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The $Z'$-gauge boson in an $U(1)_{L_\mu-L_\tau}$ gauge symmetry has twointeresting features: one is its vector couplings to the charged leptons, andthe other is the decoupling from the electron. Based on these properties, weinvestigate the feasibility to simultaneously resolve the $R_{K^{(*)}} =BR(B\to K^{(*)} \mu^+ \mu^-)/BR(B\to K^{(*)} e^+ e^-)$ and $R_{D^{(*)}} =BR(\bar B\to D^{(*)} \tau \bar\nu_\tau)/BR(\bar B\to D^{(*)} \ell\bar\nu_\ell)$ anomalies in an $U(1)_{L_\mu-L_\tau}$ model, where the former isexpected to arise from the $Z'$-penguin-induced $b\to s \mu^+ \mu^-$ processand the latter from the tree-level $b\to c \tau \bar\nu_\tau$ decay. In orderto achieve the intended purpose, we employ one vector-like doublet lepton andone singlet scalar leptoquark (LQ), in which the new particles all carry the$U(1)_{L_\mu-L_\tau}$ charges; the $b\to sZ'$ effective interaction isgenerated from the vector-like lepton and LQ loop, and the $b\to c \tau\bar\nu_\tau$ decay is induced from the LQ. When the constraints from the $b\tos \gamma$, $B^+\to K^+ \nu \bar\nu$, $B^-_c\to \tau \bar \nu_\tau$, $\DeltaF=2$, and $\tau\to \mu \ell \bar \ell$ processes are included, it is found that$R_D$ and $R_{D^*}$ can be enhanced to fit the experimental data, and theWilson coefficient $C_9$ from the LQ-loop can reach $C^{LQ,\mu}_9\sim -1$,which can explain the $R_{K}$ and $R_{K^*}$ anomalies. In addition, in thissimple model, the Higgs lepton-flavor violating $h\to \mu \tau$ decay can occurat the tree level, and its branching ratio can be as large as the currentexperimental upper limit.

Penguin $b \to s\ell'^+ \ell'^-$ and $B$-meson anomalies in a gauged ${L_\mu -L_\tau}$