Dark matter contribution to $bto s mu^+ mu^-$ anomaly in local $U(1)_{L_mu-L_tau}$ model

We propose a local $U(1)_{L_mu-L_tau}$ model to explain $b to s mu^+ mu^-$ anomaly observed at the LHCb and Belle experiments. The model also has a natural dark matter candidate $N$. We introduce $SU(2)_L$-doublet colored scalar $widetilde{q}$ to mediate $b to s$ transition at one-loop level. The $U(1)_{L_mu-L_tau}$ gauge symmetry is broken spontaneously by the scalar $S$. All the new particles are charged under $U(1)_{L_mu-L_tau}$. We can obtain $C_9^{mu,{rm NP}} sim -1$ to solve the $b to smu^+mu^-$ anomaly and can explain the correct dark matter relic density of the universe, $Omega_{rm DM} h^2 approx 0.12$, simultaneously, while evading constraints from electroweak precision tests, neutrino trident experiments and other quark flavor-changing loop processes such as $b to s gamma$ and $B_s-overline{B}_s$ mixing. Our model can be tested by searching for $Z$ and new colored scalar at the LHC and $B to K^* u overline{ u}$ process at Belle-II.