Constraints on the rare tau decays from μ → eγ in the supersymmetric see-saw model

It is now a firmly established fact that all family lepton numbers are violated in Nature. In this paper we discuss the implications of this observation for future searches for rare tau decays in the supersymmetric see-saw model. Using the two loop renormalization group evolution of the soft terms and the Yukawa couplings we show that there exists a lower bound on the rate of the rare process μ → eγ of the form BR(μ → eγ) ≳ C × BR(τ → μγ)BR(τ → eγ), where C is a constant that depends on supersymmetric parameters. Our only assumption is the absence of cancellations among the high-energy see-saw parameters. We also discuss the implications of this bound for future searches for rare tau decays. In particular, for large regions of the mSUGRA parameter space, we show that present B-factories could discover either τ → μγ or τ → eγ, but not both.