### Speaker

### Description

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