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The muon $(g-2)_\mu$ and $b\to s \bar \mu \mu$ induced $B$ anomalies as hintsof new physics beyond the standard model (SM) have attracted much attention.These two anomalies indicate that there may exist new interaction specificallyrelated to muon. A lot of theoretical ideas have been proposed to explain theseanomalies. Gauged flavor specific $U(1)_{B_q-L_\mu}$ is among the promisingones. The new gauge boson $Z'$ from $U(1)_{B_q-L_\mu}$ interacts with muon andprovides necessary ingredient to solve the $(g-2)_\mu$ anomaly. The $Z'$-quarkcoupling can generate flavor changing interactions after diagonalization ofquark mass matrix between weak eigen-state and mass eigen-state basis. Werevisit challenges for such models attempting to explain the $(g-2)_\mu$ and$B$ anomalies separately or simultaneously. We find although for$U(1)_{B_q-L_\mu}$ models there is still parameter space to provide solutionsfor separately explaining the $(g-2)_\mu$ and $B$ anomalies, there exists noparameter space for such models to solve both the anomalies simultaneously,after taking into account existing constraints from $\tau \to \mu \gamma$,$\tau \to 3 \mu$, neutrino trident and $B_s - \bar B_s$ data. Among themleptonic processes restrict $Z^\prime$ mass to be less than a few hundred MeVif required to solve the $(g-2)_\mu$ anomaly, which causes conflict betweendata from $B_s - \bar B_s$, $D^0 - \bar D^0$ mixing and also hadron decays with$Z^\prime$ in the final states. The effects of $U(1)_Y$ and $U(1)_{B_q-L_\mu}$kinetic mixing on these anomalies are also studied. We find that neither canthese effects do much to bring the two anomalies together to be solvedsimultaneously.

Flavor Specific $U(1)_{B_q-L_\mu}$ Gauge Model for Muon $g-2$ and $b \to s \bar \mu \mu$ Anomalies