Lepton flavor changing in neutrinoless $\tau$ decays

Neutrino oscillations, as recently reported by the Super-Kamiokande collaboration, imply that lepton numbers could be violated, and $\tau^{\pm}\to \mu^{\pm}+\ell^{+}+\ell^{-},\tau^{\pm}\to\mu^{\pm}+\rho^0$ are some typical examples. We point out that in these neutrinoless modes, the GIM cancelation is much milder with only a logarithmic behavior $\log (m_j /m_k)$ where $m_{j, k}$ are the neutrino masses. This is in sharp contrast with the vanishingly small amplitude $\tau^{\pm}\to \mu^{\pm}+\gamma$ strongly suppressed by the quadratic power $(m_j^2-m_k^2)/ M_{\rm W}^2$. In comparison with the hopelessly small branching ratio B$(\tau^{\pm}\to \mu^{\pm}+\gamma)\approx 10^{-40}$, the B$(\tau^{\pm}\to\mu^{\pm}+\ell^{+}+\ell^{-})$ could be larger than $10^{-14}$. The latter mode, if measurable, could give one more constraint to the lepton mixing angle $\sin 2\theta_{jk}$ and the neutrino mass ratio $m_j/m_k$, and therefore is complementary to neutrino oscillation experiments.