Dark matter, $Z^{\prime}$, vector-like quark at the LHC and $b\to s \mu\mu$ anomaly

Combining the $b\to s\mu^+\mu^-$ anomaly and dark matter observables, we study the capability of LHC to test dark matter, $Z^{\prime}$, and vector-like quark. We focus on a local $U(1)_{L_\mu-L_\tau}$ model with a vector-like $SU(2)_L$ doublet quark $Q$ and a complex singlet scalar whose lightest component $X_I$ is a candidate of dark matter. After imposing relevant constraints, we find that the $b\to s\mu^+\mu^-$ anomaly and the relic abundance of dark matter favor $m_{X_I}< 350$ GeV and $m_{Z^{\prime}}< 450$ GeV for $m_Q<$ 2 TeV and $m_{X_R}<$ 2 TeV (the heavy partner of $m_{X_I}$). The current searches for jets and missing transverse momentum at the LHC sizably reduce the mass ranges of the vector-like quark, and $m_Q$ is required to be larger than 1.7 TeV. Finally, we discuss the possibility of probing these new particles at the high luminosity LHC via the QCD process $pp \to D\bar{D}$ or $pp \to U\bar{U}$ followed by the decay $D\to s (b) Z'X_I$ or $U \to u (c) Z' X_I$ and then $Z'\to\mu^+\mu^-$. Taking a benchmark point of $m_Q$=1.93 TeV, $m_{Z^\prime}=170$ GeV, and $m_{X_I}=$ 145 GeV, we perform a detailed Monte Carlo simulation, and find that such benchmark point can be accessible at the 14 TeV LHC with an integrated luminosity 3000 fb$^{-1}$.