  Heavy neutral gauge boson $Z^\prime$ is proposed in many new physics models.
It has rich phenomena at the future muon collider. We study the properties of
$Z^\prime$ boson with the process of $\mu^+ \mu^- \rightarrow q \bar{q}$,
$\mu^+ \mu^- \rightarrow l^+ l^-$, $\mu^+ \mu^- \rightarrow Z H$ and $\mu^+
\mu^- \rightarrow W^+ W^-$. The discrepancy of $Z^\prime$ coupling to different
types of particles can be shown in the cross section distributions around the
resonance peak of various decay modes. Angular distributions of the final quark
or lepton in $\mu^+ \mu^- \rightarrow q \bar{q}/l^+ l^- $ process are sensitive
to the parameters such as mass of $Z^\prime$ and the $Z-Z^\prime$ mixing angle.
The interaction of new gauge boson coupling to the standard model gauge
particles and Higgs boson are also studied through $\mu^+ \mu^- \rightarrow Z H
\rightarrow l^+l^- b \bar{b}$ and $\mu^+ \mu^- \rightarrow W^+W^- \rightarrow
l^+l^- \nu_l \bar{\nu}_l$. The cross section and the final particles' angular
distributions with the contribution of $Z^\prime$ boson differ from those
processes with only standard model particles. A forward-backward asymmetry
defined by the angular distribution is provided to show the potential of
searching for new physics at the muon collider. Especially, the beam
polarization with certain value can effectively enlarge the forward-backward
asymmetry.
