Self-dualities and Galois symmetries in Feynman integrals

TL;DR

This paper explores additional symmetries in Feynman integrals, such as self-duality and Galois symmetries, which relate different master integrals and constrain their differential equations, revealing deeper geometric structures.

Contribution

It demonstrates that sectors with multiple master integrals can exhibit self-duality and Galois symmetries, extending known symmetry concepts to broader classes of Feynman integrals.

Findings

Self-duality often extends beyond Calabi--Yau related integrals.

Galois symmetries relate integrals through square root substitutions.

Symmetries impose constraints on the differential equations of integrals.

Abstract

It is well-known that all Feynman integrals within a given family can be expressed as a finite linear combination of master integrals. The master integrals naturally group into sectors. Starting from two loops, there can exist sectors made up of more than one master integral. In this paper we show that such sectors may have additional symmetries. First of all, self-duality, which was first observed in Feynman integrals related to Calabi--Yau geometries, often carries over to non-Calabi--Yau Feynman integrals. Secondly, we show that in addition there can exist Galois symmetries relating integrals. In the simplest case of two master integrals within a sector, whose definition involves a square root $r$, we may choose a basis $(I_1,I_2)$ such that $I_2$ is obtained from $I_1$ by the substitution $r \rightarrow -r$. This pattern also persists in sectors, which a priori are not related to any square root with dependence on the kinematic variables. We show in several examples that in such cases a suitable redefinition of the integrals introduces constant square roots like $\sqrt{3}$. The new master integrals are then again related by a Galois symmetry, for example the substitution $\sqrt{3} \rightarrow -\sqrt{3}$. To handle the case where the argument of a square root would be a perfect square we introduce a limit Galois symmetry. Both self-duality and Galois symmetries constrain the differential equation.