Renormalizable $A_4$ Model for Lepton Sector

We revisit $A_4$ flavor symmetry for the lepton sector with minimal Higgs sector, namely just one $A_4-$triplet SU(2)-doublet scalar. To increase predictivity even further, we impose the constraint of renormalizability. A geometric interpretation of an $A_4-$triplet aids our understanding of tribimaximal mixing. To increase predictivity even further, we impose the constraint of renormalizability. We investigate the neutrino mass hierarchy in such a minimal $A_4$ model and find there are two solutions: one with $m_2 \gg m_1 = m_3$ is phenomenologically unacceptable; the other with $m_3 \gg m_1 = m_2$ is a normal hierarchy. An inverted hierarchy is impossible without addition of more parameters by either more Higgs scalars or higher-order irrelevant operators.