A translational flavor symmetry in the mass terms of Dirac and Majorana fermions

Abstract

Requiring the effective mass term for a category of fundamental Dirac or Majorana fermions of the same electric charge to be invariant under the translational transformations ψ αL(R) → ψ αL(R) + n α z ψL(R) in the flavor space, where n α and z ψL(R) stand respectively for the flavor-dependent complex numbers and a constant spinor field anticommuting with the fermion fields, we show that n α can be identified as the elements U αi in the ith column of the unitary matrix U used to diagonalize the corresponding Hermitian or symmetric fermion mass matrix M ψ , and m i = 0 holds accordingly. We find that the reverse is also true. Now that the mass spectra of charged leptons, up- and down-type quarks are all strongly hierarchical and current experimental data allow the lightest neutrino to be massless, we argue that the zero mass limit for the first-family fermions and the translational flavor symmetry behind it should be a natural starting point for building viable fermion mass models.