Effective $\overline{K}N$ interaction based on chiral SU(3) dynamics

The effective $\overline{K}N$ interaction based on chiral SU(3) coupled-channel dynamics is derived and its extrapolation below the $\overline{K}N$ threshold is studied in detail. Starting from the coupled-channel scattering equations, we eliminate the channels other than $\overline{K}N$ and obtain an effective interaction in the single $\overline{K}N$ channel. An equivalent local potential in coordinate space is constructed such as to reproduce the full scattering amplitude of the chiral SU(3) coupled-channel framework. We discuss several realistic chiral SU(3)-based models in comparison to reach conclusions about the uncertainties involved. It turns out that, in the region relevant to the discussion of deeply bound $\overline{K}$-nuclear few-body systems, the resulting energy-dependent, equivalent local potential is substantially less attractive than the one suggested in previous purely phenomenological treatments.

Figure 1

Diagrammatic representation of Eqs. (4, 5), and (7). The black blob stands for $T^{\mathrm{eff}}=T_{11}$, shaded blobs stand for $V^{\mathrm{eff}}$, and white blobs denote $T_{22}^{\mathrm{single}}$.

Figure 2

The single-channel $\pi \Sigma$ scattering amplitude $F_{\pi \Sigma }^{\mathrm{single}}=-(4\pi M_{\Sigma }/\sqrt{s})T_{22}^{\mathrm{single}}$ (left panel) and the interaction term ${\tilde{V}}_{11}$ (right panel). Real parts are shown as solid lines; imaginary parts are represented as dashed lines. The thin lines are the result of the two-channel model and the thick lines are the result of the full (four-channel) model. The imaginary parts of ${\tilde{V}}_{11}$ in the two-channel and full models are indistinguishable in this figure.

Figure 3

The $\overline{K}N$ $(I=0)$ interaction at tree level given by the Weinberg-Tomozawa term (dotted lines), the effective interaction in the two-channel model (thin lines), and the effective interaction in the full model (thick lines). The real parts are shown by the solid lines, and the imaginary parts are depicted by the dashed lines. The right panel details the upper part of the left panel. The imaginary parts of $V^{\mathrm{eff}}$ in the two-channel and full models are indistinguishable in this figure.

Figure 4

Scattering amplitudes $F_{\overline{K}N}$ channel (left) and $F_{\pi \Sigma }$ (right) in the $(I=0)$ two-channel model (thin lines) and in the full model (thick lines). Real parts are shown as solid lines and imaginary parts as dashed lines.

Figure 5

Pole positions of the $\overline{K}N(I=0)$ scattering amplitude resulting from the single-channel, two-channel, and full (four-channel) models.

Figure 6

The effective interaction $V^{\mathrm{eff}}$ of the $\overline{K}N(I=0)$ channel (left) and that of the $\overline{K}N(I=1)$ channel (right) in different models. The real parts are shown as solid lines and imaginary parts as dashed lines. The lines correspond to the models as indicated in the figure. The dotted line is the tree-level WT interaction.

Figure 7

Scattering amplitude $F_{\overline{K}N}=-(4\pi M_N/\sqrt{s})T^{\mathrm{eff}}$ of the $\overline{K}N(I=0)$ channel (left) and of the $\overline{K}N(I=1)$ channel (right) in different models. The real parts are shown as solid lines and imaginary parts as dashed lines.

Figure 8

Pole positions of the $\overline{K}N(I=0)$ scattering amplitude in different models. The dotted line denotes the threshold of the $\overline{K}N$ channel.

Figure 9

Imaginary part of the $\pi \Sigma (I=0)$ amplitude in different models. The dashed histogram indicates data from Ref. [41] (${\pi }^{-}{\Sigma }^{+}$ spectrum) and the dotted histogram indicates data from Ref. [40] (sum of ${\pi }^{\pm }{\Sigma }^{\mp }$ spectra).

Figure 10

Scattering amplitudes $F_{\overline{K}N}$ from the local potential $U(r,E)$ (thick lines) and from the amplitude $T^{\mathrm{eff}}$ in the original chiral coupled-channel approach (thin lines) obtained by using the HNJH model for the $I=0$ channel (left) and the $I=1$ channel (right). Real parts are shown as solid lines and imaginary parts as dashed lines.

Figure 11

Upper panels: Strength of the fitted potential at $r=0$ (thick lines) and the strength without correction [Eq. (11); dotted lines] with the HNJH model. Lower panels: Scattering amplitude $f$ from the local potential (thick lines) and the amplitude $T_{\mathrm{eff}.}$ in the original chiral unitary approach (thin lines) with the HNJH model. The real parts are shown by the solid lines, and the imaginary parts are depicted by the dotted lines. Left: $I=0$ channel. Right: $I=1$ channel.

Figure 12

Strength of the local $I=0$ $\overline{K}N$ potential $U(r,E)$ at $r=0$ obtained by using different models as explained in the text. Left: Uncorrected potentials, with real parts shown as dotted lines. Right: Corrected potentials, with real parts shown as solid lines. Imaginary parts are depicted as dashed lines. The ordering of model assignments is the same in both panels.

Figure 13

Strength of the local $I=1$ $\overline{K}N$ potential $U(r,E)$ at $r=0$ obtained by using different models as explained in the text. Left: Uncorrected potentials, with real parts shown as dotted lines. Right: Corrected potentials, with real parts shown as solid lines. Imaginary parts are depicted as dashed lines. The ordering of model assignments is the same in both panels.

Figure 14

Scattering amplitudes $F_{\overline{K}N}$ from the corrected local potentials $U(r,E)$ for the four different models referred to in Figs. 12 and 13. Real parts are shown as solid lines and imaginary parts as dashed lines. Left: $I=0$ channel. Right: $I=1$ channel.

Figure 15

Comparison of the scattering amplitudes with phenomenological potential. Thick lines denote the results of the potentials derived in this work and thin lines denote the results of the phenomenological potential [21]. The real parts are shown by solid lines and the imaginary parts are depicted by dotted lines. Left: $I=0$ channel. Right: $I=1$ channel.