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A light $Z'$ vector boson coupled to the second and third lepton generationsthrough $L_{\mu}-L_{\tau}$ current with mass below 200 MeV provides a veryviable explanation in terms of new physics to the recently confirmed$(g-2)_\mu$ anomaly. This boson can be produced in the bremsstrahlung reaction$\mu N \rightarrow \mu N Z'$ after a high energy muon beam collides with atarget. NA64$\mu$ is a fixed-target experiment using a 160 GeV muon beam fromthe CERN Super Proton Synchrotron accelerator looking for the $Z'$ productionand its subsequent decays, $Z'\rightarrow invisible$. In this paper, we presentthe study of the NA64$\mu$ sensitivity to search for such a $Z'$. This includesa realistic beam simulation, the detailed detectors description and adiscussion about the main potential background sources. A pilot run isscheduled in order to validate the simulation results. If those are confirmed,NA64$\mu$ will be able to explore all the remaining phase space which couldprovide an explanation for the $g-2$ muon anomaly.

Prospects in the search for a new light $Z'$ boson with the NA64$\mu$ experiment at the CERN SPS