The translational $\mu$-$\tau$ reflection symmetry of Majorana neutrinos

The present neutrino oscillation data allow $m^{}_1 = 0$ (or $m^{}_3 = 0$) for the neutrino mass spectrum and support $\theta^{}_{23} \simeq \pi/4$ and $\delta \simeq -\pi/2$ 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} $\mu$-$\tau$ reflection symmetry -- the effective Majorana neutrino mass term keeps invariant under the transformations $\nu^{}_{e \rm L} \to (\nu^{}_{e \rm L})^c + U^*_{e i} z^{c}_\nu$, $\nu^{}_{\mu \rm L} \to (\nu^{}_{\tau \rm L})^c + U^*_{\tau i} z^{c}_\nu$ and $\nu^{}_{\tau \rm L} \to (\nu^{}_{\mu \rm L})^c + U^*_{\mu i} z^{c}_\nu$ (for $i = 1$ or $3$), where $z^{c}_\nu$ is the charge conjugation of a constant spinor field $z^{}_\nu$. 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.