Abstract
The flavor puzzle of the Standard Model quark sector is formulated in a non-perturbative way, using basis invariants that are independent of the choice of quark field basis. To achieve this, we first derive the algebraic ring of 10 CP even (primary) and 1 CP odd (secondary) basis invariants, using the Hilbert series and plethystic logarithm. An orthogonal basis in the ring of basis invariants is explicitly constructed, using hermitian projection operators derived via birdtrack diagrams. The thereby constructed invariants have well defined CP transformation behavior and give the most direct access to the flavor symmetric alignments of basis covariants. We firstly “measure” the orthogonal basis invariants from experimental data and characterize their location in the available parameter space. The experimentally observed orthogonal basis invariants take very close to maximal values and are highly correlated. Explaining the location of the invariants at close to maximal points, including the associated miniscule and highly correlated deviations, corresponds to solving the flavor puzzle in the invariant language. Once properly normalized, the orthogonal basis invariants are close to scale (RGE) invariant, hence, provide exquisite targets for fits of both, low- and high-scale (bottom-up and top-down) flavor models. Our result provides an entirely new angle on the flavor puzzle, and opens up ample opportunities for its ultimate exploration.
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Acknowledgments
We would like to thank Renato Fonseca for discussions on related projects. AT is grateful to Claudia Hagedorn for discussions on related projects and to Maximilian Berbig for an insightful observation. All birdtrack diagrams of this work have been generated with JaxoDraw [113].
The work of M.P.B. is supported in part by the Portuguese Fundação para a Ciência e Tecnologia (FCT) under contract SFRH/BD/146718/2019. The work of M.P.B. and J.P.S. supported in part by FCT under Contracts CERN/FIS-PAR/0008/2019, PTDC/FISPAR/29436/2017, UIDB/00777/2020, and UIDP/00777/2020; these projects are partially funded through POCTI (FEDER), COMPETE, QREN, and the EU.
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Bento, M.P., Silva, J.P. & Trautner, A. The basis invariant flavor puzzle. J. High Energ. Phys. 2024, 24 (2024). https://doi.org/10.1007/JHEP01(2024)024
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DOI: https://doi.org/10.1007/JHEP01(2024)024