Dynamical coupled-channels model of $K^{-}p$ reactions: Determination of partial-wave amplitudes

We develop a dynamical coupled-channels model of $K^{-}p$ reactions, aiming at extracting the parameters associated with hyperon resonances and providing the elementary antikaon-nucleon scattering amplitudes that can be used for investigating various phenomena in the strangeness sector such as the production of hypernuclei from kaon-nucleus reactions. The model consists of (a) meson-baryon $\left(MB\right)$ potentials $v_{M^{\prime }{B}^{\prime },MB}$ derived from the phenomenological SU(3) Lagrangian, and (b) vertex interactions ${\Gamma }_{MB,Y^{*}}$ for describing the decays of the bare excited hyperon states $\left(Y^{*}\right)$ into $MB$ states. The model is defined in a channel space spanned by the two-body $\overline{K}N,\pi \Sigma ,\pi \Lambda ,\eta \Lambda$, and $K\Xi$ states and also the three-body $\pi \pi \Lambda$ and $\pi \overline{K}N$ states that have the resonant $\pi {\Sigma }^{*}$ and ${\overline{K}}^{*}N$ components, respectively. The resulting coupled-channels scattering equations satisfy the multichannel unitarity conditions and account for the dynamical effects arising from the off-shell rescattering processes. The model parameters are determined by fitting the available data of the unpolarized and polarized observables of the $K^{-}p\to \overline{K}N,\pi \Sigma ,\pi \Lambda ,\eta \Lambda ,K\Xi$ reactions in the energy region from the threshold to invariant mass $W=2.1$ GeV. Two models with equally good ${\chi }^2$ fits to the data have been constructed. The partial-wave amplitudes obtained from the constructed models are compared with the results from a recent partial-wave analysis by the Kent State University group. We discuss the differences between these three analysis results. Our results at energies near the threshold suggest that the higher partial waves should be treated on the same footing as the $S$ wave if one wants to understand the nature of $\Lambda \left(1405\right)1/2^{-}$ using the data below the $\overline{K}N$ threshold, as will be provided by the J-PARC E31 experiment.

Figure 1

Exchange mechanisms in $v_{M^{\prime }{B}^{\prime },MB}$.

Figure 2

Total cross sections of the $K^{-}p$ reactions up to $W=2.1$ GeV. The red solid curves (blues dashed curves) are the fitted results of Model A (Model B). The same applies to Figs. 3–22 below.

Figure 3

$d\sigma /d\Omega$ of $K^{-}p\to K^{-}p$.

Figure 4

$d\sigma /d\Omega$ of $K^{-}p\to K^{-}p$ (continued).

Figure 5

$d\sigma /d\Omega$ of $K^{-}p\to {\overline{K}}^{0}n$.

Figure 6

$d\sigma /d\Omega$ of $K^{-}p\to {\overline{K}}^{0}n$ (continued).

Figure 7

$P$ of $K^{-}p\to K^{-}p$.

Figure 8

$d\sigma /d\Omega$ of $K^{-}p\to {\pi }^{-}{\Sigma }^{+}$.

Figure 9

$d\sigma /d\Omega$ of $K^{-}p\to {\pi }^0{\Sigma }^0$.

Figure 10

$d\sigma /d\Omega$ of $K^{-}p\to {\pi }^{+}{\Sigma }^{-}$.

Figure 11

$P\times d\sigma /d\Omega$ of $K^{-}p\to {\pi }^{-}{\Sigma }^{+}$.

Figure 12

$P$ of $K^{-}p\to {\pi }^{-}{\Sigma }^{+}$.

Figure 13

$P\times d\sigma /d\Omega$ of $K^{-}p\to {\pi }^0{\Sigma }^0$.

Figure 14

$P$ of $K^{-}p\to {\pi }^0{\Sigma }^0$. Filled (open) circles are the data from Ref. [79] (Ref. [75]).

Figure 15

$d\sigma /d\Omega$ of $K^{-}p\to {\pi }^0\Lambda$.

Figure 16

$d\sigma /d\Omega$ of $K^{-}p\to {\pi }^0\Lambda$ (continued).

Figure 17

$P\times d\sigma /d\Omega$ of $K^{-}p\to {\pi }^0\Lambda$.

Figure 18

$P$ of $K^{-}p\to {\pi }^0\Lambda$.

Figure 19

$d\sigma /d\Omega$ of $K^{-}p\to \eta \Lambda$.

Figure 20

$P$ of $K^{-}p\to \eta \Lambda$.

Figure 21

$d\sigma /d\Omega$ of $K^{-}p\to K^0{\Xi }^0$.

Figure 22

$d\sigma /d\Omega$ of $K^{-}p\to K^{+}{\Xi }^{-}$.

Figure 23

Determined partial-wave amplitudes of $\overline{K}N\to \overline{K}N$ with isospin $I=0$. Upper (lower) panels are for real (imaginary) parts of the amplitudes. Results of Model A (Model B) are shown in red solid (blue dashed) curves. Our results are compared with the single-energy solution (filled circles) given in Ref. [19].

Figure 24

Determined partial-wave amplitudes of $\overline{K}N\to \overline{K}N$ with isospin $I=1$. See the caption of Fig. 23 for the description of the figure.

Figure 25

Determined partial-wave amplitudes of $\overline{K}N\to \pi \Sigma$ with isospin $I=0$. See the caption of Fig. 23 for the description of the figure.

Figure 26

Determined partial-wave amplitudes of $\overline{K}N\to \pi \Sigma$ with isospin $I=1$. See the caption of Fig. 23 for the description of the figure.

Figure 27

Determined partial-wave amplitudes of $\overline{K}N\to \pi \Lambda$. See the caption of Fig. 23 for the description of the figure.

Figure 28

Spin-rotation angle $\beta$ predicted from Model A (red solid curves) and Model B (blue dashed curves). The results are shown for the $K^{-}p\to \overline{K}N,\pi \Sigma ,\pi \Lambda$ reactions. Our predictions are compared with the $\beta$ calculated by using the partial-wave amplitudes of the KSU single-energy solution [19] (black dotted curves). Note that $\beta$ is modulo $2\pi$.

Figure 29

Determined partial-wave amplitudes of $\overline{K}N\to K\Xi$ with isospin $I=0$. See the caption of Fig. 23 for the description of the figure.

Figure 30

Determined partial-wave amplitudes of $\overline{K}N\to K\Xi$ with isospin $I=1$. See the caption of Fig. 23 for the description of the figure.

Figure 31

Determined partial-wave amplitudes of $\overline{K}N\to \eta \Lambda$. See the caption of Fig. 23 for the description of the figure.

Figure 32

The $S$-wave contribution to the $K^{-}p\to \eta \Lambda$ total cross sections near the threshold. Left (right) panel is the result of Model A (Model B). The solid curves are the full results, while the dotted curves are the contribution of the $S_{01}$ partial wave only. The data are taken from the latest BNL result [81].

Figure 33

The angular dependence of the near-threshold $K^{-}p\to \eta \Lambda$ differential cross section for Model B. The result at $W=1672$ MeV is presented. The solid curve is the full result, while the dotted curve is the result in which the contribution of the $P_{03}$ partial wave is turned off.

Figure 34

The $K^{-}p$ reaction total cross sections in the threshold region. Solid (dashed) curves are the full results from Model A (Model B), which are the same as shown in Fig. 2, while dotted (dashed-dotted) curves are the $S$-wave contribution from Model A (Model B).

Figure 35

The predicted ${\pi }^{\pm }{\Sigma }^{\mp }\to {\pi }^{\pm }{\Sigma }^{\mp }$ total cross sections from the threshold up to $W=1.55$ GeV. The vertical dotted line indicates the $\overline{K}N$ threshold. The meaning of each predicted curve is the same as in Fig. 34.

Figure 36

Predicted $K^{-}p$ reaction total cross section. The upper (lower) row is the results of Model A (Model B). (Left) Comparison of our predicted ${\sigma }_{K^{-}p}^{\text{tot}}$ (solid curve) with the data (open circles). The data are taken from Ref. [9]. (Right) The curves showing how the predicted contributions from each channel are added up to the total ${\sigma }_{K^{-}p}^{\text{tot}}$.