Charged Lepton Mass Relations in a SUSY Scenario

The observed charged lepton masses satisfy the relations $K \equiv (m_e +m_\mu+m_\tau)/(\sqrt{m_e} +\sqrt{m_\mu} +\sqrt{m_\tau})^2 =2/3$ and $\kappa \equiv \sqrt{m_e m_\mu m_\tau}/(\sqrt{m_e} +\sqrt{m_\mu} +\sqrt{m_\tau})^3 =1/486$ with great accuracy. These parameters are given as $K=( {\rm Tr}[\Phi \Phi])/ ({\rm Tr}[\Phi ])^2$ and $\kappa = {\rm det} \Phi/({\rm Tr}[\Phi ])^3$ if the charged lepton masses $m_{ei}$ are given by $m_{ei} \propto \sum_k \Phi_i^{\ k} \Phi_k^{\ i}$ where $\Phi$ is a U(3)-family nonet scalar. Simple scalar potential forms to realize the relations have been already proposed in non-supersymmetric scenarios, but the potential forms are not stable against the renormalization group effects. In this paper, we examine supersymmetric scenarios and find that the parameters $K$ and $\kappa$ are made stable against the effects in a very nontrivial way, even though the superpotential itself (in the canonical basis) suffers the usual corrections. We also show possible simple superpotential forms for the relations.