Search for the Lepton-Flavor-Violating Decay τ^- to μ^- η at Belle

We have searched for the Lepton Flavor Violating decay $\tau^- \to \mu^- \eta$ using a data sample of 84.3 fb$^{-1}$ accumulated with the Belle detector at KEK. The $\eta$-meson was detected through the decay modes: $\eta \to \gamma\gamma$ and $\pi^+\pi^-\pi^0$. No signal candidates are found, and we obtain an upper limit for the branching fraction ${\cal B}(\tau^- \to \mu^- \eta)<3.4\times 10^{-7}$ at the 90% confidence level.