Intersection theory rules symbology

TL;DR

This paper introduces a new method for determining the structure of symbols in polylogarithmic Feynman integrals, streamlining calculations of differential equations and revealing geometric insights.

Contribution

It presents a novel approach using d log-bases and intersection numbers to efficiently compute symbol letters and coefficients in Feynman integral analysis.

Findings

Method simplifies computation of symbol letters and coefficients.

Provides a rule to identify zero matrix elements.

Connects symbol letters to integrand poles and Newton polytope geometry.

Abstract

We propose a novel method to determine the structure of symbols for any family of polylogarithmic Feynman integrals. Using the d log-bases and simple formulas for the leading order and next-to-leading contributions to the intersection numbers, we give a streamlined procedure to compute the entries in the coefficient matrices of canonical differential equations, including the symbol letters and the rational coefficients. We also provide a selection rule to decide whether a given matrix element must be zero. The symbol letters are deeply related to the poles of the integrands and also have interesting connections to the geometry of Newton polytopes. Our method can be applied to many cutting-edge multi-loop calculations. The simplicity of our results also hints at the possible underlying structure in perturbative quantum field theories.