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Flavor Symmetry

Flavor symmetry conjectures propose underlying symmetries governing the different quark and lepton flavors within the Standard Model, though their precise form remains elusive.
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The statement of the theorem

Let GFG_F be a proposed flavor symmetry group acting on the flavor indices i,j→i′,j′i, j \to i', j'. The Lagrangian density L\mathcal{L} must be invariant under this transformation: L→L′=L\mathcal{L} \to \mathcal{L}' = \mathcal{L}. This invariance implies that the mass matrices MijM_{ij} for quarks and leptons must transform covariantly under GFG_F. Specifically, if MM is the mass matrix, then GFG_F dictates that MM must be proportional to the identity matrix or satisfy specific relations derived from the symmetry generators TaT^a: \n∑aTaMTa=M\sum_{a} T^a M T^a = M \nThis constrains the structure of the Yukawa couplings YijY_{ij} such that Yij=Yij(GF)Y_{ij} = Y_{ij}(G_F).
Source: Wikipedia