Discrete symmetries of Feynman integrals
- 1Universität Bonn
Abstract
A bstract : 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.
Related
BibTeX
@article{duhr2026discrete,
title = {{Discrete symmetries of Feynman integrals}},
author = {Claude Duhr and Sara Maggio and Cathrin Semper and Sven F. Stawinski},
journal = {Journal of High Energy Physics},
year = {2026},
month = oct,
doi = {10.1007/jhep10(2026)031},
url = {https://link.springer.com/10.1007/JHEP10(2026)031},
note = {CC BY 4.0},
}
RIS
TY - JOUR
TI - Discrete symmetries of Feynman integrals
AU - Claude Duhr
AU - Sara Maggio
AU - Cathrin Semper
AU - Sven F. Stawinski
PY - 2026
DA - 2026/10/02
JO - Journal of High Energy Physics
DO - 10.1007/jhep10(2026)031
UR - https://link.springer.com/10.1007/JHEP10(2026)031
AB - A bstract : 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.
SN - 1029-8479
ER -