Secluded U(1) below the weak scale

A secluded U(1) sector with weak admixture to photons, $O({10}^{-2}-{10}^{-3})$, and the scale of the breaking below 1 GeV represents a natural yet poorly constrained extension of the standard model. We analyze $g-2$ of muons and electrons together with other precision QED data, as well as radiative decays of strange particles to constrain the mass-mixing angle ($m_V-\kappa$) parameter space. We point out that $m_V\simeq 214\mathrm{MeV}$ and ${\kappa }^2>3\times {10}^{-5}$ can be consistent with the hypothesis of the HyperCP Collaboration, which seeks to explain the anomalous energy distribution of muon pairs in the ${\Sigma }^{+}\to p{\mu }^{+}{\mu }^{-}$ process by a resonance, without direct contradiction to the existing data on radiative kaon decays. The same parameters lead to an $O(\mathrm{\text{few}}\times {10}^{-9})$ upward correction to the anomalous magnetic moment of the muon, possibly relaxing some tension between the experimental value and theoretical determinations of $g-2$. The ultrafine energy resolution scan of the $e^{+}{e}^{-}\to {\mu }^{+}{\mu }^{-}$ cross section and dedicated analysis of lepton spectra from $K^{+}\to {\pi }^{+}{e}^{+}{e}^{-}$ decays should be able to provide a conclusive test of this hypothesis and improve the constraints on the model.