From $\mathrm{d} \! \log$ to $\mathrm{d} \mathcal{E}$: Canonical Elliptic Integrands and Modular Symbol Letters with Pure eMPLs

TL;DR

We address the extension of canonical bases and symbol letters from polylogarithmic to elliptic Feynman integrals by introducing $d\mathcal{E}$-forms, built from pure eMPLs, as canonical integrand building blocks. A generalized IBP framework enables expressing these forms in terms of Abelian differentials, yielding a canonical, $\varepsilon$-factorized differential equation with a connection matrix whose entries are written in modular $\omega$-forms. An extended basis that treats all marked points equally reveals a hidden symmetry and allows a fiber transformation to ensure modular covariance, enabling efficient $\bar{q}$-expansion-based numerics and a unified description of symbol letters for both MPLs and elliptic cases. The results provide a new, purely integrand-level perspective on canonical elliptic bases and their symbol letters, with potential impact on symbol bootstrap, numerical evaluation, and multivariate/higher-genus generalizations.

Abstract

We propose '$\mathrm{d} \mathcal{E}$-forms' as fundamental building blocks of canonical integrands for elliptic Feynman integrals, which lead to Kronecker-Eisenstein $ω$-form symbol letters. Built upon pure elliptic multiple polylogarithms, they provide a natural extension of the '$\mathrm{d} \! \log$-form' integrands and $\mathrm{d} \! \log$ letters for polylogarithmic cases. By introducing an extended basis treating all marked points equally, we manifest a hidden symmetry structure in the canonical connection matrix, and demonstrate its covariance under modular transformations. Our result provides a novel perspective on describing canonical bases and symbol letters in a unified language of pure functions.