Toward establishing low-lying $\mathrm{\Lambda }$ and $\mathrm{\Sigma }$ hyperon resonances with the $\overline{K}+d\to \pi +Y+N$ reaction

A model for the $\overline{K}d\to \pi YN$ reactions with $Y=\mathrm{\Lambda },\mathrm{\Sigma }$ is developed, aiming at establishing the low-lying $\mathrm{\Lambda }$ and $\mathrm{\Sigma }$ hyperon resonances through analyzing the forthcoming data from the J-PARC E31 experiment. The off-shell amplitudes generated from the dynamical coupled-channels (DCC) model, which was developed in Kamano et al. [Phys. Rev. C 90, 065204 (2014)], are used as input to the calculations of the elementary $\overline{K}N\to \overline{K}N$ and $\overline{K}N\to \pi Y$ subprocesses in the $\overline{K}d\to \pi YN$ reactions. It is shown that the cross sections for the J-PARC E31 experiment with a rather high incoming-$\overline{K}$ momentum, $|{\vec{p}}_{\overline{K}}|=1$ GeV, can be predicted reliably only when the input $\overline{K}N\to \overline{K}N$ amplitudes are generated from a $\overline{K}N$ model, such as the DCC model used in this investigation, which describes the data of the $\overline{K}N$ reactions at energies far beyond the $\overline{K}N$ threshold. We find that the data of the threefold differential cross section $d\sigma /\left(d{M}_{\pi \mathrm{\Sigma }}d{\mathrm{\Omega }}_{p_n}\right)$ for the $K^{-}d\to \pi \mathrm{\Sigma }n$ reaction below the $\overline{K}N$ threshold can be used to test the predictions of the resonance poles associated with $\mathrm{\Lambda }\left(1405\right)$. We also find that the momentum dependence of the threefold differential cross sections for the $K^{-}d\to {\pi }^{-}\mathrm{\Lambda }p$ reaction can be used to examine the existence of a low-lying $J^P=1/2^{+} \mathrm{\Sigma }$ resonance with a pole mass $M_R=1457-i39$ MeV, which was found from analyzing the $K^{-}p$ reaction data within the employed DCC model.

Figure 1

Kinematics of the $\overline{K}d\to \pi YN$ reaction considered in this work. The outgoing $N$ (outgoing $\pi Y$ pair) momentum is in the direction (opposite direction) of the incoming-$\overline{K}$ momentum.

Figure 2

The outgoing nucleon momentum $p_N\equiv \left|{\vec{p}}_N\right|$ (solid curve) as a function of the kinematically allowed $\pi Y$ invariant mass $M_{\pi Y}$ for the incoming-$\overline{K}$ momentum $|{\vec{p}}_{\overline{K}}|=1$ GeV. Here the case that $Y=\mathrm{\Sigma }$ is presented. Dashed vertical line indicates the $\pi \mathrm{\Sigma }$ invariant mass at the $\overline{K}N$ threshold.

Figure 3

Diagrammatical representation of the $\overline{K}d\to \pi YN$ reaction processes considered in this work: (a) the impulse process; (b) the $\overline{K}$-exchange process. The deuteron wave function (open circles) is taken from the one constructed with the Argonne V18 potential [35], while the off-shell amplitudes describing the meson-baryon subprocesses (solid squares) are taken from our DCC model developed in Ref. [2].

Figure 4

Total cross sections for $K^{-}p\to \pi Y$ reactions near the threshold. The blue solid curves are Model B in Ref. [2], the green dot-dashed curves are the E-dep. model in Ref. [34], and the black dotted curves are from the model developed in Ref. [38] that was used for the calculation in Refs. [31, 32, 33].

Figure 5

(Top) Total cross section for $K^{-}p\to {\overline{K}}^{0}n$ in the energy region relevant to the $\overline{K}{N}_2\to {\overline{K}}^{\mathrm{ex}}N$ subprocess in the $\overline{K}$-exchange process [Fig. 3]. The blue solid curve is Model B in Ref. [2], the green dot-dashed curve is the E-dep. model in Ref. [34], and the black dotted curve is from the model developed in Ref. [38] that was used for the calculation in Refs. [31, 32, 33]. (Bottom) Allowed ranges of the invariant mass $W_{1\mathrm{st}}^{\mathrm{ex}}$ for the $\overline{K}{N}_2\to {\overline{K}}^{\mathrm{ex}}N$ subprocess as $p\equiv |-\vec{p}|$ is varied. Here the incoming-$\overline{K}$ momentum and the scattering angle of outgoing $N$ are fixed as $|{\vec{p}}_{\overline{K}}|=1$ GeV and ${\theta }_{p_N}=0$, respectively.

Figure 6

Threefold differential cross section $d\sigma /\left(d{M}_{\pi \mathrm{\Sigma }}d{\mathrm{\Omega }}_{p_n}\right)$ for the $K^{-}d\to \pi \mathrm{\Sigma }n$ reactions at $|{\vec{p}}_{K^{-}}|=1$ GeV and ${\theta }_{p_n}=0$. Solid curves (dashed curves) are the full results for which the off-shell partial-wave amplitudes of Model A (Model B) of our DCC model [2] are used for the two-body meson-baryon subprocesses. Dotted vertical lines indicate the $\pi \mathrm{\Sigma }$ invariant mass at the $\overline{K}N$ threshold.

Figure 7

Threefold differential cross section $d\sigma /\left(d{M}_{\pi \mathrm{\Sigma }}d{\mathrm{\Omega }}_{p_n}\right)$ for the $K^{-}d\to \pi \mathrm{\Sigma }n$ reactions with $|{\vec{p}}_{K^{-}}|=1$ GeV and ${\theta }_{p_n}=0$. Top, middle, and bottom panels are the results for $K^{-}d\to {\pi }^{-}{\mathrm{\Sigma }}^{+}n, K^{-}d\to {\pi }^{+}{\mathrm{\Sigma }}^{-}n$, and $K^{-}d\to {\pi }^0{\mathrm{\Sigma }}^{0}n$, respectively. The results from Model A (Model B) are presented in left panels (right panels). Each of the curves and points is the full results (solid curves) and the results in which only the $S$-wave amplitude (solid squares), the $S_{01}$ amplitude (dashed curves), or the $S_{11}$ amplitude (dashed-dotted curves) is included in ${\overline{K}}^{\mathrm{ex}}{N}_1\to \pi \mathrm{\Sigma }$ of the $\overline{K}$-exchange process. Dotted vertical lines indicate the $\pi \mathrm{\Sigma }$ invariant mass at the $\overline{K}N$ threshold.

Figure 8

Threefold differential cross section $d\sigma /\left(d{M}_{\pi \mathrm{\Sigma }}d{\mathrm{\Omega }}_{p_n}\right)$ for the $K^{-}d\to \pi \mathrm{\Sigma }n$ reactions with $|{\vec{p}}_{K^{-}}|=1$ GeV and ${\theta }_{p_n}=0$. Top, middle, and bottom panels are the results for $K^{-}d\to {\pi }^{-}{\mathrm{\Sigma }}^{+}n, K^{-}d\to {\pi }^{+}{\mathrm{\Sigma }}^{-}n$, and $K^{-}d\to {\pi }^0{\mathrm{\Sigma }}^{0}n$, respectively. The results from Model A (Model B) are presented in left panels (right panels). Solid curves are the full results, while dashed curves are the same as solid curves, except that only the nonresonant contribution is included for the $S_{01}$ amplitude of the ${\overline{K}}^{\mathrm{ex}}{N}_1\to \pi \mathrm{\Sigma }$ subprocess. Dotted vertical lines indicate the $\pi \mathrm{\Sigma }$ invariant mass at the $\overline{K}N$ threshold.

Figure 9

Pole positions of $S_{01}$ ($J^P=1/2^{-}$) $\mathrm{\Lambda }$ resonances located below the $\overline{K}N$ threshold [3], which were extracted within the DCC models developed in Ref. [2]. Red triangles (blue diamonds) are the resonance pole positions obtained from Model A (Model B).

Figure 10

Threefold differential cross section $d\sigma /\left(d{M}_{\pi \mathrm{\Sigma }}d{\mathrm{\Omega }}_{p_p}\right)$ for the $K^{-}d\to {\pi }^{-}{\mathrm{\Sigma }}^{0}p$ reaction at $|{\vec{p}}_{K^{-}}|=1$ GeV and ${\theta }_{p_p}=0$. The results from Model A (Model B) are presented in the left panel (right panel). Solid curves are the full results, while solid squares are the results in which only the $S_{11}$ amplitude is included for ${\overline{K}}^{\mathrm{ex}}{N}_1\to \pi \mathrm{\Sigma }$ of the $\overline{K}$-exchange process. Dotted vertical lines indicate the $\pi \mathrm{\Sigma }$ invariant mass at the $\overline{K}N$ threshold.

Figure 11

Threefold differential cross section $d\sigma /\left(d{M}_{\pi \mathrm{\Lambda }}d{\mathrm{\Omega }}_{p_p}\right)$ for the $K^{-}d\to {\pi }^{-}\mathrm{\Lambda }p$ reaction at $|{\vec{p}}_{K^{-}}|=1$ GeV and ${\theta }_{p_p}=0$. The results from Model A (Model B) are presented in the left panel (right panel). Each of curves and points is the full results (solid curves) and the results in which only the $S_{11}$ amplitude (dashed curves) or the $S_{11}$ and $P_{13}$ amplitudes (solid squares) are included for ${\overline{K}}^{\mathrm{ex}}{N}_1\to \pi \mathrm{\Lambda }$ of the $\overline{K}$-exchange process. Dotted vertical lines indicate the $\pi \mathrm{\Lambda }$ invariant mass at the $\overline{K}N$ threshold.

Figure 12

Threefold differential cross section $d\sigma /\left(d{M}_{\pi \mathrm{\Lambda }}d{\mathrm{\Omega }}_{p_p}\right)$ for the $K^{-}d\to {\pi }^{-}\mathrm{\Lambda }p$ reaction at ${\theta }_{p_p}=0$, computed with Model B. The left (right) panel is the result at $|{\vec{p}}_{\overline{K}}|=0.7$ GeV ($|{\vec{p}}_{\overline{K}}|=1$ GeV). Solid curves are the full results, while dashed curves are the results in which the $P_{11}$ amplitude for ${\overline{K}}^{\mathrm{ex}}{N}_1\to \pi \mathrm{\Lambda }$ in the $\overline{K}$-exchange process is turned off. Dotted vertical lines indicate the $\pi \mathrm{\Lambda }$ invariant mass at the $\overline{K}N$ threshold.

Figure 13

Threefold differential cross section $d\sigma /\left(d{M}_{\pi \mathrm{\Sigma }}d{\mathrm{\Omega }}_{p_n}\right)$ for the $K^{-}d\to \pi \mathrm{\Sigma }n$ reactions at $|{\vec{p}}_{K^{-}}|=1$ GeV and ${\theta }_{p_n}=0$. Solid curves represent the full result, while dashed curves represent the results in which only the $S$-wave amplitudes are included for $\overline{K}{N}_2\to {\overline{K}}^{\mathrm{ex}}N$ of the $\overline{K}$-exchange process. Dotted vertical lines indicate the $\pi \mathrm{\Sigma }$ invariant mass at the $\overline{K}N$ threshold.

Figure 14

Threefold differential cross section $d\sigma /\left(d{M}_{\pi \mathrm{\Sigma }}d{\mathrm{\Omega }}_{p_n}\right)$ for the $K^{-}d\to \pi \mathrm{\Sigma }n$ reactions at $|{\vec{p}}_{K^{-}}|=0.6$ GeV and ${\theta }_{p_n}=0$. Solid curves are the full results from our Model B, while dashed curves are the results from Model B in which only the the $S$-wave amplitudes are included for all meson-baryon subprocesses. Dotted curves are the results in Ref. [31], where the $S$-wave $\overline{K}N$ model developed in Ref. [38] are used for calculating the meson-baryon subprocesses. Dotted vertical lines indicate the $\pi \mathrm{\Sigma }$ invariant mass at the $\overline{K}N$ threshold.

Figure 15

(Top) Total cross section for $K^{-}p\to {\overline{K}}^{0}n$ in the energy region relevant to the $\overline{K}{N}_2\to {\overline{K}}^{\mathrm{ex}}N$ subprocess in the $\overline{K}$-exchange process [Fig. 3] for the case of $|{\vec{p}}_{\overline{K}}|=0.6$ GeV and ${\theta }_{p_N}=0$. The solid (dashed) curve is the full ($S$-wave only) result from Model B of Ref. [2], while the dotted curve is from the model developed in Ref. [38] that was used for the calculation in Refs. [31, 32, 33]. (Middle) Same as the top panel but for $K^{-}n\to {\overline{K}}^{-}n$. (Bottom) Allowed ranges of the invariant mass $W_{1\mathrm{st}}^{\mathrm{ex}}$ for the $\overline{K}{N}_2\to {\overline{K}}^{\mathrm{ex}}N$ subprocess as $p\equiv |-\vec{p}|$ is varied. Here the incoming-$\overline{K}$ momentum and the scattering angle of outgoing $N$ are fixed as $|{\vec{p}}_{\overline{K}}|=0.6$ GeV and ${\theta }_{p_N}=0$.