Restrained Dark $U(1)_d$ at Low Energies

We investigate a spontaneously broken $U(1)_d$ gauge symmetry with a muon-specific dark Higgs. Our first goal is to verify how the presence of a new dark Higgs, $\phi$, and a dark gauge boson, $V$, can simultaneously face the anomalies from the muon magnetic moment and the proton charge radius. Secondly, by assuming that $V$ must decay to an electron-positron pair, we explore the corresponding parameter space determined with the low energy constraints coming from $K \to \mu X$, electron $(g-2)_e$, $K \to \mu \nu_\mu e^+ e^-$, $K \to \mu \nu_\mu \mu^+ \mu^-$, $\tau \to \nu_\tau \mu \nu_\mu e^+ e^-$. We focus in the scenario where the $V$ mass is below $\sim 2 m_\mu$ and the $\phi$ mass runs from few MeV till $250$ MeV, with V-photon mixing of the order $\sim\mathcal{O}(10^{-3})$. Among weak process at low energies, we check the influence of the new light vector on kaon decays as well as on the scattering $e^+ e^- \rightarrow \mu^+ \mu^- e^+ e^-$ and discuss the impact of the dark Higgs on $e^+ e^- \rightarrow \mu^+ \mu^- \mu^+ \mu^-$. Finally, we consider contributions of the V-photon mixing in the decays $\pi^0 \to \gamma e^+ e^-$, $\eta \to \gamma e^+ e^-$, $\rho \to \pi e^+ e^-$, $K^* \to K e^+ e^-$ and $\phi (1020) \to \eta e^+ e^-$.