Fermion Mass and Mixing Patterns from Fusion Algebras

The large hierarchy of fermion masses in the Standard Model has long posed a challenge to be explained. One approach is to try to invoke a symmetry principle to dictate the structure ("texture") of the matrices of fermion Yukawa couplings to the Higgs field. Here we invoke noninvertible symmetries, specifically, fusion algebras, to provide such textures. In particular, the structure of the fusion-algebra can be used to assign labels to each field, which determines how suppressed any given composite operator is as a term in the Lagrangian. In UV completions, this can be interpreted as specifying a hierarchy of operators that in the IR effective theory become the Yukawa couplings to the Higgs field. Remaining agnostic of a specific UV model, we perform a Monte Carlo numerical scan over unknown Wilson coefficients, and we identify the rank-8 fusion algebra $\rm{FR}^{8,6}_8$ as the one providing the best fit for the textures of the up-quark, down-quark and charged-lepton Yukawa matrices. We also study the predictions for the CKM and PMNS mixing matrices, with the conclusion that none of the fusion algebras we study provide an accurate fit for the masses and mixing angles simultaneously. The fusion algebra structure ensures that the textures that we find are stable under RG running and persist down to low energies.