第2コマ K中間子原子核 Exotic system with strangeness?

第2コマ K中間子原子核 Exotic system with strangeness?
Nucleus containing K- meson

Kaonic nuclei Part 1 Ｋ原子核の基礎知識現象論的 KbarN相互作用反対称化分子動力学法 (AMD)による研究
まとめ

What is Kaonic nucleus? K- Kaonic atom Nucleus Atomic orbit
Mainly bound by Coulomb force Nucleus K- ～10 fm Atomic orbit Only Coulomb force 4He　～ 238Uまで幅広い原子核で kaonic atom は測定。純粋なクーロン力だけの場合からのエネルギーのズレ　　1 keV = 100 eV以下 K-と核子間に働く強い相互作用のため S. Hirenzaki et al., Phys. Rev. C61, (2000) E. Friedman et al., Nucl. Phys. A579, 518 (1994)

What is Kaonic nucleus? K- K- Kaonic atom Kaonic nucleus Nucleus
Mainly bound by Coulomb force Bound by strong interaction Inside of nucleus Nucleus K- K- ～10 fm The nuclear structure may be changed, if the interaction is so attractive. Atomic orbit Deeply bound below πΣ threshold (main decay channel) Possible to exist as a quasi-bound state with narrow width KNNN… ΣπNN… K nuclear state

Actors in Kbar nuclei Key person Leading actors
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] Leading actors I=0 Proton-K- bound state with 30MeV binding energy? 　Not 3 quark state? 　　←　can’t be explained with 　　　　　a simple quark model… Key person Excited state of Λ Supporting players

Actors in Kbar nuclei Key person Leading actors Σπ channel is open
940 p,n 1115 Λ 1190 Σ p + K- 1435 Λ(1405) 1405 1250 Λ + π Σ + π 1325 Energy [MeV] Leading actors Σπ channel is open at about 100 MeV below the Proton-K- threshold. Key person Excited state of Λ Supporting players

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 ＝ KbarN potential 27 MeV Λ（１４０５） KbarN interaction is strongly attractive. Deeply bound kaonic nucleus ?

KbarN interaction Chiral SU(3) Dynamics K+ K-
T. Waas, N. Kaiser & W. Weise, Phys. Lett. B379 (1996) 34. 1 2 3 1.5 1.0 0.5 0.0 K+ K- 200 MeV G/100 MeV

Julich KbarN Quasi-potential
A. Mueller-Groeling, K. Holinde & J. Spethe, Nucl. Phys. A513 (1990) 557. N Λ,Σ ω, ρ σ N Λ,Σ Λ Σ π ρ Δ π, ρ

Julich KbarN Quasi-potential
A. Mueller-Groeling, K. Holinde & J. Spethe, Nucl. Phys. A513 (1990) 557. N Λ,Σ ω, ρ σ Dominant Minor KN interaction KbarN interaction G-parity 変換 KbarNの場合、全てのメソンが引力的に働く。 Λ（１４０５）の形成

まとめ

現象論的相互作用 … AY KbarN interaction
Y. Akaishi and T. Yamazaki, PRC 65 (2002) 現象論的相互作用　　　　…　AY KbarN interaction 1, Free KbarN scattering data A.D.Martin, Nucl.Phys.B179(1981)33 aI=0= i 0.46 fm, aI=1= i 0.60 fm Exp. : aI=0= i 0.68 fm, aI=1= i 0.60 fm 2, X-ray data of kaonic hydrogen atom M.Iwasaki et al., Phys.Rev.Lett.78(1997)3067 aK-p= ( aI=0 +aI=1) /2 = i 0.53 fm I=0 K-p quasi-bound state Exp. : aK-p= (-0.78 ± 0.15 ± 0.03) + i (0.49 ± 0.25 ± 0.12) fm 3, Binding energy and decay width of Λ(1405) B.E.= MeV, Γ= 40 MeV cf) Λ(1405): 27 MeV below K-+p threshold

現象論的相互作用 … AY KbarN interaction
Y. Akaishi and T. Yamazaki, PRC 65 (2002) 現象論的相互作用　　　　…　AY KbarN interaction Not single channel, but coupled-channel potential. I=0 ch. I=1 ch. アイソスピン０のチャンネルで非常に強い引力をもつ。

Pioneering work by Akaishi and Yamazaki
方法: Brueckner Hartree-Fock Effective NN interaction: Hasegawa-Nagata interaction Bare KbarN interaction: AY KbarN interaction ppnK-(T=0)は I=0 KbarN interactionの割合が大きい非常に深い束縛。 Σπ-decay modeが閉じ、discreteな状態。 ppnK- (T=0): E(K)=108 MeV, Γ=20 MeV

“Contraction” B.E. = 70 MeV B.E. = 86 MeV K- は原子核の構造を大きく変える？
原子核が縮むことにより、束縛エネルギーをさらに稼ぐ。 K- は原子核の構造を大きく変える？多体系のdynamicsが重要？？

まとめ

AMD法を用いた考察反対称化分子動力学法（AMD法）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。 (井坂君の話を参照)
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。 (井坂君の話を参照) 　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。

エネルギー変分…摩擦冷却法 Negative 一種の勾配法基底状態へ時間発展とともに系のエネルギーは下がるパリティ射影した波動関数で
Hamiltonianの期待値を計算。時間発展とともに系のエネルギーは下がる基底状態へ

Essence of AMD Shell Cluster 0s 0p エネルギー変分のみによって構造は決まる。
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 エネルギー変分のみによって構造は決まる。

AMD法を用いた考察 K- 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再編するか？
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 ? ?? Kaonic nucleus Normal nucleus p p n n Highly dense? Peculiar structure? K- deeply bound?

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 2 packets
p-n mixing 2 packets Total wave function 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. I comment on superposition of wave packets. I think kaon is widely spread in a Kbar nucleus. So in my study, each nucleon is represented with two wave packets, while Kaon is represented with five wave packets. Thus, kaon is possible to more widely spread than nucleon. 5 packets

: Eigen state of angular momentum J
J & T projections (VBP) : Eigen state of angular momentum J and isospin T J projection T projection Various quantities are calculated with

Merits on the extended framework
By expressing a nucleon and a kaon with superposition of wave packets, we can represent the difference between their distributions. Ex) Nucleon: 2 packets, Kaon: 5 packets By the charge-number projection, we can remove unnecessary charge states. Ex) ppnK- Charge (p+n)(p+n)(p+n)(K-+K0) = pppK0 + pppK- + ppnK0 + ppnK- + pnnK0 + pnnK- + nnnK0 + nnnK- 3 2 1 -1 We can extract only charge 1 state by the charge number projection.

Hamiltonian G-matrix method Effective interaction
Bare NN int = Tamagaki potential (OPEG) Bare KbarN int = AY potential NN/KN effective interactions have a 10-range Gaussian form.

Antisymmetrized Molecular Dynamics
Collaboration with Akaishi-san and Yamazaki-san According to the study with Antisymmetrized Molecular Dynamics + G-matrix + Phenomenological KbarN interaction Total system is treated in a fully microscopic way. NN repulsive core is adequately smoothed out by following conventional nuclear physics. Strongly attractive, especially in I=0 channel Kaonic nuclei has interesting properties…

AMD + G-matrix + AY KbarN interaction 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 Kbar nuclei. maximum density > 4ρ0 averaged density 2～4ρ0 Proton satellite in pppK- A. D., H. Horiuchi, Y. Akaishi and T. Yamazaki, PLB 590 (2004) 51; PRC 70 (2004)

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 Kbar nuclei. maximum density > 4ρ0 averaged density 2～4ρ0 Proton satellite in pppK- A. D., H. Horiuchi, Y. Akaishi and T. Yamazaki, PLB 590 (2004) 51; PRC 70 (2004)

Rrms = fm β = Central density = /fm^3 8Be Density (/fm^3) Rrms = fm β = Central density = /fm^3 8BeK- Density (/fm^3) 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 Kbar nuclei. maximum density > 4ρ0 averaged density 2～4ρ0 Proton satellite in pppK- A. D., H. Horiuchi, Y. Akaishi and T. Yamazaki, PLB 590 (2004) 51; PRC 70 (2004)

Isovector deformation
AMD + G-matrix + AY KbarN interaction studies revealed … Isovector deformation 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 Kbar nuclei. maximum density > 4ρ0 averaged density 2～4ρ0 Proton satellite in pppK- A. D., H. Horiuchi, Y. Akaishi and T. Yamazaki, PLB 590 (2004) 51; PRC 70 (2004)

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 Kbar nuclei. maximum density > 4ρ0 averaged density 2～4ρ0 Proton satellite in pppK- A. D., H. Horiuchi, Y. Akaishi and T. Yamazaki, PLB 590 (2004) 51; PRC 70 (2004)

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 Kbar nuclei. maximum density > 4ρ0 averaged density 2～4ρ0 Proton satellite in pppK- pppK- Proton satellite A. D., H. Horiuchi, Y. Akaishi and T. Yamazaki, PLB 590 (2004) 51; PRC 70 (2004)

Nuclear density distribution
ppnK- pppK- pppnK- proton satellite 3 fm ρave = 3.1ρ0 ρave = 3.9ρ0 ρave = 2.5ρ0 6BeK- 9BK- 4 fm ρave = 2.2ρ0 ρave = 1.9ρ0

Saturation of nucleon number around K- ?
One-center like Two-center like ppnK- Nucleons Strange structure Single K- meson can interact with limited numbers of nucleons? Saturation of E(K) Kaon

まとめ

AMD+G-matrix+現象論的 KbarN pot.によると…
原子核の構造変化高密度状態の形成奇妙な構造など。。。 2つになったら ??? Double kaonic nucleus Symmetrized Wave func.: :とりあえず KbarKbar 間の相互作用はなし K- 中間子は boson AMD for Double kaonic nuclei

Double kaonic nucleus - ppnK-K- -
Total B.E. = 6.0 MeV Central density = fm-3 Rrms= 1.59 fm ppn 4 fm Density [fm-3] Total B.E. = 118 MeV Central density = 1.5 fm-3 Rrms= 0.72 fm ppnK- Density [fm-3] E(K) = 110 MeV 4 fm Total B.E. = 221 MeV Central density = fm-3 Rrms= 0.69 fm ppnK-K- Density [fm-3] E(2K) = 213 MeV 4 fm T. Yamazaki, A. Doté and Y. Akaishi, PLB578, 167(2004)

Double kaonic nucleus - ppnK-K- -
Double kaonic nuclei 4 fm 4 fm 4 fm E(K) = 110 MeV E(2K) = 213 MeV Density [fm-3] Density [fm-3] Density [fm-3] ppn ppnK- ppnK-K- Total B.E. = 6.0 MeV Central density = fm-3 Rrms= 1.59 fm Total B.E. = 118 MeV Central density = 1.5 fm-3 Rrms= 0.72 fm Total B.E. = 221 MeV Central density = fm-3 Rrms= 0.69 fm T. Yamazaki, A. Doté and Y. Akaishi, PLB578, 167(2004)

相対論的平均場（RMF）による研究これらの方程式を場を場の期待値として解く。普通の原子核への適用変分原理
（Euler-Lagrange eq.） Lagrangian density 核子（ψ）, 中間子（σ, ω, ρ） σ中間子は非線形なポテンシャル中を運動していると仮定各場に対する運動方程式核子（ψ）： σ ： ω ： ρ ：光子： Scalar density： Mass current density： Isovector current density： Proton current density：これらの方程式を場を場の期待値として解く。 Y. K. Gambhier, P. Ring, and A. Thimet, Annals of Physics, 198, 132 (1990)

相対論的平均場（RMF）による研究普通の原子核への適用：例 )TM1 Lagrangian density
核子（ψ）, 中間子（σ, ω, ρ） ω中間子にも非線形なポテンシャルを導入 Y. Sugahara and H. Toki, Nucl. Phys., A579, 557 (1994)

相対論的平均場（RMF）による研究 L = LN + LK Ｋ原子核への適用
D. Gazda, E. Friedman, A. Gal and J. Mares, PRC76, (2007); PRC77, (2008) - T. Muto, T. Maruyama and T. Tatsumi, PRC79, (2009) L = LN + LK

D. Gazda, E. Friedman, A. Gal and J. Mares, PRC76, (2007); PRC77, (2008) - T. Muto, T. Maruyama and T. Tatsumi, PRC79, (2009) L = LN + LK Lagrangian density for anti-kaon Anti-kaon density Klein-Gordon eq. for Anti-kaon

相対論的平均場（RMF）による研究 K-の Separation energy (BK)や原子核の中心密度 (ρN at r=0) が
Ｋ原子核への適用 D. Gazda, E. Friedman, A. Gal and J. Mares, PRC76, (2007); PRC77, (2008) - T. Muto, T. Maruyama and T. Tatsumi, PRC79, (2009) K-の Separation energy (BK)や原子核の中心密度 (ρN at r=0) が K-の数に対して飽和している。

D. Gazda, E. Friedman, A. Gal and J. Mares, PRC76, (2007); PRC77, (2008) - T. Muto, T. Maruyama and T. Tatsumi, PRC79, (2009) L = LN + LK カイラル対称性を尊重 (Non-linear chiral Lagrangian)

相対論的平均場（RMF）による研究 K-一つ当りのK-の束縛エネルギー (B / |S|)や、原子核の中心密度 (ρB(0) ) が
Ｋ原子核への適用 D. Gazda, E. Friedman, A. Gal and J. Mares, PRC76, (2007); PRC77, (2008) - T. Muto, T. Maruyama and T. Tatsumi, PRC79, (2009) Kaon’s binding energy / kaon (-B / |S|)max = 44MeV at |S| = 2 (-B / |S|)max = 106MeV at |S| = 7 -B(A, Z, |S|) / |S| = [ E(A, Z, |S|) – (E(A, Z, 0) + |S| mK) ] / |S| K-一つ当りのK-の束縛エネルギー (B / |S|)や、原子核の中心密度 (ρB(0) ) が K-の数に対して飽和。原子核の中心密度 Saturated to 3 ρ0

相対論的平均場（RMF）による研究Ｋ原子核への適用
D. Gazda, E. Friedman, A. Gal and J. Mares, PRC76, (2007); PRC77, (2008) - T. Muto, T. Maruyama and T. Tatsumi, PRC79, (2009) Density profile of 15O+xK- UK = -80MeV UK = -120MeV

まとめ

Summary of Part 1 KbarN相互作用 – 実験事実と Λ(1405) –
Kaonic hydrogen puzzleが解けた結果、実験事実はKbarN相互作用は斥力的であることを示唆。 “3 quark状態では記述できないΛ(1405) = K- と陽子の準束縛状態” KbarN 相互作用は引力的現象論的 KbarN相互作用 (AY KbarN potential) 1.低エネルギーKbarN散乱データ（散乱長）、 Kaonic hydrogen atomの1sレベルシフト、　 Λ(1405) = K--proton が27MeVで束縛した状態アイソスピン0 (I=0) のチャネルで非常に引力的 3HeK- (T=0) は100MeV以上束縛、主崩壊チャネル(πΣ)が閉じて準安定な状態に。（崩壊幅～20MeV） Deeply bound kaonic nucleiの存在? K- が加わることで4Heが収縮”Contraction”, 8Beが構造変化？ AMD + G-matrix + AY KbarN potential 構造になんら仮定を置かないAMD法を用いて、軽いＫ原子核（3HeK- ～ 11CK-）を系統的に研究ほとんどの原子核で K- は100MeV以上束縛、狭い崩壊幅（20～40 MeV） K-中間子からの強い引力高密度状態の形成 8Beでの激しい構造変化（αクラスター構造の消失） Isovector deformation, proton satelliteなどの面白い構造 K-中間子が相互作用する核子数に制限？どの原子核でも E(K)=100MeV と飽和してる理由？？

Theoretical studies of nuclear system with anti-kaons
Light nuclei with a single antikaon 3HeK- ～ 11CK- studied with AMD + G-matrix + AY potential E(K)≒100MeV Light nuclei with double antikaons 3HeK-K- etc studied with AMD + G-matrix + AY potential E(2K)≒200MeV Medium to heavy nuclei with multi-antikaons Studied with Relativistic Mean Field - T. Muto, T. Maruyama and T. Tatsumi, PRC79, (2009) D. Gazda, E. Friedman, A. Gal and J. Mares, PRC76, (2007); PRC77, (2008) Repulsive KbarKbar interaction Saturation for the number of antikaons Central nuclear density and Separation energy of anti-kaon (or Kaon’s binding energy / kaon) are saturated. Nuclear matter with antikoans Neutron star, kaon condensation…

Kaonic nuclei ＝ exotic system
①　Deeply bound and Dense ②　Drastic change of structure 8Be 8BeK- Kaonic nuclei ＝ exotic system ③　Isovector deformation ④　proton satellite pppK-

中性子星内部に匹敵する状態が地球上で作れてまう！！！？？？平均密度２ρ０以上補足：Ｋ原子核の“エキゾチック”な性質
高密度状態中性子星内部で数倍のρ0 「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 中性子星内部に匹敵する状態が地球上で作れてまう！！！？？？しかしＫ原子核では平均密度　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）
補足：雑な見積もり通常核密度（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

第2コマ K中間子原子核 Exotic system with strangeness?

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