Search for the dark photon and the dark Higgs boson at Belle

The dark photon, $A^\prime$, and the dark Higgs boson, $h^\prime$, are hypothetical constituents featured in a number of recently proposed Dark Sector Models. Assuming prompt decays of both dark particles, we search for their production in the so-called Higgs-strahlung channel, $e^+e^- \rightarrow A^\prime h'$, with $h^\prime \rightarrow A^\prime A^\prime$. We investigate ten exclusive final-states with $A^\prime \rightarrow e^+e^-$, $\mu^+\mu^-$, or $\pi^+\pi^-$, in the mass ranges $0.1$~GeV/$c^2$~$< m_{A^\prime} < 3.5$~GeV/$c^2$ and $0.2$~GeV/$c^2$~$< m_{h'} < 10.5$~GeV/$c^2$. We also investigate three inclusive final-states, $2(e^+e^-)X$, $2(\mu^+\mu^-)X$, and $(e^+e^-)(\mu^+\mu^-)X$, where $X$ denotes a dark photon candidate detected via missing mass, in the mass ranges $1.1$~GeV/$c^2$~$< m_{A^\prime} < 3.5$~GeV/$c^2$ and $2.2$~GeV/$c^2$~$< m_{h'} < 10.5$~GeV/$c^2$. Using the entire $977\,\mathrm{fb}^{-1}$ data set collected by Belle, we observe no significant signal. We obtain individual and combined 90$\%$ confidence level upper limits on the branching fraction times the Born cross section, $\cal B \times \sigma_{\mathrm{Born}}$, on the Born cross section, $\sigma_{\mathrm{Born}}$, and on the dark photon coupling to the dark Higgs boson times the kinetic mixing between the Standard Model photon and the dark photon, $\alpha_D \times \epsilon^2$. These limits improve upon and cover wider mass ranges than previous experiments. The limits from the final-states $3(\pi^+\pi^-)$ and $2(e^+e^-)X$ are the first placed by any experiment. For $\alpha_D$ equal to 1/137, $m_{h'}<$ 8 GeV/$c^2$, and $m_{A^\prime}<$ 1 GeV/$c^2$, we exclude values of the mixing parameter, $\epsilon$, above $\sim 8 \times 10^{-4}$