Discrete symmetries of Feynman integrals

TL;DR

This paper investigates discrete symmetries of Feynman integrals, revealing their mathematical structure and implications for master integrals, with applications to banana integrals up to four loops.

Contribution

It introduces a comprehensive mathematical framework for discrete symmetries of Feynman integrals, linking permutations, cohomology, and topology, and applies it to compute master integrals.

Findings

Symmetries correspond to permutations of Lee-Pomeransky polynomials.

The symmetry group acts on twisted cohomology, with characters related to Euler characteristics.

Derived formulas for counting master integrals, exemplified on banana integrals.

Abstract

We perform a comprehensive study of a certain class of discrete symmetries of families of Feynman integrals, defined as affine changes of variables that map different sectors of the family into each other. We show that these transformations are always encoded into permutations of the Feynman parameters that relate the Lee-Pomeransky polynomials of the two sectors, irrespective of the integral representation used to define the Feynman integrals. We then construct an affine map in loop-momentum space that encodes such a permutation. We also show that these symmetries can be naturally embedded into the framework of twisted cohomology theories, and the period and intersection parings are invariant under the symmetry transformations. If we focus on symmetries within a fixed sector, we obtain a group acting on the twisted cohomology group, and we study the decomposition of this action into irreducible representations. One of our main mathematical results is that the character of this representation is proportional to the Euler characteristic of the corresponding fixed-point set. We then study the implications for Feynman integrals, in particular for the intersection matrix in a canonical basis. We also present a formula for the number of master integrals in a given sector in the presence of a non-trivial symmetry group in terms of the Euler characteristics of fixed-point sets. As an application, we obtain the numbers of master integrals for banana integrals with up to four loops for arbitrary configurations of non-zero masses. In order to achieve our results, we had to combine tools from various different areas of mathematics, including graph theory, group theory and algebraic topology.