ストレンジでエキゾチックな Ｋ中間子原子核 ＫＥＫ　土手昭伸 イントロダクション Ｋ原子核の面白さ　 　　　　　ー　反対称化分子動力学法（ＡＭＤ法）による研究　ー Ｋ原子核の“エキゾチック”な性質 Ｋ原子核研究の現状 このスクールとの関係 まとめ ＫＥＫサマースクール 「エキゾチック原子核実践講座　－あなたも計算できる－」 ９月１３日　＠　ＫＥＫ　４号館３階輪講室１（３４５）

１．イントロダクション Isospin (RIBF) Strangeness (J-PARC) Ｋ中間子原子核 ダブルΛハイパー核 Ξハイパー核 YY相互作用 ΞＮ相互作用、 　　より複雑なカップリング 　　（ΞＮ－ΣΣ－ΛΛ） Ｋ中間子原子核 （Ｋ原子核、Kaonic nuclei） K-中間子が束縛された原子核 ストレンジネスを持つ。 バリオンでなく、中間子を構成粒子 　として含む。 YN相互作用 ΛＮ－ΣＮ coupling (coherent) 5ΛHeにおけるαの変化 Λハイパー核 Isospin (RIBF) 不安定核（中性子・陽子過剰核） skin、haloといった新しい構造 shell structureの変化、新しい魔法数 N=16 安定核 Ｎ＝Ｚ Ｓ＝０

Ｋ原子核とは？ K- K- Kaonic atom Kaonic nucleus Nucleus 原子核内部に、 強い相互作用によって 束縛 クーロン力によって束縛 原子核内部に、 強い相互作用によって 束縛 Nucleus K- K- ～数十fm 原子核自体を 変化させる可能性あり。 Atomic orbit Σπ　threshold（主崩壊チャンネル） 　　より深く束縛 準安定状態として 存在する可能性あり。 KNNN… ΣπNN… K nuclear state

この話での登場人物 主役 準主役 Proton-K-が３０ MeV束縛した アイソスピン 0の状態？ 脇役 940 p,n 1115 Λ 1190 Σ p + K- 1435 Λ(1405) 1405 1250 Λ + π Σ + π 1325 Energy [MeV] 940 p,n 1115 Λ 1190 Σ p + K- 1435 Λ(1405) 1405 1250 Λ + π Σ + π 1325 Energy [MeV] 主役 Proton-K-が３０ MeV束縛した 　　アイソスピン 0の状態？ 　３クォーク状態ではない？ 　　←　単純なクォーク模型では 　　　　　説明できない… 準主役 ハイパー核に入っているΛの励起状態 脇役

Kaonic hydrogen atom KbarN interaction is “Repulsive”. K- stays on an atomic orbit around a proton. ☆ Exp. by M. Iwasaki (RIKEN) PRL 78, 3067(1997) Repulsive Attractive ??? 1s Coulomb potential + KbarN potential Kaonic hydrogen puzzle Low energy KbarN scattering　　　　Repulsive vs. Level shift of atomic 1s　　　　Attractive Repulsive KbarN interaction is “Repulsive”.

“Repulsion” ＝ Strong attraction K- + p nuclear bound state = Λ（１４０５） V0 Nuclear resonance state Nuclear bound state atomic state Y. Akaishi, M. Iwasaki R. Seki PRC 5, 1196 (1972) -8 MeVでnuclear bound state Attractive Repulsive 1s Coulomb potential 14 keV ＝ KN potential 27 MeV Λ（１４０５） KbarN interaction is strongly attractive. Deeply bound kaonic nucleus ?

... Deeply bound kaonic nuclei 赤石さん、山崎さんの先駆的研究 現象論的 KbarN potential (Akaishi-Yamazaki KbarN potential) Strongly attractive. free KN scattering data 1s level shift of kaonic hydrogen atom binding energy and width of Λ(1405) = K- + proton Y. Akaishi and T. Yamazaki, PRC 52 (2002) 044005 Deeply bound; Binding energy of K- > 100 MeV Discrete state; Below Σπ threshold Very attractive I=0 KN interaction makes … ... Deeply bound kaonic nuclei

２．Ｋ原子核の面白さ ー ＡＭＤ法による研究 ー ２．Ｋ原子核の面白さ　ー　ＡＭＤ法による研究　ー 反対称化分子動力学法（AMD法） 一核子波動関数　＝　ガウス波束 　Ａ体系をＡ体系として、 　　完全に微視的に取り扱う。 　構造に関してなんら仮定を置かない。 　　…変形、対称性、クラスターの存在など。 　シェル的構造からクラスター的構造まで 　　一つの枠組みで記述できる。 反対称化 I’d like to know the structure of Kbar nuclei well. So I am investigating them with anti-symmetrized molecular dynanics method. AMD method treat the system in a fully microscopic way. And it has no assumption on nuclear structure, existence of cluster, deformation, axial symmetry and so on. The system self-organizes only following energy variation. This is a normal nucleus. If I put one K- meson into this, how does this nucleus changes? I want to know what kind of structure this A+1 system favor. パリティ・角運動量射影 多彩な構造の現れる軽い 安定核・不安定核の構造の説明に成功。 (延与さん、木村君) 　摩擦冷却法によりエネルギー変分。 　　その際、試行関数は　　　　　　を取る。

Essence of AMD Shell Cluster 0s 0p The structure is determined AMD wave function can describe not only shell-model-like structure but also cluster-like one. 0s 0p Shell Gaussian wave packet Cooling Here, I explain about AMD method briefly. This method has been developed to study light stable and unstable nuclei. In this method, we express each nucleon with a single Gaussian wave packet whose center is a variational parameter. Of course, since nucleon is a Fermion, Total system is anti-symmetrized. We determine the position of Gaussian wave packet by frictional cooling method, which is a kind of gradient method. If the system favor a shell-model-like structure, all wave packets come to near the origin. This is just corresponding to shell structure due to the anti-symmetrization. On the other hand, if the system favors clustering structure, wave packets are separated to two groups. This correspond to clustering structure. Which case the system chooses depends on the Hamiltonian. The structure is determined by only the energy-variation. Cluster The structure is determined by only the energy-variation.

K- ＡＭＤ法をＫ原子核に適用 Kaonic nucleus K- 中間子 ＝ 強い引力の種 疑問 Ａ＋１体系は自己をどのように Ａ核子系の中に、突如強い引力の種である K- 中間子が入ると、どうなるか？ Ａ＋１体系は自己をどのように 再編するか？ 完全に微視的に取り扱うＡＭＤなら 答えることが出来るであろう… 疑問 normal nucleus ? ?? Kaonic nucleus Normal nucleus I’d like to know the structure of Kbar nuclei well. So I am investigating them with anti-symmetrized molecular dynanics method. AMD method treat the system in a fully microscopic way. And it has no assumption on nuclear structure, existence of cluster, deformation, axial symmetry and so on. The system self-organizes only following energy variation. This is a normal nucleus. If I put one K- meson into this, how does this nucleus changes? I want to know what kind of structure this A+1 system favor. p p n n

Ｋ原子核を扱うためのＡＭＤの改善 p n + Charge projection Ｋ原子核研究において、Ｉ＝０　KbarN 相互作用は非常に重要 p n + Charge projection Charge-mixed single-particle state ＡＭＤのようなCharge baseを用いる方法では、 ＫbarＮ 相互作用は電荷混合を引き起こす。 これによりＫ原子核の系統的研究が可能に。

Wave function Essence of mixing Nucleon’s wave function Total wave function Nucleon’s wave function p-n mixing Anti-kaon’s wave function mixing Charge projection as a trial function The detail of my wave function is as follows: This is a nucleon wave function. This time, I express a single nucleon with superposition of several Gaussian wave packets. These are spin and isospin wave functions. Isospin wave function can represent proton-neutron mixed state as I said. Kaon’s wave function is similar to nucleon one. Anti-symmetrizing nucleon wave functions, and combining kaon’s wave function by product, And then projecting onto eigen-state of parity. And we perform charge projection to this total wave function. We employ this wave function as a trial wave function.

Hamiltonian in AMD calculation : effective NN int. 　　　　　　　　 Tamagaki potential (OPEG) : effective KN int. 　　　　　　　　 AY KN potential Y. Akaishi and T. Yamazaki, PRC 52 (2002) 044005 The Hamiltonian used in AMD calculation is as follows: Kinetic energy part, effective NN interaction, effective KN interaction and Coulomb interaction. Center of Mass motion energy is subtracted. The effective NN and KN interaction is derived from Tamagaki potential and Akaishi-Yamazaki interaction by G-matrix method, respectively. At the early stage of our study, namely before the discovery of strange tribaryons, we used only central potential as NN effective interaction. G-matrix method

A. D., H. Horiuchi, Y. Akaishi and T. Yamazaki, AMD studies revealed … Nucleus-K-　threshold Σπ threshold (simple AMD) Width (Σπ, Λπ) E(K) > 100 MeV for various light nuclei Drastic change of the structure of 8Be, isovector deformation in 8BeK- Highly dense state is formed in K nuclei. maximum density > 4ρ0 averaged density 2～4ρ0 Proton satellite in pppK- OK, now let’s show my results of systematic study of various light nuclei with AMD method. First, it is found that K- meson can be deeply bound in various light nuclei. Blue point means the binding energy, violet bar means decay width. Red line is the Sigma-pi threshold. Their binding energy is about 100MeV measured from each nucleus-K- threshold. In most cases, it can be deeply bound below SigmaPi threshold. So they can be discrete state. Second, we can see the drastic change of 8Be structure. This is typical density distribution of 8Be. As you can see, 8Be has well-developed clustering structure. Then, I put a K- meson into this, its structure changes drastically. Clustering structure almost disappears. As a result of shrinkage, highly dense state is formed. In addition, we can see interesting structure in this 8BeK-. As you know, the original 8Be is N=Z nucleus. So proton-neutron distributions are identical to each other. But I=0 KN interaction is much more attractive than I=1 one. This means K-p interaction is Much more attractive than K-n one. Since K- attracts protons more than neutrons, proton-neutron separation occurs in 8BeK-. We call this phenomenon isovector deformation. Third, highly dense state is formed in Kbar nuclei, as a result that K- meson attracts nucleons Around itself. The maximum nuclear density is more than four times normal density, And averaged density is 2 to 4 times normal density. Last, very exotic system, three protons and K-, can be deeply bound. And it has very strange structure like this. This part is just one proton. We call this proton-satellite. A. D., H. Horiuchi, Y. Akaishi and T. Yamazaki, PLB 590 (2004) 51; PRC 70 (2004) 044313.

Isovector deformation AMD studies revealed … Rrms = 2.46 fm β = 0.63 Central density = 0.10 /fm^3 8Be Density (/fm^3) 0.0 0.10 0.20 Isovector deformation Rrms = 1.42 fm β = 0.55 Central density = 0.76 /fm^3 8BeK- Density (/fm^3) 0.0 0.41 0.83 4.5 normal density Binding energy of K- = 104 MeV E(K) > 100 MeV for various light nuclei Drastic change of the structure of 8Be, isovector deformation in 8BeK- Highly dense state is formed in K nuclei. maximum density > 4ρ0 averaged density 2～4ρ0 Proton satellite in pppK- OK, now let’s show my results of systematic study of various light nuclei with AMD method. First, it is found that K- meson can be deeply bound in various light nuclei. Blue point means the binding energy, violet bar means decay width. Red line is the Sigma-pi threshold. Their binding energy is about 100MeV measured from each nucleus-K- threshold. In most cases, it can be deeply bound below SigmaPi threshold. So they can be discrete state. Second, we can see the drastic change of 8Be structure. This is typical density distribution of 8Be. As you can see, 8Be has well-developed clustering structure. Then, I put a K- meson into this, its structure changes drastically. Clustering structure almost disappears. As a result of shrinkage, highly dense state is formed. In addition, we can see interesting structure in this 8BeK-. As you know, the original 8Be is N=Z nucleus. So proton-neutron distributions are identical to each other. But I=0 KN interaction is much more attractive than I=1 one. This means K-p interaction is Much more attractive than K-n one. Since K- attracts protons more than neutrons, proton-neutron separation occurs in 8BeK-. We call this phenomenon isovector deformation. Third, highly dense state is formed in Kbar nuclei, as a result that K- meson attracts nucleons Around itself. The maximum nuclear density is more than four times normal density, And averaged density is 2 to 4 times normal density. Last, very exotic system, three protons and K-, can be deeply bound. And it has very strange structure like this. This part is just one proton. We call this proton-satellite. A. D., H. Horiuchi, Y. Akaishi and T. Yamazaki, PLB 590 (2004) 51; PRC 70 (2004) 044313.

A. D., H. Horiuchi, Y. Akaishi and T. Yamazaki, AMD studies revealed … E(K) > 100 MeV for various light nuclei Drastic change of the structure of 8Be, isovector deformation in 8BeK- Highly dense state is formed in K nuclei. maximum density > 4ρ0 averaged density 2～4ρ0 Proton satellite in pppK- OK, now let’s show my results of systematic study of various light nuclei with AMD method. First, it is found that K- meson can be deeply bound in various light nuclei. Blue point means the binding energy, violet bar means decay width. Red line is the Sigma-pi threshold. Their binding energy is about 100MeV measured from each nucleus-K- threshold. In most cases, it can be deeply bound below SigmaPi threshold. So they can be discrete state. Second, we can see the drastic change of 8Be structure. This is typical density distribution of 8Be. As you can see, 8Be has well-developed clustering structure. Then, I put a K- meson into this, its structure changes drastically. Clustering structure almost disappears. As a result of shrinkage, highly dense state is formed. In addition, we can see interesting structure in this 8BeK-. As you know, the original 8Be is N=Z nucleus. So proton-neutron distributions are identical to each other. But I=0 KN interaction is much more attractive than I=1 one. This means K-p interaction is Much more attractive than K-n one. Since K- attracts protons more than neutrons, proton-neutron separation occurs in 8BeK-. We call this phenomenon isovector deformation. Third, highly dense state is formed in Kbar nuclei, as a result that K- meson attracts nucleons Around itself. The maximum nuclear density is more than four times normal density, And averaged density is 2 to 4 times normal density. Last, very exotic system, three protons and K-, can be deeply bound. And it has very strange structure like this. This part is just one proton. We call this proton-satellite. A. D., H. Horiuchi, Y. Akaishi and T. Yamazaki, PLB 590 (2004) 51; PRC 70 (2004) 044313.

A. D., H. Horiuchi, Y. Akaishi and T. Yamazaki, AMD studies revealed … E(K) > 100 MeV for various light nuclei Drastic change of the structure of 8Be, isovector deformation in 8BeK- Highly dense state is formed in K nuclei. maximum density > 4ρ0 averaged density 2～4ρ0 Proton satellite in pppK- pppK- Proton satellite OK, now let’s show my results of systematic study of various light nuclei with AMD method. First, it is found that K- meson can be deeply bound in various light nuclei. Blue point means the binding energy, violet bar means decay width. Red line is the Sigma-pi threshold. Their binding energy is about 100MeV measured from each nucleus-K- threshold. In most cases, it can be deeply bound below SigmaPi threshold. So they can be discrete state. Second, we can see the drastic change of 8Be structure. This is typical density distribution of 8Be. As you can see, 8Be has well-developed clustering structure. Then, I put a K- meson into this, its structure changes drastically. Clustering structure almost disappears. As a result of shrinkage, highly dense state is formed. In addition, we can see interesting structure in this 8BeK-. As you know, the original 8Be is N=Z nucleus. So proton-neutron distributions are identical to each other. But I=0 KN interaction is much more attractive than I=1 one. This means K-p interaction is Much more attractive than K-n one. Since K- attracts protons more than neutrons, proton-neutron separation occurs in 8BeK-. We call this phenomenon isovector deformation. Third, highly dense state is formed in Kbar nuclei, as a result that K- meson attracts nucleons Around itself. The maximum nuclear density is more than four times normal density, And averaged density is 2 to 4 times normal density. Last, very exotic system, three protons and K-, can be deeply bound. And it has very strange structure like this. This part is just one proton. We call this proton-satellite. A. D., H. Horiuchi, Y. Akaishi and T. Yamazaki, PLB 590 (2004) 51; PRC 70 (2004) 044313.

３．Ｋ原子核の“エキゾチック”な性質 高密度状態 従来の原子核の常識を覆す！？ 最大密度 ４ρ０以上、 平均密度 ２ρ０以上 「密度の飽和性」　…　原子核物理の常識 安定な原子核では、その内部の密度は質量数によらず、 どんな原子核でも一定である。 通常核密度（normal density） ρ0 = 0.17 fm-3 しかしＫ原子核では 最大密度　４ρ０以上、　平均密度　２ρ０以上 「密度の飽和性」に反している。 従来の原子核の常識を覆す！？

通常核密度（normal density） ３．Ｋ原子核の“エキゾチック”な性質 通常核密度（normal density） ρ0 = 0.17 fm-3 　平均核子間距離　＝　2.2 fm （球を占有してる場合） 陽子 硬い“芯” pionの雲 0.8 fm 　核子の硬い“芯”のサイズ　＝　半径 0.5 fm程度 　（クォークからなる？） 0.5 fm 　陽子の荷電半径　＝　0.8 fm 通常原子核内部 芯と芯がふれ合う状態 　…　平均核子間距離　約1 fm 2.2 fm 0.5 fm まだ芯と芯は余裕を持って離れている。 原子核にはまだ隙間がある。 ～1 fm ＝　通常原子核の約半分 ～ 8 ρ０

３．Ｋ原子核の“エキゾチック”な性質 予想外の状態が基底状態？ pn pp NNKbar（二核子＋Kbar 中間子）の場合 + + 重陽子 アイソスピン ０ pp 陽子二つ アイソスピン 1 + +

? ３．Ｋ原子核の“エキゾチック”な性質 予想外の状態が基底状態？ + K- + K- pn pp NNKbar（二核子＋Kbar 中間子）の場合 pn 重陽子 アイソスピン ０ + K- pp 陽子二つ アイソスピン 1 + K- + + ? 2.2 MeV 束縛 束縛状態なし 元々、重陽子という束縛状態を作ってた pn に K- が加わった方が強く束縛しそう…

??? ３．Ｋ原子核の“エキゾチック”な性質 予想外の状態が基底状態？ + K- + K- pn pp NNKbar（二核子＋Kbar 中間子）の場合 pn 重陽子 アイソスピン ０ + K- pp 陽子二つ アイソスピン 1 + K- ??? 元々、重陽子という束縛状態を作ってた pn に K- が加わった方が強く束縛しそう… 否、元々束縛状態を形成しない pp に K- が加わった方が強く束縛！

３．Ｋ原子核の“エキゾチック”な性質 Kbarが全体を支配する！ 予想外の状態が基底状態？ アイソスピン状態 Very attractive ppK- の方が deuteron + K- よりも、I=0 KbarNの成分を多く含む。 Λ（１４０５）：　 非常に強い引力である I=0 KbarN 相互作用の寄与が ppK- では大きくなる。 ppK- ：　 核子系のみの時と逆に、ppK- の方が deuteron + K- より深く束縛する。 核子系アイソスピン = 1 Kbarが全体を支配する！ Deuteron + K- ：　 核子系アイソスピン = 0

Invariant mass of p and Λ ４．Ｋ原子核研究の現状 = 実験 = Invariant mass of p and Λ H. Fujioka et al. @ FINUDA 4He (stopped K-, n) ppnK- M. Iwasaki et al. @ KEK ppnK- (T=0) B.E. = 169 MeV Γ < 25 MeV ppK- B.E. = 116 MeV Γ = 67 MeV 16O (in-flight K-, n) 15OK- T. Kishimoto et al. @ BNL Heavy ion collision N. Herrmann et al. @ GSI 15OK- B(K) = 90 MeV ppnK- B.E. = 150 MeV Γ ～ 100MeV

Invariant mass of p and Λ ４．Ｋ原子核研究の現状 = 実験 = Invariant mass of p and Λ H. Fujioka et al. @ FINUDA 4He (stopped K-, n) ppnK- M. Iwasaki et al. @ KEK 新しい実験…現在解析中 統計を１０倍 にアップ！ ターゲット別 にデータを収集！ ppnK- (T=0) B.E. = 169 MeV Γ < 25 MeV ppK- B.E. = 116 MeV Γ = 67 MeV データを再解析したところ、 ＹＮ に強い相関？ これが意味するのは…？ 追試で確認されず。 12C (in-flight K-, N ) T. Kishimoto et al. @ KEK N = proton N = neutron 16O (in-flight K-, n) 15OK- T. Kishimoto et al. @ BNL 15OK- B(K) = 90 MeV Heavy ion collision N. Herrmann et al. @ GSI 束縛領域にシグナルあり。 スペクトルを再現するには N=nの場合、190MeV N=pの場合、160MeV の深さのＫbarＮポテンシャルが必要。 T. Kishimoto et. al. PTP118, 181(2007) Very preliminary ppnK- B.E. = 150 MeV Γ ～ 100MeV

４．Ｋ原子核研究の現状 = 理論 = ppK- = “Prototype of Kbar nuclei” 「最も基本的な K 原子核」 FINUDAグループの実験結果　…　B. E. = 116 MeV, Γ = 67 MeV 3体系なので非常に簡単な系 … 様々な手法でアプローチできる。 　　特に精密に解くことが出来る。 いろいろな NN, KbarN 相互作用を試す。 … 赤石・山崎の現象論的 KbarN 相互作用以外では？

４．Ｋ原子核研究の現状 = 理論 = ppK- = “Prototype of Kbar nuclei” 「最も基本的な K 原子核」 Λ（１４０５）を再現する元では…

５．このスクールとの関係 構造 ハドロン 反応 K 原子核 “エキゾチック原子核の性質を解明” 新しい「質」の発見！ （エキゾチック原子核） 形、サイズ（密度） 量子数（角運動量・パリティ・アイソスピン） 束縛エネルギー、束縛機構 新しい「質」の発見！ K 原子核 （エキゾチック原子核） ハドロン “相互作用が分からないことには 　構造・反応計算のしようがない！” 反応 “人間の目で直接構造を見ることは 　出来ない！” NN のように膨大な実験データがあれば現象論的にでも 相互作用を作ることが出来る。 しかしYN, YY （ハイパー核）、KbarN （Ｋ原子核）では そうも行かない。 “実験で作らないことには話は始まらない！” …　上手く作るにはどういう反応がいいか？ ハドロン物理の助けが必要。 QCD及びその有効理論によって ハドロン間の相互作用の情報を得る。

６．まとめ 通常原子核では見られない“エキゾチック”な性質！ … K 中間子原子核　…　K- 中間子が原子核内部に強い相互作用によって束縛した系。 　　　　　　　　　　　　　　 中間子という形でストレンジネスを原子核に持ち込む。 KbarN 相互作用　…　非常に強い引力の可能性、特にアイソスピン 0 のチャンネルで。 Λ（１４０５）＝ K- p アイソスピン 0 の束縛状態 反対称化分子動力学法 (AMD 法)＋現象論的ＫbarＮ 相互作用による研究の結果 １． 深い束縛、狭い幅 ２． 高密度状態の形成 ３． 面白い構造 軽い原子核でK- は100 MeV程度束縛 … 最大密度　4ρ0以上、平均密度　2～4ρ0 8Beでの激しい構造変化、アイソベクトル変形 pppK-のproton satellite構造 通常原子核では見られない“エキゾチック”な性質！ 「密度の飽和性」を破る。 変な状態がエネルギー的に得する場合がある。

６．まとめ J-PARCに期待！ 3He (inflight-K-, n) ppK- → p + Λ 現象論的 KbarN 相互作用以外？ Non-mesonic decay （KbarNN → YN ； 二核子吸収）による崩壊幅？ 今の議論には Mesonic decay （KbarN → Yπ ； 二核子吸収）しか入ってない… 現在、もっとも基本的なK原子核 “ppK- ”の研究が盛んに行われている。 理論 ・ 実験共に更なる研究が必要。 J-PARCに期待！ 3He (inflight-K-, n) ppK- → p + Λ

Kbar nuclei … Exotic system ! KbarN interaction Λ（１４０５） カイラル対称性の回復 Kaonic atom Kbar nuclei K凝縮 Cold and Dense Strange quark matter 通常核では見られない構造 … related to various fields