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

TL;DR

This paper introduces $ ext{d} ext{E}$-forms as fundamental components for elliptic Feynman integrals, extending the concept of $ ext{d} ext{log}$ forms to include modular symbol letters, revealing symmetry and covariance properties.

Contribution

It develops a new framework of pure elliptic multiple polylogarithms and $ ext{d} ext{E}$-forms, unifying canonical bases and symbol letters in elliptic integrals.

Findings

Introduction of $ ext{d} ext{E}$-forms as fundamental integrand blocks

Demonstration of covariance under modular transformations

Revealing hidden symmetry in the canonical connection matrix

Abstract

We propose '$\mathrm{d} \mathcal{E}$-forms' as fundamental building blocks of canonical integrands for elliptic Feynman integrals, which lead to Kronecker-Eisenstein $\omega$-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.