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Considering the constraints on the lepton flavor violating (LFV) couplings ofthe new gauge boson $Z'$ to ordinary leptons from the experimental upper limitfor the LFV process $\ell\rightarrow 3\ell'$, we calculate the contributions of$Z'$ to the LFV decays $V\rightarrow\ell_{i}\ell_{j}$ with$V\in\{\phi,J/\Psi,\Psi(2S),\Upsilon^{(n)}\}$ and $\tau\to\mu(e)\phi$ in thecontext of several $Z'$ models. We find that all $Z'$ models considered in thispaper can produce significant contributions to these decay processes and makethe value of the branching ratio $Br(\tau\to e\phi)$ above its experimentalupper limit. The experimental upper limit of $\tau\to e\phi$ can give moresevere constraints on these $Z'$ models than those given by the rare decayprocess $\tau\to3e$.

New gauge boson Z' and lepton flavor violating decays and production of vector mesons