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Light Mediators in Anomaly Free $U(1)_X$ Models II - Constraints on Dark Gauge Bosons

We consider experimental constraints in the MeV region in order to determine the parameter space for the $U(1)_X$ extension of the Standard Model, presented in the first part of our work. In particular, we focus on the model UV-completed by cold WIMPs. We conclude that the electron anomalous magnetic moment, the neutrino trident production and the relic abundance $Ω_{CDM}$ provide the most stringent bounds and, in particular cases, they are sufficient to exclude dark-photon ($A'$) models. By allowing the axial-vector coupling of the dark gauge boson $Z'$, the interference effect with the SM gauge bosons may reduce the bounds coming from the trident neutrino production. At the same time, such coupling allows a region of the parameter space already favored both by the relic abundance and by the discrepancy between experimental result and theoretical prediction for the muon anomalous magnetic moment. We emphasize that light-$Z'$ interactions, non-universal for the two first lepton families, necessarily create a difference in the proton charge radius measured in the Lamb shift of the $e$-hydrogen and $μ$-hydrogen. Finally, we determine the effects of the new gauge boson on the forward-backward asymmetry in $e^+ e^- \rightarrow \bar{f} f$, $f = μ, τ$, and on the leptonic decays $M \rightarrow j ν_j l^+ l^-$, where $M = π, K, D, D_s, B$ and $j,l = e, μ$.