Nonzero $\theta_{13}$ in $SO(3) \rightarrow A_4$ 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 $m_\tau/m_\mu$.