  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 $\tau$ lepton, the muon and the $Z$ boson.
  Using these formulae, we calculate the branching fractions of various
leptonic and hadronic LFV $\tau$ decays, and the LFV $Z$ boson decay: $Z\to
\tau\mu$. 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 $\tau\to3\mu$
and $\tau\to\mu\rho$ decays. We also calculate the branching fractions of the
processes, $\tau\to \mu\pi$, $\tau\to \mu\eta^{(\prime)}$ and $\tau\to\mu a_1$,
which are dominated by the $Z$-boson exchanges due to the spin and the parity
of the hadrons.
