12009, May 11th

Tokyo Institute of Technologya

Tetsuo Hyodoa

Structure of the Λ(1405) and kaon-nucleon dynamics

supported by Global Center of Excellence Program“Nanoscience and Quantum Physics”

2

Contents

Chiral unitary approachStructure of Λ(1405)

Effective single-channel KN interaction

On the KNN (strange dibaryon) system

Contents

T. Hyodo and W. Weise, Phys. Rev. C 77, 035204 (2008)

T. Hyodo, D. Jido, L. Roca, Phys. Rev. D77, 056010 (2008). L. Roca, T. Hyodo, D. Jido, Nucl. Phys. A809, 65 (2008).

T. Hyodo, D. Jido, A. Hosaka, Phys. Rev. C78, 025203 (2008).

・Nc Behavior and quark structure

・Dynamical or CDD pole (genuine quark state) ?

・Electromagnetic propertiesT. Sekihara, T. Hyodo, D. Jido, Phys. Lett. B669, 133-138 (2008).

3

Description of S = -1, KN s-wave scattering : Λ(1405) in I=0Chiral dynamics

- Interaction <-- chiral symmetry

- Amplitude <-- unitarity (coupled channel)R.H. Dalitz, T.C. Wong, G. Rajasekaran, PR153, 1617 (1967)

Y. Tomozawa, Nuovo Cim. 46A, 707 (1966); S. Weinberg, Phys. Rev. Lett. 17, 616 (1966)

Introduction

T =1

1− V GV

T

= +T

N. Kaiser, P. B. Siegel, W. Weise, Nucl. Phys. A594, 325 (1995),E. Oset, A. Ramos, Nucl. Phys. A635, 99 (1998),J. A. Oller, U. G. Meissner, Phys. Lett. B500, 263 (2001),M.F.M. Lutz, E. E. Kolomeitsev, Nucl. Phys. A700, 193 (2002), .... many others

works successfully, also in S=0 sector, meson-meson scattering sectors, systems including heavy quarks, ...

chiral

4

T. Hyodo, S.I. Nam, D. Jido, A. Hosaka, Phys. Rev. C68, 018201 (2003),T. Hyodo, S.I. Nam, D. Jido, A. Hosaka, Prog. Theor. Phys. 112, 73 (2004)

200

150

100

50

0

!T [

mb]

300200100

K-p

70

60

50

40

30

20

10

0300200100

"0#

200

150

100

50

0300200100

"+$%

60

50

40

30

20

10

0

!T [

mb]

300200100Plab [MeV/c]

K0n

60

50

40

30

20

10

0300200100

Plab [MeV/c]

"0$0

80

60

40

20

0300200100

Plab [MeV/c]

"%$+

How it works? vs experimental dataChiral unitary approach

14401420140013801360s [MeV]

!" m

ass d

istrib

utio

n

γ Rc Rn

exp. 2.36 0.664 0.189

theo. 1.80 0.624 0.225

Total cross sections threshold ratios

πΣ spectrum

==> KN interaction in this framework

5

Two poles for one resonancePoles of the amplitude in the complex plane : resonance

∼

Tij(√

s) ∼ gigj√s − MR + iΓR/2

Real part MassImaginary part Width/2

Residues Couplings

1360

1380

1400

1420

1440

-20

-40

-60

-80

0.5

1.0

1.5

0.5

1.0

1.5

!(1405)

Re[z]Im[z]

|T|

|Λ(1405)〉 = + b|Λ∗2〉a|Λ∗

1〉D. Jido, J.A. Oller, E. Oset, A. Ramos, U.G. Meissner, Nucl. Phys. A 723, 205 (2003);T. Hyodo, W. Weise, Phys. Rev. C 77, 035204 (2008)

Physical state: superposition

Chiral unitary approach

6

Dynamical state and CDD poleResonances in two-body scattering・Knowledge of interaction (potential)

(b) CDD pole: elementary, independent, ...

(a) dynamical state: molecule, quasi-bound, ...

L. Castillejo, R.H. Dalitz, F.J. Dyson, Phys. Rev. 101, 453 (1956)

・Experimental data (cross section, phase shift,...)

Resonances in chiral unitary approach -> (a) dynamical?

Structure of Λ(1405) resonance

MB

e.g.) Deuteron in NN, positronium in e+e-, (σ in π π), ...

e.g.) J/Ψ in e+e-, (ρ in π π), ...

7

CDD pole contribution in chiral unitary approachAmplitude in chiral unitary model

Known CDD pole contribution(1) Explicit resonance field in V(2) Contracted resonance propagator in V

V : interaction kernel (potential)G : loop integral (Greenʼs function)

Defining “natural renormalization scheme”, we find CDD pole contribution in G (subtraction constant).

Structure of Λ(1405) resonance

T. Hyodo, D. Jido, A. Hosaka, Phys. Rev. C78, 025203 (2008).

N(1535) in πN scattering --> dynamical + CDD poleΛ(1405) in KN scattering --> mostly dynamical !"#

!$#

!%#

!&#

#'()*)+,

-./

0$1#0$##011#01##0%1#0%##021#3-)*)+,-./

*0!4)5)!60%#17

*84)5)8601217

*&!4)5)!60%#17

8

Nc scaling in the modelNc : number of color in QCDHadron effective theory / quark structure

The Nc behavior is known from the general argument.<-- introducing Nc dependence in the model, analyze the resonance properties with respect to NcJ.R. Pelaez, Phys. Rev. Lett. 92, 102001 (2004)

Nc scaling of (excited) qqq baryon

-250-200-150-100

-500

Im W

[MeV

]2001000-100

Re W - MN - mK [MeV]

z1(Nc=3)

z2(Nc=3)

z2(12)

z1(12)

Result : ~ non-qqq (i.e. dynamical) structure

T. Hyodo, D. Jido, L. Roca, Phys. Rev. D77, 056010 (2008). L. Roca, T. Hyodo, D. Jido, Nucl. Phys. A809, 65 (2008).

Structure of Λ(1405) resonance

9

Electromagnetic propertiesAttaching photon to resonance--> em properties : rms, form factors,...

Structure of Λ(1405) resonance

large (em) size of the Λ(1405) : c.f. -0.12 [fm2] for neutron--> meson-baryon picture

T. Sekihara, T. Hyodo, D. Jido, Phys. Lett. B669, 133-138 (2008).

result of mean squared radii :|〈r2〉E| = 0.33 [fm2]

10

Summary 1 : Structure of Λ(1405)

Dynamical or CDD?

Analysis of Nc scaling

Electromagnetic properties

=> dominance of the MB components

=> large e.m. size

We study the structure of the Λ(1405)

=> non-qqq structure

Structure of Λ(1405) resonance

11

Summary 1 : Structure of Λ(1405)

Dynamical or CDD?

Analysis of Nc scaling

Electromagnetic properties

Independent analyses consistently support the meson-baryon molecule picture of the Λ(1405)

=> dominance of the MB components

=> large e.m. size

We study the structure of the Λ(1405)

=> non-qqq structure

Structure of Λ(1405) resonance

MB

12

Effective single-channel KN interaction

Effective interaction based on chiral SU(3) dynamics

Coupled-channel BS eq. + real valued interaction

Result of chiral dynamics --> single channel potential

few-body K-nuclei

(exact transformation)

Single-channel BS eq.+ complex interaction

(with approximation)

Schrödinger equation + local, complex, and energy-dependent potential

4

2

0

-2

F KN

[fm

]

1440140013601320!s [MeV]

Re F, full

KN(I=0)

1.5

1.0

0.5

0.0

-0.5

-1.0

F !" [f

m]

1440140013601320!s [MeV]

!"(I=0)

Im F, full

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(Diagonal) scattering amplitude in KN and πΣEffective single-channel KN interaction

Binding energy : B = 15 MeV <--> 30 MeV

~1405 MeV

Resonance in KN channel : at around 1420 MeV<-- consequence of strong πΣ dynamics (coupled-channel)

~1420 MeV

Very strong attraction in KN (higher energy) --> bound stateStrong attraction in πΣ (lower energy) --> resonance

-120

-100

-80

-60

-40

-20

0

Im z

[MeV

]1440140013601320

Re z [MeV]

!" resonance

KN bound stateWithout off-diagonal

KNπΣ

14

Origin of the two-pole structure

KN πΣ

Chiral interaction

-120

-100

-80

-60

-40

-20

0

Im z

[MeV

]1440140013601320

Re z [MeV]

!" resonance

z2(full)

z1(full)

KN bound state

KNπΣ

Two poles : natural consequence of chiral interaction(pole position is model dependent)

Effective single-channel KN interaction

15

Schematic illustration : AY vs Chiral

KN

πΣ

Feshbach resonance

Feshbach resonance on resonating continuum

continuum

bound state

Λ(1405)experiment

AY

Effective single-channel KN interaction

KN

πΣresonance

bound state

Chiral (Dalitzʼs coupled -channel model)

16

Summary 1 : KN interactionWe study the consequence of chiral SU(3) dynamics in KN phenomenology.

Resonance structure in KN appears at around 1420 MeV <-- strong πΣ dynamicsTwo attractive interactions in KN and πΣ

Application to K-pp system (without πΣN)

Effective single-channel KN interaction

T. Hyodo and W. Weise, Phys. Rev. C 77, 035204 (2008)

Doté-san’s talk

--> two poles for the Λ(1405)--> weaker effective KN interaction

observed in ΛN inv. mass effect of decay channel coupling

-120

-100

-80

-60

-40

-20

0

Im z

[MeV

]

1440140013601320Re z [MeV]

!" resonance

z2(full)

z1(full)

KN bound state

KNπΣ

17

Which channel is relevant?On the KNN (strange dibaryon) system

Theoretical studiesKNN - (πΣN) channels

πΣN

K̄NN

ΛN

ΣN

!"##

!$%

!&%

!"#$%

&"'(!)

πΛN

!&%

*+,-./ ,012

Experimental candidates- energy ≦ πΣN

18

Λ*N state in chiral dynamicsChiral dynamics --> two Λ* states : Λ*1, Λ*2

On the KNN (strange dibaryon) system

|Λ(1405)〉 = a|Λ∗1〉 + b|Λ∗

2〉B=2 system : Λ*1N, Λ*2N ?

Y. Ikeda, RCNP workshop, Dec. 25, 2008

|B = 2, S = −1〉 = a′|Λ∗1N〉 + b′|Λ∗

2N〉

T. Uchino, T. Hyodo, M. Oka, in preparationlevel repulsion --> Λ*1N becomes light?

K̄NNΛ∗

1N

Λ∗2N

!"##$%&'()$*+

+

,-$./0/12 (/34$./0/12

5

5

**

mixing of Λ*1N <--> Λ*2N

19

Summary 2 : KNN systemKNN or strange dibaryon system

To compare with observed candidates, we should choose relevant channel(s).

If there are two states for Λ*(1405), there could be two states in KNN-πΣN system

On the KNN (strange dibaryon) system

importance of πΣN channel?

Mixing of two Λ*--> level repulsion?

actual decay into ΛN channel--> change the spectrum?