Few-body approach to structure of $\bar{K}$-nuclear quasi-bound states

TL;DR

This study employs full few-body calculations to analyze light antikaon-nuclear quasi-bound states, revealing increased central densities and the dependence of ground state spin on the $ar{K}N$ interaction strength.

Contribution

It provides the first comprehensive few-body analysis of these systems using realistic potentials constrained by experimental data.

Findings

Central nucleon densities increase with antikaon injection.

The $ar{K}NNNN$ system has the highest central density among studied systems.

Ground state spin depends on the $ar{K}N$ attraction strength.

Abstract

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

Tables

Table 1. Table 1: Total binding energies B 𝐵 B , point-proton rms radii ⟨ r p 2 ⟩ delimited-⟨⟩ superscript subscript 𝑟 𝑝 2 \sqrt{\langle r_{p}^{2}\rangle} , point-neutron rms radii ⟨ r n 2 ⟩ delimited-⟨⟩ superscript subscript 𝑟 𝑛 2 \sqrt{\langle r_{n}^{2}\rangle} and rms matter radii ⟨ r N 2 ⟩ delimited-⟨⟩ superscript subscript 𝑟 𝑁 2 \sqrt{\langle r_{N}^{2}\rangle} of ordinary nuclei. The experimental data of the binding energies B 𝐵 B are taken from Ref.  Audi  et al. ( 2002 ) . The nuclear charge radii ⟨ r c ​ h 2 ⟩ delimited-⟨⟩ superscript subscript 𝑟 𝑐 ℎ 2 \sqrt{\langle r_{ch}^{2}\rangle} in Ref.  Angeli and Marinova ( 2013 ) are converted into point-proton rms radii ⟨ r p 2 ⟩ delimited-⟨⟩ superscript subscript 𝑟 𝑝 2 \sqrt{\langle r_{p}^{2}\rangle} by using the formula: ⟨ r c ​ h 2 ⟩ = ⟨ r p 2 ⟩ + ⟨ r p ​ c 2 ⟩ + N Z ​ ⟨ r n ​ c 2 ⟩ + 3 ​ ℏ 4 ​ m p 2 ​ c 2 delimited-⟨⟩ superscript subscript 𝑟 𝑐 ℎ 2 delimited-⟨⟩ superscript subscript 𝑟 𝑝 2 delimited-⟨⟩ superscript subscript 𝑟 𝑝 𝑐 2 𝑁 𝑍 delimited-⟨⟩ superscript subscript 𝑟 𝑛 𝑐 2 3 Planck-constant-over-2-pi 4 superscript subscript 𝑚 𝑝 2 superscript 𝑐 2 {\langle r_{ch}^{2}\rangle}={\langle r_{p}^{2}\rangle}+{\langle r_{pc}^{2}\rangle}+\frac{N}{Z}{\langle r_{nc}^{2}\rangle}+\frac{3\hbar}{4m_{p}^{2}c^{2}}   Friar  et al. ( 1997 ) , where ⟨ r p ​ c 2 ⟩ = 0.878 2 delimited-⟨⟩ superscript subscript 𝑟 𝑝 𝑐 2 superscript 0.878 2 \langle r_{pc}^{2}\rangle=0.878^{2} [fm 2 ], ⟨ r n ​ c 2 ⟩ = − 0.115 delimited-⟨⟩ superscript subscript 𝑟 𝑛 𝑐 2 0.115 \langle r_{nc}^{2}\rangle=-0.115 [fm 2 ], 3 ​ ℏ 4 ​ m p 2 ​ c 2 = 0.0332 3 Planck-constant-over-2-pi 4 superscript subscript 𝑚 𝑝 2 superscript 𝑐 2 0.0332 \frac{3\hbar}{4m_{p}^{2}c^{2}}=0.0332 [fm 2 ] are proton mean-square charge radius, neutron mean-square charge radius and Darwin-Foldy term, respectively.

	AV4’	Expt.	AV4’	Expt.	AV4’
	$B$ [MeV]		$\sqrt{⟨r_p^2⟩}$ [fm]		$\sqrt{⟨r_n^2⟩}$ [fm]	$\sqrt{⟨r_N^2⟩}$ [fm]
2H	$2.24$	$2.22$	$2.02$	$1.97$	$2.02$	$2.02$
3H	$8.99$	$8.48$	$1.59$	$1.59$	$1.70$	$1.67$
3He	$8.33$	$7.72$	$1.73$	$1.77$	$1.60$	$1.69$
4He	$32.1$	$28.3$	$1.39$	$1.46$	$1.38$	$1.39$
6He	$32.2$	$29.3$	$2.00$	$1.93$	$2.95$	$2.67$
6Li	$35.8$	$32.0$	$2.43$	$2.45$	$2.42$	$2.43$

Table 2. Table 2: Properties of the calculation for K − ​ p ​ p superscript 𝐾 𝑝 𝑝 K^{-}pp - K ¯ 0 ​ p ​ n superscript ¯ 𝐾 0 𝑝 𝑛 \bar{K}^{0}pn system with J π = 0 − superscript 𝐽 𝜋 superscript 0 J^{\pi}=0^{-} . See text for details.

$K^{-}pp$-${\overline{K}}^{0}pn\left(J^{\pi }=0^{-}\right)$
Model	SIDDHARTA		AY
	Type I	Type II
$B$ [MeV]	$27.9$	$26.1$	$48.7$
$\Gamma$ [MeV]	$30.9$	$59.3$	$61.9$
$\delta \sqrt{s}$ [MeV]	$-61.0-i25.0$	$-30.2-i23.7$
$P_{K^{-}}$	$0.65$	$0.65$	$0.64$
$P_{{\overline{K}}^0}$	$0.35$	$0.35$	$0.36$
$\sqrt{⟨r_{NN}^2⟩}$ [fm]	$2.16$	$2.07$	$1.84$
$\sqrt{⟨r_{\overline{K}N}^2⟩}$ [fm]	$1.80$	$1.73$	$1.55$
$\sqrt{⟨r_N^2⟩}$ [fm]	$1.12$	$1.08$	$0.958$
$\sqrt{⟨r_{\overline{K}}^2⟩}$ [fm]	$1.14$	$1.10$	$0.988$
${⟨T⟩}_G^{K^{-}}$ [MeV]	$117+i28.8$	$124+i53.1$	$102+i31.4$
${⟨V⟩}_G^{K^{-}}$ [MeV]	$-113-i33.7$	$-120-i63.9$	$-102-i47.0$
${⟨T⟩}_G^{{\overline{K}}^0}$ [MeV]	$74.3+i18.4$	$76.3+i33.1$	$63.1+i15.5$
${⟨V⟩}_G^{{\overline{K}}^0}$ [MeV]	$-62.0-i19.1$	$-64.3-i35.6$	$-48.6-i21.6$
$2{⟨V⟩}_G^{K^{-}{\overline{K}}^0}$ [MeV]	$-44.1-i9.76$	$-41.9-i16.4$	$-64.0-i9.24$
$P_{\overline{K}N}^{I=0}$	$0.72$	$0.73$	$0.73$
$P_{\overline{K}N}^{I=1}$	$0.28$	$0.27$	$0.27$

Table 3. Table 3: Properties of the calculation for K − ​ p ​ n superscript 𝐾 𝑝 𝑛 K^{-}pn - K ¯ 0 ​ n ​ n superscript ¯ 𝐾 0 𝑛 𝑛 \bar{K}^{0}nn system with J π = 0 − superscript 𝐽 𝜋 superscript 0 J^{\pi}=0^{-} .

$K^{-}pn$-${\overline{K}}^{0}nn\left(J^{\pi }=0^{-}\right)$
Model	SIDDHARTA		AY
	Type I	Type II
$B$ [MeV]	$27.6$	$25.3$	$48.1$
$\Gamma$ [MeV]	$31.6$	$59.4$	$61.6$
$\delta \sqrt{s}$ [MeV]	$-60.2-i25.6$	$-29.4-i23.8$
$P_{K^{-}}$	$0.38$	$0.38$	$0.37$
$P_{{\overline{K}}^0}$	$0.62$	$0.62$	$0.63$
$\sqrt{⟨r_{NN}^2⟩}$ [fm]	$2.18$	$2.10$	$1.85$
$\sqrt{⟨r_{\overline{K}N}^2⟩}$ [fm]	$1.82$	$1.75$	$1.56$
$\sqrt{⟨r_N^2⟩}$ [fm]	$1.13$	$1.09$	$0.963$
$\sqrt{⟨r_{\overline{K}}^2⟩}$ [fm]	$1.15$	$1.11$	$0.993$
${⟨T⟩}_G^{K^{-}}$ [MeV]	$67.6+i20.6$	$81.4+i34.9$	$65.3+i16.1$
${⟨V⟩}_G^{K^{-}}$ [MeV]	$-44.5-i10.2$	$-69.9-i38.0$	$-51.3-i22.6$
${⟨T⟩}_G^{{\overline{K}}^0}$ [MeV]	$112+i28.7$	$118+i52.2$	$99.5+i30.8$
${⟨V⟩}_G^{{\overline{K}}^0}$ [MeV]	$-107-i33.3$	$-112-i62.2$	$-97.3-i45.8$
$2{⟨V⟩}_G^{K^{-}{\overline{K}}^0}$ [MeV]	$-44.5-i10.2$	$-42.2-i16.7$	$-64.3-i9.40$
$P_{\overline{K}N}^{I=0}$	$0.72$	$0.72$	$0.73$
$P_{\overline{K}N}^{I=1}$	$0.28$	$0.28$	$0.27$

Table 4. Table 4: Properties of the calculation for H K ¯ 3 superscript subscript H ¯ 𝐾 3 {}^{3}_{\bar{K}}\text{H} system with J π = 1 / 2 − superscript 𝐽 𝜋 1 superscript 2 J^{\pi}=1/2^{-} .

${}_{\overline{K}}{}^3\text{H}\left(J^{\pi }=1/2^{-}\right)$
Model	SIDDHARTA		AY
	Type I	Type II
$B$ [MeV]	$45.3$	$49.7$	$72.6$
$\Gamma$ [MeV]	$25.5$	$69.4$	$78.6$
$\delta \sqrt{s}$ [MeV]	$-70.4-i20.7$	$-26.1-i18.6$
$P_{K^{-}}$	$0.53$	$0.53$	$0.51$
$P_{{\overline{K}}^0}$	$0.47$	$0.47$	$0.49$
$\sqrt{⟨r_{NN}^2⟩}$ [fm]	$1.99$	$1.90$	$1.87$
$\sqrt{⟨r_{\overline{K}N}^2⟩}$ [fm]	$1.79$	$1.68$	$1.63$
$\sqrt{⟨r_N^2⟩}$ [fm]	$1.17$	$1.11$	$1.09$
$\sqrt{⟨r_{\overline{K}}^2⟩}$ [fm]	$1.17$	$1.08$	$1.03$
${⟨T⟩}_G^{K^{-}}$ [MeV]	$114+i17.4$	$126+i42.2$	$107+i27.6$
${⟨V⟩}_G^{K^{-}}$ [MeV]	$-118-i22.4$	$-135-i54.0$	$-114-i44.6$
${⟨T⟩}_G^{{\overline{K}}^0}$ [MeV]	$103+i15.9$	$113+i39.8$	$101+i26.1$
${⟨V⟩}_G^{{\overline{K}}^0}$ [MeV]	$-105-i20.2$	$-118-i50.0$	$-107-i42.1$
$2{⟨V⟩}_G^{K^{-}{\overline{K}}^0}$ [MeV]	$-39.0-i3.56$	$-36.0-i12.7$	$-59.6-i6.33$
$P_{\overline{K}N}^{I=0}$	$0.50$	$0.50$	$0.50$
$P_{\overline{K}N}^{I=1}$	$0.50$	$0.50$	$0.50$

Table 5. Table 5: Properties of the calculation for He K ¯ 4 superscript subscript He ¯ 𝐾 4 {}^{4}_{\bar{K}}\text{He} system with J π = 0 − superscript 𝐽 𝜋 superscript 0 J^{\pi}=0^{-} .

${}_{\overline{K}}{}^4\text{He}\left(J^{\pi }=0^{-}\right)$
Model	SIDDHARTA		AY
	Type I	Type II
$B$ [MeV]	$67.9$	$72.7$	$85.2$
$\Gamma$ [MeV]	$28.3$	$74.1$	$86.5$
$\delta \sqrt{s}$ [MeV]	$-67.6-i23.0$	$-18.4-i15.0$
$P_{K^{-}}$	$0.08$	$0.06$	$0.16$
$P_{{\overline{K}}^0}$	$0.92$	$0.94$	$0.84$
$\sqrt{⟨r_{NN}^2⟩}$ [fm]	$1.98$	$1.91$	$2.07$
$\sqrt{⟨r_{\overline{K}N}^2⟩}$ [fm]	$1.83$	$1.72$	$1.81$
$\sqrt{⟨r_N^2⟩}$ [fm]	$1.22$	$1.18$	$1.27$
$\sqrt{⟨r_{\overline{K}}^2⟩}$ [fm]	$1.22$	$1.12$	$1.14$
${⟨T⟩}_G^{K^{-}}$ [MeV]	$32.6+i6.75$	$26.7+i16.2$	$50.3+i7.22$
${⟨V⟩}_G^{K^{-}}$ [MeV]	$-25.1-i6.74$	$-20.0-i15.9$	$-42.3-i12.0$
${⟨T⟩}_G^{{\overline{K}}^0}$ [MeV]	$214+i27.8$	$240+i66.0$	$183+i52.7$
${⟨V⟩}_G^{{\overline{K}}^0}$ [MeV]	$-265-i38.4$	$-300-i94.3$	$-232-i88.3$
$2{⟨V⟩}_G^{K^{-}{\overline{K}}^0}$ [MeV]	$-24.8-i3.66$	$-20.2-i9.13$	$-43.9-i2.82$
$P_{\overline{K}N}^{I=0}$	$0.28$	$0.27$	$0.31$
$P_{\overline{K}N}^{I=1}$	$0.72$	$0.73$	$0.69$

Table 6. Table 6: Properties of the calculation for H K ¯ 4 superscript subscript H ¯ 𝐾 4 {}^{4}_{\bar{K}}\text{H} system with J π = 0 − superscript 𝐽 𝜋 superscript 0 J^{\pi}=0^{-} .

${}_{\overline{K}}{}^4\text{H}\left(J^{\pi }=0^{-}\right)$
Model	SIDDHARTA		AY
	Type I	Type II
$B$ [MeV]	$69.6$	$75.5$	$87.4$
$\Gamma$ [MeV]	$28.0$	$74.5$	$87.2$
$\delta \sqrt{s}$ [MeV]	$-68.7-i22.4$	$-19.1-i14.9$
$P_{K^{-}}$	$0.93$	$0.94$	$0.86$
$P_{{\overline{K}}^0}$	$0.07$	$0.06$	$0.14$
$\sqrt{⟨r_{NN}^2⟩}$ [fm]	$1.96$	$1.89$	$2.04$
$\sqrt{⟨r_{\overline{K}N}^2⟩}$ [fm]	$1.82$	$1.71$	$1.79$
$\sqrt{⟨r_N^2⟩}$ [fm]	$1.21$	$1.17$	$1.26$
$\sqrt{⟨r_{\overline{K}}^2⟩}$ [fm]	$1.21$	$1.11$	$1.13$
${⟨T⟩}_G^{K^{-}}$ [MeV]	$216+i28.1$	$244+i66.6$	$188+i53.4$
${⟨V⟩}_G^{K^{-}}$ [MeV]	$-269-i39.1$	$-306-i95.7$	$-241-i90.0$
${⟨T⟩}_G^{{\overline{K}}^0}$ [MeV]	$30.5+i5.50$	$25.1+i14.7$	$47.0+i6.54$
${⟨V⟩}_G^{{\overline{K}}^0}$ [MeV]	$-23.0-i5.54$	$-18.5-i14.2$	$-38.7-i10.9$
$2{⟨V⟩}_G^{K^{-}{\overline{K}}^0}$ [MeV]	$-24.4-i3.00$	$-19.9-i8.71$	$-42.7+i2.64$
$P_{\overline{K}N}^{I=0}$	$0.28$	$0.27$	$0.30$
$P_{\overline{K}N}^{I=1}$	$0.72$	$0.73$	$0.70$

Table 7. Table 7: Total binding energies B 𝐵 B and decay widths Γ Γ \Gamma of ordinary nuclei and kaonic nuclei. SIDDHARTA potential Type I is employed for K ¯ ​ N ¯ 𝐾 𝑁 \bar{K}N interaction.

	$B$ [MeV]				$(B,\Gamma )$ [MeV]
	AV4’	ATS3	MN		AV4’	ATS3	MN
2H	$2.24$	$2.22$	$2.20$	$K^{-}pp$-${\overline{K}}^{0}pn$	$(27.9,30.9)$	$(26.6,32.0)$	$(28.4,23.9)$
3He	$8.33$	$8.11$	$7.72$	${}_{\overline{K}}{}^{}{}_{}{}^3$H	$(45.3,25.5)$	$(44.6,25.0)$	$(45.7,19.3)$
4He	$32.1$	$30.8$	$30.0$	${}_{\overline{K}}{}^{}{}_{}{}^4$H	$(69.6,28.0)$	$(68.7,26.0)$	$(68.6,24.6)$

Table 8. Table 8: Properties of the calculation for Li K ¯ 6 superscript subscript Li ¯ 𝐾 6 {}^{6}_{\bar{K}}\text{Li} system with J π = 0 − superscript 𝐽 𝜋 superscript 0 J^{\pi}=0^{-} .

${}_{\overline{K}}{}^6\text{Li}\left(J^{\pi }=0^{-}\right)$
Model	SIDDHARTA		AY
	Type I	Type II
$B$ [MeV]	$69.8$	$79.7$	$103$
$\Gamma$ [MeV]	$23.7$	$75.6$	$88.0$
$\delta \sqrt{s}$ [MeV]	$-76.2-i18.1$	$-15.1-i10.2$
$P_{K^{-}}$	$0.70$	$0.72$	$0.64$
$P_{{\overline{K}}^0}$	$0.30$	$0.28$	$0.36$
$\sqrt{⟨r_{NN}^2⟩}$ [fm]	$2.80$	$2.75$	$2.57$
$\sqrt{⟨r_{\overline{K}N}^2⟩}$ [fm]	$2.52$	$2.40$	$2.27$
$\sqrt{⟨r_N^2⟩}$ [fm]	$1.82$	$1.78$	$1.66$
$\sqrt{⟨r_{\overline{K}}^2⟩}$ [fm]	$1.61$	$1.49$	$1.43$
${⟨T⟩}_G^{K^{-}}$ [MeV]	$186+i19.3$	$213+i47.9$	$184+i34.4$
${⟨V⟩}_G^{K^{-}}$ [MeV]	$-220-i28.5$	$-258-i69.3$	$-224-i61.7$
${⟨T⟩}_G^{{\overline{K}}^0}$ [MeV]	$89.5+i5.08$	$88.9+i26.5$	$110+i17.5$
${⟨V⟩}_G^{{\overline{K}}^0}$ [MeV]	$-95.4-i8.43$	$-97.6-i33.4$	$-121-i31.7$
$2{⟨V⟩}_G^{K^{-}{\overline{K}}^0}$ [MeV]	$-30.1+i0.703$	$-25.6-i9.55$	$-52.4-i2.59$
$P_{\overline{K}N}^{I=0}$	$0.41$	$0.41$	$0.41$
$P_{\overline{K}N}^{I=1}$	$0.59$	$0.59$	$0.59$

Table 9. Table 9: Properties of the calculation for He K ¯ 6 superscript subscript He ¯ 𝐾 6 {}^{6}_{\bar{K}}\text{He} system with J π = 0 − superscript 𝐽 𝜋 superscript 0 J^{\pi}=0^{-} .

${}_{\overline{K}}{}^6\text{He}\left(J^{\pi }=0^{-}\right)$
Model	SIDDHARTA		AY
	Type I	Type II
$B$ [MeV]	$70.6$	$80.0$	$103$
$\Gamma$ [MeV]	$23.9$	$75.5$	$88.0$
$\delta \sqrt{s}$ [MeV]	$-75.6-i18.3$	$-14.9-i10.2$
$P_{K^{-}}$	$0.41$	$0.41$	$0.40$
$P_{{\overline{K}}^0}$	$0.59$	$0.59$	$0.60$
$\sqrt{⟨r_{NN}^2⟩}$ [fm]	$2.79$	$2.74$	$2.56$
$\sqrt{⟨r_{\overline{K}N}^2⟩}$ [fm]	$2.51$	$2.40$	$2.27$
$\sqrt{⟨r_N^2⟩}$ [fm]	$1.81$	$1.77$	$1.66$
$\sqrt{⟨r_{\overline{K}}^2⟩}$ [fm]	$1.61$	$1.49$	$1.43$
${⟨T⟩}_G^{K^{-}}$ [MeV]	$115+i10.5$	$125+i31.1$	$121+i21.1$
${⟨V⟩}_G^{K^{-}}$ [MeV]	$-129-i15.7$	$-144-i41.5$	$-137-i37.7$
${⟨T⟩}_G^{{\overline{K}}^0}$ [MeV]	$161+i14.2$	$177+i43.4$	$173+i30.6$
${⟨V⟩}_G^{{\overline{K}}^0}$ [MeV]	$-188-i21.3$	$-211-i60.9$	$-208-i55.4$
$2{⟨V⟩}_G^{K^{-}{\overline{K}}^0}$ [MeV]	$-30.7+i0.330$	$-26.2-i9.77$	$-52.8-i2.72$
$P_{\overline{K}N}^{I=0}$	$0.41$	$0.41$	$0.40$
$P_{\overline{K}N}^{I=1}$	$0.59$	$0.59$	$0.60$

Table 10. Table 10: Properties of the calculation for Li K ¯ 6 superscript subscript Li ¯ 𝐾 6 {}^{6}_{\bar{K}}\text{Li} system with J π = 1 − superscript 𝐽 𝜋 superscript 1 J^{\pi}=1^{-} .

${}_{\overline{K}}{}^6\text{Li}\left(J^{\pi }=1^{-}\right)$
Model	SIDDHARTA		AY
	Type I	Type II
$B$ [MeV]	$70.8$	$77.5$	$92.9$
$\Gamma$ [MeV]	$26.4$	$75.2$	$88.0$
$\delta \sqrt{s}$ [MeV]	$-70.2-i21.5$	$-13.2-i10.4$
$P_{K^{-}}$	$0.07$	$0.06$	$0.16$
$P_{{\overline{K}}^0}$	$0.93$	$0.94$	$0.84$
$\sqrt{⟨r_{NN}^2⟩}$ [fm]	$2.95$	$2.90$	$2.83$
$\sqrt{⟨r_{\overline{K}N}^2⟩}$ [fm]	$2.54$	$2.45$	$2.37$
$\sqrt{⟨r_N^2⟩}$ [fm]	$1.91$	$1.88$	$1.83$
$\sqrt{⟨r_{\overline{K}}^2⟩}$ [fm]	$1.54$	$1.45$	$1.40$
${⟨T⟩}_G^{K^{-}}$ [MeV]	$33.3+i5.24$	$27.3+i16.9$	$57.1+i8.22$
${⟨V⟩}_G^{K^{-}}$ [MeV]	$-26.4-i5.55$	$-21.1-i16.8$	$-50.3-i13.7$
${⟨T⟩}_G^{{\overline{K}}^0}$ [MeV]	$240+i24.7$	$268+i61.3$	$219+i46.1$
${⟨V⟩}_G^{{\overline{K}}^0}$ [MeV]	$-294-i35.2$	$-332-i89.9$	$-275-i81.3$
$2{⟨V⟩}_G^{K^{-}{\overline{K}}^0}$ [MeV]	$-23.8-i2.40$	$-19.2-i9.03$	$-43.3-i3.31$
$P_{\overline{K}N}^{I=0}$	$0.27$	$0.27$	$0.29$
$P_{\overline{K}N}^{I=1}$	$0.73$	$0.73$	$0.71$

Table 11. Table 11: Properties of the calculation for He K ¯ 6 superscript subscript He ¯ 𝐾 6 {}^{6}_{\bar{K}}\text{He} system with J π = 1 − superscript 𝐽 𝜋 superscript 1 J^{\pi}=1^{-} .

${}_{\overline{K}}{}^6\text{He}\left(J^{\pi }=1^{-}\right)$
Model	SIDDHARTA		AY
	Type I	Type II
$B$ [MeV]	$72.8$	$80.7$	$95.6$
$\Gamma$ [MeV]	$26.0$	$75.6$	$88.5$
$\delta \sqrt{s}$ [MeV]	$-71.6-i20.8$	$-13.8-i10.4$
$P_{K^{-}}$	$0.93$	$0.94$	$0.86$
$P_{{\overline{K}}^0}$	$0.07$	$0.06$	$0.14$
$\sqrt{⟨r_{NN}^2⟩}$ [fm]	$2.95$	$2.89$	$2.81$
$\sqrt{⟨r_{\overline{K}N}^2⟩}$ [fm]	$2.53$	$2.44$	$2.36$
$\sqrt{⟨r_N^2⟩}$ [fm]	$1.91$	$1.87$	$1.82$
$\sqrt{⟨r_{\overline{K}}^2⟩}$ [fm]	$1.53$	$1.44$	$1.39$
${⟨T⟩}_G^{K^{-}}$ [MeV]	$242+i25.0$	$272+i61.9$	$223+i46.8$
${⟨V⟩}_G^{K^{-}}$ [MeV]	$-298-i35.9$	$-339-i91.3$	$-284-i83.0$
${⟨T⟩}_G^{{\overline{K}}^0}$ [MeV]	$31.2+i3.96$	$25.6+i15.4$	$53.5+i7.40$
${⟨V⟩}_G^{{\overline{K}}^0}$ [MeV]	$-24.2-i4.36$	$-19.7-i15.2$	$-46.5-i12.4$
$2{⟨V⟩}_G^{K^{-}{\overline{K}}^0}$ [MeV]	$-23.3-i1.69$	$-18.8-i8.64$	$-42.3-i3.10$
$P_{\overline{K}N}^{I=0}$	$0.27$	$0.26$	$0.29$
$P_{\overline{K}N}^{I=1}$	$0.73$	$0.74$	$0.71$

Equations

$$H$$

$$+\sum _{i<j}^{\mathcal{N}-1}{V}_{ij}^{(NN)}+\sum _{i=1}^{\mathcal{N}-1}{V}_{i\mathcal{N}}^{(\bar{K}N)}+\sum _{i<j}^{\mathcal{N}}{V}_{ij}^{\text{Coul.}}.$$

$$T_{\text{cm}}=\frac{({\sum }_{i=1}^{\mathcal{N}}{\mathbf{p}}_i{)}^2}{2\{(\mathcal{N}-1)m_N+m_{\bar{K}}\}},$$

$$V_{ij}$$

$$=\frac{1}{2}(V_{ij}^{I=0}+V_{ij}^{I=1})-\frac{1}{2}(V_{ij}^{I=0}-V_{ij}^{I=1}){\hat{P}}_{\tau }^{ij},$$

$$⟨\hat{\mathcal{O}}⟩$$

$$1$$

$$⟨\hat{\mathcal{O}}{⟩}_G$$

$$1$$

$$V^{(\bar{K}N)}(r,E)$$

$$\times \frac{{\omega }_{\bar{K}}+E_N}{{\omega }_{\bar{K}}{E}_N}\left[\sum _i{K}_i{\left(\frac{E}{100\text{MeV}}\right)}^i\right],$$

$$E$$

$$E_N$$

$${\omega }_{\bar{K}}$$

$$-B_{\bar{K}}\equiv ⟨H{⟩}_G-⟨H_N{⟩}_G,$$

$$H_N=\sum _{i=1}^{\mathcal{N}-1}{T}_i$$

$$T_{\text{cm}}^N$$

$$\sqrt{s}$$

$$\text{Type I:}\delta \sqrt{s}=-B_{\bar{K}},$$

$$\text{Type II:}\delta \sqrt{s}=-B_{\bar{K}}/(\mathcal{N}-1).$$

$${\Psi }_{JM{M}_T}(\mathbf{x})=\sum _{k=1}^K{c}_k{\Phi }_{JM{M}_T}(\mathbf{x},A_k),$$

$${\Phi }_{S{M}_S{M}_T}(\mathbf{x},A)$$

$${\mathbf{x}}_i$$

$$U$$

$$\displaystyle\chi_{SM_{S}}=\big{|}\big{[}\cdots\big{[}\big{[}\tfrac{1}{2}\tfrac{1}{2}\big{]}_{S_{12}}\tfrac{1}{2}\big{]}_{S_{123}}\cdots\big{]}_{SM_{S}}\big{\rangle}.$$

$${\eta }_{M_T}={\eta }_{\frac{1}{2}{m}_{{\tau }_1}}\cdots {\eta }_{\frac{1}{2}{m}_{{\tau }_{\mathcal{N}}}}.$$

$$m_{{\tau }_k}=\{\begin{array}{ll}\frac{1}{2}& (k=1,\dots ,M_T+\frac{\mathcal{N}}{2}-1,\mathcal{N})\\ -\frac{1}{2}& (\text{otherwise}),\end{array}$$

$$m_{{\tau }_k}=\{\begin{array}{ll}\frac{1}{2}& (k=1,\dots ,M_T+\frac{\mathcal{N}}{2})\\ -\frac{1}{2}& (\text{otherwise}),\end{array}$$

$$r_{NN}^2$$

$$r_{\bar{K}N}^2$$

$$r_N^2$$

$$r_{\bar{K}}^2$$

$${\rho }_N^{N\text{cm}}(r)$$

$${\mathbf{r}}_i^{N\text{cm}}$$

$${\rho }_N(r)$$

$${\rho }_{\bar{K}}(r)$$

Full text

I Introduction

In recent years, properties of the antikaon($\bar{K}$)-nuclear quasi-bound states, so-called kaonic nuclei, have been studied actively. Since the nominal location of the $\Lambda(1405)$ mass is slightly below the $K^{-}p$ threshold Patrignani et al. (2016), the $\Lambda(1405)$ is considered as a $\bar{K}N$ quasi-bound state embedded in the $\pi\Sigma$ continuum Dalitz and Tuan (1959, 1960). Motivated by such a picture, phenomenological $\bar{K}N$ interaction models were constructed so that they reproduce the $\Lambda(1405)$ nominal mass together with two-body scattering data Akaishi and Yamazaki (2002); Shevchenko (2012). The strong attraction of the phenomenological potential models predicts deeply-bound $\bar{K}$ states in light nuclei with binding energy larger than $100$ MeV, and extremely dense systems about ten times higher than the ordinary nuclear density Akaishi and Yamazaki (2002); Yamazaki and Akaishi (2002); Dote et al. (2004, 2004). It should, however, be noted that the few-body problem was not accurately solved to predict such high-density systems, but the optical potential model or the $g$-matrix approach were adopted. The validity of those approaches should be examined with care, at least in the few-body systems.

The $\bar{K}N$ interactions are essential for determining the structure of the kaonic nuclei. The $\bar{K}$ belongs to a part of the pseudscalar octet of Nambu-Goldstone bosons associated with the spontaneous symmetry breaking of chiral SU(3)${}_{L}\times$SU(3)R in low-energy QCD. Thus, the chiral SU(3) effective field theory based on the symmetry breaking mechanism is a more systematic framework to obtain the $\bar{K}N$ interaction, and has succeeded in dealing with the $\bar{K}N$ interaction with $\bar{K}N$-$\pi\Sigma$ couplings Kaiser et al. (1995); Oset and Ramos (1998); Oller and Meissner (2001); Hyodo and Jido (2012). In fact, including the next-to-leading order (NLO) contributions, the chiral SU(3) approach reproduces all existing experimental data at the level of $\chi^{2}/$d.o.f $\sim 1$ Ikeda et al. (2011, 2012). Among others, the precise measurement of the kaonic hydrogen by the SIDDHARTA collaboration Bazzi et al. (2011, 2012) gives strong constraint at the $\bar{K}N$ threshold, with which the uncertainty in the subthreshold extrapolation of the $\bar{K}N$ amplitude is significantly reduced. The equivalent single-channel $\bar{K}N$ potential to the NLO chiral dynamics including the SIDDHARTA constraint is constructed in Ref. Miyahara and Hyodo (2016) based on the framework presented in Ref. Hyodo and Weise (2008). Thus, the realistic $\bar{K}N$ potential is now available.

The $\bar{K}N$-$\pi\Sigma$ scattering amplitude from the chiral SU(3) dynamics has two poles in the $\Lambda(1405)$ energy region Oller and Meissner (2001); Jido et al. (2003); Meissner and Hyodo (2016): one is located around $1420$ MeV, while the other exhibits a broad resonant structure above the $\pi\Sigma$ threshold. The pole located around $1420$ MeV corresponds to the $\bar{K}N$ quasi-bound state with the binding energy of $15$ MeV, about a half of the binding energy assumed in the phenomenological $\bar{K}N$ interactions. This different pole structure comes from different off-shell properties of the $\bar{K}N$ interactions. The $\bar{K}N$ interaction based on the chiral SU(3) dynamics is energy-dependent, and that in the subthreshold becomes less attractive than the one proposed by the energy-independent phenomenological potential Hyodo and Weise (2008). These different off-shell properties also appear in how the $\Lambda(1405)$ resonance shows up in the differential cross section of the $K^{-}d\rightarrow\pi\Sigma n$ reaction Ohnishi et al. (2016). These differences are further enhanced in the light kaonic nuclei. For the lightest kaonic nuclei so-called strange dibaryons in the $\bar{K}NN$-$\pi YN$ ($Y=\Sigma$, $\Lambda$) coupled system, the energy-dependent potential models Dote et al. (2008, 2009); Ikeda et al. (2010); Barnea et al. (2012); Dote et al. (2015) give resonance energies higher than the energy-independent ones Yamazaki and Akaishi (2002, 2007); Wycech and Green (2009); Shevchenko et al. (2007, 2007); Ikeda and Sato (2007, 2009). How a possible signature of this strange dibaryon resonance shows up in the resonance production reaction is also of interest as it reflects the two-body dynamics of the $\bar{K}N$ system Ohnishi et al. (2013).

Given the background described above, we raise three questions to be discussed in this paper; 1) What are the structure of light kaonic nuclei when the reliable $NN$ and $\bar{K}N$ interactions are used? 2) Can the high-density $\bar{K}$ nuclei be realized within the accurate few-body treatment? 3) How the off-shell dependence of the $\bar{K}N$ interaction affect the few-body systems? To answer these questions, we perform fully microscopic few-body calculations for three- to seven-body systems including an antikaon. Here, the systems with a $\bar{K}$ and $(\mathscr{N}-1)$ nucleons are accurately described by employing the stochastic variational method (SVM) with correlated Gaussian (CG) basis Varga and Suzuki (1995, 1997); Suzuki and Varga (1998). We employ the $\bar{K}N$ interaction based on the chiral SU(3) dynamics with the SIDDHARTA constraint Miyahara and Hyodo (2016) as a realistic $\bar{K}N$ force. Combining them with the reliable nuclear forces, we present quantitative predictions of the structure of the light kaonic nuclei. Next, we perform the same few-body calculations with the phenomenological $\bar{K}N$ interaction, so-called Akaishi-Yamazaki (AY) potential Akaishi and Yamazaki (2002); Yamazaki and Akaishi (2007), in order to examine the validity of the many-body approximations used in the prediction of the high-density states. Furthermore, the comparison of the results with two $\bar{K}N$ potentials serves as a study of the off-shell dependence of the interactions. In this way, we systematically study the structure of kaonic nuclei and discuss how the nuclear structure is changed by $\bar{K}$.

In Sec. II, we briefly review the two-body interactions used in this work. The SVM with the CG for the $\mathscr{N}$-body systems is explained in Sec. III. We summarize the quantities to analyze the structures of the few-body systems in Sec. IV. Numerical results of the properties of the light kaonic nuclei are presented in Sec. V. A summary is given in Sec. VI.

II Two-Body Interactions

II.1 Hamiltonian and expectation values

The Hamiltonian for $(\mathscr{N}-1)$ nucleons and an antikaon takes the form

[TABLE]

Here $T_{i}$ is the kinetic energy of the $i$-th particle. The particle label, $i=\mathscr{N}$, always indicates an antikaon and the others are for nucleons. $T_{\text{cm}}$ is the energy of the center-of-mass (c.m.) motion

[TABLE]

where the isospin-averaged nucleon and antikaon masses, $m_{N}=939$ MeV and $m_{\bar{K}}=496$ MeV, are used in this paper. $V_{ij}^{(NN)}$, $V_{ij}^{(\bar{K}N)}$ , and $V_{ij}^{\rm Coul.}$ are the $NN$, $\bar{K}N$, and Coulomb interactions between the $i$- and $j$-th particles, respectively. The $NN$ and $\bar{K}N$ interactions depend on isospin of two-particles, and they can be written as

[TABLE]

where $\hat{P}^{I=0}_{ij}=\frac{1-\bm{\tau}_{i}\cdot\bm{\tau}_{j}}{4}$ and $\hat{P}^{I=1}_{ij}=\frac{3+\bm{\tau}_{i}\cdot\bm{\tau}_{j}}{4}$ are isospin-projection operators for $I=0$ and $1$, and $\hat{P}_{\tau}^{ij}=\frac{1+\bm{\tau}_{i}\cdot\bm{\tau}_{j}}{2}$ is the isospin-exchange operator for the $i$- and $j$-th particles. The isospin-exchange operator $\hat{P}_{\tau}$ acts on particle basis as $\hat{P}_{\tau}|nn\rangle=|nn\rangle$, $\hat{P}_{\tau}|pp\rangle=|pp\rangle$ and $\hat{P}_{\tau}|pn\rangle=|np\rangle$ for $NN$, and $\hat{P}_{\tau}|K^{-}n\rangle=|K^{-}n\rangle$, $\hat{P}_{\tau}|\bar{K}^{0}p\rangle=|\bar{K}^{0}p\rangle$, $\hat{P}_{\tau}|K^{-}p\rangle=-|\bar{K}^{0}n\rangle$ and $\hat{P}_{\tau}|\bar{K}^{0}n\rangle=-|K^{-}p\rangle$ for $\bar{K}N$. We have to treat the $K^{-}p$-$\bar{K^{0}}n$ channel coupling explicitly in the particle basis calculation.

The single-channel $\bar{K}N$ potential $V^{(\bar{K}N)}$ has an imaginary part which represents the decay processes into the lower energy $\pi\Sigma$ and $\pi\Lambda$ channels. Because of the complex nature of the potential, the Hamiltonian is non-Hermite and can have an eigenstate with a complex eigenvalue, called a quasi-bound state. In order to discuss the structure of the quasi-bound state, we need to evaluate the expectation values of some operators. For a stable bound state, the expectation value of an operator $\hat{\mathcal{O}}(\bm{x})$ with the wavefunction $\Psi_{JMM_{T}}(\bm{x})$ is given by (notation of the wavefunction will be explained in Sec. III)

[TABLE]

with the normalization of the wavefunction

[TABLE]

However, since the eigenfunctions of a non-Hermite Hamiltonian do not form an orthogonal set Hokkyo (1965), we should introduce the Gamow states to treat an unstable state. The expectation value with the Gamow states is

[TABLE]

with the normalization

[TABLE]

With the normalization of Eq. (7), expectation values are in general obtained as complex numbers, which are not straightforwardly interpreted. In some cases, however, we can extract a real-valued quantity. As explained in Appendix of Ref. Miyahara and Hyodo (2016), for a quasi-bound state whose real part of the eigenenergy is negative, the damping of the wavefunction outside the potential can be extracted from the standard expectation values with the normalization (4). In this paper, we calculate the root-mean-square (rms) distances $\sqrt{\langle r^{2}\rangle}$, density distributions $\rho(r)$, and the probabilities of finding various channels in the wave functions $P$ by using the standard expectation value (4). For the other operators such as Hamiltonian and its decomposition, the expectation values are calculated by using Gamow state normalization (7).

II.2 $NN$ interactions

As a nucleon-nucleon interaction $V_{NN}$ we employ the Argonne V4’ potential Wiringa and Pieper (2002). AV4’ potential is obtained by simplifying the full AV18 potential by suppressing the small electro-magnetic, the spin-orbit, and the tensor terms and readjusting the central spin- and isospin- dependent interactions. TABLE 1 lists the binding energies and radii of two- to six-nucleon systems calculated with the AV4’ potential. The AV4’ potential model reasonably reproduces the properties of light nuclei.

II.3 $\bar{K}N$ interactions

As a realistic $\bar{K}N$ interaction, $V^{(\bar{K}N)}$, we employ the SIDDHARTA potential, which is the energy-dependent effective interaction based on the chiral SU(3) dynamics constructed in Ref. Miyahara and Hyodo (2016):

[TABLE]

where $E$, $E_{N}$ and $\omega_{\bar{K}}$ are the non-relativistic two-body energy, the energy of the nucleon and the energy of the antikaon:

[TABLE]

The coefficients $K_{i}$ of the energy dependent strength and the range parameter $b$ are determined so as to reproduce the $\bar{K}N$ amplitude Ikeda et al. (2011, 2012) calculated based on the NLO chiral SU(3) dynamics (see Ref. Miyahara and Hyodo (2016)). The SIDDHARTA potential is the single channel $\bar{K}N$ interaction model where the meson-baryon channel coupling effect with strangeness $S=-1$ is renormalized, and thus the coefficients $K_{i}$ are the complex numbers. The origin of the energy dependence is two-fold. The coupled-channel interaction depends on the energy through the time derivatives in the chiral Lagrangians, and the construction of the equivalent single-channel potential introduces additional energy dependence. By solving the Schrödinger equation, the pole positions of the $\bar{K}N(I=0)$ amplitude are found to be $1424-26i$ and $1381-81i$ MeV.

For the use of the energy-dependent potential, it is necessary to determine the $\bar{K}N$ two-body energies in the $\mathscr{N}$-body systems. Though the two-body energies in the $\mathscr{N}$-body systems cannot be determined uniquely, we follow the same way as used in Refs. Dote et al. (2008, 2009) to determine the $\bar{K}N$ two-body energies for practical calculations.111We also examine the prescription of the two-body energy suggested in Ref. Barnea et al. (2012). The results of the few-body systems turn out to be in between the two choices shown in this paper. First, we introduce an “antikaon binding energy” $B_{\bar{K}}$ as

[TABLE]

where $H_{N}$ is the Hamiltonian for $(\mathscr{N}-1)$ nucleons defined by

[TABLE]

with

[TABLE]

Note that $-B_{\bar{K}}$ is in general complex. We employ the following three types of the $\bar{K}N$ two-body energy as

[TABLE]

Type I corresponds to the picture in which the $\bar{K}$ field collectively surrounds the $(\mathscr{N}-1)$ nucleons, and Type II corresponds to the picture in which the $\bar{K}$ energy is distributed equally to the $(\mathscr{N}-1)$ nucleons Dote et al. (2008, 2009). The eigenstate is determined in a self-consistent manner; the two-body energy calculated by the expectation values in Eq. (15) should equal to the energy variable in the $\bar{K}N$ interaction in $V^{(\bar{K}N)}(r,E)$.

For comparison, we also examine the Akaishi-Yamazaki (AY) potential. The potential was originally constructed in the coupled-channel $\bar{K}N$-$\pi\Sigma$-$\pi\Lambda$ system by fitting the old data of the scattering lengths in Ref. Martin (1981) and the nominal pole position of $\Lambda(1405)$ Akaishi and Yamazaki (2002). Here we adopt the single-channel version presented in Ref. Yamazaki and Akaishi (2007) in which the energy dependence of the potential through the Feshbach projection method is eliminated by hand.

We note that the imaginary parts of the SIDDHARTA and AY potentials represent the decay into the $\pi\Sigma$ and $\pi\Lambda$ channels. In the few-body kaonic nuclei, there are two types of the decay processes, the mesonic decays with a pion emission and the nonmesonic decays with multi-nucleon absorptions. In this work, the imaginary part of the eigenenergy corresponds only to the mesonic decay width, reflecting the imaginary part of the two-body potential. When the nonmesonic decays are taken into account, such effect would increase the decay width of the kaonic nuclei by several tens of MeV Dote et al. (2009); Sekihara et al. (2009, 2012); Bayar and Oset (2012).

III Stochastic variational method with correlated Gaussian basis

We investigate the structure of the kaonic nuclei with a powerful few-body approach, that is, the SVM with the CG Varga and Suzuki (1995, 1997); Suzuki and Varga (1998). The method is flexible to cope with strongly correlated few-particle quantum systems as exemplified in Ref. Mitroy et al. (2013).

The wavefunction for the $\mathscr{N}$-body system is expanded as a combination of the basis functions:

[TABLE]

where $J$ is the total angular momentum, $M$ ($M_{T}$) is the $z$-component of the total angular momentum (isospin). Since we employ central $NN$ and $\bar{K}N$ interactions, no channel coupling occurs between states with different $L$. In this paper, we consider total orbital momentum $L=0$ state by taking the basis functions with total spin $J(=S)$ to have the form

[TABLE]

where the operator $\mathscr{A}$ is an antisymmetrizer for the nucleons; $M_{S}(=M)$ is the $z$-components of the total spin; ${\bm{x}}$ is an $(\mathscr{N}-1)$-dimensional column vector, whose $i$-th element is a $3$-dimensional Jacobi coordinate ${\bm{x}}_{i}$; the symbol $\widetilde{\bm{x}}$ stands for a transpose of ${\bm{x}}$; $A$ is an $(\mathscr{N}-1)\times(\mathscr{N}-1)$-positive-definite-symmetric matrix. The Jacobi coordinate, ${\bm{x}}_{i}$, including the center-of-mass coordinate ${\bm{x}}_{\mathscr{N}}$ are related to the $i$-th single-particle coordinate ${\bm{r}}_{i}$ by a linear transformation:

[TABLE]

with

[TABLE]

We can easily apply the CG to the present three- to seven-body model because the CG keeps its functional form under any linear transformation between different coordinate sets for any number of particles. The CG basis in Eq. (19) can only be applicable to states with total orbital angular momentum $L=0$. It should be noted that the higher partial waves for each coordinate are taken into account through the cross terms, ${\bm{x}}_{i}\cdot{\bm{x}}_{j}$.

The spin wavefunction $\chi_{SM_{S}}$ is expressed using the basis of successive coupling:

[TABLE]

Here we take all possible intermediate spins $(S_{12},~{}S_{123},~{}\dots)$ for a given $S$. For the isospin wavefunction, $\eta_{M_{T}}$, we employ the particle basis, which is given as the product of single-particle isospin wavefunctions:

[TABLE]

The sets of the single-particle isospins $(m_{\tau_{1}},~{}\dots,~{}m_{\tau_{\mathscr{N}}})$ take the values

[TABLE]

for the states with $\bar{K}^{0}$ and

[TABLE]

for the states with $K^{-}$.

Each basis function has $\mathscr{N}(\mathscr{N}-1)/2$ nonlinear parameters $(A_{k})_{ij}$ and also spin and isospin quantum numbers. The adequate choice of these parameters is crucial to determine accuracy of the variational calculation. The SVM offers efficient and economical ways to find optimal sets of the variational parameters Varga and Suzuki (1995, 1997); Suzuki and Varga (1998), in which we increase the basis size one-by-one by searching for the best among many random trials for the basis function. For the Hermitian Hamiltonian, the eigenvalues for the trial wavefunctions are larger than or equal to the exact eigenvalue. Since we use the complex $\bar{K}N$ potential in this work, the eigenvalues of the trial wavefunctions give no longer the lower limit. Practically, we apply the SVM for the real part of the Hamiltonian to obtain the energy curve, and then we diagonalize the full Hamiltonian by using the basis optimized for the ground state with the real Hamiltonian. The validity of this method can be confirmed in the two-body sector where the exact value of the pole position can be obtained. Examples of the few-body calculations for $\bar{K}NN$ ($K^{-}pp$), $\bar{K}NNN$ (${}^{3}_{\bar{K}}$H), $\bar{K}NNNN$ (${}^{4}_{\bar{K}}$H) and $\bar{K}NNNNNN$ [${}^{6}_{\bar{K}}$He ($J^{\pi}$)] are shown in Fig. 1 with the AY potential. The eigenvalues with the real part of the Hamiltonian are shown in Fig. 1 (a). Corresponding complex energy curves of the full Hamiltonian are plotted in Figs. 1 (b) and (c). We find that if the energy convergence is reached with the real part of the Hamiltonian, the eigenvalues with the full Hamiltonian are also converged. The obtained energies in this method are consistent with other calculations for two- and three-body systems Akaishi and Yamazaki (2002); Yamazaki and Akaishi (2002, 2007).

For the $\bar{K}NN$, $\bar{K}NNN$, $\bar{K}NNNN$ and $\bar{K}NNNNNN$ systems, the basis sizes are $200$, $1000$, $4000$ and $10000$, respectively. The binding energies and widths change less than $0.0001$ MeV when the numbers of basis increase by one from these basis numbers.

IV Structure of few-body systems

The internal structure of the the $\bar{K}$ nuclei is reflected in the obtained wavefunction $\Psi_{JMM_{T}}$. Here we define several quantities which are useful to investigate the structure of the few-body systems.

We first define the $NN$ root-mean-square (rms) distances $\sqrt{\langle r_{NN}^{2}\rangle}$, $\bar{K}N$ rms distances $\sqrt{\langle r_{\bar{K}N}^{2}\rangle}$, $N$ rms radii $\sqrt{\langle r_{N}^{2}\rangle}$ and $\bar{K}$ rms radii $\sqrt{\langle r_{\bar{K}}^{2}\rangle}$ by using the following operators:

[TABLE]

where ${\bm{r}}_{i}$ and ${\bm{r}}_{\bar{K}}={\bm{r}}_{\mathscr{N}}$ are the single-particle coordinates of the $i$-th nucleon and the antikaon. The rms distances represent the averaged distance of the two-body subsystems, and the rms radii measure the averaged distance of the particle from the center-of-mass of the total system $\bm{x}_{\mathscr{N}}$. As discussed in Sec. II, we calculate the expectation values of these operators $\sqrt{\langle r^{2}\rangle}$ using the standard normalization condition (4).

To investigate how the nuclear system shrinks by adding an antikaon, we define the nucleon density distributions

[TABLE]

where ${\bm{r}}_{i}^{N\text{cm}}$ denotes the $i$-th nucleon coordinate measured from the center-of-mass system of nucleons; $\rho_{N}^{N\text{cm}}(r)$ is normalized as $\int 4\pi r^{2}\rho_{N}^{N\text{cm}}(r)=\mathscr{N}-1$. Here, again, we adopt the standard normalization condition (4). The comparison of $\rho_{N}^{N\text{cm}}(r)$ with the corresponding quantity of the normal nuclei with $\mathscr{N}-1$ nucleons shows the effect of the modification of the distribution of the nucleons by the presence of the antikaon.

We also calculate the nucleon and antikaon density distribution $\rho_{N}$ and $\rho_{\bar{K}}$ measured from the total center-of-mass system defined as:

[TABLE]

where ${\bm{r}}_{i(\bar{K})}^{\text{cm}}={\bm{r}}_{i(\bar{K})}-{\bm{x}}_{\mathscr{N}}$ is $i$-th nucleon (antikaon) coordinate from the total center-of-mass coordinate ${\bm{x}}_{\mathscr{N}}$.

It is also instructive to estimate the fractions of different components in the wavefunctions. We define the projections onto the component with $K^{-}$ [Eq. (25)] and that with $\bar{K}^{0}$ [Eq. (24)] as

[TABLE]

with $\hat{P}_{K^{-}}+\hat{P}_{\bar{K}^{0}}=1$. By taking the expectation value of Eq. (4), we obtain the probability of finding each component in the wavefunction

[TABLE]

The projection operators can also be used to decompose the eigenenergy into different contributions from each term of the Hamiltonian. For this purpose, we use the Gamow state normalization (7). The expectation values of the kinetic energy and potential energy of the diagonal $K^{-}$ channel $\langle T\rangle^{K^{-}}_{G}$ and $\langle V\rangle^{K^{-}}_{G}$, and of the diagonal $\bar{K}^{0}$ channel $\langle T\rangle_{G}^{\bar{K}^{0}}$ and $\langle V\rangle_{G}^{\bar{K}^{0}}$, and of the off-diagonal $K^{-}$-$\bar{K}^{0}$ channel $\langle V\rangle_{G}^{K^{-}\bar{K}^{0}}$ are given by

[TABLE]

With these definitions, the eigenenergy is decomposed as

[TABLE]

We also investigate the probability of finding each $\bar{K}N$ isospin component in the wavefunction by using the following expectation values,

[TABLE]

V Results and Discussion

V.1 Structure of strange dibaryon resonances $\bar{K}NN$

We proceed now to investigate the structure of the kaonic nuclei. For the three-body systems, we investigate the structure of the $I=1/2$ quasi-bound states for the strange dibaryon resonances $\bar{K}NN$, $K^{-}pp$-$\bar{K}^{0}pn$ and $K^{-}pn$-$\bar{K}^{0}nn$ systems, with $J^{\pi}=0^{-}$ and $1^{-}$ in the charge-basis representation. As in the previous studies Shevchenko et al. (2007, 2007); Ikeda and Sato (2007, 2009); Yamazaki and Akaishi (2007); Dote et al. (2008, 2009); Wycech and Green (2009); Ikeda et al. (2010); Barnea et al. (2012); Dote et al. (2015), we find one quasi-bound state below the $\Lambda(1405)+N$ threshold for $J^{\pi}=0^{-}$, but we could not find any states below the threshold for $J^{\pi}=1^{-}$. We summarize the detailed properties of the $K^{-}pp$-$\bar{K}^{0}pn$ and $K^{-}pn$-$\bar{K}^{0}nn$ systems with $J^{\pi}=0^{-}$ in Tables 2 and 3, respectively.

We first compare the results with different choices of the two-body energy (Types I and II) discussed in Sec. II.3. The real part of $\delta\sqrt{s}$ with Type II is about a half of that of Type I. The binding energy of $K^{-}pp$-$\bar{K}^{0}pn$ system is not sensitive to the choice of the two-body energy and the values are around $27$ MeV. Meanwhile, the decay width of Type I ($\sim 31$ MeV) becomes about a half of that of Type II ($\sim 59$ MeV). The rms distances $\sqrt{\langle r_{NN}^{2}\rangle}$, $\sqrt{\langle r_{\bar{K}N}^{2}\rangle}$, $\sqrt{\langle r_{N}^{2}\rangle}$, and $\sqrt{\langle r_{K}^{2}\rangle}$ of Type II are slightly smaller than those of Type I. The probabilities of finding several components, $P_{K^{-}}$, $P_{\bar{K}^{0}}$, $P_{\bar{K}N}^{I=0}$ and $P_{\bar{K}N}^{I=1}$ are not sensitive to the choice of the two-body energy.

We obtain the binding energies at 20-30 MeV, which are consistent with recent experimental measurement of the ${}^{3}{\rm He}(K^{-},\Lambda p)n$ reaction by J-PARC E15 Sada et al. (2016) and its theoretical analysis Sekihara et al. (2016). The obtained binding energies with the SIDDHARTA potential are $5$-$10$ MeV larger than the values obtained in Refs. Dote et al. (2009, 2008); Barnea et al. (2012) with the chiral potential constructed in Ref. Hyodo and Weise (2008). This difference mainly comes from the different treatment of the imaginary part of the potential as well as the difference of the potential model. In Refs. Dote et al. (2009, 2008); Barnea et al. (2012), the Schrödinger equation is solved only with the real part of Hamiltonian and estimate the decay width by taking the expectation value with the imaginary part of the potential, while we solve the Schrödinger equation by direct diagonalization with the full complex Hamiltonian. If we do not take into account the energy dependence of the $\bar{K}N$ potential, the binding energies obtained by direct diagonalization of the complex potential $V_{\bar{K}N}(r,E)$ become smaller than those obtained by using the real part of the potential $\text{Re}[V_{\bar{K}N}(r,E)]$. However, the $\bar{K}N$ two-body energy $E$ in the three-body system also becomes a complex value with the complex potential $V_{\bar{K}N}(r,E)$. Considering the energy dependence of the potential, the $\bar{K}N$ interaction $V_{\bar{K}N}(r,E)$ becomes more attractive than the $\bar{K}N$ interaction $V_{\bar{K}N}(r,\text{Re}[E])$ on the real energy axis. This is found in Fig. 2 which plots the real part of $V_{\bar{K}N}$ on the complex energy plane. As a result of self-consistent calculation, the binding energies of the complex potential $V_{\bar{K}N}(r,E)$ become larger than those of only the real part of potential $\text{Re}[V_{\bar{K}N}(r,E)]$. Therefore, we obtain the binding energies $5$-$10$ MeV larger than those obtained in Refs. Dote et al. (2009, 2008); Barnea et al. (2012).

We then investigate the origin of the binding using the decomposition in Eq. (39). From Tables 2 and 3, we see that Re $\langle T\rangle^{K^{-},\bar{K}^{0}}_{G}$ almost cancels out Re $\langle V\rangle^{K^{-},\bar{K}^{0}}_{G}$ in both $K^{-}$ and $\bar{K}^{0}$ channels. Therefore the $K^{-}$-$\bar{K}^{0}$ channel coupling is essential for the energy gain. Meanwhile, both of the diagonal and the off-diagonal components contribute to the decay width.

It is also instructive to decompose the wavefunction into the isospin components. The dominant isospin component of the $\bar{K}NN$ ground state is considered to have $I_{NN}=1$ and total isospin $I=1/2$. This is because the $I_{\bar{K}N}=0$ channel has stronger attraction than that of the $I_{\bar{K}N}=1$ channel, and the $I_{NN}=1$ channel gives more $I_{\bar{K}N}=0$ component than that of the $I_{NN}=0$ channel Yamazaki and Akaishi (2002). This can easily be explained by recoupling the isospins in the following way:

[TABLE]

[TABLE]

If the ground state is a pure $I=1$ $NN$ state, the ratio of the probabilities of finding the $\bar{K}N$ $I=0$ and $I=1$ channels is given by $P_{\bar{K}N}^{I=0}:P_{\bar{K}N}^{I=1}=3:1$. We can further decompose this state by the third component of the isospin of the antikaon as

[TABLE]

where the first (second) term corresponds to $K^{-}pp$ ($\bar{K}^{0}pn$). This leads to the ratio of the probabilities of the $K^{-}pp$ and $\bar{K}^{0}pn$ components as $P_{K^{-}}:P_{\bar{K}^{0}}=2:1$. The obtained $P_{\bar{K}N}^{I}$s in Tables 2 and 3 well satisfy these relations. The small deviations from the ideal ratios $2:1$ and $3:1$ come from the contributions of isospin-singlet $NN$ component with odd wave and the Coulomb interaction that induces the isospin mixing.

Because the Coulomb interaction is included in our formalism, the energy splitting of the two members of the isospin doublet, $K^{-}pp$-$\bar{K}^{0}pn$ and $K^{-}pn$-$\bar{K}^{0}nn$, appears. The splitting between these two systems is very small, $\Delta B=B(K^{-}pp$-$\bar{K}^{0}pn)-B(K^{-}pn$-$\bar{K}^{0}nn)\sim 0.5$ MeV. There are two attractive and one repulsive Coulombic pairs in the $K^{-}pp$ channel, and one attractive Coulombic pairs in the $K^{-}pn$ channel. Because $P_{K^{-}}=0.65$ in $K^{-}pp$-$\bar{K}^{0}pn$ is much larger than $P_{K^{-}}=0.38$ in $K^{-}pn$-$\bar{K}^{0}nn$, the Coulomb interaction affects more attractively the $K^{-}pp$-$\bar{K}^{0}pn$ system than the $K^{-}pn$-$\bar{K}^{0}nn$ system.

Next, we show the particle density distributions of the $K^{-}pp$-$\bar{K}^{0}pn$ system. Figure 3 plots the density distributions of nucleons $\rho_{N}^{N\text{cm}}(r)$ defined in Eq. (30). The density distribution of the deuteron is also plotted for comparison. The central nucleon density $(r\lesssim 0.3$ fm) is suppressed due to the repulsive core of the nuclear force employed. The nuclear system in the $\bar{K}NN$ system becomes more compact than in the deuteron. The shrinkage effect of nucleons with Type II is slightly stronger than that with Type I. This is because the $\bar{K}N$ interactions with Type II is more attractive than those with Type I due to different $\delta\sqrt{s}$. In Fig. 4, we also show the nucleon and antikaon density distribution $\rho_{N}$ and $\rho_{\bar{K}}$ defined in Eqs. (32) and (33). The nucleon density is suppressed at around the origin, while the antikaon density distribution is not suppressed since there are no repulsive core in the $\bar{K}N$ potential.

We also perform the same calculations by employing the phenomenological potential model, so-called Akaishi-Yamazaki (AY) potential Akaishi and Yamazaki (2002); Yamazaki and Akaishi (2007). The results are shown in Tables 2 and 3 and Figs. 3 and 4. The AY potential is more attractive than the SIDDHARTA potential in the subthreshold energy region. The binding energies of the $\bar{K}NN$ systems are approximately twice of those with the SIDDHARTA potential, and the decay widths are $\Gamma\sim 62$ MeV. The particle density distributions become more compact, and the rms radii are about $0.85$ times smaller than those for the SIDDHARTA potential. Our results for AY potential are comparable with the results in Ref. Yamazaki and Akaishi (2007).

V.2 Structure of $\bar{K}NNN$ quasi-bound state

Next, we investigate the structure of the four-body system, strange tribaryon $\bar{K}NNN$ system with $J^{\pi}=1/2^{-}$. We find a quasi-bound state in the $K^{-}ppn$-$\bar{K}^{0}pnn$ coupled system ($\equiv~{}^{3}_{\bar{K}}\text{H}$). Our results are listed in Table 4. We also investigate the $K^{-}ppp$-$\bar{K}^{0}ppn$ coupled system ($\equiv~{}^{3}_{\bar{K}}\text{He}$) with $J^{\pi}=1/2^{-}$, but we do not find any states below the $(\bar{K}NN)+N$ threshold.

We see a similar trend, shown in the three-body sector, in the dependence of the choice of the two-body energy (Types I and II). For Type II the real part of $\delta\sqrt{s}$ is less than a half of those with Type I. Therefore the $\bar{K}N$ attraction with Type II is stronger than that with Type I, and the binding energy with Type II is larger than that with Type I. The decay width with Type II ($\sim 69$ MeV) becomes around three times as large as that with Type I ($\sim 26$ MeV). The obtained binding energies by using the SIDDHARTA potential are $15$-$20$ MeV larger than the values obtained in Ref. Barnea et al. (2012). As discussed in the three-body sector, this difference mainly comes from the different treatment of the imaginary part of the potential as well as the difference of the potential model. Since the number of $\bar{K}N$ pairs is larger, the effects of those differences are stronger in the four-body systems than those in the three-body systems.

The rms distances $\sqrt{\langle r_{NN}^{2}\rangle}$, $\sqrt{\langle r_{\bar{K}N}^{2}\rangle}$, $\sqrt{\langle r_{N}^{2}\rangle}$, and $\sqrt{\langle r_{K}^{2}\rangle}$ with Type II are slightly smaller than those with Type I, in accordance with the larger binding. The probabilities of finding the $K^{-}ppn$ ($P_{K^{-}}$) and $\bar{K}^{0}pnn$ ($P_{\bar{K}^{0}}$) channels are not sensitive to the choice of the two-body energies. A large contribution to the real part from $2\langle V\rangle_{G}^{K^{-}\bar{K}^{0}}$ indicates that the $K^{-}$-$\bar{K}^{0}$ channel coupling is essential for gaining the binding energy for four-body systems, while the diagonal channels also give contributions to the decay width.

The $K^{-}ppn$ and $\bar{K}^{0}pnn$ channels are isospin mirror states. Therefore, the probabilities of these two components follow $P_{K^{-}}:P_{\bar{K}^{0}}=1:1$ with the isospin symmetric $\bar{K}N$ and $NN$ interactions. In fact, the numerical results in Table 4 are consistent with this expectation within a small isospin mixing by the Coulomb interaction. There are two attractive and one repulsive Coulombic pairs in the $K^{-}ppn$ channel, while there are no Coulomb interacting pair in the $\bar{K}^{0}pnn$ channel. The Coulomb interaction in total affects attractive in the $K^{-}ppn$ channel, and the $P_{K^{-}}$ becomes slightly larger than the $P_{\bar{K}^{0}}$.

Figure 5 displays $\rho_{N}^{N\text{cm}}(r)$ and $r^{2}\rho_{N}^{N\text{cm}}(r)$ of the ${}^{3}_{\bar{K}}\text{H}$ system. In the $\bar{K}NNN$ system, the nuclear system becomes more compact, and the central density becomes about two times larger than that in the 3He. Since the $\bar{K}N$ interaction with Type II is more attractive than that with Type I due to small magnitude of the real part of $\delta\sqrt{s}$, the shrinkage effect of nucleons with Type II is slightly stronger than that with Type I. In Fig. 6, we show the nucleon and antikaon density distribution, $\rho_{N}$ and $\rho_{\bar{K}}$. Those density distributions are similar to each other, and the antikaon rms radius $\sqrt{\langle r_{\bar{K}}^{2}\rangle}$ is well comparable with the nucleon rms radius $\sqrt{\langle r_{N}^{2}\rangle}$. The antikaon moves in the whole region of the nuclear system in order to gain the energy from the strong $\bar{K}N$ interaction.

With the AY potential, the binding energy of the $\bar{K}NNN$ system is about 30 MeV larger than the binding energies with the SIDDHARTA potential, and the decay width is $\Gamma\sim 79$ MeV. The particle density distributions and the rms radii are similar to those obtained with the SIDDHARTA potential with Type II. It is now clear that the deeply bound ($B>100$ MeV in Ref. Yamazaki and Akaishi (2002); Dote et al. (2004)) and high density (8.2 times of the normal density shown in Ref. Dote et al. (2004)) $\bar{K}NNN$ state is not realized in the accurate few-body calculation. Such an extreme result can be regarded as an artifact due to the approximated treatment of the few-body systems. We, however, emphasize that the existence of the bound state is confirmed also with the realistic SIDDHARTA $\bar{K}N$ interaction, and the central density of nucleons can be about two times larger than that in the 3He.

V.3 Structure of $\bar{K}NNNN$ quasi-bound state

Next, we investigate the structure of the five-body systems, strange tetrabaryon $\bar{K}NNNN$ systems with $J^{\pi}=0^{-}$. We find isospin-doublet quasi-bound states in the $K^{-}pppn$-$\bar{K}^{0}ppnn$ ($\equiv~{}^{4}_{\bar{K}}\text{He}$) and $K^{-}ppnn$-$\bar{K}^{0}pnnn$ ($\equiv~{}^{4}_{\bar{K}}\text{H}$) systems. Our results are listed in Tables 5 and 6.

The binding energy with Type II is 5 MeV larger than that with Type I. The decay width with Type II ($\sim 75$ MeV) is about three times larger than that with Type I ($\sim 28$ MeV). The rms distances $\sqrt{\langle r_{NN}^{2}\rangle}$, $\sqrt{\langle r_{\bar{K}N}^{2}\rangle}$, $\sqrt{\langle r_{N}^{2}\rangle}$, and $\sqrt{\langle r_{K}^{2}\rangle}$ with Type II are smaller than those with Type I. The probabilities $P_{K^{-}}$ and $P_{\bar{K}^{0}}$ are not sensitive to the choice of Types I and II. In contrast to these features, the energy decomposition of the $\bar{K}NNNN$ exhibits different characteristics from the three- and four-body systems. The incomplete cancellation of the kinetic energy and potential energy in the dominant component ($\langle T\rangle^{\bar{K}^{0}}_{G}+\langle K\rangle^{\bar{K}^{0}}_{G}$ in ${}^{4}_{\bar{K}}\text{He}$ and $\langle T\rangle^{K^{-}}_{G}+\langle K\rangle^{K^{-}}_{G}$ in ${}^{4}_{\bar{K}}\text{H}$) leaves sizable contributions to the binding energy, which are comparable with the off-diagonal components $2\langle V\rangle_{G}^{K^{-}\bar{K}^{0}}$. For the decay widths, the diagonal channels are also important as in the four-body systems.

From the results of $P_{K^{-}}$ and $P_{\bar{K}^{0}}$, we see that the dominant component in the ${}^{4}_{\bar{K}}\text{He}$ (${}^{4}_{\bar{K}}\text{H}$) system is the $\bar{K}^{0}ppnn$ ($K^{-}ppnn$) channel, although the $K^{-}pppn$ ($\bar{K}^{0}pnnn$) channel contains more $\bar{K}N$ $I=0$ components than the $\bar{K}^{0}ppnn$ ($K^{-}ppnn$) channel. This is because the nucleon contribution of the $\bar{K}^{0}ppnn$ ($K^{-}ppnn$) channel, which can form an $\alpha$-particle configuration giving the binding energy about 30 MeV, is larger than the nucleon contribution of the $K^{-}pppn$ ($\bar{K}^{0}pnnn$) channel, and thus the $\bar{K}^{0}ppnn$ ($K^{-}ppnn$) channel is favored. It is also for this reason that the $\bar{K}^{0}$ ($K^{-}$) diagonal component gains the binding energy in the ${}^{4}_{\bar{K}}\text{He}$ (${}^{4}_{\bar{K}}\text{H}$) system, as we see above.

The Coulomb splitting between these two systems is larger than that in the $\bar{K}NN$ systems, $\Delta B=B(^{4}_{\bar{K}}\text{He})-B(^{4}_{\bar{K}}\text{H})\sim 2$ MeV. There are one repulsive Coulombic pair in the $\bar{K}^{0}ppnn$ channel which is the dominant component of the ${}^{4}_{\bar{K}}\text{He}$ system, and two attractive and one repulsive Coulombic pairs in the $K^{-}ppnn$ channel which is the dominant component of the ${}^{4}_{\bar{K}}\text{H}$ system. Therefore, Coulomb interaction is repulsive in the ${}^{4}_{\bar{K}}\text{He}$ system and attractive in the ${}^{4}_{\bar{K}}\text{H}$ system, and the Coulomb splitting becomes larger than that of the three-body systems.

Figures 7 and 8 plot the particle density distributions in the ${}^{4}_{\bar{K}}\text{H}$ system. The nucleons in the $\bar{K}NNNN$ system become more compact, and the central density increases to about 1.3-1.5 times higher than that in the 4He. As in the three- and four-body systems, the shrinkage effect of nucleons with Type II is slightly stronger than that with Type I. The antikaon density distribution is similar to the nucleon density distribution.

When we use the AY potential, the binding energies of the $\bar{K}NNNN$ system are about 12-24 MeV larger than those with the SIDDHARTA potential. Because the quasi-bound state appears above the $\pi\Sigma NNN$ threshold, it has a sizable decay width of about $\Gamma\sim 87$ MeV, in contrast to the narrow state predicted in Ref. Akaishi and Yamazaki (2002). The probability $P_{K^{-}}$ ($P_{\bar{K}^{0}}$) in the ${}^{4}_{\bar{K}}\text{He}$ (${}^{4}_{\bar{K}}\text{H}$) system becomes larger than the result with the SIDDHARTA potential because the AY potential model has more attractive $\bar{K}N$ $I=0$ interaction than that in the SIDDHARTA potential. As a result, $P_{\bar{K}N}^{I=0}$ is enhanced. The rms radius of the antikaon is smaller than the corresponding result of the SIDDHARTA potential with Type I, whereas the nucleon radius is slightly larger.

V.4 Nuclear force dependence

Here we discuss the $NN$ interaction dependence of our results by comparing the results with the AV4’ potential and those with other $NN$ interaction models such as Afnan-Tang S3 (ATS3) Afnan and Tang (1968) and Minnesota (MN) potential models Thompson et al. (1977). These potential models are often used in studying light nuclei, and well reproduce the binding energy of the $s$-shell nuclei. It is noted that the strengths of the repulsive core are quite different between these three models as displayed in Fig. 9. The AV4’ potential has the strongest repulsive core, which is comparable to the realistic nuclear forces such as the Argonne V18 potential model Wiringa et al. (1995). The ATS3 potential model has also a strong short-range repulsion at around the origin. The repulsive core of the MN potential is quite soft. Since the $\bar{K}N$ interaction is strongly attractive, and the kaonic nuclei become more compact than ordinary nuclei. Therefore, there is a possibility that these different repulsive cores affect the results of the kaonic nuclei.

Figure 10 displays the binding energies and decay widths of the $K^{-}pp$-$\bar{K}^{0}pn$ ($J^{\pi}=0^{-}$), ${}_{\bar{K}}^{3}$H ($J^{\pi}=1/2^{-}$) and ${}_{\bar{K}}^{4}$H ($J^{\pi}=0^{-}$) systems with those three types of nuclear forces. Here, we use the SIDDHARTA potential as the $\bar{K}N$ potential, and the energy dependence of the potential is determined by Type I. The binding energies and decay widths are almost the same in those with three nuclear forces as well as the binding energies of ordinary $s$-shell nuclei without the antikaon as listed in Table 7.

The qualitative difference becomes apparent in the density distributions. Figures 11 and 12 plot the nucleon and antikaon density distributions of the $K^{-}pp$-$\bar{K}^{0}pn$ ($J^{\pi}=0^{-}$), ${}^{3}_{\bar{K}}\text{H}$ ($J^{\pi}=1/2^{-}$) and ${}^{4}_{\bar{K}}\text{H}$ ($J^{\pi}=0^{-}$) systems. The density distributions with the AV4’ potential model are similar to those with the ATS3 model, while the central densities with the MN potential with repulsive core are significantly higher than those for the other potential models. Since the short-range repulsive core of the MN potential is not as strong as those of the other potential models, nucleons can come very close to each other due to the strong $\bar{K}N$ attraction. In the ${}^{3}_{\bar{K}}\text{H}$ system, the central density obtained with the MN potential model becomes $\rho_{N}^{N\text{cm}}(r=0)\sim 1.2$ fm*-3*, approximately two times larger than those with the AV4’ and ATS3 potential models. The value is close to $1.4$ fm*-3* predicted in Ref. Dote et al. (2004, 2004) by using the antisymmetrized-molecular-dynamics calculation with the effective treatment of the $\bar{K}N$ and $NN$ interactions with the $g$-matrix. Since the $\bar{K}N$ interaction is very strong, the nucleons can be compressed too much with a soft core potential and form such an unrealistically high density state. Use of the realistic nuclear force is necessary in order to avoid such an artificial solution.

V.5 $\bar{K}NNNNN$ quasi-bound state

For the six-body $\bar{K}NNNNN$ systems, we could not find any states below the strange tetrabaryon and a nucleon $(\bar{K}NNNN)+N$ threshold energy in the $L=0$ state. Since the 5He ground state is observed as a resonant state with $J^{\pi}=3/2^{-}$, the orbital angular momentum of the ground state is expected to be $L=1$.

Investigation with $L>0$ states is possible by introducing the global vectors that efficiently describe the rotational motion of the system with any $L^{\pi}$ Varga and Suzuki (1995); Suzuki and Varga (1998); Suzuki et al. (2008); Aoyama et al. (2012), but this extension is beyond the scope of this paper.

V.6 Structure of $\bar{K}NNNNNN$ quasi-bound state

Finally, we investigate the structure of the seven-body systems, strange hexabaryon $\bar{K}NNNNNN$. The ground state of the six-nucleon systems without an antikaon is 6Li with $J^{\pi}=1^{+}$, and the $J^{\pi}=0^{+}$ isospin-triplet states, 6He, 6Li and 6Be, are the excited states. By adding an antikaon, we can construct two isospin doublets with $J^{\pi}=1^{-}$ and $0^{-}$. We find quasi-bound states in these quantum numbers, while we do not find any states below the $(\bar{K}NNNN)+2N$ threshold for $I=3/2$ with $J^{\pi}=0^{-}$, such as $K^{-}ppnnnn$-$\bar{K}^{0}pnnnnn$ ($\equiv~{}^{6}_{\bar{K}}\text{H}$) system.

Tables 8, 9, 10, and 11 list our results of $K^{-}ppppnn$-$\bar{K}^{0}pppnnn$ ($\equiv~{}^{6}_{\bar{K}}\text{Li}$) system with $J^{\pi}=0^{-}$, $K^{-}pppnnn$-$\bar{K}^{0}ppnnnn$ ($\equiv~{}^{6}_{\bar{K}}\text{He}$) system with $J^{\pi}=0^{-}$, ${}^{6}_{\bar{K}}\text{Li}$ system with $J^{\pi}=1^{-}$, and ${}^{6}_{\bar{K}}\text{He}$ system with $J^{\pi}=1^{-}$, respectively. The binding energies for the $J^{\pi}=0^{-}$ ($1^{-}$) states with Type II is 9-10 MeV (7-8 MeV) larger than those with Type I. The decay widths with Type II are three times larger than those with Type I. The rms distances $\sqrt{\langle r_{NN}^{2}\rangle}$, $\sqrt{\langle r_{\bar{K}N}^{2}\rangle}$, $\sqrt{\langle r_{N}^{2}\rangle}$, and $\sqrt{\langle r_{K}^{2}\rangle}$ with Type II are slightly smaller than those with Type I. The probabilities $P_{K^{-}}$ and $P_{\bar{K}^{0}}$ are not sensitive to the choice of Types I and II. As in the case of four- and five-body systems, the diagonal channels give important contributions to the decay width. In contrast, both of the diagonal and the off-diagonal components produce about a half of the binding energy. The diagonal components of the channel with ${}^{6}\text{Li}$, that is, the $K^{-}$ channel in ${}^{6}_{\bar{K}}\text{He}$ and the $\bar{K}^{0}$ channel in ${}^{6}_{\bar{K}}\text{Li}$, are important especially for the $J^{\pi}=1^{-}$ states.

The dominant component of the ${}^{6}_{\bar{K}}\text{Li}$ (${}^{6}_{\bar{K}}\text{He}$) system with $J^{\pi}=0^{-}$ is the $K^{-}ppppnn$ ($\bar{K}^{0}ppnnnn$) channel, while that of the ${}^{6}_{\bar{K}}\text{Li}$ (${}^{6}_{\bar{K}}\text{He}$) system with $J^{\pi}=1^{-}$ is the $\bar{K}^{0}pppnnn$ ($K^{-}pppnnn$) channel. For the spin-singlet states ($J^{\pi}=0^{-}$), the core nuclei in ${}^{6}_{\bar{K}}\text{Li}$ and ${}^{6}_{\bar{K}}\text{He}$ are the isospin-triplet states of 6Be, 6Li and 6He, and the channels with larger fraction of the $\bar{K}N$ $I=0$ components are favored. Meanwhile, for the spin-triplet states ($J^{\pi}=1^{-}$), the core nucleus with $J^{\pi}=1^{+}$ ($\sim^{6}$Li) is the isospin-singlet state. The spin-triplet 6Li is the ground state of the six-nucleon systems, while 6Be and 6He with $J^{\pi}=1^{+}$ are not bound. Therefore, the nucleons in the $\bar{K}^{0}pppnnn$ ($K^{-}pppnnn$) channel feel larger attraction than that in the other channel. This determines the dominant component in the $J^{\pi}=1^{-}$ state. This is also the reason why the $\bar{K}^{0}$ ($K^{-}$) diagonal component gains the large binding energy in the spin-singlet ${}^{6}_{\bar{K}}\text{Li}$ (${}^{6}_{\bar{K}}\text{He}$) system as similar to the $\bar{K}NNNN$ system.

The Coulomb splitting between ${}^{6}_{\bar{K}}\text{Li}$ and ${}^{6}_{\bar{K}}\text{He}$ in the $J^{\pi}=0^{-}$ channel ($0.3$-$0.8$ MeV) is smaller than the splitting in $J^{\pi}=1^{-}$ ($2.0$-$3.2$ MeV). In the dominant $K^{-}ppppnn$ component of the ${}^{6}_{\bar{K}}\text{Li}$ with $J^{\pi}=0^{-}$, there are four attractive and six repulsive Coulombic pairs, while the dominant $\bar{K}^{0}ppnnnn$ channel in ${}^{6}_{\bar{K}}\text{He}$ contains one repulsive pair. Therefore, the Coulomb interaction in ${}^{6}_{\bar{K}}\text{Li}$ system is expected to be slightly repulsive than ${}^{6}_{\bar{K}}\text{He}$ system. On the other hand, in the dominant $\bar{K}^{0}pppnnn$ component of the ${}^{6}_{\bar{K}}\text{Li}$ system with $J^{\pi}=1^{-}$, there are three repulsive Coulombic pairs, and the dominant $K^{-}pppnnn$ channel in ${}^{6}_{\bar{K}}\text{He}$ has three attractive and three repulsive pairs. In the ${}^{6}_{\bar{K}}\text{He}$ system, the rms distance $\sqrt{\langle r^{2}_{\bar{K}N}\rangle}$ is smaller than $\sqrt{\langle r^{2}_{NN}\rangle}$, and therefore, the $\bar{K}N$ Coulomb attraction is stronger than the $NN$ repulsion, and the Coulomb interaction works in total attractively in the $K^{-}pppnnn$ channel. Therefore, the Coulomb splitting between ${}^{6}_{\bar{K}}\text{Li}$ and ${}^{6}_{\bar{K}}\text{He}$ in $J^{\pi}=1^{-}$ becomes larger than the splitting in $J^{\pi}=0^{-}$.

Next, we compare the spin-singlet and triplet states. With Type I, the binding energy of ${}^{6}_{\bar{K}}\text{Li}$ with $J^{\pi}=1^{-}$ is 1 MeV larger than the $J^{\pi}=0^{-}$ state, and ${}^{6}_{\bar{K}}\text{He}$ with $J^{\pi}=1^{-}$ is 2.2 MeV larger than in $J^{\pi}=0^{-}$. With Type II, the binding energy of ${}^{6}_{\bar{K}}\text{Li}$ with $J^{\pi}=1^{-}$ is 2.2 MeV smaller than the $J^{\pi}=0^{-}$ state, and ${}^{6}_{\bar{K}}\text{He}$ with $J^{\pi}=1^{-}$ is 0.8 MeV larger than the $J^{\pi}=0^{-}$ state. Except for ${}^{6}_{\bar{K}}\text{Li}$ with Type II, the binding energies of the spin-triplet states are larger than the spin-singlet states. This is in accordance with the level structure of the six-nucleon systems. With Type II, the magnitude of the real part of $\delta\sqrt{s}$ used in the two-body $\bar{K}N$ interaction is smaller, and the $\bar{K}N$ interaction becomes more attractive than that with Type I. Because the $J^{\pi}=0^{-}$ state contains larger fraction of the $I=0$ $\bar{K}N$ components than the $J^{\pi}=1^{-}$ state, as in the case of the strange dibaryon $\bar{K}NN$ systems, the $\bar{K}N$ interaction with Type II is so strong that the binding energy of ${}^{6}_{\bar{K}}\text{Li}$ with $J^{\pi}=0^{-}$ becomes larger than the $J^{\pi}=1^{-}$ state. In other words, by adding an antikaon, the spin of the ground state of the six-nucleon system may change depending on the strength of the $\bar{K}N$ interaction. The inversion of the level structure of the ground state and the first excited state by the antikaon is also seen in the two-nucleon systems (strange dibaryon $\bar{K}NN$). The ground state of the two-nucleon sector without the antikaon is the spin-triplet deuteron, while the spin-singlet channel is unbound. As discussed in Sec. V.1, by injecting an antikaon, the ground state is spin singlet which maximizes the fraction of the $\bar{K}N(I=0)$ component. Recalling that the 6He and 6Li are well approximated by an $\alpha+N+N$ three-body model (See, for example, Ref. horiuchi07 and references therein), the difference of $J^{\pi}=1^{-}$ and $J^{\pi}=0^{-}$ states of seven-body systems can be essentially caused by the difference of the $\bar{K}NN$ subsystems. Similar to the three-body systems, the level inversion can take place also in the seven-body systems, which is driven by the balance between the nuclear structure and the attraction in the $\bar{K}N$ system.

This property is more pronounced if the strength of the $\bar{K}N$ attraction is further increased. For instance, when we use the AY potential which is more attractive than the SIDDHARTA potential, the binding energies of ${}^{6}_{\bar{K}}\text{Li}$ and ${}^{6}_{\bar{K}}\text{He}$ with $J^{\pi}=0^{-}$ become $10$ and 7 MeV larger than those with $J^{\pi}=1^{-}$. Namely, the stronger $\bar{K}N$ attraction leads to more drastic level inversion in the seven-body systems. In this way, there is a possibility to extract the information on the $\bar{K}N$ interaction not only from the binding energies but also from the ground state quantum number $J^{\pi}$ and from the splitting between $J^{\pi}=0^{-}$ and $1^{-}$ states.

Finally, we show, in Figs. 13,  14, 15, and 16, the particle density distributions of the ${}^{6}_{\bar{K}}\text{He}$ system with $J^{\pi}=0^{-}$ and $1^{-}$. The nucleon density distributions of the 6He ($J^{\pi}=0^{+}$) and 6Li ($J^{\pi}=1^{+}$) systems are also plotted for comparison. In the $\bar{K}NNNNNN$ system with $J^{\pi}=0^{-}$, the central nucleon density becomes slightly larger than that with $J^{\pi}=1^{-}$ and about two times larger than that in the 6Li. Meanwhile, the central antikaon density with $J^{\pi}=0^{-}$ is slightly smaller than that with $J^{\pi}=1^{-}$.

In Fig. 17, we summarize the nucleon and antikaon density distributions of various kaonic nuclei from three- to seven-body systems. The central nucleon densities of the seven-body systems become about a half, and antikaon densities become one third of the densities of the four- and five-body systems. The central nucleon densities of the five-body systems are highest in the light kaonic nuclei up to seven-body systems, while the densities are not as high as those suggested by using the effective interaction based on $g$-matrix approach in Refs. Dote et al. (2004, 2004). The large nucleon density in the five-body system is mainly caused by the formation of an $\alpha$-particle configuration, rather than maximizing the $\bar{K}N(I=0)$ pairs. For the antikaon distribution, the central densities become smaller as the number of nucleons increases. Because the antikaon feels attraction from all the nucleons, its spatial extent increases in a large nucleus.

VI Summary

We have studied structure of the light kaonic nuclei, $\bar{K}NN$, $\bar{K}NNN$, $\bar{K}NNNN$, and $\bar{K}NNNNNN$ with a powerful few-body approach, the correlated Gaussian (CG) method. Fully converged three- to seven-body solutions are obtained by the stochastic variational method (SVM). As a realistic $\bar{K}N$ interaction, we employ the SIDDHARTA potential constructed based on the NLO chiral SU(3) dynamics with the SIDDHARTA constraint obtained from Refs. Bazzi et al. (2011, 2012).

We find one quasi-bound state in the $\bar{K}NN$, $\bar{K}NNN$, and $\bar{K}NNNN$ systems, and two quasi-bound states with $J^{\pi}=0^{-}$ and $1^{-}$ in the $\bar{K}NNNNNN$ system. All the states are found above the $\pi\Sigma$ emission threshold. The central densities of nucleons are enhanced by an injected antikaon, and become about two times larger than those without an antikaon. The central nucleon density reaches its maximum in the $\bar{K}NNNN$ system with $J^{\pi}=0^{-}$, where the nucleons can form an $\alpha$-particle configuration. The rms radius of the antikaon increases along with the nucleon rms radius when the mass number is increased.

By decomposing the eigenenergy into different contributions, we find that the $K^{-}$-$\bar{K}^{0}$ channel coupling is important for the binding of the light kaonic nuclei. For the $\bar{K}NN$, $\bar{K}NNN$ and $\bar{K}NNNNNN$ with $J^{\pi}=0^{-}$, the core nuclei belong to same isospin multiplet, and the mixing between $K^{-}$ and $\bar{K}^{0}$ channels and the energy gains from the off-diagonal components are large. Meanwhile, for the $\bar{K}NNNN$ and $\bar{K}NNNNNN$ with $J^{\pi}=1^{-}$, the channel with core nucleus 4He or 6Li is dominant, and the energy gains from both of the off-diagonal and diagonal components with 4He or 6Li become large.

To take into account the energy dependence of the SIDDHARTA potential, we examine two methods to determine the $\bar{K}N$ two-body energy in $\mathscr{N}$-body systems (Types I and II). Quantitatively, the binding energies with Type II become gradually larger than those with Type I as the number of particles increase, and the decay widths with Type II become 2-3 times larger than those with Type I. The qualitative features of the kaonic nuclei are not sensitive to the choice of the method.

In order to examine the predictions of deeply-bound and high-density kaonic nuclei Akaishi and Yamazaki (2002); Dote et al. (2004, 2004), we also use the AY potential model. When we employ the AY potential, the binding energies are about $20$-$30$ MeV larger than those with the SIDDHARTA potential for each system, and decay widths become around $60$-$80$ MeV. Even in this case, the obtained binding energies up to seven-body systems except that $\bar{K}NNNNNN$ with $J^{\pi}=0^{-}$ are smaller than $100$ MeV predicted by using the optical potential approach Akaishi and Yamazaki (2002). The central density of the $\bar{K}NNNN$ system is not as high as those suggested by using effective $\bar{K}N$ and $NN$ interactions based on the $g$-matrix approach in Ref. Dote et al. (2004, 2004).

The comparison of the two $\bar{K}N$ potential models (SIDDHARTA and AY) leads to interesting results in the seven-body systems. If the $\bar{K}N$ attraction is not so strong, we see the spin-triplet ground ($J^{\pi}=1^{-}$) and the spin-singlet ($J^{\pi}=0^{-}$) excited states reflecting the lightest core nucleus, 6Li with $J^{\pi}=1^{+}$. If the $\bar{K}N$ interaction is strong enough as the AY potential, the level ordering of the $J^{\pi}=0^{-}$ and $J^{\pi}=1^{-}$ states is inverted. Therefore, it is possible to extract the information on the $\bar{K}N$ interaction from the ground state quantum number $J^{\pi}$ as well as the energy splitting between $J^{\pi}=0^{-}$ and $1^{-}$ of the seven-body kaonic states.

In this work, we employ the single channel $\bar{K}N$ interaction where the $\pi\Sigma$ channel coupling effect are renormalized into its imaginary part. This potential model reproduces the two-pole structure of $\Lambda(1405)$, while we could not find two-pole structure in the kaonic nuclei. One of these poles with the large binding energy and width is originated from the $\pi\Sigma$ resonance pole. In order to study the effect of the other pole in the kaonic nuclei, it may be necessary to take into account the channel coupling effect of $\bar{K}N$-$\pi\Sigma$ explicitly. The work in this direction is underway and will be reported elsewhere.

Acknowledgements.

The authors thank A. Gal, A. Ohnishi, A. Doté and Y. Ikeda for helpful comments and discussions. The numerical calculation has been performed on supercomputers (NEC SX-ACE) at the Research Center for Nuclear Physics, Osaka University and (CRAY XC40) at the Yukawa Institute for Theoretical Physics, Kyoto University. This work was partly supported by the Grants-in-Aid for Scientific Research on Innovative Areas from MEXT (Grant No. 2404:24105008), by JSPS KAKENHI Grant No. 24740152, and by the Yukawa International Program for Quark-Hadron Sciences (YIPQS).