A Symbol and Coaction for Higher-Loop Sunrise Integrals

TL;DR

This paper develops a symbol and coaction framework for higher-loop sunrise integrals involving Calabi-Yau threefolds, extending the mathematical tools used for complex Feynman integrals.

Contribution

It introduces the first concrete symbols and coactions for integrals with Calabi-Yau geometries, using a unipotent differential equation approach and an augmented basis of master integrals.

Findings

Constructed symbols and coactions for $l$-loop sunrise integrals.

Recast differential equations as unipotent equations for finite-length symbols.

Connected the construction to polylogarithms and elliptic polylogarithms.

Abstract

We construct a symbol and coaction for $l$-loop sunrise integrals, both for the equal-mass and generic-mass cases. These constitute the first concrete examples of symbols and coactions for integrals involving Calabi-Yau threefolds and higher. In order to achieve a symbol of finite length, we recast the differential equations satisfied by the master integrals of this topology in the form of a unipotent differential equation. We augment the basis of master integrals in a natural way by including ratios of maximal cuts $\tau_i$. We discuss the relationship of this construction to constructions of symbols and coactions for multiple polylogarithms and elliptic multiple polylogarithms, in particular its connection to notions of transcendental weight.