The translational $μ$-$τ$ reflection symmetry of Majorana neutrinos

Abstract

The present neutrino oscillation data allow $m^{}_1 = 0$ (or $m^{}_3 = 0$) for the neutrino mass spectrum and support $θ^{}_{23} \simeq π/4$ and $δ\simeq -π/2$ as two good approximations for the PMNS lepton flavor mixing matrix $U$. We show that these intriguing possibilities can be a very natural consequence of the {\it translational} $μ$-$τ$ reflection symmetry -- the effective Majorana neutrino mass term keeps invariant under the transformations $ν^{}_{e \rm L} \to (ν^{}_{e \rm L})^c + U^*_{e i} z^{c}_ν$, $ν^{}_{μ\rm L} \to (ν^{}_{τ\rm L})^c + U^*_{τi} z^{c}_ν$ and $ν^{}_{τ\rm L} \to (ν^{}_{μ\rm L})^c + U^*_{μi} z^{c}_ν$ (for $i = 1$ or $3$), where $z^{c}_ν$ is the charge conjugation of a constant spinor field $z^{}_ν$. Extending such a working flavor symmetry to the canonical seesaw mechanism at a superhigh-energy scale, we calculate its soft breaking effects at the electroweak scale by using the one-loop renormalization-group equations.