  A search for a narrow Z$'$ gauge boson with a mass between 5 and 70 GeV
resulting from an $L_{\mu}$ $-$ $L_{\tau}$ $U(1)$ local gauge symmetry is
reported. Theories that predict such a particle have been proposed as an
explanation of various experimental discrepancies, 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
flavor universality violation in B meson decays. A data sample of proton-proton
collisions at a center-of-mass energy of 13 TeV is used, corresponding to an
integrated luminosity of 77.3 fb$^{-1}$ recorded in 2016 and 2017 by the CMS
detector at the LHC. Events containing four muons with an invariant mass near
the standard model Z boson mass are analyzed, and the selection is further
optimized to be sensitive to the events that may contain Z $\to$ Z$'\mu\mu$
$\to$ 4$\mu$ decays. The event yields are consistent with the standard model
predictions. Upper limits of 10$^{-8}$-10$^{-7}$ at 95% confidence level are
set on the product of branching fractions $\mathcal{B}$(Z $\to$ Z$'\mu\mu$)
$\mathcal{B}$(Z$'$ $\to$ $\mu\mu$), depending on the Z$'$ mass, which excludes
a Z$'$ boson coupling strength to muons above 0.004-0.3. These are the first
dedicated limits on $L_{\mu}$ $-$ $L_{\tau}$ models at the LHC and result in a
significant increase in the excluded model parameter space. The results of this
search may also be used to constrain the coupling strength of any light Z$'$
gauge boson to muons.
