K-pp studied with Coupled-channel Complex Scaling method

Workshop on “Hadron and Nuclear Physics (HNP09)” ’09.11.19 @ Arata hall, Osaka univ., Ibaraki, Osaka, Japan

1. Introduction

2. Question --- Variational calculation and Faddeev ---

3. Complex Scaling Method

4. Result

5. Summary and Future plan

A. Doté (KEK) and T. Inoue (Tsukuba univ.)

Two-body K-p system … Λ(1405)studied with AY potential

Not new calculation but confirmation of the past work

Kbar nuclei (Nuclei with anti-koan) = Exotic system !?I=0 KbarN potential … very attractive

Deeply bound (Total B.E. ～ 100MeV)Highly dense state formed in a nucleusInteresting structures that we have never seen in normal nuclei…

To make the situation more clear …

“K-pp” = Prototype of Kbar nuclei(KbarNN, Jp=1/2-, T=1/2)

3He

0.0 0.15

K- ppn

0.0 2.0

K- ppp

3 x 3 fm2

0.0 2.0 [fm-3]

Calculated with Antisymmetrized Molecular Dynamics method employing a phenomenological KbarN potential

A. Doté, H. Horiuchi, Y. Akaishi and T. Yamazaki, PRC70, 044313 (2004)

1. Introduction

1. Introduction

Variational calculation of K-pp with a chiral SU(3)-based KbarN potential

Av18 NN potential … a realistic NN potential with strong repulsive core.

Effective KbarN potential based on Chiral SU(3) theory

Variational method… Investigate various properties with the obtained wave function.

… reproduce the original KbarN scattering amplitude obtained with coupled channel chiral dynamics.

Single channel, Energy dependent, Complex, Gaussian-shape potential

Four variants of chiral unitary models 2

N K

N K

M m B Ks

M m B K

×

Total B. E. : 20 ± 3 MeVG （ KbarN→pY ） : 40 ～ 70 MeV

Shallow binding and large decay width

NN distance = 2.2 fmKbarN distance = 2.0 fm

～　 NN distance in normal nuclei

A. Doté, T. Hyodo and W. Weise,Nucl. Phys. A804, 197 (2008)Phys. Rev. C79, 014003 (2009)

I=0 KbarN resonance “Λ(1405)”appears at 1420 MeV, not 1405 MeV

I=0 KbarN resonance “Λ(1405)”appears at 1420 MeV, not 1405 MeV

50 60 70 80 90 100 110 120 130

-140

-120

-100

-80

-60

-40

-20

0

Width (KbarNN→πYN) [MeV]

- B.E

. [M

eV]

Doté, Hyodo, Weise [1](Variational, Chiral SU(3))

Akaishi, Yamazaki [2](Variational, Phenomenological)

Exp. : FNUDA [5]

Exp. : DISTO [6](Preliminary)

Using S-wave KbarN potentialconstrained by experimental data. … KbarN scattering data, Kaonic hydrogen atom data, “Λ(1405)” etc.

Ikeda, Sato [4](Faddeev, Chiral SU(3))

Shevchenko, Gal [3](Faddeev, Phenomenological)

[1] PRC79, 014003 (2009)[2] PRC76, 045201 (2007)[3] PRC76, 044004 (2007)[4] PRC76, 035203 (2007)

[5] PRL94, 212303 (2005)[6] arXiv:0810.5182 [nucl-ex]

Recent results of calculation of K-pp Recent results of calculation of K-pp and related experiments

50 60 70 80 90 100 110 120 130

-140

-120

-100

-80

-60

-40

-20

0

Width (KbarNN→πYN) [MeV]

- B.E

. [M

eV]

Doté, Hyodo, Weise [1](Variational, Chiral SU(3))

Akaishi, Yamazaki [2](Variational, Phenomenological)

Exp. : FNUDA [5]

Exp. : DISTO [6](Preliminary)

Recent results of calculation of K-pp and related experiments

Using S-wave KbarN potentialconstrained by experimental data. … KbarN scattering data, Kaonic hydrogen atom data, “Λ(1405)” etc.

Ikeda, Sato [4](Faddeev, Chiral SU(3))

Shevchenko, Gal [3](Faddeev, Phenomenological)

[1] PRC79, 014003 (2009)[2] PRC76, 045201 (2007)[3] PRC76, 044004 (2007)[4] PRC76, 035203 (2007)

[5] PRL94, 212303 (2005)[6] arXiv:0810.5182 [nucl-ex]

2. Question --- Variational calc. and Faddeev ---

The KbarN potentials used in both calculations are constrained with Chiral SU(3) theory, but …

Total B. E. = 20±3 MeV, Decay width = 40 ～ 70 MeV

A. Doté, T. Hyodo and W. Weise,Phys. Rev. C79, 014003 (2009)

Variational calculation(DHW)

Faddeev calculation(IS)

Total B. E. = 60 ～ 95 MeV, Decay width = 45 ～ 80 MeV

Y. Ikeda, and T. Sato,Phys. Rev. C76, 035203 (2007)

??? Discrepancy between Variational calc. and Faddeev calc.

Separable potential used in Faddeev calculation? Non-relativistic (semi-relativistic) vs relativistic? Energy dependence of two-body system (KbarN) in the three-body system (KbarNN)? …???

Why ?

A possible reason is

πΣN thee-body dynamics Y. Ikeda and T. Sato, Phys. Rev. C79, 035201(2009)

• This is correct for the two-body system.• The effective KbarN potential has energy dependence due to the elimination of πΣ channel. • It reproduces the original KbarN scattering amplitude calculated with KbarN-πΣ coupled channel model.

Two-body system

In the variational calculation (DHW), πΣ channel is eliminated and incorporated into the effective KbarN potential.

K

N+ … + + …

…=

Σ

πK

N

K

N

K

N

π

Σ

K

N

K

N

V11 V1i Vj1

2. Question --- Variational calc. and Faddeev ---

Three-body system calculated with the effective KbarN potential

Σ Σ

K

N+ … + + …

…=

πK

N

K

N

K

N

πK

N

K

N

N N N N N N

The energy of the intermediate πΣ state is fixed to the initial KbarN energy.

Not true!

Apply the effective KbarN potential to the three-body system…

2. Question --- Variational calc. and Faddeev ---

Three-body system calculated with the effective KbarN potential

Σ Σ

K

N+ … + + …

…=

πK

N

K

N

K

N

πK

N

K

N

N N N N N N

Due to the third nucleon, the intermediate πΣ energy can differ from the initial KbarN energy.

N N

Apply the effective KbarN potential to the three-body system…

2. Question --- Variational calc. and Faddeev ---

Three-body system calculated with the effective KbarN potential

Σ Σ

K

N+ …

…=

πK

N

πK

N

K

N

N N N N

Due to the third nucleon, the intermediate πΣ energy can differ from the initial KbarN energy.

N N

Only the total energy of the three-body system should be conserved. The intermediate energy of the πΣ state can be variable, due to the third nucleon.

barK NNE

conserved

πΣN thee-body dynamics

Apply the effective KbarN potential to the three-body system…

2. Question --- Variational calc. and Faddeev ---

πΣN three-body dynamics is taken into account in the Faddeev calculation, because the πΣ channel is directly treated in their calculation.

Dr. Ikeda‘s talk in KEK theory center workshop “Nuclear and Hadron physics”(KEK, 11-13 Aug. ‘09)

2. Question --- Variational calc. and Faddeev ---

Coupled channel calculation

Direct treatment of the “πΣ” degree of freedom

• The obtained state is possibly to appear below KbarNN threshold, but above πΣN threshold.

• The actual calculation is done in multi channels such as KbarN and πΣ.

A bound state for KbarNN channel, Kbar + N + N

“Kbar N N”

π + Σ + N

but a resonant state for πΣN channel

Resonant state can’t be treated with a variational method. Here, we employ

“Complex Scaling method”

to deal with resonant state.

3. Complex Scaling method

3. Complex Scaling method

The resonant state can be obtained by diagonalizing 　 with Gaussian basis, quite in the same way as calculating bound state.

When 　　　　　　　　 , the wave function of a resonant state changes to a dumping function. 　

Wave function of a resonant state (two-body system)

Negative!

Boundary condition is the same as that for a bound state.

Complex rotation of coordinate

Advised by Dr. Myo (RCNP)

The energy of bound and resonant states is independent of scaling angle θ. (ABC theorem)

• Two-body system: KbarN-πΣ coupled system with s-wave and isospin zero state … Λ(1405)

Here, we consider just two-channel problem; KbarN and πΣ channels

Employ Akaishi-Yamazaki potential

Y. Akaishi and T. Yamazaki, PRC 52 (2002) 044005

Schrödinger equation to be solved

: complex parameters to be determined

Wave function expanded with Gaussian base

Complex-rotate , then diagonalize with Gaussian base.

4. Result

4. Resultq trajectory

• # Gauss base (n) = 30• Max range (b) = 10 [fm]

EG /

2

[MeV]

q = 0 deg.

2q

4. Result

EG /

2

[MeV]

q = 5 deg.

q trajectory• # Gauss base (n) = 30• Max range (b) = 10 [fm]

2q

4. Result

EG /

2

[MeV]

q =10 deg.

q trajectory• # Gauss base (n) = 30• Max range (b) = 10 [fm]

2q

4. Result

EG /

2

[MeV]

q =15 deg.

q trajectory• # Gauss base (n) = 30• Max range (b) = 10 [fm]

2q

4. Result

EG /

2

[MeV]

q =20 deg.

q trajectory• # Gauss base (n) = 30• Max range (b) = 10 [fm]

2q

4. Result

EG /

2

[MeV]

q =25 deg.

q trajectory• # Gauss base (n) = 30• Max range (b) = 10 [fm]

4. Result

EG /

2

[MeV]

q =30 deg.

q trajectory• # Gauss base (n) = 30• Max range (b) = 10 [fm]

2q

4. Result

EG /

2

[MeV]

q =35 deg.

q trajectory• # Gauss base (n) = 30• Max range (b) = 10 [fm]

2q

4. Result

EG /

2

[MeV]

q =40 deg.

q trajectory• # Gauss base (n) = 30• Max range (b) = 10 [fm]

2q

4. Result

EG /

2

[MeV]

q trajectory

2q q =30 deg.

pS KbarN

pS continuum

KbarN continuum

Resonance!(E, Γ/2) = (75.8, 20.0)

Measured from KbarN thr.,

B. E. (KbarN) = 28.2 MeVΓ = 40.0 MeV… L(1405) !

4. Result

b trajectory• # Gauss base (n) = 30• Rotation angle q = 30 [deg]

n trajectory• Max range (b) = 10 [fm]• Rotation angle q = 30 [deg] b = 50 [fm]

Stability of the resonance for other parameters

• # Gauss base (n) = 30• Max range (b) = 10 [fm]

Analysis of θ trajectory with

Resonance found at B. E. (KbarN) = 28.2 MeV

Γ = 40.0 MeV

5. Summary and Future plan

• No other resonant state (no interaction between π and Σ)

Solve the Coupled Channel problem = KbarN + πΣ

Complex Scaling Method applying to Λ(1405)

Resonance found at B. E. (KbarN) = 28.2 MeV

Γ = 40.0 MeV

In case of AY potential,

Comment

• In case of only VKN-KN, a bound state appears at B. E. = 3.2 MeV.

• Coupling with πΣ continuum state increases drastically the binding energy. Also, it gives the decay width.

KbarN thr.

VKN-πΣ

VKN-KN

5. Summary and Future planFuture plan

Keep doing calculations!

2. Calculate three-body system … KbarNN-πΣN system corresponding to “K-pp”

1. Calculate two-body system … KbarN-πΣ system corresponding to Λ(1405)

• See what happen if we use a chiral SU(3)-based KbarN potential (HW potential, energy-dependent).

T. Hyodo and W. Weise, PRC77, 035204(2008)

• For a test calculation, a phenomenological KbarN potential (AY potential, energy-independent) will be employed.

Y. Akaishi and T. Yamazaki, PRC 52 (2002) 044005 Done

Investigate the property of

the obtained wave function

and thinking…

J-PARC will give us lots of interesting data!

E15: A search for deeply bound kaonic nuclear states by 3He(inflight K-, n) reaction --- Spokespersons: M. Iwasaki (RIKEN), T. Nagae (Kyoto)

E17: Precision spectroscopy of kaonic 3He atom 3d→2p X-rays --- Spokespersons: R. Hayano (Tokyo), H. Outa (Riken)

Thank you very much!

Acknowledgement:Dr. Myo (RCNP) taught me CSM and supplied a subroutine for diagonalization of a complex matrix.