Kaon-baryon coupling schemes and kaon condensation in hyperon-mixed matter

The effective chiral Lagrangian for kaons and baryons in the CI scheme kn86 is given by

	$\displaystyle{\cal L}_{K,B}$	$\displaystyle=$	$\displaystyle\frac{1}{4}f^{2}\ {\rm Tr}\partial^{\mu}U^{\dagger}\partial_{\mu}U+\frac{1}{2}f^{2}\Lambda_{\chi{\rm SB}}({\rm Tr}M(U-1)+{\rm h.c.})+{\rm Tr}\overline{\Psi}(i{\gamma^{\mu}\partial_{\mu}}-M_{B})\Psi$		(9)
		$\displaystyle+$	$\displaystyle{\rm Tr}\overline{\Psi}i\gamma^{\mu}[V_{\mu},\Psi]+D{\rm Tr}\overline{\Psi}\gamma^{\mu}\gamma^{5}\{A_{\mu},\Psi\}+F{\rm Tr}\overline{\Psi}\gamma^{\mu}\gamma^{5}[A_{\mu},\Psi]$		(10)
		$\displaystyle+$	$\displaystyle a_{1}{\rm Tr}\overline{\Psi}(\xi M^{\dagger}\xi+{\rm h.c.})\Psi+a_{2}{\rm Tr}\overline{\Psi}\Psi(\xi M^{\dagger}\xi+{\rm h.c.})+a_{3}({\rm Tr}MU+{\rm h.c.}){\rm Tr}\overline{\Psi}\Psi\ ,$		(11)

where the first and second terms on the r. h. s. of Eq. (11) are the kinetic and mass terms of kaons $U$ in the nonlinear representation, respectively, and $\Lambda_{\chi{\rm SB}}$ is the chiral symmetry breaking scale ($\sim$ 1 GeV), $M$ the quark mass matrix, $M\equiv{\rm diag}(m_{u},m_{d},m_{s})$ with the quark masses $m_{i}$kn86 . The free kaon mass is identified with $m_{K}=[\Lambda_{\chi{\rm SB}}(m_{u}+m_{s})]^{1/2}$ and is set to be the empirical value (=494 MeV). The third term in Eq. (11) denotes the kinetic and mass terms for baryons, where $M_{\rm B}$ is a baryon mass generated by spontaneous breaking of chiral symmetry. The fourth term in Eq. (11) gives the $s$-wave $K$-$B$ vector interaction corresponding to the Tomozawa-Weinberg term with $V_{\mu}$ being the mesonic vector current defined by $V_{\mu}\equiv 1/2(\xi^{\dagger}\partial_{\mu}\xi+\xi\partial_{\mu}\xi^{\dagger})$ with $\xi\equiv U^{1/2}$. The fifth and sixth terms are the $K$-$B$ axial-vector interaction with the mesonic axial-vector current defined by $A_{\mu}\equiv i/2(\xi^{\dagger}\partial_{\mu}\xi-\xi\partial_{\mu}\xi^{\dagger})$. Throughout this paper, we simply omit these axial-vector coupling terms and retain only the $s$-wave $K$-$B$ vector interaction in order to figure out the consequences from $s$-wave kaon condensation. The last three terms in Eq. (11) give the $s$-wave $K$-$B$ scalar interaction, which explicitly breaks chiral symmetry.

By the use of Eq. (1), the Lagrangian density (11) is separated into the baryon part ${\cal L}_{B}$ and kaon part ${\cal L}_{K}$ in the mean-field approximation. For ${\cal L}_{B}$ one obtains

$${\cal L}_{B}=\sum_{b=p,n,\Lambda,\Sigma^{-},\Xi^{-}}\overline{\psi_{b}}(i\gamma^{\mu}\partial_{\mu}-M_{b}^{\ast})\psi_{b}\ ,$$

(12)

where the $s$-wave $K$-$B$ scalar interaction is absorbed into the effective baryon mass $M_{b}^{\ast}$ :

$$M_{b}^{\ast}=M_{b}-\Sigma_{Kb}(1-\cos\theta)$$

(13)

with $M_{b}$ ($b$=$p,n,\Lambda,\Sigma^{-},\Xi^{-}$) being the baryon mass defined by

	$\displaystyle M_{p}$	$\displaystyle=$	$\displaystyle M_{B}-2(a_{1}m_{u}+a_{2}m_{s})-2a_{3}(m_{u}+m_{d}+m_{s})\ ,$		(14)
	$\displaystyle M_{n}$	$\displaystyle=$	$\displaystyle M_{B}-2(a_{1}m_{d}+a_{2}m_{s})-2a_{3}(m_{u}+m_{d}+m_{s})\ ,$		(15)
	$\displaystyle M_{\Lambda}$	$\displaystyle=$	$\displaystyle M_{B}-1/3\cdot(a_{1}+a_{2})(m_{u}+m_{d}+4m_{s})-2a_{3}(m_{u}+m_{d}+m_{s})\ ,$		(16)
	$\displaystyle M_{\Sigma^{-}}$	$\displaystyle=$	$\displaystyle M_{B}-2(a_{1}m_{d}+a_{2}m_{u})-2a_{3}(m_{u}+m_{d}+m_{s})\ ,$		(17)
	$\displaystyle M_{\Xi^{-}}$	$\displaystyle=$	$\displaystyle M_{B}-2(a_{1}m_{s}+a_{2}m_{u})-2a_{3}(m_{u}+m_{d}+m_{s})\ ,$		(18)

and $\Sigma_{Kb}$ being the “$K$-baryon sigma term” which simulates the $K$-$B$ attractive interaction in the scalar channelm93 ; m08 :

	$\displaystyle\Sigma_{Kp}$	$\displaystyle=$	$\displaystyle\Sigma_{K\Xi^{-}}=-(a_{1}+a_{2}+2a_{3})(m_{u}+m_{s})\ ,$		(19a)
	$\displaystyle\Sigma_{Kn}$	$\displaystyle=$	$\displaystyle\Sigma_{K\Sigma^{-}}=-(a_{2}+2a_{3})(m_{u}+m_{s})\ ,$		(19b)
	$\displaystyle\Sigma_{K\Lambda}$	$\displaystyle=$	$\displaystyle-\left(\frac{5}{6}a_{1}+\frac{5}{6}a_{2}+2a_{3}\right)(m_{u}+m_{s})\ .$		(19c)

(See Appendix 12.1.) In the following, $M_{b}$ ($b=p,n,\Lambda,\Sigma^{-},\Xi^{-}$) [Eq. (18)] are identified with the empirical baryon masses, i. e. , $M_{p}$=938.27 MeV, $M_{n}$=939.57 MeV, $M_{\Lambda}$=1115.68 MeV, $M_{\Sigma^{-}}$=1197.45 MeV, and $M_{\Xi^{-}}$=1321.71 MeV.

Following Ref. kn86 , the quark masses $m_{i}$ are chosen to be $m_{u}$ = 6 MeV, $m_{d}$ = 12 MeV, and $m_{s}$ = 240 MeV. Together with these values, the parameters $a_{1}$ and $a_{2}$ are fixed to be $a_{1}$ = $-$0.28, $a_{2}$ = 0.56 so as to reproduce the empirical octet baryon mass splittingskn86 . The remaining parameter $a_{3}$ is fixed so as to give the $KN$ sigma term, $\Sigma_{KN}$. Throughout this paper, we consider two cases as allowable values of $\Sigma_{Kn}$: $\Sigma_{Kn}$ = 300 MeV and 400 MeV, for which $a_{3}=-0.89$ and $-1.1$, respectively. In these cases, one also obtains, from Eqs. (19a)$-$(19c), $\Sigma_{K\Sigma^{-}}$ = 300 MeV, $\Sigma_{Kp}$ = $\Sigma_{K\Xi^{-}}$ = 369 MeV, and $\Sigma_{K\Lambda}$ = 380 MeV for $a_{3}=-0.89$, and $\Sigma_{K\Sigma^{-}}$ = 400 MeV, $\Sigma_{Kp}$ = $\Sigma_{K\Xi^{-}}$ = 469 MeV, and $\Sigma_{K\Lambda}$ = 480 MeV for $a_{3}=-1.1$. According to the lattice QCD result, the current quark masses have been fixed to be rather smaller values, $(m_{u},m_{d},m_{s})$ = (2.2, 4.7, 95) MeV PDG2020 , than adopted in this paper. However, it can be shown that the $\Sigma_{KN}$ is little altered by the use of different quark masses as far as $(m_{u}+m_{s})/\hat{m}$ ($\sim$14) is almost the same in both set of quark masses mmt2021 .

For ${\cal L}_{K}$ one obtainste97

	$\displaystyle{\cal L}_{K}$	$\displaystyle=$	$\displaystyle\frac{1}{2}\Bigg{\{}1+\left(\frac{\sin\theta}{\theta}\right)^{2}\Bigg{\}}\partial^{\mu}K^{+}\partial_{\mu}K^{-}+\frac{1-\left(\frac{\sin\theta}{\theta}\right)^{2}}{2f^{2}\theta^{2}}\Big{\{}(K^{+}\partial_{\mu}K^{-})^{2}+(K^{-}\partial_{\mu}K^{+})^{2}\Big{\}}$		(20)
		$\displaystyle-$	$\displaystyle m_{K}^{2}\left(\frac{\sin(\theta/2)}{\theta/2}\right)^{2}K^{+}K^{-}+iX_{0}\left(\frac{\sin(\theta/2)}{\theta/2}\right)^{2}\left(K^{+}\partial_{0}K^{-}-\partial_{0}K^{+}K^{-}\right)$		(21)
		$\displaystyle=$	$\displaystyle\frac{1}{2}(f\mu_{K}\sin\theta)^{2}-f^{2}m_{K}^{2}(1-\cos\theta)+2\mu_{K}X_{0}f^{2}(1-\cos\theta)\ .$		(22)

The last term on the r.h.s. of Eq. (22) stands for the $s$-wave $K$-$B$ vector interaction with $X_{0}$ being given by

$$X_{0}\equiv\frac{1}{2f^{2}}\sum_{b=p,n,\Lambda,\Sigma^{-},\Xi^{-}}Q_{V}^{b}\rho_{b}=\frac{1}{2f^{2}}\left(\rho_{p}+\frac{1}{2}\rho_{n}-\frac{1}{2}\rho_{\Sigma^{-}}-\rho_{\Xi^{-}}\right)\ ,$$

(23)

where $Q_{V}^{b}$ $\equiv\frac{1}{2}\left(I_{3}^{(b)}+\frac{3}{2}Y^{(b)}\right)$ is the V-spin charge with $I_{3}^{(b)}$ and $Y^{(b)}$ being the third component of the isospin and hypercharge for baryon species $b$, respectively. The form of Eq. (23) for $X_{0}$ is specified model-independently within chiral symmetry. From Eqs. (22) and (23), one can see that the $s$-wave $K$-$B$ vector interaction works attractively for protons and neutrons, while repulsively for $\Sigma^{-}$ and $\Xi^{-}$ hyperons, as far as $\mu_{K}>0$.

In Ref. mmt2015 , we improved our model for the $s$-wave $K$-$B$ interaction by introduction of the range terms and a pole contribution from $\Lambda$(1405) (denoted as $\Lambda^{\ast}$) so as to reproduce the on-shell $s$-wave $KN$ scattering lengthsm81 . It has been shown that the range terms work repulsively and that the onset density of kaon condensation realized from hyperon-mixed matter is slightly pushed up to a higher density as compared with the case without the range terms. Nevertheless, the repulsive effect from the range terms ($\propto\mu_{K}^{2}$) on the EOS of the ($Y+K$) phase is tiny over the relevant densities as far as $\mu_{K}\lesssim O(m_{\pi})$. The contribution from the $\Lambda^{\ast}$ pole to the energy is also negligible over the baryon density $\rho_{B}\gtrsim\rho_{0}$, since the kaon energy in matter lies well below the pole position of $M_{\Lambda^{\ast}}-M_{N}$. Therefore, these effects of the range terms and the pole contribution from $\Lambda^{\ast}$ are omitted in this paper.

In the RMF framework, the $B$-$B$ interaction is mediated by scalar ($\sigma$, $\sigma^{\ast}$) and vector ($\omega$, $\rho$, $\phi$) mesons. The scalar meson $\sigma^{\ast}$ ($\sim\bar{s}s$) and the vector meson $\phi$ ($\sim\bar{s}\gamma^{\mu}s$), both of which carry the strangeness and couple only to hyperons ($Y$), are introduced according to the extension of baryons to include hyperons. The quark structure of the $\phi$ meson comes from the assumption of an ideal mixing between $\omega$ and $\phi$ mesons. The scalar meson-baryon couplings lead to modification of the effective baryon masses (13) to

$$\widetilde{M}_{b}^{\ast}=M_{b}-g_{\sigma b}\sigma-g_{\sigma^{\ast}b}\sigma^{\ast}-\Sigma_{Kb}(1-\cos\theta)$$

(24)

with $g_{\sigma b}$ ($b=p,n,\Lambda,\Sigma^{-},\Xi^{-}$) and $g_{\sigma^{\ast}Y}$ ($Y=\Lambda,\Sigma^{-},\Xi^{-}$) being the scalar meson-baryon coupling constants. The vector meson-baryon couplings are introduced by the covariant derivatives at the baryon kinetic terms as $\partial_{\mu}$$\rightarrow$$D^{(b)}_{\mu}\equiv\partial_{\mu}+ig_{\omega b}\omega_{\mu}+i\widetilde{g}_{\rho b}{\vec{I}}^{\ (b)}\cdot{\vec{R}}_{\mu}+ig_{\phi b}\phi_{\mu}$, where $\omega^{\mu}$, $\vec{R}^{\mu}$, and $\phi^{\mu}$ denote the vector meson fields for $\omega$, $\rho$, and $\phi$ mesons, respectively. The arrow attached to the $\rho$-meson field $\vec{R}_{\mu}$ refers to an isovector with the isospin operator $\vec{I}^{\ (b)}$ for baryon $b$. The $g_{\omega b}$, $\widetilde{g}_{\rho b}$, and $g_{\phi b}$ in the covariant derivative are the vector meson-baryon coupling constants. It is to be noted that the $\rho$-meson-baryon coupling constant, $g_{\rho b}$, is factorized as $g_{\rho b}\equiv\widetilde{g}_{\rho b}\cdot|I_{3}^{(b)}|$ with the third component of the isospin for the baryon $b$. The $\rho$-meson-baryon coupling constants utilized in the other RMF models mostly correspond to $\widetilde{g}_{\rho b}$ in our model.

The baryon part of the Lagrangian density ${\cal L}_{B}$ is then modified from Eq. (12) to

$${\cal L}_{B}=\sum_{b=p,n,\Lambda,\Sigma^{-},\Xi^{-}}\overline{\psi_{b}}(i\gamma^{\mu}D^{(b)}_{\mu}-\widetilde{M}_{b}^{\ast})\psi_{b}\ .$$

(25)

In addition, the meson part of the Lagrangian density, ${\cal L}_{M}$, including the $\sigma$ self-interaction is introduced:

	$\displaystyle{\cal L}_{M}$	$\displaystyle=$	$\displaystyle\frac{1}{2}\left(\partial^{\mu}\sigma\partial_{\mu}\sigma-m_{\sigma}^{2}\sigma^{2}\right)-U_{\sigma}(\sigma)+\frac{1}{2}\left(\partial^{\mu}\sigma^{\ast}\partial_{\mu}\sigma^{\ast}-m_{\sigma^{\ast}}^{2}\sigma^{\ast 2}\right)$		(26)
		$\displaystyle-$	$\displaystyle\frac{1}{4}\omega^{\mu\nu}\omega_{\mu\nu}+\frac{1}{2}m_{\omega}^{2}\omega^{\mu}\omega_{\mu}-\frac{1}{4}{\vec{R}}^{\mu\nu}\cdot{\vec{R}}_{\mu\nu}+\frac{1}{2}m_{\rho}^{2}{\vec{R}}^{\mu}\cdot{\vec{R}}_{\mu}-\frac{1}{4}\phi^{\mu\nu}\phi_{\mu\nu}+\frac{1}{2}m_{\phi}^{2}\phi^{\mu}\phi_{\mu}\ .$		(27)

The second term on the r. h. s. of Eq. (27) is the scalar self-interaction potential given by $U_{\sigma}(\sigma)$=$bM_{N}(g_{\sigma N}\sigma)^{3}/3+c(g_{\sigma N}\sigma)^{4}/4$ with $b$=0.008659 and $c$=$-$0.002421g85 ; mmt09 . $U_{\sigma}(\sigma)$ is introduced so as to set the incompressibility at the nuclear saturation density to be 240 MeV, which is consistent with an empirical value. In this paper, the $U_{\sigma}(\sigma)$ is solely considered as the NLSI term. The kinetic terms of the vector mesons are given in terms of $\omega^{\mu\nu}\equiv\partial^{\mu}\omega^{\nu}-\partial^{\nu}\omega^{\mu}$, $\vec{R}^{\mu\nu}\equiv\partial^{\mu}{\vec{R}}^{\nu}-\partial^{\nu}{\vec{R}}^{\mu}$, $\phi^{\mu\nu}\equiv\partial^{\mu}\phi^{\nu}-\partial^{\nu}\phi^{\mu}$. Throughout this paper, only time components of the vector meson mean fields and the third component of the isovector $\rho$ mean field are considered for description of the ground state of the system. We simply denote these components as $\omega_{0}$, $R_{0}$, and $\phi_{0}$. Then the $\rho$-$B$ coupling term in the covariant derivative is rewritten as $i\widetilde{g}_{\rho b}I_{3}^{(b)}R_{0}=ig_{\rho b}\hat{I}_{3}^{(b)}R_{0}$ with $g_{\rho b}$ (=$\widetilde{g}_{\rho b}\cdot|I_{3}^{(b)}|$) and $\hat{I}_{3}^{(b)}\equiv I_{3}^{(b)}/|I_{3}^{(b)}|$, where $\hat{I}_{3}^{(b)}$ is assigned as $\hat{I}_{3}^{(p)}$=+1, $\hat{I}_{3}^{(n)}=\hat{I}_{3}^{(\Sigma^{-})}=\hat{I}_{3}^{(\Xi^{-})}=-1$.

In the CI scheme, there is no direct kaon-meson ($m$) coupling ($m$ = $\sigma$, $\sigma^{\ast}$, $\omega$, $\rho$, $\phi$) [see Eqs. (22) and (27)].

The values of $g_{\sigma N}$, $g_{\omega N}$, $g_{\rho N}$, which are related to the $N$-$N$ interaction, are determined so as to reproduce not only the properties of normal nuclear matter with saturation density $\rho_{0}$ = 0.153 fm^-3, the binding energy (=16.3 MeV), and the symmetry energy (=32.8 MeV), but also proton-mixing ratio and density distributions of proton and neutron for normal nucleimmt09 . One obtains $g_{\sigma N}$ = 6.39, $g_{\omega N}$ = 8.72, and $g_{\rho N}$ = 4.27. The $\sigma^{\ast}$-$N$ and $\phi$-$N$ couplings are not taken into account since they should be suppressed due to the OZI rule.

The vector meson couplings for the hyperon $Y$ are obtained from the relations in the SU(6) symmetrysdg94 :

	$\displaystyle g_{\omega\Lambda}$	$\displaystyle=$	$\displaystyle g_{\omega\Sigma^{-}}=2g_{\omega\Xi^{-}}=\frac{2}{3}g_{\omega N}\ ,$		(28a)
	$\displaystyle g_{\rho\Lambda}$	$\displaystyle=$	$\displaystyle 0\ ,\ g_{\rho\Sigma^{-}}=2g_{\rho\Xi^{-}}=2g_{\rho N}\ ,$		(28b)
	$\displaystyle g_{\phi\Lambda}$	$\displaystyle=$	$\displaystyle g_{\phi\Sigma^{-}}=\frac{1}{2}g_{\phi\Xi^{-}}=-\frac{\sqrt{2}}{3}g_{\omega N}\ .$		(28c)

The scalar ($\sigma$, $\sigma^{\ast}$) mesons-hyperon couplings are determined from the phenomenological analyses of recent hypernuclear experiments as much as possible. The scalar $\sigma$ meson coupling for the hyperon $Y$, $g_{\sigma Y}$, is related with the potential depth of the hyperon $Y$ ($Y=\Lambda$, $\Sigma^{-}$, $\Xi^{-}$) at $\rho_{\rm B}=\rho_{0}$ in SNM, $V_{Y}^{N}$, which is written in the RMF as

$$V_{Y}^{N}=-g_{\sigma Y}\langle\sigma\rangle_{0}+g_{\omega Y}\langle\omega_{0}\rangle_{0}$$

(29)

with $\langle\sigma\rangle_{0}$ and $\langle\omega_{0}\rangle_{0}$ being the meson mean fields at $\rho_{B}=\rho_{0}$ in SNM. For $Y=\Lambda$, the single $\Lambda$ orbital energies in ordinary hypernuclei are fitted well with the $\Lambda$-nucleus single particle potential with the depth $\sim$ $-27$ MeVmdg88 . Based on this result, $V_{\Lambda}^{N}$ is set to be $-27$ MeV. One then obtains $g_{\sigma\Lambda}$ = 3.84 from Eq. (29).

The depth of the $\Sigma^{-}$ potential $V_{\Sigma^{-}}^{N}$ has been shown to be repulsive, according to the recent theoretical calculationskf00 ; fk01 and phenomenological analyses on the ($K^{-}$, $\pi^{\pm}$) reactions at BNLb99 ; d99 , ($\pi^{-}$, $K^{+}$) reactions at KEKn02 ; dr04 ; hh05 , and the $\Sigma^{-}$ atom datamfgj95 . Following Ref. d99 , the $V_{\Sigma^{-}}^{N}$ is set to be 23.5 MeV as a typical value, from which one obtains $g_{\sigma\Sigma^{-}}$ = 2.28 from Eq. (29).

The depth of the $\Xi^{-}$ potential in nuclear matter is set to be attractive, $V_{\Xi^{-}}^{N}$ = $-14$ MeV with reference to the experimental results deduced from the ($K^{-}$, $K^{+}$) reactions, ($-$14)$-$($-$20) MeVf98 ; k00 . One then obtains $g_{\sigma\Xi^{-}}$ = 1.94 from Eq. (29).

The $\sigma^{\ast}$ meson couplings for the hyperon $Y$, $g_{\sigma^{\ast}Y}$, are relevant to the $Y$-$Y$ interaction as well as binding energy of hypernuclei. From recent detection of the double $\Lambda$ hypernuclei in the KEK-E176-E373 experiments, separation energies for two $\Lambda$’s, $B_{\Lambda\Lambda}$, have been obtained for several double $\Lambda$ hypernucleighm16 . For the “Hida event”, the experimental value of the separation energy for the ground state of the ${}^{\ 11}_{\Lambda\Lambda}{\rm Be}$ has been estimated to be 20.83$\pm$1.27 MeVghm16 ; h2010 . We determined $g_{\sigma^{\ast}\Lambda}$ so as to reproduce the empirical values of $B_{\Lambda\Lambda}$(${}^{\ \ 11}_{\Lambda\Lambda}$Be) with use of the present $B$-$B$ interaction model in the RMF extended to finite nucleimmt09 ; mmt14 . One finds $g_{\sigma^{\ast}\Lambda}$ = 7.2, for which $B^{\rm th}_{\Lambda\Lambda}(^{\ 11}_{\Lambda\Lambda}{\rm Be})\equiv B^{\rm th}(^{\ 11}_{\Lambda\Lambda}{\rm Be})-B^{\rm exp}(^{9}{\rm Be})$ = 20.7 MeV with $B^{\rm exp}(^{9}{\rm Be})$ = 58.16 MeVawt03 . [Throughout this paper, the superscript, “exp” (“th”), denotes an experimental value (a theoretical value obtained in our $B$-$B$ interaction model).] In this case, the $\Lambda$-separation energy for ${}^{10}_{\ \Lambda}{\rm Be}$ is estimated to be $B^{\rm th}_{\Lambda}(^{10}_{\ \Lambda}{\rm Be})\equiv B^{\rm th}(^{10}_{\ \Lambda}{\rm Be})-B^{\rm exp}(^{9}{\rm Be})$ = 9.95 MeV, whereas the experimental value for the ground state peak (mixture of 1^- and 2^- states) at J-Lab is reported as 8.55 MeVg2016 . It is to be noted that our $B$-$B$ interaction model applied to finite nuclei assumes local density approximation and spherical symmetry for profiles of baryon density distributions, whereas the ${}^{\ 11}_{\Lambda\Lambda}{\rm Be}$ nucleus has a clustering structureh2010 . Furthermore a quantitatively accurate estimation in our model may not be expected for a few-body system such as the ${}^{\ \ 6}_{\Lambda\Lambda}$He (the “Nagara event”), although a precise extraction of the $\Lambda\Lambda$ binding energy $B_{\Lambda\Lambda}$(${}^{\ \ 6}_{\Lambda\Lambda}$He) and the bond energy $\Delta B_{\Lambda\Lambda}$(${}^{\ \ 6}_{\Lambda\Lambda}$He)$\equiv B_{\Lambda\Lambda}$(${}^{\ \ 6}_{\Lambda\Lambda}$He)$-$2$B_{\Lambda}$(${}^{5}_{\Lambda}$He) has been done experimentallyht01 ; a13 .

For the $\sigma^{\ast}$-$\Xi^{-}$ coupling, the bound $\Xi^{-}$ hypernucleus was detected through the “Kiso” event, $\Xi^{-}$ + ¹⁴N $\rightarrow$ ${}^{15}_{\ \Xi}$C $\rightarrow$ ${}^{10}_{\ \Lambda}$Be + ${}^{5}_{\Lambda}$Hen15 . In Ref. sh16 , ${}^{15}_{\ \Xi}$C was assumed to be an excited state with the $\Xi^{-}$ being in the 1$p$ state, and the estimated separation energies of $\Xi^{-}$ for both ${}^{\ \ 15}_{\ \Xi(p)}$C and the ground state of ${}^{\ \ 12}_{\ \Xi(s)}$Be were shown to be consistent with the empirical values, i.e., $B^{\rm exp}_{\Xi}(^{\ \ 15}_{\ \Xi(p)}$C) = 1.11$\pm$0.25 MeV and $B^{\rm exp}_{\Xi}(^{\ \ 12}_{\ \Xi(s)}$Be) $\approx$ 5 MeV. In this case, the $B_{\Xi}(^{\ \ 15}_{\ \Xi(s)}$C) for the ground state of the ${}^{15}_{\ \Xi}$C was estimated to be (8.0 $-$ 9.4) MeV within the RMF calculationsh16 . This interpretation concerning the energy-level structure and the binding energy of the $\Xi^{-}$-¹⁴N system has been confirmed by the recent observation of the twin-$\Lambda$ hypernuclei (the “IBUKI” event) at the J-PARC E07 experimenthayakawa2021 ; yoshimoto2021 . In our $B$-$B$ interaction model, $g_{\sigma^{\ast}\Xi^{-}}$ is taken to be 4.0, for which one obtains $B^{\rm th}_{\Xi}(^{\ \ 15}_{\ \Xi(s)}$C) = 8.1 MeV and $B^{\rm th}_{\Xi}(^{\ \ 12}_{\ \Xi(s)}$Be) = 5.1 MeV, which are consistent with the empirical values.

The remaining parameter for the $\sigma^{\ast}$-$\Sigma^{-}$ coupling, $g_{\sigma^{\ast}\Sigma^{-}}$, is simply set to be zero since there is little experimental information on $\Sigma$ hypernuclei. As is seen in the numerical result (Sec. LABEL:sec:eos), the $\Sigma^{-}$ hyperons are not mixed over the relevant densities due to the strong repulsion of the $V_{\Sigma^{-}}^{N}$. Therefore it may safely be said that such simplification, $g_{\sigma^{\ast}\Sigma^{-}}=0$, little affects the results in this paper. It is to be noted that $g_{\sigma^{\ast}Y}$ ($Y=\Lambda,\Sigma^{-},\Xi^{-}$) is also related with the $s$-wave $K$-$B$ scalar attraction in the ME scheme [see Eq. (62) in Sec. 6.1].

Together with these coupling constants, the meson masses are taken to be $m_{\sigma}$ = 400 MeV, $m_{\sigma^{\ast}}$ = 975 MeV, $m_{\omega}$ = 783 MeV, $m_{\rho}$ = 769 MeV, and $m_{\phi}$ = 1020 MeV. The parameters relevant to the meson-baryon interaction used in our RMF model are listed in Table 1 and 2.

The total Lagrangian density ${\cal L}$ in the CI scheme for the description of the ($Y$+$K$) phase is given by Eqs. (22), (25), (27), together with the electron part ${\cal L}_{e}$ : ${\cal L}$=${\cal L}_{B}$+${\cal L}_{M}$+${\cal L}_{K}$+${\cal L}_{e}$. The energy density of the ($Y+K$) phase, ${\cal E}$ (=${\cal E}_{B}+{\cal E}_{M}+{\cal E}_{K}+{\cal E}_{e}$), is obtained from the ground-state expectation value of the Hamiltonian density, ${\cal H}={\cal H}_{B}+{\cal H}_{M}+{\cal H}_{K}+{\cal H}_{e}$ with

	$\displaystyle{\cal H}_{B}$	$\displaystyle=$	$\displaystyle\sum_{b}(\partial{\cal L}_{B}/\partial\dot{\psi}_{b})\dot{\psi}_{b}-{\cal L}_{B}\ ,$		(30a)
	$\displaystyle{\cal H}_{M}$	$\displaystyle=$	$\displaystyle\sum_{m}(\partial{\cal L}_{M}/\partial\dot{\varphi}_{m})\dot{\varphi}_{m}-{\cal L}_{M}\ ,$		(30b)
	$\displaystyle{\cal H}_{K}$	$\displaystyle=$	$\displaystyle(\partial{\cal L}_{K}/\partial\dot{K}^{-})\dot{K}^{-}+(\partial{\cal L}_{K}/\partial\dot{K}^{+})\dot{K}^{+}-{\cal L}_{K}\ ,$		(30c)
	$\displaystyle{\cal H}_{e}$	$\displaystyle=$	$\displaystyle(\partial{\cal L}_{e}/\partial\dot{\psi}_{e})\dot{\psi}_{e}-{\cal L}_{e}\ ,$		(30d)

where $\varphi_{m}$ ($=\sigma,\sigma^{\ast},\omega_{0},R_{0},\phi_{0}$) is the meson field mediating the $B$-$B$ interaction, and $\psi_{e}$ the electron field. From Eqs. (30a) and (12), one obtains

$${\cal E}_{B}=\sum_{b=p,n,\Lambda,\Sigma^{-},\Xi^{-}}\Bigg{\{}\frac{2}{(2\pi)^{3}}\int_{|{\bf p}|\leq p_{F}(b)}d^{3}|{\bf p}|(|{\bf p}|^{2}+\widetilde{M}_{b}^{\ast 2})^{1/2}+\rho_{b}\left(g_{\omega b}\omega_{0}+g_{\rho b}\hat{I}_{3}^{(b)}R_{0}+g_{\phi b}\phi_{0}\right)\Bigg{\}}\ ,$$

(31)

where $p_{F}(b)$ is the Fermi momentum of baryon $b$. From Eqs. (30b) and (27), one obtains

$${\cal E}_{M}=\frac{1}{2}m_{\sigma}^{2}\sigma^{2}+U(\sigma)+\frac{1}{2}m_{\sigma^{\ast}}^{2}\sigma^{\ast 2}-\frac{1}{2}m_{\omega}^{2}\omega_{0}^{2}-\frac{1}{2}m_{\rho}^{2}R_{0}^{2}-\frac{1}{2}m_{\phi}^{2}\phi_{0}^{2}\ .$$

(32)

The kaon part of the energy density, ${\cal E}_{K}$, is expressed from Eq. (30c) as

$${\cal E}_{K}=\mu_{K}\rho_{K^{-}}-{\cal L}_{K}\ ,$$

(33)

where the first term is obtained by rewriting the first two terms on the r. h. s. of Eq. (30c) by the use of the time dependence of the classical kaon field, $\dot{K}^{\pm}=\pm i\mu_{K}K^{\pm}$, which follows from Eq. (1)kn86 ; bc79 , and the number density of kaon condensates,

$$\rho_{K^{-}}=-iK^{-}(\partial{\cal L}_{K}/\partial\dot{K^{-}})+iK^{+}(\partial{\cal L}_{K}/\partial\dot{K^{+}})\ .$$

(34)

Substituting Eqs. (1) and (22) into Eq. (34), one obtains

$$\rho_{K^{-}}=\mu_{K}f^{2}\sin^{2}\theta+2f^{2}X_{0}(1-\cos\theta)\ ,$$

(35)

where the first term is the kaon kinetic part and the second term comes from the $s$-wave $K$-$B$ vector interaction. With Eqs. (35) and (22), Eq. (33) reads

$${\cal E}_{K}=\frac{1}{2}(\mu_{K}f\sin\theta)^{2}+f^{2}m_{K}^{2}(1-\cos\theta)\ .$$

(36)

The electron part, Eq. (30d), is simply written as

$${\cal E}_{e}\simeq\mu_{e}^{4}/(4\pi^{2})$$

(37)

for the ultra-relativistic electrons.