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

Abstract

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

5+6 pages, 2 tables, 1 appendix, 1 ancillary file for the new example