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We study the implications of the recent anomalies in the $W$-boson mass andthe anomalous magnetic moment of the muon on $U(1)_{L_\mu-L_\tau}$ model. Weshow that the introduction of vector-like leptons which mix with muon can solveboth anomalies. Contrary to the conventional wisdom the electroweak scale$Z'$-boson is allowed without conflict with the trident neutrino productionexperiments.

Implications of CDF $W$-mass and $(g-2)_\mu$ on $U(1)_{L_\mu-L_\tau}$ model