Implications of a Light "Dark Higgs" Solution to the $g_\mu-2$ Discrepancy

A light scalar $\phi$ with mass $\lesssim 1$ GeV and muonic coupling $\mathcal{O}(10^{-3})$ would explain the 3.5 $\sigma$ discrepancy between the Standard Model (SM) muon $g-2$ prediction and experiment. Such a scalar can be associated with a light remnant of the Higgs mechanism in the "dark" sector. We suggest $\phi\to l^+l^-$ bump hunting in $\mu\to e\nu\bar\nu\phi$, $\mu^-p\to\nu_\mu n\phi$ (muon capture), and $K^\pm\to \mu^\pm\nu\phi$ decays as direct probes of this scenario. In a general setup, a potentially observable muon electric dipole moment $\lesssim 10^{-23}\ e \cdot\textrm{cm}$ and lepton flavor violating decays $\tau\to\mu (e) \phi$ or $\mu \to e \phi$ can also arise. Depending on parameters, a deviation in BR($H\to\mu^+\mu^-$) from SM expectations, due to Higgs coupling misalignment, can result. We illustrate how the requisite interactions can be mediated by weak scale vector-like leptons that typically lie within the reach of future LHC measurements.