The Simplest and Most Predictive Model of Muon $g-2$ and Thermal Dark Matter

The long-standing $4.2 \, \sigma$ muon $g-2$ anomaly may be the result of a new particle species which could also couple to dark matter and mediate its annihilations in the early universe. In models where both muons and dark matter carry equal charges under a $U(1)_{L_\mu-L_\tau}$ gauge symmetry, the corresponding $Z^\prime$ can both resolve the observed $g-2$ anomaly and yield an acceptable dark matter relic abundance, relying on annihilations which take place through the $Z^\prime$ resonance. Once the value of $(g-2)_{\mu}$ and the dark matter abundance are each fixed, there is very little remaining freedom in this model, making it highly predictive. We provide a comprehensive analysis of this scenario, identifying a viable range of dark matter masses between approximately 10 and 100 MeV, which falls entirely within the projected sensitivity of several accelerator-based experiments, including NA62, NA64$\mu$, $M^3$, and DUNE. Furthermore, portions of this mass range predict contributions to $\Delta N_{\rm eff}$ which could ameliorate the tension between early and late time measurements of the Hubble constant, and which could be tested by Stage 4 CMB experiments.