On symbology and differential equations of Feynman integrals from Schubert analysis

TL;DR

This paper extends Schubert analysis to general dimensions and masses, enabling the derivation of symbol letters and differential equations for Feynman integrals using embedding space and maximal cut solutions.

Contribution

It introduces a generalized Schubert analysis method in embedding space for analyzing Feynman integrals across dimensions and masses, linking symbol letters to maximal cut solutions.

Findings

All symbol letters derived as cross-ratios from maximal cuts.

Reproduction of entries in canonical differential equations.

Application to one-loop and two-loop integral families.

Abstract

We take the first step in generalizing the so-called "Schubert analysis", originally proposed in twistor space for four-dimensional kinematics, to the study of symbol letters and more detailed information on canonical differential equations for Feynman integral families in general dimensions with general masses. The basic idea is to work in embedding space and compute possible cross-ratios built from (Lorentz products of) maximal cut solutions for all integrals in the family. We demonstrate the power of the method using the most general one-loop integrals, as well as various two-loop planar integral families (such as sunrise, double-triangle and double-box) in general dimensions. Not only can we obtain all symbol letters as cross-ratios from maximal-cut solutions, but we also reproduce entries in the canonical differential equations satisfied by a basis of dlog integrals.