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 transformation $\psi^{}_{\alpha \rm L (R)} \to \psi^{}_{\alpha \rm L (R)} + n^{}_{\alpha} z^{}_\psi$ in the flavor space, where $n^{}_\alpha$ and $z^{}_\psi$ stand respectively for the flavor-dependent complex numbers and the spacetime- and flavor-independent element of the Grassmann algebra, we show that $n^{}_\alpha$ can be identified as the elements $U^{}_{\alpha i}$ in the $i$-th column of the unitary matrix $U$ used to diagonalize the corresponding Hermitian or symmetric fermion mass matrix $M^{}_\psi$, and $m^{}_i = 0$ holds accordingly. We find that the reverse is also true. Given the very facts that the charged leptons, up- and down-type quarks all have a strong mass hierarchy and current experimental data allow the lightest neutrino to be (almost) massless, the zero mass limit for the first-family fermions and the translational flavor symmetry behind it are expected to be a natural starting point for model building.