We study a model with $U(1)_{L_mu - L_tau}$ gauge symmetry and discuss collider searches for a scalar boson, which breaks $U(1)_{L_mu - L_tau}$ symmetry spontaneously, decaying into light $Z$ gauge boson. In this model, the new gauge boson, $Z$, with a mass lighter than $mathcal{O}(100)$ MeV, plays a role in explaining the anomalous magnetic moment of muon via one-loop contribution. For the gauge boson to have such a low mass, the scalar boson, $phi$ with $mathcal{O}(100)$ GeV mass appears associated with the symmetry breaking. We investigate experimental constraints on $U(1)_{L_mu - L_tau}$ gauge coupling, kinetic mixing, and mixing between the SM Higgs and $phi$. Then collider search is discussed considering $phi$ production followed by decay process $phi to Z Z$ at the large hadron collider and the international linear collider. We also estimate discovery significance at the linear collider taking into account relevant kinematical cut effects.
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