Structure of $\mathrm{\Lambda }\left(1405\right)$ and construction of $\overline{K}N$ local potential based on chiral SU(3) dynamics

We develop the single-channel local potential for the $\overline{K}N$ system, which is applicable to quantitative studies of $\overline{K}$ bound states in nuclei. Because the high-precision measurement of the kaonic hydrogen by SIDDHARTA reduces the uncertainty of the $\overline{K}N$ amplitude below the $\overline{K}N$ threshold, the local potential should be calibrated in a wide energy region. We establish a new method to construct the local potential focusing on the behavior of the scattering amplitude in the complex energy plane. Applying this method, we construct the $\overline{K}N$ potential based on the chiral coupled-channel approach with the SIDDHARTA constraint. The wave function from the new potential indicates the $\overline{K}N$ molecular structure of $\mathrm{\Lambda }\left(1405\right)$.

Figure 1

Scattering amplitudes from the local potentials $F_{\overline{K}N}$ (thick lines) and the amplitudes directly from chiral unitary approach $F_{\overline{K}N}^{\mathrm{Ch}}$ (thin lines) with models ORB [68], HNJH [66, 67], BNW [56, 57], and BMN [58]. The real (imaginary) parts are shown by the solid (dotted) lines.

Figure 2

The contour plot of $\mathrm{\Delta }F$ of the HNJH potential in Ref. [55]. The unfilled region corresponds to large deviation, $\mathrm{\Delta }F>2$. The precise region is defined as $\mathrm{\Delta }F<0.2$. The crosses represent the original pole positions of $\mathrm{\Lambda }\left(1405\right)$.

Figure 3

The contour plot of $\mathrm{\Delta }F$ of Potential I. The unfilled region corresponds to large deviation, $\mathrm{\Delta }F>2$. The precise region is defined as $\mathrm{\Delta }F<0.2$. The crosses represent the original pole positions of $\mathrm{\Lambda }\left(1405\right)$.

Figure 4

The contour plot of $\mathrm{\Delta }F$ of Potential II. The unfilled region corresponds to large deviation, $\mathrm{\Delta }F>2$. The precise region is defined as $\mathrm{\Delta }F<0.2$. The crosses represent the original pole positions of $\mathrm{\Lambda }\left(1405\right)$.

Figure 5

The result of the isospin symmetric $\overline{K}N$ amplitude in the $I=0$ channel (thick lines) and the combination of the original amplitudes $(F_{K^{-}p{K}^{-}p}+2F_{K^{-}p{\overline{K}}^{0}n}+F_{{\overline{K}}^{0}n{\overline{K}}^{0}n})/2$ with isospin breaking [48, 49] (thin lines). The real parts are shown by the solid lines, and the imaginary parts are shown by the dotted lines.

Figure 6

The contour plot of $\mathrm{\Delta }F$ of SIDDHARTA potential $(I=0)$. The precise region is defined as $\mathrm{\Delta }F<0.2$. The crosses represent the original pole positions of $\mathrm{\Lambda }\left(1405\right)$.

Figure 7

Strength of SIDDHARTA potential $(I=0) U(r,E)$ at $r=0$. The real part is shown by the solid line, and the imaginary part is shown by the dotted line.

Figure 8

Strength of SIDDHARTA potential $(I=1) U(r,E)$ at $r=0$. The real part is shown by the solid line, and the imaginary part is shown by the dotted line.

Figure 9

$I=0 \overline{K}N$-wave function ${\psi }_G$ with the normalization (16) at the $\mathrm{\Lambda }\left(1405\right)$ pole energy, $1423.97-26.28i$ MeV. The real part is shown by the solid line, and the imaginary part is shown by the dotted line.

Figure 10

$\overline{K}N$ density distribution $\rho$ in Eq. (23) (thick solid line), the real part of SIDDHARTA potential $(I=0)$ (thin solid line), and the imaginary part (dotted line) at the $\mathrm{\Lambda }\left(1405\right)$ pole energy, $1423.97-26.28i$ MeV.