Abstract
The $\mu$-$\tau$ exchange symmetry in the neutrino mass matrix and its breaking as a perturbation are discussed. The exact $\mu$-$\tau$ symmetry restricts the 2-3 and 1-3 neutrino mixing angles as $\theta_{23} = \pi/4$ and $\theta_{13} = 0$ at a zeroth order level. We claim that the $\mu$-$\tau$ symmetry breaking prefers a large CP violation to realize the observed value of $\theta_{13}$ and to keep $\theta_{23}$ nearly maximal, though an artificial choice of the $\mu$-$\tau$ breaking can tune $\theta_{23}$, irrespective of the CP phase. We exhibit several relations among the deviation of $\theta_{23}$ from $\pi/4$, $\theta_{13}$ and Dirac CP phase $\delta$, which are useful to test the $\mu$-$\tau$ breaking models in the near future experiments. We also propose a concrete model to break the $\mu$-$\tau$ exchange symmetry spontaneously and its breaking is mediated by the gauge interactions radiatively in the framework of the extended gauge model with $B-L$ and $L_\mu - L_\tau$ symmetries. As a result of the gauge mediated $\mu$-$\tau$ breaking in the neutrino mass matrix, the artificial choice is unlikely, and a large Dirac CP phase is preferable.
Highlights
The long baseline neutrino oscillation experiments are ongoing [1,2], and it is expected that the 2-3 neutrino mixing and a CP phase will be measured more accurately [3,4]
II, we describe the separation of the μ-τ symmetric and breaking pieces in the neutrino mass matrix, and we discuss how the large CP phase is preferred by exhibiting the relations among δθ23, θ13, and δ
We have shown that the description of μ-τ symmetric and μ-τ breaking pieces is a good base to consider the size of δθ23 and the Dirac CP phase δ, which will be measured more accurately at the near-future long-baseline neutrino oscillation experiments
Summary
The long baseline neutrino oscillation experiments are ongoing [1,2], and it is expected that the 2-3 neutrino mixing and a CP phase will be measured more accurately [3,4]. We claim that the separation of μ-τ symmetric and μ-τ breaking pieces is a good parametrization of the neutrino mass matrix to describe the deviation from the maximal angle of the 2-3 mixing (δθ23), 1-3 mixing θ13, and the CP phase δ (even if there is not an underlying μ-τ symmetry in the Lagrangian). If there is μ-τ exchange symmetry and the symmetry is spontaneously broken in the neutrino sector, the deviation δθ is, very naively, of similar size to θ13 In this sense, the observed 1-3 mixing angle is too large to explain the nearly maximal angle θ23, which may be the reason why there is less interest in the μ-τ symmetry. This paper is organized as follows: In Sec. II, we describe the separation of the μ-τ symmetric and breaking pieces in the neutrino mass matrix, and we discuss how the large CP phase is preferred by exhibiting the relations among δθ, θ13, and δ. In Appendix B, we discuss the relation of our description with the μ-τ reflection symmetry
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