Observation of a $\overline{K}NN$ bound state in the ${}^3\mathrm{He}(K^{-},\mathrm{\Lambda }p)n$ reaction

We have performed an exclusive measurement of the $K^{-}+{}^3\mathrm{He}\to \mathrm{\Lambda }pn$ reaction at an incident kaon momentum of $1\mathrm{GeV}/c$. In the $\mathrm{\Lambda }p$ invariant mass spectrum, a clear peak was observed below the mass threshold of $\overline{K}+N+N$, as a signal of the kaonic nuclear bound state, $\overline{K}NN$. The binding energy, decay width, and $S$-wave Gaussian reaction form factor of this state were observed to be $B_K=42\pm 3{\left(\mathrm{stat}.\right)}_{-4}^{+3}\left(\mathrm{syst}.\right)\mathrm{MeV}, {\mathrm{\Gamma }}_K=100\pm 7{\left(\mathrm{stat}.\right)}_{-9}^{+19}\left(\mathrm{syst}.\right)\mathrm{MeV}$, and $Q_K=383\pm 11{\left(\mathrm{stat}.\right)}_{-1}^{+4}\left(\mathrm{syst}.\right)\mathrm{MeV}/c$, respectively. The total production cross section of $\overline{K}NN$, determined by its $\mathrm{\Lambda }p$ decay mode, was ${\sigma }_K^{\mathrm{tot}}B{R}_{\mathrm{\Lambda }p}=9.3\pm 0.8{\left(\mathrm{stat}.\right)}_{-1.0}^{+1.4}\left(\mathrm{syst}.\right)\mu \mathrm{b}$. We estimated the branching ratio of the $\overline{K}NN$ state to the $\mathrm{\Lambda }p$ and ${\mathrm{\Sigma }}^{0}p$ decay modes as $B{R}_{\mathrm{\Lambda }p}/B{R}_{{\mathrm{\Sigma }}^{0}p}\sim 1.7$, by assuming that the physical processes leading to the $\mathrm{\Sigma }NN$ final states are analogous to those of $\mathrm{\Lambda }pn$.

Figure 1

(a) 2D plot of $m_{R^0}$ and $l\left(\mathit{x}\right)$, where $m_{R^0}$ is the missing mass of the ${}^3\mathrm{He}(K^{-},pp{\pi }^{-})R^0$ reaction and $l\left(\mathit{x}\right)$ is the event consistency with $\mathrm{\Lambda }pn$. (b) Projected spectrum of the $m_{R^0}$ axis by selecting $l\left(\mathit{x}\right)<30$. The vertical black dashed lines are the masses of $n$ ($m_n$), $N+{\pi }^0$ ($m_N+m_{\pi }$), and $\mathrm{\Lambda }$ ($m_{\mathrm{\Lambda }}$). The $\mathrm{\Lambda }pn$ event was selected below the red line in the 2D plot. The projection of selected events is shown by the red histogram in (b).

Figure 2

(a) 2D plot of $m_{p{\pi }^{-}}$ and $l\left(\mathit{x}\right)$. (b) Projected spectrum on the $m_{p{\pi }^{-}}$ axis. Events in the $\mathrm{\Lambda }pn$-selection window [shown in Fig. 1] are plotted. The vertical black dashed line is the $\mathrm{\Lambda }$ mass.

Figure 3

Distributions of (a) $m_{R^0}$ of ${}^3\mathrm{He}(K^{-},{\pi }^{-}pp)R^0$ [the same as Fig. 1], (b) $m_{R^{-}}$ of ${}^3\mathrm{He}(K^{-},pp)R^{-}$, and (c) $l\left(\mathit{x}\right)$. For the $m_{R^0}$ and $m_{R^{-}}$ spectra, $l\left(\mathit{x}\right)<30$ was selected. These three distributions were simultaneously fitted by simulated spectra shown by colored lines. The fitting $\chi$-square and number of degrees of freedom were 917 and 506, respectively. For mesonic ($YNN+\pi$), the final states all of possible charged states and combinations were summed.

Figure 4

2D plot on the $m_X$ and $q_X$ plane after acceptance correction. The black dotted line shows the kinematical limit of the reaction. The vertical gray dotted line and blue dotted curve are $M_{\overline{K}NN}$ and $M_F\left(q\right)$, respectively. The gray hatched regions indicate where the experimental efficiency is $<0.5\%$.

Figure 5

(a) Simulated spectra of experimental efficiency $\varepsilon (m_X,q_X)$ for $\mathrm{\Lambda }pn$ final states. $\varepsilon (m_X,q_X)$ includes geometrical acceptance of CDS and analysis efficiency (decay branching ratio of $\mathrm{\Lambda }$ is also taken into account). The efficiency is calculated bin by bin. The hatched regions are insensitive in the present setup, where $\varepsilon <0.5\%$. (b) Lorentz-invariant $\mathrm{\Lambda }pn$ phase space $\rho (m_X,q_X)$ taking into account the kaon beam momentum bite. The ratio is normalized by one generated event. The roughness of the contours in both (a) and (b) is due to the limited statistics of the simulation. The vertical gray dotted lines and blue dotted curves are the same as in Fig. 4.

Figure 6

2D spectral functions for (a) the $\overline{K}NN$ bound state $f_K(m_X,q_X)$ [see Eq. (8)], (b) the ${\mathrm{QF}}_{\overline{K}-\mathrm{abs}}$ process $f_F(m_X,q_X)$ [see Eq. (10)], and (c) a broad distribution $f_B(m_X,q_X)$ [see Eq. (12)]. For all the figures, the function strength is given in a logarithmic scale, where the contours are in the steps of 10% (red–orange), 1% (orange–cyan), and 0.1% (cyan–blue) compared to the maximum density of each function. The vertical gray dotted lines and blue dotted curves are the same as in Fig. 4.

Figure 7

$m_X$ distribution (integrated by $q_X$ over the whole kinematically allowed region) of (a) data and (b) model functions. The model function is limited to the $\mathrm{\Lambda }pn$ final state; i.e., the $\mathrm{\Sigma }NN$ contribution is excluded. The colored lines are spectra of three processes. The magenta thick curve is the sum of all the processes with an error band of the 95% confidence level. The vertical gray dotted lines are $M_{\overline{K}NN}$.

Figure 8

Experimental acceptance for each $\mathrm{\Sigma }NN$ contamination: (a) ${\mathrm{\Sigma }}^{0}pn$ final state and (b) ${\mathrm{\Sigma }}^{-}pp$ final state. The vertical and horizontal axes for ${\mathrm{\Sigma }}^{0}pn$ (${\mathrm{\Sigma }}^{-}pp$) are the momentum transfer and invariant mass of the ${\mathrm{\Sigma }}^{0}p$ (${\mathrm{\Sigma }}^{-}p$) system. The hatched regions are insensitive in the present setup, where $\varepsilon <0.5\%$. The roughness of the contours is due to the limited statistics of the simulation. The efficiency is calculated bin by bin. The vertical gray dotted lines and blue dotted curves are the same as in Fig. 4.

Figure 9

Expected spectral shapes of $m_X$ for the (a) ${\mathrm{\Sigma }}^{0}pn$ final state and (b) ${\mathrm{\Sigma }}^{-}pp$ final state. The horizontal axis of (a) is the $\mathrm{\Lambda }p$ invariant mass, which is a partial invariant mass of ${\mathrm{\Sigma }}^{0}p$, where $\gamma$ of ${\mathrm{\Sigma }}^0\to \gamma \mathrm{\Lambda }$ is missing. The axis of (b) is the $\mathrm{\Lambda }p$ invariant mass after misidentification as the $\mathrm{\Lambda }pn$ final state; i.e., $X$ is a pair of pseudo-$\mathrm{\Lambda }$ and $p$. The vertical gray dotted lines are $M_{\overline{K}NN}$. Note that the full scale of the differential cross section is different from that of Fig. 7.

Figure 10

$m_X$ spectra for various intervals of $q_X$: (a) $q_X\le 0.3\mathrm{GeV}/c$, (b) $0.3<q_X\le 0.6\mathrm{GeV}/c$, (c) $0.6<q_X\le 0.9\mathrm{GeV}/c$, and (d) $0.9\mathrm{GeV}/c<q_X$. The dotted lines correspond to the $m_X$-slice regions given in Fig. 11.

Figure 11

$q_X$ spectra for various intervals of $m_X$: (a) $m_X\le 2.27\mathrm{GeV}/c^2$, (b) $2.27<m_X\le 2.37\mathrm{GeV}/c^2$, (c) $2.37<m_X\le 2.6\mathrm{GeV}/c^2$, and (d) $2.6\mathrm{GeV}/c^2<m_X$, with the fitting results shown as colored lines. The dotted lines correspond to the $q_X$-slice regions given in Fig. 10.

Figure 12

$m_X$ spectra for (a) $m_{R^0}\le m_n$ and (b) $m_{R^0}>m_n$, with the fitting result. The selected $q_X$ region is the same as for Fig. 10 ($0.3<q_X\le 0.6\mathrm{GeV}/c$). The vertical dotted line is $M_{\overline{K}NN}$. Note that the bin width is exceptional, $\mathrm{\Delta }{m}_X=20$ MeV/$c^2$, for these figures to have sufficient statistics.