Constraints on $Z^\prime$ Boson within Minimal Flavor Violation

We explore a $Z^\prime$ boson coupled only with the Standard Model (SM) fermions ($Z^\prime f\bar{f}$) in the framework of Minimal Flavor Violation. We study its effects on the processes with lepton flavor violation $\ell_j\to \ell_i\ell_k\bar{\ell}_l$, $\ell_j\to\ell_i\gamma$, $\mu^{-}N\to e^- N$, quark flavor changing neutral currents $b\to s \ell^+\ell^-$, neutral $B$ and $K$ meson mixing, and $e^+e^-\to f\bar{f}$ at the LEP experiment to constrain the parameter space of $Z^\prime$ mass and couplings. We find that among those relevant processes, $\mu\to e\gamma$, $\mu\to 3e$, $\mu$-$e$ conversion and $e^+e^-\to f\bar{f}$ can put more stringent bounds on $Z^\prime f\bar{f}$ couplings normalized by $Z^\prime$ mass. Its implications on various processes are obtained, such as $B$ and $K$ mixing and $B/K\to M \ell_1 \bar\ell_2$ decays. In addition, we also make analysis on $Z^\prime$ signatures at the LHC with $\sqrt{s}=8$ and 13 TeV.