Explaining lepton-flavor non-universality and self-interacting dark matter with $L_\mu-L_\tau$

Experimental hints for lepton-flavor universality violation in the muon's magnetic moment as well as neutral- and charged-current $B$-meson decays require Standard-Model extensions by particles such as leptoquarks that generically lead to unacceptably fast rates of charged lepton flavor violation and proton decay. We propose a model based on a gauged $U(1)_{L_\mu-L_\tau}$ that eliminates all these unwanted decays by symmetry rather than finetuning and efficiently explains $(g-2)_\mu$, $R_{K^{(*)}}$, $R_{D^{(*)}}$, and neutrino masses. The $U(1)_{L_\mu-L_\tau}$ furthermore acts as a stabilizing symmetry for dark matter and the light $Z'$ gauge boson mediates velocity-dependent dark-matter self-interactions that resolve the small-scale structure problems. Lastly, even the Hubble tension can be ameliorated via the light $Z'$ contribution to the relativistic degrees of freedom.