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

Abstract: The ${\rm U(1)}{L\mu - L_\tau}$ model is anomaly-free with the Standard Model (SM) fermion content, and can make substantial contributions to the muon $(g-2)$ at the level of $\Delta a_\mu \sim O(10) \times 10^{-10}$ for $M_{Z'} \sim O(10-100)$ MeV and $g_X \sim (4 - 8) \times 10^{-4}$. In this light $Z'$ region, it was claimed that the model can also incorporate thermal WIMP dark matter (DM) if $M_{\rm DM} \sim M_{Z'}/2$. This setup relies on DM particles annihilating into SM particles through a $Z'$-mediated $s$-channel. In this work, we show that this tight relationship between $M_{Z'}$ and $M_{\rm DM}$ can be evaded or nullified both for scalar and spin-1/2 DM by considering the contributions from the dark Higgs boson ($H_1$). The dark Higgs boson plays an important role, not only because it gives mass to the dark photon but also because it introduces additional DM annihilation channels, including new final states such as $H_1 H_1$, $Z' Z'$, and $Z' H_1$. As a result, the model does not require a close mass correlation between the $Z'$ boson and dark matter $M_{\rm DM} \sim M_{Z'}/2$ any longer, allowing for a broader range of mass possibilities for both scalar and fermionic dark matter types. We explore in great details various scenarios where the $U(1)$ symmetry is either fully broken or partially remains as discrete symmetries, $Z_2$ or $Z_3$. This approach broadens the model's capacity to accommodate various WIMP dark matter phenomena in the light $Z'$ region where the muon $(g-2)_\mu$ makes a sensitive probe of the model.