsilon$-form method to elliptic Feynman integrals and informs the development of elliptic generalizations of polylogarithms for precision calculations.

Abstract

Feynman integrals are easily solved if their system of differential equations is in $\varepsilon$-form. In this letter we show by the explicit example of the kite integral family that an $\varepsilon$-form can even be achieved, if the Feynman integrals do not evaluate to multiple polylogarithms. The $\varepsilon$-form is obtained by a (non-algebraic) change of basis for the master integrals.