Journal Article

PUBDB-2025-00326

http://join2-wiki.gsi.de/foswiki/pub/Main/Artwork/join2_logo100x88.png Self-dualities and Galois symmetries in Feynman integrals

Pögel, S. ; Wang, X. ; Weinzierl, S. (Corresponding author) ; Wu, K.DESY* ; Xu, X.

2024
Springer Heidelberg

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Please use a persistent id in citations: doi:10.1007/JHEP09(2024)084  doi:10.3204/PUBDB-2025-00326

Report No.: arXiv:2407.08799

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 → −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} $ → $ -\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.

Keyword(s): Differential and Algebraic Geometry ; Higher Order Electroweak Calculations ; Higher-Order Perturbative Calculations ; Scattering Amplitudes

Note: 41 pages, v2: version to be published

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Contributing Institute(s):

1. Theorie-Gruppe (T)

Research Program(s):

1. 611 - Fundamental Particles and Forces (POF4-611) (POF4-611)
2. DFG project G:(GEPRIS)390831469 - EXC 2118: Precision Physics, Fundamental Interactions and Structure of Matter (PRISMA+) (390831469) (390831469)
3. DFG project G:(GEPRIS)390783311 - EXC 2094: ORIGINS: Vom Ursprung des Universums bis zu den ersten Bausteinen des Lebens (390783311) (390783311)

Experiment(s):

1. No specific instrument

Appears in the scientific report 2024

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Linked articles:

http://join2-wiki.gsi.de/foswiki/pub/Main/Artwork/join2_logo100x88.png Preprint Pögel, S. ; Wang, X. ; Weinzierl, S. (Corresponding author) ; Wu, K.DESY* ; Xu, X.
Self-dualities and Galois symmetries in Feynman integrals
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