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${\rm U(1)}_{L_\mu - L_\tau} \equiv {\rm U(1)}_X$ model is anomaly freewithin the Standard Model (SM) fermion content, and can accommodate the muon$(g-2)$ data for $M_{Z'} \sim O(10-100)$ MeV and $g_X \sim (4 - 8) \times10^{-4}$. WIMP type thermal dark matter (DM) can be also introduced for $M_{Z'}\sim 2 M_{\rm DM}$, if DM pair annihilations into the SM particles occur onlythrough the $s$-channel $Z'$ exchange. In this work, we show that this tightcorrelation between $M_{Z'}$ and $M_{\rm DM}$ can be completely evaded both forscalar and fermionic DM, if we include the contributions from dark Higgs boson($H_1$). Dark Higgs boson plays a crucial role in DM phenomenology, not onlyfor generation of dark photon mass, but also opening new channels for DM pairannihilations into the final states involving dark Higgs boson, such as darkHiggs pair as well as $Z' Z'$ through dark Higgs exchange in the $s$-channel,and co-annihilation into $Z' H_1$ in case of inelastic DM. Thus dark Higgsboson will dissect the strong correlation $M_{Z'} \sim 2 M_{\rm DM}$, and muchwider mass range is allowed for $U(1)_X$-charged complex scalar and Diracfermion DM, still explaining the muon $(g-2)$. We consider both generic$U(1)_X$ breaking as well as ${\rm U(1)}_X \rightarrow Z_2$ (and also into$Z_3$ only for scalar DM case).

Muon $(g-2)$ and Thermal WIMP DM in ${\rm U(1)}_{L_\mu-L_\tau} $ Models