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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, $ϕ$, 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 μX$, electron $(g-2)_e$, $K \to μν_μe^+ e^-$, $K \to μν_μμ^+ μ^-$, $τ\to ν_τμν_μe^+ e^-$. We focus in the scenario where the $V$ mass is below $ \sim 2 m_μ$ and the $ϕ$ 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 $e^+ e^- \rightarrow μ^+ μ^- e^+ e^-$ and discuss the impact of the dark Higgs on $e^+ e^- \rightarrow μ^+ μ^- μ^+ μ^- $. Finally, we consider contributions of the V-photon mixing in the decays $π^0 \to γe^+ e^-$, $η\to γe^+ e^-$, $ρ\to πe^+ e^-$, $K^* \to K e^+ e^-$ and $ϕ(1020) \to ηe^+ e^-$.