Integrand Analysis, Leading Singularities and Canonical Bases beyond Polylogarithms

TL;DR

This paper explores the relationship between leading singularities and canonical bases of Feynman integrals beyond polylogarithms, introducing new transcendental functions linked to underlying geometries.

Contribution

It generalizes the concept of leading singularities in dimensional regularization and connects them to new transcendental functions and differential equations beyond polylogarithmic cases.

Findings

Integrals with unit leading singularities satisfy ε-factorized differential equations.

New transcendental functions correspond to differential forms in the Gauss-Manin connection.

The construction relates to splitting the period matrix into semi-simple and unipotent parts.

Abstract

In this paper, we elaborate on the connection between leading singularities and canonical bases of Feynman integrals beyond polylogarithms. We start by discussing a notion of leading singularities in dimensional regularization, which can be generalized from the Riemann sphere to more complex geometries, and use it to demonstrate how selecting Feynman integrals with unit leading singularities necessitates introducing new transcendental functions related to the periods of the underlying geometries. Integrals with unit leading singularities in this generalized sense, satisfy $\epsilon$-factorized differential equations, and the new transcendental functions are in direct correspondence to the new differential forms appearing in their Gauss-Manin connection. We argue that this construction is mathematically equivalent to the splitting of the period matrix into semi-simple and unipotent parts plus a clean-up step, and demonstrate its use with examples of increasing complexity that require the interplay of multiple geometries.