@KEK 19/11/07横断研究会

バリオンのソリトン描像から見た K中間子束縛核

西川哲夫 （東工大）

近藤良彦（國學院大学）

based on hep-ph/0703100; arXiv:0710.0948

Kaonic nuclei: Kbar-nucleus bound states formed by strong attraction in (KbarN)I=0

Experimental evidence (?) by FINUDA collaboration: peak in the invariant mass spectrum of Λp from

B.E.=115MeV, Γ=67MeV Still controversial

Future experiment planned at J-PARC:

ppK-: the lightest kaonic nucleiAkaishi&Yamazaki, 2003

K−+ A→ X + Λ + p

K−+ 3He→ n+ (ppK −)→ n+ Λ + p

Theoretical studies of ppK-

Reference Method Binding energy Width

Akaishi&Yamazaki, PLB535,70(2002)

Phenomenological potential + ATMS

48 MeV 61 MeV

Shevchenko et al., PRL98,082301(2007)

Fadeev 55-70 MeV 95-110 MeV

Dote&Weise, nucl-th/0701050

Chiral SU(3) based potential + AMD

Ikeda&Sato, PRC76:035203,2007

Fadeev ∼80 MeV ∼73 MeV

Arai, Oka&Yasui, arXiv:0705.3936

Λ(1405)-hyper nucleus

depends on unkown meson-baryon coupling constants

%< 50 MeV

Large Nc QCD

QCD at Nc→∞: weakly interacting meson theory Fundamental degrees of freedom: mesons (Diagrams representing meson propagation are dominant.) Interaction between mesons〜 O(1/ Nc)

(t’Hooft, 1974)

Baryons emerge as topological solitons (“Skyrmion”) of the meson field. (Skyrme, 1961; Witten, 1983)

c.f. derived from the string theory via AdS/CFT correspondence (Sakai&Sugimoto, 2005; Nawa, Suganuma, Kojo, 2007)

QCD at Nc→∞: weakly interacting meson theory Fundamental degrees of freedom: mesons (Diagrams representing meson propagation are dominant.) Interaction between mesons〜 O(1/ Nc)

(t’Hooft, 1974)

Baryons emerge as topological solitons (“Skyrmion”) of the meson field. (Skyrme, 1961; Witten, 1983)

c.f. derived from the string theory via AdS/CFT correspondence (Sakai&Sugimoto, 2005; Nawa, Suganuma, Kojo, 2007)

Anit-kaon nucleus bound states?Anit-kaon nucleus bound states?

Why soliton model?

The action, written in terms of NG boson fields U, respects chiral symmetry and reproduces anomaly (Wess-Zumino-Witten term).

KN interaction is unambiguously determined, once Fπ and e are fixed, e.g. fitted to MN and MΔ.

Hyperons can be well described as kaon-soliton bound states.

Skyrme termWess-Zumino-Witten term

NL-sigma model

Skyrmion

Skyrmion: topological soliton of the pion field

Ansatz for pion field:

　 Isospin is oriented to the radial direction, 　

Skyrmion (Skyrme, 1961)

U =exp(irτ ⋅r̂F(r))

r̂.

“Hedgehog ansatz”

Skyrmion

Hedgehog represents a mapping:

classified by n (“winding number”). Winding number: conserved

∵ Mappings with different n cannot be smoothly connected with each other

Winding number = Baryon number

, … aS3 S3 S3or

n=1 n=2

S3(phys.) a S3(int .)

Zero mode (collective coordinate) quantization Zero mode: displacement without changing the energ

y invalidates the semi-classical approx. full quantum mechanical treatment is necessary.

The Skyrme lagrangian is invariant under

Regarding A(t) as collective coordinate, quantize the rigid body rotation of the Skyrmion,

Nucleon spectrum as rotational spectrum of a Skyrmion

U → AUA† , A∈SU(2)

M =Msoliton +12I

J (J +1)

Bound kaon approachto the strangeness in the Skyrme model

Kaon’s equation of motion under the b.g. of Skyrmion

Bound states of K-Skyrmion

Quantize the collective rotation of the bound system

Hyperon mass spectrum

Bound kaon approach in the Skyrme model

bound kaon

(Callan and Klebanov, 1985)

U → A(t)UA(t)† , K → A(t)K

Hyperon

Baryon masses Callan and Klebanov, 1985;Rho, Riska and Scoccola, 1992

I JP Set I Set II Exp.

1/2 1/2+ fitted 1003 N (939)

3/2 3/2+ fitted fitted Δ(1232)

0 1/2+ 1105 1202 Λ(1116)

0 1/2- 1325 fitted Λ(1405)

1 1/2+ 1203 1295 Σ(1193)

1 3/2+ 1349 1384 Σ(1385)

1/2 1/2+ 1332 1471 Ξ(1318)

Set I: Fπ and e are fitted to MN and MΔ 　 (Adkins, Nappi&Witten).Set II: fitted to MΛ(1405) and MΔ (present study)

(mπ=0, 　 FK/Fπ=1.23)

Description of ppK- systemin the bound kaon approach

Description of the ppK- system

bound kaon

Hedgehog skyrmion

Description of the ppK- system Kaon’s EoM for Skyrmions at fixed positions (adiabatic approximation)

kaon’s energy

Solve the pp radial motion rough estimate of the binding energy of ppK-

R

ωK (R)

VNN+ωKR

Derivation of the kaon’s EoM

Ansatz for chiral field

U(1) and U(2): hedgehogs centered at r =±R/2

UK: kaon field

U =U(1)UKU(2)

UK =exp(iK̂ ), K̂ = 22Fπ

0 0 K +

0 0 K 0

K − K0 0

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟

U(1)→ A1U(1)A1†

U(2)→ A2U(2)A2†

(A1,2 : SU(2) collective coordinate)

R

Derivation of the kaon’s EoM

Substitute the ansatz into the action

Expand up to O(K2) and neglect O(1/Nc) terms

Lagrangian for K under the background B=2 Skyrmion

(KN interaction is unambiguously determined, once the ansatz for the chiral field is given.)

U =U(1)UKU(2)

Derivation of the kaon’s EoM Collective coordinate quantization

projection of the skyrmion rotation onto (pp)S=0

Average the orientation of

Spherical partial wave analysis is allowed:

L[K ] = dΩ∫ (pp)S=0 L (pp)S=0 ,

(pp)S=0 =1N

p↑p↓ − p↓p↑( ), NI3 , J 3=

12π

(−1)I3 +1/2 D−I3 J 31/2 (Ω)

rR

EoM for the kaon in S-, P-,..wave

Results

Energy of K-

R=2.0 fmBK= 139MeV

normal nuclei

Dependence on the choices, Set I or II, is weak.

S-wave K- is strongly bound even for relatively large R, e.g. BK= 233MeV (R=1.5 fm) BK= 139MeV (R=2.0 fm)

Distribution of K- and baryon # density

Molecular nature (R %

> 2.0 fm)

R

z (fm) r (fm)

R =2.0fm

Veff (r, z :R)

Molecular states ⇒ deep binding of K-

Suppose Λ(1405) isan “atomic” state

K-

proton

approaching

ppK-: “molecular” state

K-

z (fm) r (fm)

R =2.0fm

Veff (r, z :R)

Origin of the strong binding: WZW term

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ǙDZÇÃÉsÉNÉ`ÉÉÇ å©ÇÈÇΩÇflÇ…ÇÕïKóvÇ≈Ç∑ÅB

Role of the Wess-Zumino-Witten term Origin of the WZW term: anomaly in QCD

Effective theory should reproduce anomaly in QCD ⇒ WZW term Effects ：

An extra symmetry of the chiral Lagrangian is removed. Skyrmion behaves like a fermion.

KN Interaction from the WZW term,

attractive potential VWZW to negative strangeness states Correct mass spectrum of hyperons

Λ(1405) is bound owing to the VWZW alone.

　　　　　　　　　　 gives a double-well potential for K- coupled with pp

LWZW =−

iNc

Fπ2 Bμ[K †DμK −(DμK )†K ] (Tomozawa-Weinber term)

VWZW ∝ ρBωK

p-p potential with and without K-

pp potential without K- (VNN)

pp potential in ppK- (VNN+VK)

Energy of K- bound to pp(VK=ωK-mK)

VNN =− d3x∫ LSkyrme[U(1)U(2)] −2MSoliton( )

p-p potential with and without K-

pp potential without K- (VNN)

pp potential in ppK- (VNN+VK)

Energy of K- (VK=ωK-mK)

Akaishi&Yamazaki,2007

p-p radial motion

Assume p-p radial motion is governed by the Hamiltonian:

From

H =P2

2μN

+VNN (R) +ωK (R)−mK

HΨ =EΨ,VNN+ωK-mK

R

VNN =− d3x∫ LSkyrme[U(1)U(2)] −2MS

ωK (R) : energy of kaon coupled to pp

binding energy of ppK- =−E

Binding energy of ppK- and its decomposition

Parameter set

TNN (MeV)

VNN

(MeV)ωK-MK

(MeV)Total(MeV)

p-p distance(fm)

Set I 42.0 74.5 -239.2 -125.5 1.63

Set II 36.2 73.7 -211.3 -104.0 1.80

Parameter set of Fπ and e Set I: fitted to MN and MΔ. Set II: fitted to MΛ(1405) and MΔ

Conclusion

π中間子場のソリトンとして表された陽子２個に結合した K-のエネルギーは著しく小さくなり得る。（ Wess-Zumino-Witten 項が大きな役割を果たす。）

K-の空間分布は、 ppK-が分子状態であることを示唆する。

K-が生む非常に強い引力が、斥力的な ppポテンシャルに勝って、 ppK-を深く束縛させる。

　〜我々のアプローチでのシナリオ〜

Λ(1405) は必ずしも KbarN 束縛状態ではない。 K-soliton 全体の回転を量子化して得られた状態。

我々のアプローチ1. ２個のソリトンを独立に回して量子化し、 ppに射影

2. K-を束縛させる。

より適切には、新たに集団座標を導入し、

A comment

K-

pp

πΣ p

U → A(t)UA(t)† ,K → A(t)K

U(1)→ A1(t)U(1)A1†(t), U(2)→ A2 (t)U(2)A2

†(t)

U(1)→ A1(t)U(1)A1†(t),

U(2)→ A2 (t)U(2)A2†(t),

K → A3(t)K

K-

pπΣ

The action of the Skyrme model

Skyrme term (included by hand) stabilizes solitons.

Wess-Zumino-Witten term remove an extra symmetry of the chiral Lagrangian makes a Skyrmion behave like a fermion

Skyrme termWess-Zumino-Witten term

ρ (size)

E

ρmin

total

Skyrme term

NL-sigma term

NL-sigma model

ENL-sigma ∝ ρ, ESkyrme ∝1ρ

(ρ : size of soliton)