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

When combined the recent measurement $g-2$ of muon from Fermilab and the previous result from BNL, the deviation from the standard model (SM) prediction increased to 4.2$\sigma$ level. This is a strong hint of new physics which generates extensive studies to find possible underlying new physics to explain the anomaly. There are also recent new data enforcing anomaly in the measurement of $R_{K}$ from $B \to K \bar \mu \mu$ and increasing the confidence level to 3.1$\sigma$ for $b\to s \bar \mu \mu$ induced anomalies. These anomalies indicate that there may be new interaction specific to muon making all these anomalies related. The new interactions are originated in new physics related to the second generation. We propose a flavor specific $U(1)_{B_q-L_\mu}$ gauged interaction to explain these anomalies. The new gauge boson $Z'$ has a vector-like interaction to muon. This provides a solution to the muon $g-2$ anomaly. Requiring the new gauge coupling to be below non-perturbative unitary bound, $Z'$ mass is constrained to be less than 700 GeV. $b\to s \bar \mu \mu$ induced rare B decay anomalies are solved by a small mixing due to $Z'$ coupling for $b\to s$ transition.