A search for a new $Z'$ gauge boson predicted by $L_{\mu}-L_{\tau}$ models, based on charged-current Drell-Yan production, $pp \rightarrow W^{\pm(*)} \rightarrow Z' \mu^{\pm} \nu \rightarrow \mu^{\pm}\mu^{\mp}\mu^{\pm}\nu$, is presented. The data sample used corresponds to an integrated luminosity of 140 fb$^{-1}$ of proton-proton collisions at $\sqrt{s} = 13$ TeV recorded by the ATLAS detector at the Large Hadron Collider. The search examines a final state of $3\mu$ plus large missing transverse momentum. Upper limits are set on the $Z'$ production cross-section times branching ratio in the mass range of 5-81 GeV. After combining with the previous $Z'$ search using the neutral-current Drell-Yan production with a $4\mu$ final state, the most stringent exclusion limits to date are achieved in the parameter space of the $Z'$ coupling strength and mass.
Observed and expected upper limits at 95% CL on the production cross-section times branching fraction of the process $pp\to W\to Z^{\prime}$ $\mu \nu \to \mu \mu \mu \nu$ as a function of $m_{Z^{\prime}}$.
Observed and expected upper limits at 95% CL on the coupling parameter $g_{Z^{\prime}}$ as a function of $m_{Z^{\prime}}$ from the statistical combination of the $3\mu$ and $4\mu$ channels.
Exclusion contour compared to the limits from the Neutrino Trident and the $B_{S}$ mixing experimental results.
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