Nonzero ${\theta }_{13}$ in $\mathrm{SO}(3)\to A_4$ lepton models

The simplest neutrino mass models based on $A_4$ symmetry predict ${\theta }_{13}=0$ at tree level, a value that contradicts recent data. We study models that arise from the spontaneous breaking of an SO(3) symmetry to its $A_4$ subgroup, and find that such models can naturally accommodate a nonzero ${\theta }_{13}$ at tree level. Standard Model charged leptons mix with additional heavy ones to generate a ${\theta }_{13}$ that scales with the ratio of the $A_4$-breaking to SO(3)-breaking scales. A suitable choice of energy scales hence allows one to reproduce the correct lepton mixing angles. We also consider an alternative approach where we modify the alignment of flavons associated with the charged lepton masses, and find that the effects on ${\theta }_{13}$ are enhanced by a factor that scales as $m_{\tau }/m_{\mu }$.

Figure 1

Graphs of $\sin ({\theta }_{13})$ against $\sup \{v/v_T,v_5/v_T\}$ for two collections (a) ${\mathcal{C}}_1$ and (b) ${\mathcal{C}}_2$ of random parameter sets. Various lines have been included for reference. Both graphs demonstrate the linear dependence predicted in our analysis. Collection ${\mathcal{C}}_2$ has a charged-lepton mass spectrum much closer to the actual hierarchy than ${\mathcal{C}}_1$, as a result of which the constant of proportionality is significantly enhanced by the amplification effect.

Figure 2

Graphs of $\sin ({\theta }_{12})$ and $\sin ({\theta }_{23})$ against $\sin ({\theta }_{13})$. The error ellipses for the mixing angles are roughly derived from the $1\sigma$ to $3\sigma$ ranges in Ref. [62].

Figure 3

Graphs of $\sin ({\theta }_{13})$ against the change in flavon alignment $\chi$ for collections (a) ${\mathcal{C}}_3$, (b) ${\mathcal{C}}_4$, (c) ${\mathcal{C}}_5$ and (d) ${\mathcal{C}}_6$. The collections differ in the conditions imposed on the ratio of mass eigenvalues.