We conjecture that all $CP$ violations (both Dirac and Majorana types) arise from a common origin in the neutrino seesaw. With this conceptually attractive and simple conjecture, we deduce that $\mu \mathrm{\text{−}}\tau$ breaking shares the common origin with all $CP$ violations. We study the common origin of $\mu \mathrm{\text{−}}\tau$ and $CP$ breaking in the Dirac mass matrix of seesaw Lagrangian (with right-handed neutrinos being $\mu \mathrm{\text{−}}\tau$ blind), which uniquely leads to inverted mass ordering of light neutrinos. We then predict a very different correlation between the two small $\mu \mathrm{\text{−}}\tau$ breaking observables ${\theta }_{13}-0°$ and ${\theta }_{23}-45°$, which can saturate the present experimental upper limit on ${\theta }_{13}$. This will be tested against our previous normal mass-ordering scheme by the ongoing oscillation experiments. We also analyze the correlations of ${\theta }_{13}$ with Jarlskog invariant and neutrinoless $\beta \beta$-decay observable. From the common origin of $CP$ and $\mu \mathrm{\text{−}}\tau$ breaking in the neutrino seesaw, we establish a direct link between the low energy $CP$ violations and the cosmological $CP$ violation for baryon asymmetry. With these we further predict a lower bound on ${\theta }_{13}$, supporting the ongoing probes of ${\theta }_{13}$ at Daya Bay, Double Chooz, and RENO experiments. Finally, we analyze the general model-independent ${\mathbb{Z}}_2\otimes {\mathbb{Z}}_2$ symmetry structure of the light neutrino sector, and map it into the seesaw sector, where one of the ${\mathbb{Z}}_2$’s corresponds to the $\mu \mathrm{\text{−}}\tau$ symmetry ${\mathbb{Z}}_2^{\mu \tau }$ and another the hidden symmetry ${\mathbb{Z}}_2^s$ (revealed in our previous work) which dictates the solar mixing angle ${\theta }_{12}$. We derive the physical consequences of this ${\mathbb{Z}}_2^s$ and its possible partial violation in the presence of $\mu \mathrm{\text{−}}\tau$ breaking (with or without the neutrino seesaw), regarding the ${\theta }_{12}$ determination and the correlation between $\mu \mathrm{\text{−}}\tau$ breaking observables.

• Received 6 May 2011

DOI:https://doi.org/10.1103/PhysRevD.84.033009