$U(1)_{L_mu-L_tau}$ for Light Dark Matter, $g_mu-2$, the $511$ keV excess and the Hubble Tension

In this paper we introduce a light Dirac particle $psi$ as thermal dark matter candidate in a $U(1)_{L_{mu}-L_{tau}}$ model. Together with the new gauge boson $X$, we find a possible parameter space with $m_X simeq 20$ MeV, $U(1)_{L_{mu}-L_{tau}}$ coupling $g_X simeq 5 cdot 10^{-4}$ and $m_psi gtrsim m_X/2$ where the $(g-2)_mu$ anomaly, dark matter, the Hubble tension, and (part of) the excess of $511$ keV photons from the region near the galactic center can be explained simultaneously. This model is safe from current experimental and astrophysical constraints, but can be probed by the next generation of neutrino experiments as well as low-energy $e^+e^-$ colliders.