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

The $Z^{\mathrm{\prime }}$-gauge boson in an $U(1{)}_{L_{\mu }-L_{\tau }}$ gauge symmetry has two interesting features: one is its vector couplings to the charged leptons, and the other is the decoupling from the electron. Based on these properties, we investigate the feasibility to simultaneously resolve the $R_{K^{()}} = BR(B\to K^{()} \mu^+ \mu^-)/BR(B\to K^{()} e^+ e^-)$and$R_{D^{()}} = BR(\bar B\to D^{()} \tau \bar\nu_\tau)/BR(\bar B\to D^{()} \ell \bar\nu_\ell)$anomaliesinan$U(1){L\mu-L_\tau}$ model, where the former is expected to arise from the $Z^{\mathrm{\prime }}$-penguin-induced $b\to s{\mu }^{+}{\mu }^{-}$ process and the latter from the tree-level $b\to c\tau {\bar{\nu }}_{\tau }$ decay. In order to achieve the intended purpose, we employ one vector-like doublet lepton and one singlet scalar leptoquark (LQ), in which the new particles all carry the $U(1{)}_{L_{\mu }-L_{\tau }}$ charges; the $b\to s{Z}^{\mathrm{\prime }}$ effective interaction is generated from the vector-like lepton and LQ loop, and the $b\to c \tau \bar\nu_\tau$decayisinducedfromtheLQ\mathrm{.}Whentheconstraintsfromthe$b\to s \gamma$,$B^+\to K^+ \nu \bar\nu$,$B^-c\to \tau \bar \nu\tau$,$\Delta F=2$,and$\tau\to \mu \ell \bar \ell$processesareincluded,itisfoundthat$R_D$and$R_{D^*}$ can be enhanced to fit the experimental data, and the Wilson coefficient $C_9$ from the LQ-loop can reach $C_9^{LQ,\mu }\sim -1$, which can explain the $R_K$ and $R_{K^{\ast }}$ anomalies. In addition, in this simple model, the Higgs lepton-flavor violating $h\to \mu \tau$ decay can occur at the tree level, and its branching ratio can be as large as the current experimental upper limit.