Permutation Symmetry S₃ and VEV Structure of Flavor-Triplet Higgs Scalars

A model with flavor-triplet Higgs scalars \phi_i (i=1,2,3)is investigated under a permutation symmetry S_3 and its symmetry breaking. A possible S_3 breaking form of the Higgs potential whose vacuum expectation values v_i=< \phi_i> satisfy a relation v_1^2 +v_2^2 +v_3^2 ={2/3}(v_1 +v_2 +v_3)^2 is investigated, because if we suppose a seesaw-like mass matrix model M_e = m M^{-1} m with m_{ij} \propto \delta_{ij} v_i and M_{ij} \propto \delta_{ij}, such a model can lead to the well-known charged lepton mass relation m_e +m_\mu +m_\tau = {2/3} (\sqrt{m_e}+\sqrt{m_\mu} +\sqrt{m_\tau})^2.