The $L_{\mu}-L_{\tau}$ extension of the standard model predicts the existence of a lepton-flavor-universality-violating $Z^{\prime}$ boson that couples only to the heavier lepton families. We search for such a $Z^\prime$ through its invisible decay in the process $e^+ e^- \to \mu^+ \mu^- Z^{\prime}$. We use a sample of electron-positron collisions at a center-of-mass energy of 10.58GeV collected by the Belle II experiment in 2019-2020, corresponding to an integrated luminosity of 79.7fb$^{-1}$. We find no excess over the expected standard-model background. We set 90$\%$-confidence-level upper limits on the cross section for this process as well as on the coupling of the model, which ranges from $3 \times 10^{-3}$ at low $Z^{\prime}$ masses to 1 at $Z^{\prime}$ masses of 8$GeV/c^{2}$.
Observed 90% CL upper limits on the cross section $\sigma (e^+ e^- \to \mu^+ \mu^- Z', Z' \to $ invisible) as functions of the $Z'$ mass for the cases of negligible $\Gamma_{Z'}$ and for $\Gamma_{Z'} = 0.1 M_{Z'}$. Also shown are previous limits from Belle II.
Observed 90% CL upper limits on the coupling $g'$ for the fully invisible $L_\mu − L_\tau$ model as functions of the $Z'$ mass for the cases of negligible $\Gamma_{Z'}$ and for $\Gamma_{Z'} = 0.1 M_{Z'}$. Also shown are previous limits from NA64-e and Belle II searches. The red band shows the region that explains the muon anomalous magnetic moment $(g - 2)_\mu \pm 2 \sigma$. The vertical dashed line indicates the limit beyond which the hypothesis $B(Z' \to \chi\bar{\chi}) \approx 1$ is not respected in the negligible $\Gamma_{Z'}$ case.
Observed 90% CL upper limits on the coupling $g'$ for the vanilla $L_\mu − L_\tau$ model as functions of the $Z'$ mass. Also shown are previous limits from Belle II and NA64-e searches for invisible $Z'$ decays, and from Belle, BaBar and CMS searches for $Z'$ decays to muons (at 95% CL). The red band shows the region that explains the muon anomalous magnetic moment $(g - 2)_\mu \pm 2 \sigma$.
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