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 Higgstrahlung channel $e^{+}{e}^{-}\to A^{\prime }{h}^{\prime }$, with $h^{\prime }\to A^{\prime }{A}^{\prime }$. We investigate ten exclusive final states with $A^{\prime }\to e^{+}{e}^{-}$, ${\mu }^{+}{\mu }^{-}$, or ${\pi }^{+}{\pi }^{-}$ in the mass ranges $0.1\mathrm{GeV}/c^2<m_{A^{\prime }}<3.5\mathrm{GeV}/c^2$ and $0.2\mathrm{GeV}/c^2<m_{h^{\prime }}<10.5\mathrm{GeV}/c^2$. 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 $1.1\mathrm{GeV}/c^2<m_{A^{\prime }}<3.5\mathrm{GeV}/c^2$ and $2.2\mathrm{GeV}/c^2<m_{h^{\prime }}<10.5\mathrm{GeV}/c^2$. Using the entire $977{\mathrm{fb}}^{-1}$ data set collected by Belle, we observe no significant signal. We obtain individual and combined 90% credibility level upper limits on the branching fraction times the Born cross section, $\mathcal{B}\times {\sigma }_{\text{Born}}$, on the Born cross section ${\sigma }_{\text{Born}}$, 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 $1/137$, $m_{h^{\prime }}<8\mathrm{GeV}/c^2$, and $m_{A^{\prime }}<1\mathrm{GeV}/c^2$, we exclude values of the mixing parameter $\epsilon$ above $\sim 8\times {10}^{-4}$.

Figure 1

Example $m_{A_{\text{cand}}^{\prime }}^1-m_{A_{\text{cand}}^{\prime }}^3$ distribution for the $A^{\prime }{h}^{\prime }\to A^{\prime }{A}^{\prime }{A}^{\prime }\to 6\pi$ channel, for $m_{A_{\text{cand}}^{\prime }}^1=2.0\pm 0.1\mathrm{GeV}/c^2$, where $m_{A_{\text{cand}}^{\prime }}^1$ and $m_{A_{\text{cand}}^{\prime }}^3$ are the dark photon candidates with the highest and lowest mass, respectively. The same-sign distributions (blue), where at least one $A^{\prime }$ candidate is reconstructed from ${\pi }^{+}{\pi }^{+}$ or ${\pi }^{-}{\pi }^{-}$, are normalized to the opposite-sign $3({\pi }^{+}{\pi }^{-})$ distributions (red) in the sidebands, and are used to predict the background in the signal region.

Figure 2

(a) Signal candidates observed versus dark photon candidate mass $m_{A_{\text{cand}}^{\prime }}$ and dark Higgs boson candidate mass $m_{A_{\text{cand}}^{\prime }{A}_{\text{cand}}^{\prime }}$ for the 13 final states. There are three entries per event. (b),(c) Projection of signal candidates onto $m_{A_{\text{cand}}^{\prime }{A}_{\text{cand}}^{\prime }}$ and $m_{A_{\text{cand}}^{\prime }}$ (red points) with the predicted background (blue squares) from the scaled same-sign distributions for comparison. The dark photon candidate mass distribution has been scaled by $1/3$. (d) Normalized residuals between the signal candidate distribution and the predicted background, versus dark photon candidate mass (red points) and dark Higgs boson candidate mass (black squares). The same-sign error bars contain statistical and systematic errors. For empty bins, the systematic error is one event.

Figure 3

Left: 90% C.L. upper limit on the product $\mathcal{B}\times {\sigma }_{\text{Born}}$ for each of the 13 final states considered versus dark photon mass for different hypotheses for the dark Higgs boson mass. Black, red, green, blue, and yellow curves correspond to $m_{h^{\prime }}=1,3,5,7$, and $9\mathrm{GeV}/c^2$, respectively, for exclusive channels and $m_{h^{\prime }}=3,4,5,7$, and $9\mathrm{GeV}/c^2$, respectively, for inclusive channels. Right: 90% C.L. upper limit on the cross section of $e^{+}{e}^{-}\to A^{\prime }{h}^{\prime }$, $h^{\prime }\to A^{\prime }{A}^{\prime }$ versus dark photon and dark Higgs boson mass.

Figure 4

90% C.L. upper limit on the product ${\alpha }_D\times {\epsilon }^2$ versus dark photon mass (top row) and dark Higgs boson mass (bottom row) for Belle (solid red curve) and BABAR [29] (dashed black curve). BABAR limits should be divided by ($1+\delta$) before being compared with Belle limits. The blue dotted curve, which coincides more or less with the solid red curve, shows the expected Belle limit.