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 ${\theta }_{23}^{}\simeq \pi \mathrm{/}4$ and $\delta \simeq -\pi \mathrm{/}2$ 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 ${\nu }_{e\mathrm{L}}^{}\to ({\nu }_{e\mathrm{L}}^{}{)}^c+U_{ei}^{\ast }{z}_{\nu }^c$, ${\nu }_{\mu \mathrm{L}}^{}\to ({\nu }_{\tau \mathrm{L}}^{}{)}^c+U_{\tau i}^{\ast }{z}_{\nu }^c$ and ${\nu }_{\tau \mathrm{L}}^{}\to ({\nu }_{\mu \mathrm{L}}^{}{)}^c+U_{\mu i}^{\ast }{z}_{\nu }^c$ (for $i=1$ or $3$), where $z_{\nu }^c$ 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.