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

TL;DR

This paper introduces a translational flavor symmetry in fermion mass terms, linking the symmetry to the structure of the fermion mass matrices and suggesting a natural basis for modeling fermion masses, including massless first-family fermions.

Contribution

It establishes a novel translational flavor symmetry in fermion mass terms and connects it to the structure of the mass matrices, providing a new perspective for fermion mass modeling.

Findings

Zero mass limit for first-family fermions is natural.

Translational symmetry relates to the structure of the fermion mass matrices.

Supports the idea of massless lightest neutrino and hierarchical charged fermion masses.

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 $\psi^{}_{\alpha \rm L (R)} \to \psi^{}_{\alpha \rm L (R)} + n^{}_{\alpha} z^{}_{\psi \rm L(R)}$ in the flavor space, where $n^{}_\alpha$ and $z^{}_{\psi \rm L(R)}$ stand respectively for the flavor-dependent complex numbers and a constant spinor field anticommuting with the fermion fields, we show that $n^{}_\alpha$ 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. 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.