Search for an $L_{\mu}-L_{\tau}$ gauge boson using Z$\rightarrow4\mu$ events in pp collisions at $\sqrt{s}=13~\mathrm{TeV}$

A search for a narrow Z$^{\prime}$ gauge boson with a mass between 5 and 70 GeV resulting from an $L_{\mu}-L_{\tau}$ $U(1)^{\prime}$ symmetry is reported. Theories that predict such a particle have been proposed as an explanation of various experimental anomalies including the lack of a dark matter signal in direct detection experiments, tension in the measurement of the anomalous magnetic moment of the muon, and reports of possible lepton flavour universality violation in B-meson decays. The data sample consists of proton-proton collisions at $\sqrt{s}=13~\mathrm{TeV}$ collected with the CMS detector at the LHC and corresponds to an integrated luminosity of $77.3~\mathrm{fb}^{-1}$. Events containing four muons with an invariant mass near the Z boson mass are analyzed, and the selection is further optimized to be sensitive to the Z$^{\prime}$ signal that may be present in Z$\rightarrow4\mu$ decays. The observations are consistent with the standard model predictions. Upper limits of $10^{-8}$--$10^{-7}$ at $95\%$ confidence level are set on the branching fraction $\mathcal{B}(\rm{Z} \rightarrow$Z$^{\prime} \mu \mu) \times \mathcal{B}($Z$^{\prime} \rightarrow\mu \mu)$, which excludes a Z$^{\prime}$ boson coupling strength to muons above 0.004--0.3, depending on the Z$^{\prime}$ mass. These are the first dedicated limits on the $L_{\mu}-L_{\tau}$ model at the LHC.