The extension of Standard Model made by inclusion of additional $U(1)$ gauge $L_{\mu }-L_{\tau }$ symmetry can explain the difference between the measured and the predicted value of the muon magnetic moment and solve the tension in $B$ meson decays. This model predicts the existence of a new, light $Z^{\prime }$ vector boson, predominantly coupled to second and third generation leptons, whose interaction with electrons is due to a loop mechanism involving muons and taus. In this work, we present a rigorous evaluation of the upper limits in the $Z^{\prime }$ parameter space, obtained from the analysis of the data collected by the NA64-$e$ experiment at CERN SPS, that performed a search for light dark matter with $2.84\times {10}^{11}$ electrons impinging with 100 GeV on an active thick target. The resulting limits touch the muon $g-2$ preferred band for values of the $Z^{\prime }$ mass of order of 1 MeV, while the sensitivity projections for the future high-statistics NA64-$e$ runs demonstrate the power of the electrons/positron beam approach in this theoretical scenario.

• Received 10 June 2022
• Accepted 2 August 2022
• Corrected 19 January 2023

DOI:https://doi.org/10.1103/PhysRevD.106.032015

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Published by the American Physical Society