Few-body approach to the structure of $\overline{K}$-nuclear quasibound states

The structure of light antikaon-nuclear quasibound states, which consist of an antikaon $(\overline{K}=K^{-},{\overline{K}}^0)$ and a few nucleons $(N=p,n)$, such as $\overline{K}NN$, $\overline{K}NNN$, $\overline{K}NNNN$, and $\overline{K}NNNNNN$ systems, is studied with full three- to seven-body calculations. Employing a realistic $\overline{K}N$ potential based on the chiral SU(3) effective field theory with the SIDDHARTA constraint, we show that the central nucleon densities of these systems increase when the antikaon is injected, by about factor of 2 at maximum. The $\overline{K}NNNN$ system shows the largest central density, about 0.74 ${\mathrm{fm}}^{-3}$ even with the phenomenological $\overline{K}N$ potential, which is not as high as those suggested in previous studies with approximate treatments of the few-body systems. We find that the spin of the ground state of the $\overline{K}NNNNNN$ system depends on the strength of the $\overline{K}N$ attraction. Thus, the quantum number of the ground state can be another constraint on the $\overline{K}N$ interaction.

Figure 1

Energy convergence of the ground states of three- to seven-body systems including an antikaon. (a) Eigenvalues calculated only with the real part of the Hamiltonian; (b) real part of the full Hamiltonian eigenvalue; (c) imaginary part of the full Hamiltonian eigenvalue. The AY potential model is employed as the $\overline{K}N$ interaction. See text for details.

Figure 2

Real part of the Kyoto $\overline{K}N$ potential, $\mathrm{Re}{V}_{\overline{K}N}^{I=0}(r=0,E)$, on the complex energy plane.

Figure 3

Nucleon density distributions (a) ${\rho }_N^{N\mathrm{cm}}\left(r\right)$ and (b) $r^2{\rho }_N^{N\mathrm{cm}}\left(r\right)$ for the $K^{-}pp\text{−}{\overline{K}}^{0}pn$ system measured from the center-of-mass of nucleons. We employ the Kyoto $\overline{K}N$ potential with the Type-I and -II $\overline{K}N$ two-body energies and the AY potential. The dotted-dashed curves show that of deuteron for comparison. Note that the spin-parity is $J^{\pi }=0^{-}$ for $\overline{K}NN$ but $J^{\pi }=1^{+}$ for deuteron.

Figure 4

(a) Nucleon density distribution ${\rho }_N\left(r\right)$ and (b) antikaon density distribution ${\rho }_{\overline{K}}\left(r\right)$ for the $K^{-}pp\text{−}{\overline{K}}^{0}pn$ system measured from the center-of-mass of the system. The Kyoto $\overline{K}N$ and AY potentials are employed.

Figure 5

Same as Fig. 3 but for the ${}_{\overline{K}}^3\mathrm{H}$ system. The nucleon density distributions for ${}^3\mathrm{He}$ is plotted for comparison.

Figure 6

Same as Fig. 4 but for the ${}_{\overline{K}}{}^3\mathrm{H}$ system.

Figure 7

Same as Fig. 3 but for the ${}_{\overline{K}}{}^4\mathrm{H}$ system. The nucleon density distribution for ${}^4\mathrm{He}$ is plotted for comparison.

Figure 8

Same as Fig. 4 but for the ${}_{\overline{K}}{}^4\mathrm{H}$ system.

Figure 9

(a) ${}^{3}S_1$ and (b) ${}^{1}S_0$ channels of the $\mathrm{AV}{4}^{\prime }$, ATS3, and MN $NN$ potentials.

Figure 10

Energy levels of $K^{-}pp(J^{\pi }=0^{-})$, ${}_{\overline{K}}{}^3\mathrm{H}(J^{\pi }=1/2^{-})$, and ${}_{\overline{K}}{}^4\mathrm{H}(J^{\pi }=0^{-})$ systems with the $\mathrm{AV}{4}^{\prime }$, ATS3, and MN potentials. The shaded widths represent the decay widths. The Kyoto $\overline{K}N$ potential with Type I is employed for the $\overline{K}N$ interaction.

Figure 11

Nucleon density distribution measured from the center-of-mass system of nucleons for (a) $K^{-}pp$, (b) ${}_{\overline{K}}{}^3\mathrm{H}$, and (c) ${}_{\overline{K}}{}^4\mathrm{H}$ systems with the $\mathrm{AV}{4}^{\prime }$, ATS3, and MN potentials. The Kyoto $\overline{K}N$ potential with Type I is employed for the $\overline{K}N$ interaction.

Figure 12

Antikaon density distribution ${\rho }_{\overline{K}}\left(r\right)$ in the center-of-mass system for (a) $K^{-}pp$, (b) ${}_{\overline{K}}{}^3\mathrm{H}$, and (c) ${}_{\overline{K}}{}^4\mathrm{H}$ systems. The Kyoto $\overline{K}N$ potential with Type I is employed for the $\overline{K}N$ interaction.

Figure 13

Same as Fig. 3 but for the ${}_{\overline{K}}{}^6\mathrm{He}$ system with $J^{\pi }=0^{-}$. The nucleon density distributions for ${}^6\mathrm{He}$ with $J^{\pi }=0^{+}$ is plotted for comparison.

Figure 14

Same as Fig. 4 but for the ${}_{\overline{K}}{}^6\mathrm{He}$ system with $J^{\pi }=0^{-}$.

Figure 15

Same as Fig. 3 but for the ${}_{\overline{K}}{}^6\mathrm{He}$ system with $J^{\pi }=1^{-}$. The nucleon density distributions for ${}^6\mathrm{Li}$ with $J^{\pi }=1^{+}$ is plotted for comparison.

Figure 16

Same as Fig. 3 but for the ${}_{\overline{K}}{}^6\mathrm{He}$ system with $J^{\pi }=1^{-}$.

Figure 17

(a) Nucleon density distributions ${\rho }_N^{N\mathrm{cm}}\left(r\right)$ and (b) antikaon density distributions ${\rho }_{\overline{K}}\left(r\right)$ for the three-, four-, five-, and seven-body kaonic nuclear systems. Kyoto $\overline{K}N$ potential Type I is employed for the $\overline{K}N$ interaction.