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Lepton flavor violating $Z$-boson couplings from non-standard Higgs interactions

The Standard Model predicts that the Higgs boson couples to the fermions in the mass eigenstates. We consider the effects of lepton flavor violating (LFV) $Z$ boson couplings in the case where the Higgs boson has flavor non-diagonal Yukawa interactions with the muon and the tau lepton generated from physics beyond the Standard Model. We list the formulae of the couplings of the effective interactions among the $τ$ lepton, the muon and the $Z$ boson. Using these formulae, we calculate the branching fractions of various leptonic and hadronic LFV $τ$ decays, and the LFV $Z$ boson decay: $Z\to τμ$. Although the $Z$-boson contributions to LFV tau decays cannot be ignored in terms of the counting of operator dimensions or chirality flipptings, it turns out that they are not very significant for $τ\to3μ$ and $τ\toμρ$ decays. We also calculate the branching fractions of the processes, $τ\to μπ$, $τ\to μη^{(\prime)}$ and $τ\toμa_1$, which are dominated by the $Z$-boson exchanges due to the spin and the parity of the hadrons.