Workshop on Progress on J-PARC hadron physics in 2014!
（Ibaraki Quantum Beam Research Center (IQBRC), Tokai, Ibaraki,!
Nov. 30 - Dec.2, 2014) ����������������

Possible coexistence of !
kaon condensation and hyperons !
in multi-strangeness systems!
Takumi Muto (Chiba Inst. Tech.)!
in collaboration with : Toshiki Maruyama (JAEA) !
Toshitaka Tatsumi (Kyoto Univ.)�

1. Introduction
In neutron-stars�
hyperonic matter�(Λ, Σ, Ξ, ・・・!
in the ground state)!

1-1 Multi-strangeness system !
in hadronic matter!
Strange matter!

Kaon condensation�

（u, d, s quark matter ) !

Bose-Einstein�condensation of kaons (K-)�

・Softening of EOS�・Rapid cooling of neutron stars�

・ Coexistence of antikaons and hyperons�
In nuclei�

Toward unified description based on the same !
Kaon-baryon(B), B-B interaction model.�

Kaonic nuclei!

[ Y.Akaishi and T.Yamazaki, Phys.Rev. C65 (2002) 044005. ]!

Multi-Antikaonic Nuclei !

relativistic mean-field theory (RMF)�

[T. Muto, T. Maruyama and T. Tatsumi, !
Phys. Rev. C79, 035207 (2009). ]�
・Coexistence of antikaons ! [T. Muto, T. Maruyama and T. Tatsumi, !
Genshikaku Kenkyu 57 Supplement 3, 230(2013).]�
and hyperons �Finite nuclei�

A=15, Z=8 ()!

| UK | < 180 MeV!

li ( fm−3 )Ground state :� Multi-hypernuclei� No K- meson is bound.!
0.2

|S|=2

lB

0.1

ln
lp
lR

0.0
01

2

34r (fm)

Central density ! ρΒ(0) ∼ ρ0�

Finite nuclei�
Total energy !

Energy difference per unit of strangeness !
from normal nuclei ( |S| = 0 )�| UK | < 180 MeV!

For |S| >3!
Strong decay of !
multi-hypernuclei!
to !

We consider !
coexistence of kaon condensation with hyperonic matter !
in neutron stars!
based on the RMF model!
coupled with nonlinear effective chiral Lagrangian, �which is the same interaction model as used for�
multi-antikaonic nuclear bound states with hyperon-mixing !
for finite nuclei!
[T. Muto, T. Maruyama and T. Tatsumi, Phys. Rev. C79, 035207 (2009). �!
Genshikaku Kenkyu 57 Supplement 3, 230(2013). ]�������

Consistency with observation of massive neutron stars!
[ P. Demorest, T.Pennucci, S. Ransom,!
M(PSR J1614-2230) = 1.97 ± 0.04 M!! M. Roberts and J.W.T.Hessels,!
Nature 467 (2010) 1081.]�
M(PSR J0348+0432) = (2.01 ± 0.04) M!! [J. Antoniadis et al., !
Science 340, 6131 (2013).]�

2. Outline of the model�
Baryons: �Mesons:!

2-1. Baryon-Baryon interaction!
Relativistic mean-field theory�parameters�
--- NN interaction ---! gross features of normal nuclei and nuclear matter�・saturation properties of nuclear matter! (ρ0 =0.153 fm−3)�
・binding energy of nuclei and proton-mixing ratio!
・density distributions of p and n !
--- vector meson couplings for Y --- SU(6) symmetry!

--- scalar meson couplings for Y --- ! Hyperon potentials deduced !
( analysis of Λ single-particle orbitals )�from hypernuclear experiments�

・(K-, π±) at BNL!

T=3/2 state: strongly repulsive �[ J. Dabrowski, Phys. Rev. C60 (1999), 025205. ]!

・(π-, K+) at KEK!

23.5 MeV 80.4 MeV[ H. Noumi et al., , Phys. Rev. Lett. 89 (2002), 072301; ibid 90(2003), 049902(E). ]!

analysis of Σ - atoms : repulsive[ C. J. Batty, E. Friedman, A. Gal,!
Phys. Rep. 287 (1997), 385. ]!

repulsive case!

[ T. Fukuda et al., Phys. Rev. C58 (1998), 1306. ,!
P. Khaustov et al., Phys. Rev. C61 (2000), 054603. ]!

2-2 K − B, K − K interactions !
SU(3)L× SU(3)R chiral effective Lagrangian!

[ D. B. Kaplan and A. E. Nelson, !
Phys. Lett. B 175 (1986) 57. ]!

Baryons!
Ψ!

Meson fields (K± )!

(p, n, !
Λ, Ξ-, Σ-)!

Vector current !
Axial-vector current !

(Classical K- field!

Meson decay constant!

μK : kaon chemical potential �)!

kaon fields (K± )!
!!

"!!

!!"

!!

""!!

#$%$&'('))))!*+,-.+/0(scalar)!
!!

!!
""

(vector)!

!!

(scalar)!

(vector)!

!!

RMF!
""

#1)$
""

Kaonic part of the Lagrangian density�

S wave scalar int.�
S wave vector int.�

Effect of Λ(1405) and range terms�
+�Energy correction�

Effective baryon masses �
determined to reproduce !
the on-shell S-wave KN scattering lengths�parameters�
Corresponding to !
--- vector meson couplings for Kaon --- ! the Tomozawa-Weinberg term�

c.f. quark – isospin counting rule!
SU(6) symmetry�--- scalar meson couplings for Kaon --- !
: Decay of f0(975)!
K- optical potential depth :�at ρ0 ! U = ( −180, −120, �−80) MeV�K

(extremely, moderately, weakly) attractive �

3. Numerical results�3-1. EOS in β-equilibrated matter
Energy per particle�

3. Results�

!

Including !
Λ(1405) and range effect�

chemical equilibrium!
for weak processes!

3-2. Comparison with multi-strangeness nuclei !
Finite system formed in laboratory!
Antikaons do not receive much K − B attraction �ωK- >� μΛ�ー�μp!

Finite effects of nuclei!

Chemical equilibrium for strong processes!
hard to satisfy!
In neutron stars!

Antikaons receive!
much K − B attraction �dense infinite matter!

chemical equilibrium for weak processes�

K- chemical potential :! ω =�
K-= O(mπ) → < 0 !
for high densities!

Problems: Too soft EOS resulting from coexistence of kaon condensates!
and hyperons�
K- -baryon attractive interactions, especially, the S-wave scalar!
attractions lead to additional softening of the EOS as compared with !
the case of hyperonic matter.�
The soft EOS cannot give massive compact stars (~ 2 M!).�

Strong repulsion between baryons and suppression ofattractive K - B interaction at high densities are needed.!
Suppression mechanisms of K-Baryon attractions�K- field equation!

S-wave scalar int.�

S-wave vector int.�

S-wave scalar int.� --- Self-suppression of scalar int. --- �Scalar
Scalarmean
meanfields�
fields�Kaon condensation develops ⇒ θ : large ⇒�σ�mean-field : large�Effective baryon mass : decrease �Saturation of σ�mean-field �Baryon scalar density : decrease �

S-wave vector int.�

Chiral symmetry�Equations of motion for vector mean fields�

Vector interaction between K- mesons and hyperons (Σ- and Ξ- ) works !
repulsively as far as μ > 0 , unfavorable for coexistence.!
Development of kaon condensates suppresses vector attraction �

3-3. Particle fractions

Normal hyperonic matter!

!

Kaon-condensed phase�
UK= − 120 MeV�

4. Discussion and summary �
Repulsive effects on EOS�(1) Kaon-Baryon sector� -- suppression of K- - B attractions ---�(i) K- potential depth�ρBC~ 12 ρ0 �
( <N| ss |N> ~ 0 , ΣKN ~ 280 MeV ) �UK ~ − 87 MeV�

UK = − 120 MeV�

ρBC~ 4.5 ρ0 �(ii) S-wave K- - B vector int.�

Consequence from chiral symmetry�
Vector interaction between K- mesons and hyperons (Σ- and Ξ- ) works !
repulsively as far as μ > 0 , leading to suppression of kaon condensates .!
Development of kaon condensates suppresses vector attraction �But, such suppression of K- - baryon attractions is not enough !
for making the EOS stiffer. �Strong repulsion between baryons at high densities are needed.!

Making EOS stiffer at high density�
(1) Baryon-baryon sector�
(i) Phenomenological universal YNN, YYN, YYY repulsions ����[ S. Nishizaki, Y. Yamamotoand T. Takatsuka, !
Prog. Theor.Phys. 108 (2002) 703. ] )�(cf : RMF extended to BMM, MMM type diagrams) �[K. Tsubakihara and A. Ohnishi, arXiv:1211.7208.]�

(ii) relativistic Hartree-Fock�
Introduction of tensor coupling of vector mesons�Cf. for hyperonic matter, !
[T. Miyatsu, T. Katayama, K. Saito, Phys. Lett.B709 242(2012).]�

Connection to quark matter�(2) Relation between kaon condensation in hadronic matter!
and that in quark matter �Hadron phase and quark phase are connected with cross-over region�
Massive stars (~ 2 M!)�[ K. Masuda, T. Hatsuda, T. Takatsuka, Astrophys. J. Lett. 764, 12 (2013).]�

Coexistence of kaon-condensates !
and hyperons for hadronic phase�Strange quark matter !
(kaon-condensates in quark matter)�