A light $Z^{\prime }$ vector boson coupled to the second and third lepton generations through the $L_{\mu }-L_{\tau }$ current with mass below 200 MeV provides a very viable explanation in terms of new physics to the recently confirmed $(g-2{)}_{\mu }$ anomaly. This boson can be produced in the bremsstrahlung reaction $\mu N\to \mu N{Z}^{\prime }$ after a high energy muon beam collides with a target. $\mathrm{NA}64\mu$ is a fixed-target experiment using a 160 GeV muon beam from the CERN Super Proton Synchrotron accelerator looking for $Z^{\prime }$ production and its subsequent decays, $Z^{\prime }\to \mathrm{invisible}$. In this paper, we present the study of the $\mathrm{NA}64\mu$ sensitivity to search for such a boson. This includes a realistic beam simulation, a detailed description of the detectors and a discussion about the main potential background sources. A pilot run is scheduled in order to validate the simulation results. If those are confirmed, $\mathrm{NA}64\mu$ will be able to explore all the remaining parameter space which could provide an explanation for the $g-2$ muon anomaly in the $L_{\mu }-L_{\tau }$ model.

• Received 4 November 2021
• Accepted 31 January 2022

DOI:https://doi.org/10.1103/PhysRevD.105.052006

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Published by the American Physical Society