New physics with the lepton flavor violating decay $\tau\to 3\mu$

Lepton flavour violating (LFV) processes are a smoking gun signal of new physics (NP). If the semileptonic $B$ decay anomalies are indeed due to some NP, such operators can potentially lead to LFV decays involving the second and the third generation leptons, like $\tau\to 3\mu$. In this paper, we explore how far the nature of NP can be unraveled at the next generation $B$-factories like Belle-II, provided the decay $\tau\to 3\mu$ has been observed. We use four observables with which the differentiation among NP operators may be achieved to a high confidence level. Possible presence of multiple NP operators are also analysed with the Optimal Observable technique. While the analysis can be improved even further if the final state muon polarisations are measured, we present this work as a motivational tool for the experimentalists, as well as a template for the analysis of similar processes.