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 ${\psi }_{\alpha \mathrm{L}(\mathrm{R})}^{}\to {\psi }_{\alpha \mathrm{L}(\mathrm{R})}^{}+n_{\alpha }^{}{z}_{\psi \mathrm{L}(\mathrm{R})}^{}$ in the flavor space, where $n_{\alpha }^{}$ and $z_{\psi \mathrm{L}(\mathrm{R})}^{}$ 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.

Comments: 15 pages. A proof is given to show that the translation of a massless neutrino field, which assures the Dirac equation to be invariant, is equivalent to the translational transformations of three neutrino flavor states in the flavor space as a consequence of flavor mixing. More discussions are added. Accepted for publication in JPG