結合チャネル複素スケーリング法による“ K-pp” の研究

1. Introduction

2. Fully coupled-channel Complex Scaling Method

for the essential kaonic nucleus “K-pp”

3. Result Phenomenological KbarN potential

Chiral SU(3)-based KbarN potential

4. Discussion

5. Summary and future prospects

Akinobu Doté(KEK Theory Center, IPNS / J-PARC branch)

KEK理論センター研究会 『ハドロン・原子核物理の理論研究最前線 ２０１７』, 22. Nov., ’17 @ KEK (Tsukuba campus), Japan

Takashi Inoue (Nihon univ.)

Takayuki Myo (Osaka Inst. Tech.)

1. Introduction

Kaonic nuclei = Nuclear system with Kbar mesons

Nucleus

qbar

s

Kbar meson(K0bar, K-)

Jπ=0-, I=1/2

Involve strange quarks (Strangeness).

Real mesons are involved

as a constituent of the system.

Expect “Dense and Cold” state Y. Akaishi, T. Yamazaki, PRC65, 044005 (2002)

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

K-

Proton

KbarN two-body system

Excited hyperon Λ(1405) = K- proton quasi-bound state

• Low energy scattering data, 1s level shift of kaonic hydrogen atom

• Hard to describe Λ(1405) with Quark Model as a 3-quark state

• Success of Chiral Unitary Model with a meson-baryon picture

Excited hyperon Λ（１４０５）

K-

Proton

KbarN two-body system

Excited hyperon Λ(1405) = K- proton quasi-bound state

• Low energy scattering data, 1s level shift of kaonic hydrogen atom

• Hard to describe Λ(1405) with Quark Model as a 3-quark state

• Success of Chiral Unitary Model with a meson-baryon picture

Doorway to dense matter†

→ Chiral symmetry restoration in dense matter

Interesting structure†

Neutron star

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

3HeK-, pppK-, 4HeK-, pppnK-,

…, 8BeK-,…

Kaonic nuclei

Strongly attractive KbarN potential

K-

ProtonKbarN two-body system = Λ(1405)

Kaonic nuclei

= Nuclear many-body system with antikaons

P PK-

Prototype system = K- pp

P PK-

Prototype system = K- pp

K-pp bound state???

Studied with various methods,

since K-pp is a three-body system

Clear evidence of K-pp bound state

J-PARC E15; Exclusive exp. 3He(K-, Λp)nmissing

K-pp bound state!T. Sekihara, E. Oset, A. Ramos,

PTEP 2016, 123D03

Theoretical study

J-PARC E15 (2nd run)

Yamaga’s talk

(JPS symposium 2017)

P PK-

2. Fully coupled-channel

Complex Scaling Method

for

the essential kaonic nucleus “K-pp”

Resonant state of KbarN-πΣ coupled-channel system

•Hyperon resonance Λ(1405)

According to early studies ...

⇒ “coupled-channel

Complex Scaling Method”

Resonant state of

KbarNN-πΣN-πΛN coupled channel

three-body system

•Prototype system of kaonic nuclei “K-pp”

Doté, Hyodo, Weise, PRC79, 014003(2009). Akaishi, Yamazaki, PRC76, 045201(2007)

Ikeda, Sato, PRC76, 035203(2007). Shevchenko, Gal, Mares, PRC76, 044004(2007)

Barnea, Gal, Liverts, PLB712, 132(2012)

Resonance & Channel coupling

Complex Scaling Method… Powerful tool for resonance study of many-body system

S. Aoyama, T. Myo, K. Kato, K. Ikeda, PTP116, 1 (2006)

T. Myo, Y. Kikuchi, H. Masui, K. Kato, PPNP79, 1 (2014)

Hamiltonian

1,2

3,MB MB MB MB

i

V V i

... symmetric for baryon’s site (i=1,2)

Glöckle, Miyagawa, Few-body Systems 30, 241 (2001)

1

2

3

Baryon

Baryon

Meson

1

62 285 285

0 0 MeV

0

bar

I

K N

V

Y. Akaishi and T. Yamazaki,

PRC 65, 044005 (2002)

R. B. Wiringa, V. G. J. Stoks, and R. Schiavilla, PRC 51, 38 (1995) NN potential = Av18 potential

KbarN-πY potential = Akaishi-Yamazaki potential

2

exp 0.66 fmI

IV V r

0 436 412MeV

0

bar

I

K N

V

Phenomenological energy-independent potential

Ignore YN and πN potentials

( ) ( )

,bar

NN MB MB

K N

H M T V V

Wave function

Baryon-Baryon are antisymmetrized on space, spin and isospin as well as label (flavor).Glöckle, Miyagawa, Few-body Systems 30, 241 (2001)

Spatial part = Correlated Gaussian function

including 3 types of Jacobi coordinates

projected onto a parity eigenstate of B1B2,

M3

B2

B1

x1(3)

x2(3)

P PK-

3. Result

Phenomenological KbarN potential(Energy-independent)

“K-pp” =

KbarNN – πΣN – πΛN (Jπ = 0-, T=1/2)

A. Dote, T. Inoue, T. Myo, PRC 95, 062201(R) (2017)

Just diagonalize

the complex-scaled Hamiltonian matrix!

No channel elimination!!

KbarNN

- KbarNN

θKbarNN

- πΣN

KbarNN

- πΛN

πΣN

- πΣN

πΣN

- πΛN

πΛN

- πΛN

＊

＊ ＊

Hij

Complex eigenvalue distributionNN: Av18 pot.

KbarN-πY: Akaishi-Yamazaki pot.

(Pheno., E-indep.)

Scaling angle θ=30°Dimension = 6400

Λ*N cont.

Λ* pole

BKN = 28 MeV, ΓπΣ /2 = 20 MeV

The “K-pp” pole of AY potential :

BKNN = 51 MeV, ΓπYN /2 = 16 MeV

πΛN πΣN KbarNN

KbarNN cont.

πΛN cont. πΣN cont.

Property of the “K-pp” pole

Norm 1.004 - i 0.286 -0.002 + i 0.276 -0.002 + i 0.010

BB distance [fm]

1.86 + i 0.14 1.50 + i 0.48 1.10 + i 0.23

KbarN

(I=0) 0.726 - i 0.213

(I=1) 0.278 - i 0.073

πΣ

(I=0) -0.009 + i 0.261

(I=1) 0.007 + i 0.015

πΛ

(I=0) ---

(I=1) -0.002 + i 0.010

MB distance [fm]

(I=0) 1.40 - i 0.01

(I=1) 1.95 + i 0.14

(I=0) 0.52 + i 1.31

(I=1) 0.80 + i 1.22

(I=0) ---

(I=1) 0.74 + i 0.74

NN: Av18 pot.

KbarN-πY: Akaishi-Yamazaki pot.

(Pheno., E-indep.)

Property of the “K-pp” pole

Norm 1.004 - i 0.286 -0.002 + i 0.276 -0.002 + i 0.010

BB distance [fm]

1.86 + i 0.14 1.50 + i 0.48 1.10 + i 0.23

KbarN

(I=0) 0.726 - i 0.213

(I=1) 0.278 - i 0.073

πΣ

(I=0) -0.009 + i 0.261

(I=1) 0.007 + i 0.015

πΛ

(I=0) ---

(I=1) -0.002 + i 0.010

MB distance [fm]

(I=0) 1.40 - i 0.01

(I=1) 1.95 + i 0.14

(I=0) 0.52 + i 1.31

(I=1) 0.80 + i 1.22

(I=0) ---

(I=1) 0.74 + i 0.74

NN distance = 1.9 fm

⇒1.6 ρ0

NN: Av18 pot.

KbarN-πY: Akaishi-Yamazaki pot.

(Pheno., E-indep.)

Property of the “K-pp” pole

Norm 1.004 - i 0.286 -0.002 + i 0.276 -0.002 + i 0.010

BB distance [fm]

1.86 + i 0.14 1.50 + i 0.48 1.10 + i 0.23

KbarN

(I=0) 0.726 - i 0.213

(I=1) 0.278 - i 0.073

πΣ

(I=0) -0.009 + i 0.261

(I=1) 0.007 + i 0.015

πΛ

(I=0) ---

(I=1) -0.002 + i 0.010

MB distance [fm]

(I=0) 1.40 - i 0.01

(I=1) 1.95 + i 0.14

(I=0) 0.52 + i 1.31

(I=1) 0.80 + i 1.22

(I=0) ---

(I=1) 0.74 + i 0.74

cf) Λ(1405) ~ KbarN (I=0)

... 1.25 fm

NN: Av18 pot.

KbarN-πY: Akaishi-Yamazaki pot.

(Pheno., E-indep.)

P PK-

3. Result

Chiral SU(3)-based KbarN potential(Energy-dependent)

“K-pp” =

KbarNN – πΣN – πΛN (Jπ = 0-, T=1/2)

A. Dote, T. Inoue, T. Myo, arXiv: 1710.07589

Chiral SU(3)-based KbarN potential

Coupled-channel chiral dynamics

(Chiral Unitary model)N. Kaiser, P.B. Siegel, W. Weise, NPA 594 (1995) 325

E. Oset, A. Ramos, NPA 635 (1998) 99

Weinberg-Tomozawa term

of effective chiral Lagrangian

Based on Chiral SU(3) theory

→ Energy dependence

• Anti-kaon, Pion = Nambu-Goldstone boson

... governed by chiral dynamics

Constrained by KbarN scattering length

• Old data: aKN(I=0) = -1.70 + i0.67 fm, aKN(I=1) = 0.37 + i0.60 fm A. D. Martin, NPB179, 33(1979)

• SIDDHARTA K-p data with a coupled-channel chiral dynamics:

aK-p = -0.65 + i0.81 fm, aK-n = 0.57 + i0.72 fm M. Bazzi et al., NPA 881, 88 (2012)

Y. Ikeda, T. Hyodo, W. Weise, NPA 881, 98 (2012)

( 0,1)

( 0,1)

2

1

8

I

ijI

ij ii j j

i j

CV r g r

f m m

3

2

/ 2 3ex

1p

ijij

ij

g rd

r d

Non-relativistic potential

: Gaussian form

ωi: meson energy

A. Dote, T. Inoue, T. Myo, NPA 912, 66 (2013)

How to deal with E-dep. potential?

( 0,1)

( 0,1)

2

1

8

I

ijI

ij ii j j

i j

CV r g r

f m m

Chiral SU(3)-based potential is an energy-dependent potential.

How to treat energy-dependent potentials in many-body system?

Especially, in the KbarNN-πYN coupled-channel problem case???

Generalize earlier studies of the KbarNN single-channel problem

with E-dep. effective potential:A. D., T. Hyodo, W. Weise, PRC79, 014003 (2009)

A. D., T. Inoue, T. Myo, PTEP 2015, 043D02 (2015)

“Self-consistency for Meson-Baryon energy”

Self-consistency for KbarN energy

[KbarNN single-channel case]

barK NN NNB K H H

1. Kaon’s binding energy:

: Hamiltonian of 2N

2

N K

KN N K

N K

M m B KE M

M m B K

2. Define a KbarN-bond energy in two ways:

Estimation the two-body energy in the three-body system

: Field picture

: Particle picture

barK NNH : Hamiltonian of 2N+Kbar

NNH

A. D., T. Hyodo, W. Weise,

PRC79, 014003 (2009)

An interacting KbarN pair

carries

... 100% of B(K)

... 50% of B(K)

Self-consistency for MB energy

[KbarNN-πＹＮ coupled-channel case]

1 2 1 2 MMB B B BB M H H m

1. Meson’s binding energy:

: Hamiltonian of 2B

( )

2

B M

B M

B M

M m B ME MB M

M m B M

2. Define a MB-bond energy in two ways:

Estimation the two-body energy in the three-body system

A. D., T. Hyodo, W. Weise,

PRC79, 014003 (2009)

: Field picture

: Particle picture

1 2MB BH : Hamiltonian of 2B+M

1 2B BH

Generalized version of

Realization of self-consistency

Indicator Δ

[MeV]

- Re E(MB)In [MeV]

Chiral SU(3) pot.

(fπ=110 MeV, Martin)

Field picture

Indicator of self-consistency

Δ=|E(MB)Cal – E(MB)In|

Self-consistent solution :

BKNN = 23.5 MeV, ΓπYN /2 = 9.1 MeV

Result

Martin constraint SIDDHARTA constraint

Field: (B, Γ/2) = (19－36, 8－14)

Particle: (B, Γ/2) = (30－47, 12－14)

Field: (B, Γ/2) = (14－28, 8－15)

Particle: (B, Γ/2) = (22－38, 13－18)

(-BKNN, -Γ/2) [MeV]

120

100

90

(-BKNN, -Γ/2) [MeV]

fπ=120 MeV

90100

120

fπ=120 MeV

P PK-

4. Discussion

1. Dense matter or not?

P K-

P PK-

Chiral SU(3) potential

(E-dep.)A. Dote, T. Inoue and T. Myo,

NPA 912, 66 (2013)

B. E. (Λ*) ~ 15 MeV

⇒ Λ* = Λ(1420)

B. E. (“K-pp”) = 14-38 MeV

NN distance = 2.2 fm

⇒ ~ ρ0 (=0.17 fm-3)

Pheno. potential

(E-indep.)Y. Akaishi and T. Yamazaki,

PRC 65, 044005 (2002)

B. E. (Λ*) = 28 MeV

⇒ Λ* = Λ(1405)

B. E. (“K-pp”) = 51 MeV

NN distance = 1.9 fm

⇒ ~ 1.6 ρ0

2. Comparison of Theory and Experiment

0 [MeV]

KbarNN thr.

-50 [MeV]

J-PARC E15

(1st run)

B ~ 16 MeV

Γ ~ 110 MeV

2. Comparison of Theory and Experiment

0 [MeV]

KbarNN thr.

-50 [MeV]

J-PARC E15

(1st run)

B ~ 16 MeV

Γ ~ 110 MeV

J-PARC E15

(2nd run)

Full ccCSM calculation

Phenomenological pot. : BK-pp = 51 MeV(Martin)

Chiral SU(3)-based pot. : BK-pp = 14－28 MeV(SIDDHARTA, Field)

Chiral SU(3) pot.Pheno. pot.

Yamaga’s talk

(JPS symposium

2017)

5. Summary

and

Future prospects

“K-pp” studied with Fully coupled-channel Complex Scaling Method

• Chiral SU(3)-based KbarN potential constrained with the latest KbarN data (SIDDAHRTA)

K-pp (Jπ=0-, T=1/2) … (B, Γ/2) = (14--28, 8--15) MeV (Field picture)

(22--38, 13--18) MeV (Particle picture)

• Phenomenological KbarN potential (Akaishi-Yamazaki potential; Energy-independent)

K-pp (Jπ=0-, T=1/2) …. (B, Γ/2) = (51, 16) MeV

Self-consistency for meson-baryon energy is considered.

• At the moment, J-PARC E15 (2nd run) seems consistent with the result of K-pp (Jπ=0-, T=1/2) calculated with a phenomenological potential ???

• NN mean distance of “K-pp” system = 2.2 fm (Chiral pot.), 1.9 fm (AY pot.)

If KbarN potential is so attractive as Akaishi-Yamazaki potential, kaonic nuclei could be a gateway to dense matter.

Among kaonic nuclei, “K-pp” is the most essential system:

All theoretical studies predict B(K-pp) < 100 MeV.

“K-pp” = Resonance state of KbarNN-πYN coupled system

Kaonic nuclei are expected to have exotic nature, such as formation of

“Dense and cold” state, due to the strongly attractive KbarN interaction.

5. Summary

Semi-relativistic treatment... Pion mass is small.

Lower pole of K-pp?… E-dep. Chiral SU(3) pot. may have the double pole structure.

Reaction spectrum... Direct comparison with experimental data.

It can be calculated using the Green function obtained

with ccCSM.

(“Morimatsu-Yazaki Green function method”)

...

5. Future prospects

References:

• A. Dote, T. Inoue, T. Myo, PRC 95, 062201(R) (2017)• A. Dote, T. Inoue, T. Myo, arXiv: 1710.07589

Thank you very much!