Domain Walls in the $A_4$ Flavored NMSSM

In this work, we study the phenomenology of neutrinos and the formation of cosmic domain walls in the NMSSM extended by an $A_4 \times Z_3$ flavor symmetry. Neutrino masses result from the type I seesaw mechanism using only two flavon fields and the NMSSM singlet $\mathcal{S}$ while their mixing is of Trimaximal mixing form. We perform our phenomenological study in the normal mass hierarchy where we find that observables like $m_{\beta\beta}$, $m_\beta$, and $\sum_i m_i$ can be tested by future experiments. Due to the difference between the $A_4$ subgroups that undergo spontaneous breaking in both the charged lepton and neutrino sectors, the resulting domain walls in each sector exhibit distinct structures. We delve into the details of the breaking patterns within these two sectors, and we introduce a nuanced geometric representation for them. To tackle the domain wall problem, we explore a well-established method involving the explicit breaking of the flavor symmetry. This is achieved through the introduction of Planck-suppressed operators induced by supergravity.