Charged lepton mass relations in a SUSY scenario

The observed charged lepton masses satisfy the relations K≡(me+mμ+mτ)/(me+mμ+mτ)2=2/3 and κ≡memμmτ/(me+mμ+mτ)3=1/486 with great accuracy. These parameters are given as K=(Tr[ΦΦ])/(Tr[Φ])2 and κ=detΦ/(Tr[Φ])3 if the charged lepton masses mei are given by mei∝∑kΦikΦki where Φ 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 κ 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.