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Abstract We study a model with $$U(1)_{L_\mu - L_\tau }$$ U(1)Lμ-Lτ gauge symmetry and discuss collider searches for a scalar boson, which breaks $$U(1)_{L_\mu - L_\tau }$$ U(1)Lμ-Lτ symmetry spontaneously, decaying into light $$Z'$$ Z′ gauge boson. In this model, the new gauge boson, $$Z'$$ Z′ , with a mass lighter than $${\mathcal {O}}(100)$$ 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, $$\phi $$ ϕ with $${\mathcal {O}}(100)$$ O(100) GeV mass appears associated with the symmetry breaking. We investigate experimental constraints on $$U(1)_{L_\mu - L_\tau }$$ U(1)Lμ-Lτ gauge coupling, kinetic mixing, and mixing between the SM Higgs and $$\phi $$ ϕ . Then collider search is discussed considering $$\phi $$ ϕ production followed by decay process $$\phi \rightarrow Z' Z'$$ ϕ→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.