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Coherent elastic neutrino-nucleon scattering, challenged by the low nucleusrecoil energy of a few tens of keV, has been observed by the COHERENTexperiment using targets CsI and Ar. We study the contribution of a light $Z'$mediator in a gauged $U(1)_{L_\mu-L_\tau}$ symmetry. In contrast to themechanism from the kinetic mixing between $U(1)_{\rm em}$ and$U(1)_{L_\mu-L_\tau}$, we adopt a dynamical symmetry breaking of the$U(1)_{L_\mu-L_\tau}$ by employing an extra Higgs doublet. As a result, theweak charge mediated by $Z'$ only depends on the mass of light gauge boson.Since two Goldstone bosons are required to serve as the longitudinal componentsof $Z$ and $Z'$, the model does not contain a physical CP-odd scalar. Using theintroduced Higgs doublet carrying the $U(1)_{\mu-\tau}$ charge, new Higgs decaychannels $h\to Z_1 Z_1/Z_1 Z_2$ with percent-level branching fractions becomeaccessible. The $W$-mass anomaly observed by CDF II can be resolved byenhancing the oblique parameter $T$. With the flavored gauge symmetry, theYukawa couplings to fermion flavors are strictly limited. By utilizing thecharacteristic and introducing a scalar leptoquark $S^{\frac{1}{3}}=(3,1,2/3)$that uniquely couples to the $\tau$ lepton, the excesses of $R(D)$ and $R(D^*)$can be explained. Moreover, $\tau \to \mu (Z_1\to ) e^- e^+$ via the resonantlight gauge boson decay can reach the sensitivity of Belle II at an integratedluminosity of 50 ab$^{-1}$.

Compatibility of CE$\nu$NS with muon $g-2$, $W$ mass, and $R(D^{(*)})$ in a gauged ${L_\mu -L_\tau}$ with a scalar LQ