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

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 $ψ^{}_{α\rm L (R)} \to ψ^{}_{α\rm L (R)} + n^{}_α z^{}_{ψ\rm L(R)}$ in the flavor space, where $n^{}_α$ and $z^{}_{ψ\rm L(R)}$ 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 $i$-th 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.