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

A light <math display="inline"><msup><mi>Z</mi><mo>′</mo></msup></math> vector boson coupled to the second and third lepton generations through the <math display="inline"><msub><mi>L</mi><mi>μ</mi></msub><mo>-</mo><msub><mi>L</mi><mi>τ</mi></msub></math> current with mass below 200 MeV provides a very viable explanation in terms of new physics to the recently confirmed <math display="inline"><mo stretchy="false">(</mo><mi>g</mi><mo>-</mo><mn>2</mn><msub><mo stretchy="false">)</mo><mi>μ</mi></msub></math> anomaly. This boson can be produced in the bremsstrahlung reaction <math display="inline"><mrow><mi>μ</mi><mi>N</mi><mo stretchy="false">→</mo><mi>μ</mi><mi>N</mi><msup><mrow><mi>Z</mi></mrow><mrow><mo>′</mo></mrow></msup></mrow></math> after a high energy muon beam collides with a target. <math display="inline"><mrow><mi>NA</mi><mrow><mn>64</mn><mi>μ</mi></mrow></mrow></math> is a fixed-target experiment using a 160 GeV muon beam from the CERN Super Proton Synchrotron accelerator looking for <math display="inline"><msup><mi>Z</mi><mo>′</mo></msup></math> production and its subsequent decays, <math display="inline"><mrow><msup><mrow><mi>Z</mi></mrow><mrow><mo>′</mo></mrow></msup><mo stretchy="false">→</mo><mi>invisible</mi></mrow></math>. In this paper, we present the study of the <math display="inline"><mrow><mi>NA</mi><mrow><mn>64</mn><mi>μ</mi></mrow></mrow></math> 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, <math display="inline"><mrow><mi>NA</mi><mrow><mn>64</mn><mi>μ</mi></mrow></mrow></math> will be able to explore all the remaining parameter space which could provide an explanation for the <math display="inline"><mi>g</mi><mo>-</mo><mn>2</mn></math> muon anomaly in the <math display="inline"><msub><mi>L</mi><mi>μ</mi></msub><mo>-</mo><msub><mi>L</mi><mi>τ</mi></msub></math> model.