$\overline{K}+N\to K+\mathrm{\Xi }$ reaction and $S=-1$ hyperon resonances

The $\overline{K}+N\to K+\mathrm{\Xi }$ reaction is studied for center-of-momentum energies ranging from threshold to 3 GeV in an effective Lagrangian approach that includes the hyperon $s$- and $u$-channel contributions as well as a phenomenological contact amplitude. The latter accounts for the rescattering term in the scattering equation and possible short-range dynamics not included explicitly in the model. Existing data are well reproduced and three above-the-threshold resonances were found to be required to describe the data, namely, the $\mathrm{\Lambda }\left(1890\right), \mathrm{\Sigma }\left(2030\right)$, and $\mathrm{\Sigma }\left(2250\right)$. For the latter resonance we have assumed the spin-parity of $J^P=5/2^{-}$ and a mass of 2265 MeV. The $\mathrm{\Sigma }\left(2030\right)$ resonance is crucial in achieving a good reproduction of not only the measured total and differential cross sections but also the recoil polarization asymmetry. More precise data are required before a more definitive statement can be made about the other two resonances, in particular, about the $\mathrm{\Sigma }\left(2250\right)$ resonance that is introduced to describe a small bump structure observed in the total cross section of $K^{-}+p\to K^{+}+{\mathrm{\Xi }}^{-}$. The present analysis also reveals a peculiar behavior of the total cross-section data in the threshold energy region in $K^{-}+p\to K^{+}+{\mathrm{\Xi }}^{-}$, where the $P$ and $D$ waves dominate instead of the usual $S$ wave. Predictions for the target-recoil asymmetries of the $\overline{K}+N\to K+\mathrm{\Xi }$ reaction are also presented.

Figure 1

Diagrams describing the amplitude (11) in the present calculation. The labeling of the external legs of the $s$-channel diagram, $M_s$, follows the reaction equation (1); the labels apply correspondingly also to the external legs of the $u$-channel diagram, $M_u$, and the contact term, $M_c$. The intermediate hyperon exchanges, $\mathrm{\Lambda }$ and $\mathrm{\Sigma }$, indicated for $M_s$ also appear in $M_u$. The details of the contact amplitude, $M_c$, are discussed in Sec. 2.

Figure 2

Total cross section for the $K^{-}+p\to K^{+}+{\mathrm{\Xi }}^{-}$ reaction. (a) The solid blue (gray) line represents the result of the full calculation of the present model. The red (gray) dashed line shows the combined $\mathrm{\Lambda }$ hyperon contributions. The magenta (dark gray) dash-dotted line shows the combined $\mathrm{\Sigma }$ hyperon contributions. The brown (very dark gray) dotted line shows the combined $\mathrm{\Lambda }$ and $\mathrm{\Sigma }$ hyperon contributions. The green (gray) dash-dash-dotted line corresponds to the contact term. (b) The solid red (gray) line represents the combined $\mathrm{\Lambda }$ hyperon contributions, that is the same as the red dashed line in panel (a). The dotted and dashed lines show the $\mathrm{\Lambda }\left(1116\right)$ and $\mathrm{\Lambda }\left(1890\right)$ contributions, respectively. (c) The solid magenta (dark gray) line represents the combined $\mathrm{\Sigma }$ hyperon contributions, that is the same as the magenta dash-dotted line in panel (a). The dotted, dashed, dot-dashed, and dot-dot-dashed lines show the contributions from $\mathrm{\Sigma }\left(1193\right), \mathrm{\Sigma }\left(1385\right), \mathrm{\Sigma }\left(2250\right)$, and $\mathrm{\Sigma }\left(2030\right)$, respectively. The experimental data (black circles) are the digitized version as quoted in Ref. [50] from the original work of Refs. [29, 30, 31, 32, 33, 34, 36, 37, 38, 39, 41, 42, 43].

Figure 3

Same as Fig. 2 for the $K^{-}+p\to K^0+{\mathrm{\Xi }}^0$ reaction. The experimental data (black circles) are the digitized version as quoted in Ref. [50] from the original work of Refs. [30, 37, 38, 39, 40, 43].

Figure 4

Total cross-section results with individual resonances switched off (a) for $K^{-}+p\to K^{+}+{\mathrm{\Xi }}^{-}$ and (b) for $K^{-}+p\to K^0+{\mathrm{\Xi }}^0$. The blue (gray) lines represent the full result shown in Figs. 2 and 3. The red (gray) dashed lines which almost coincide with the blue lines represent the result with $\mathrm{\Lambda }\left(1890\right)$ switched off. The green (gray) dash-dotted lines represent the result with $\mathrm{\Sigma }\left(2030\right)$ switched off and the magenta (dark gray) dash-dash-dotted lines represent the result with $\mathrm{\Sigma }\left(2250\right)5/2^{-}$ switched off.

Figure 5

Kaon angular distributions in the center-of-mass frame (a) for $K^{-}+p\to K^{+}+{\mathrm{\Xi }}^{-}$ and (b) for $K^{-}+p\to K^0+{\mathrm{\Xi }}^0$. The blue (gray) lines represent the full model results. The red (gray) dashed lines show the combined $\mathrm{\Lambda }$ hyperon contributions. The magenta (dark gray) dash-dotted lines show the combined $\mathrm{\Sigma }$ hyperon contributions. The green (gray) dash-dash-dotted line corresponds to the contact term. The numbers in the upper right corners correspond to the centroid total energy of the system $W$. Note the different scales used. The experimental data (black circles) are the digitized version as quoted in Ref. [50] from the original work of Refs. [31, 32, 33, 34, 36, 37] for the $K^{-}+p\to K^{+}+{\mathrm{\Xi }}^{-}$ reaction and of Ref. [30, 36, 37, 40] for the $K^{-}+p\to K^0+{\mathrm{\Xi }}^0$ reaction.

Figure 6

Partial wave decomposition of the total cross section and the angular distribution for $K^{-}+p\to K^{+}+{\mathrm{\Xi }}^{-}$. (a) Total cross section sectioned by contributions from each partial wave $L$. The red (gray, very bottom) shaded area indicates the $S$-wave contribution, while the green (gray, second from bottom) area corresponds to the $P$-wave contribution. Magenta (dark gray, second from top) indicates the $D$-wave contribution and maroon (dark gray, top) the $F$-wave contribution. (b) $K^{+}$ angular distribution: the solid blue (gray) lines are the full results, while the dotted green (gray) lines represent the sum of $S+P$ waves, the red (gray) dashed lines represent the $S+P+D$ waves and the dash-dotted magenta (dark gray) lines represent the $S+P+D+F$ waves. For lower energies, the $S+P+D$ waves already saturate the full cross-section results so that the $F$- and higher-wave contributions cannot be distinguished from the full result.

Figure 7

Ratio of the measured total cross section $\sigma$ and the final state $K\mathrm{\Xi }$ relative momentum $p^{\prime }$ as a function of $p^{\prime 2}$. The blue (gray) square data correspond to $K^{-}+p\to K^{+}+{\mathrm{\Xi }}^{-}$, while the red (gray) circle data correspond to $K^{-}+p\to K^0+{\mathrm{\Xi }}^0$. The blue (gray) solid and red (gray) dashed curves are the present model results corresponding to $K^{-}+p\to K^{+}+{\mathrm{\Xi }}^{-}$ and $K^{-}+p\to K^0+{\mathrm{\Xi }}^0$, respectively.

Figure 8

Same as in Fig. 6 but for $K^{-}+p\to K^0+{\mathrm{\Xi }}^0$.

Figure 9

The recoil asymmetry multiplied by the cross section, $P\frac{d\sigma }{d\mathrm{\Omega }}$, for both the $K^{-}+p\to K^{+}+{\mathrm{\Xi }}^{-}$ and $K^{-}+p\to K^0+{\mathrm{\Xi }}^0$ reactions. The blue (gray) solid lines represent the full results of the current model. Data are from Refs. [33, 37].

Figure 10

Target-recoil asymmetries $K_{xx}$ (green [gray] dashed curves) and $K_{xz}$ (blue [gray] solid curves) as defined in Ref. [57] multiplied by the cross sections for the reactions (a) $K^{-}+p\to K^{+}+{\mathrm{\Xi }}^{-}$ and (b) $K^{-}+p\to K^0+{\mathrm{\Xi }}^0$. The numbers in the upper right corners represent the total energy of the system $W$ in units of GeV.

Figure 11

Same as Figs. 2 and 5 for the $K^{-}+n\to K^0+{\mathrm{\Xi }}^{-}$ reaction. The experimental data are from Refs. [30, 38].

Figure 12

Same as Figs. 2 and 5 for the $K_L+p\to K^{+}+{\mathrm{\Xi }}^0$ reaction.