Λ(1405) and “K-pp”

on a coupled-channel scheme

1. Introduction

2. Method

• Complex Scaling Method (CSM)

• Chiral SU(3)-based KbarN interaction

3. Results

• Λ(1405) as a KbarN-πΣ system with coupled-channel CSM

• “K-pp” as a KbarNN-πYN system with ccCSM+Feshbach method

4. Further analysis of “K-pp”

5. Summary and future plan

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

Takashi Inoue (Nihon university)

Takayuki Myo (Osaka Institute of Technology)

「ストレンジネス核物理の発展方向」03. Aug. ’15 @ KEK Tokai campus (Tokai 1st building), Tokai., Ibaraki, Japan

1. Introduction

Phenomenologically derived I=0 KbarN potential … Very attractive

• Deeply bound (Total B.E. ～100MeV)

• Quasi-stable (πΣ decay mode closed)

• Highly dense state … anti-kaon attracts nucleons

Excited hyperon Λ(1405) = quasi-bound state of I=0 KbarN

Y. Akaishi and T. Yamazaki, PRC65, 044005 (2002)

Kaonic nuclei = Exotic system !?

3He

0.0 0.15

K- ppn

0.0 2.0

K- ppp

3 x 3 fm2

0.0 2.0 [fm-3]

Antisymmetrized Molecular Dynamics

method with

a phenomenological KbarN potential

A. Dote, H. Horiuchi, Y. Akaishi and

T. Yamazaki, PRC70, 044313 (2004)

Restoration of chiral symmetry in dense matter

Interesting structure, Neutron star, …

Relate to various physics:

“Prototype of

kaonic nuclei”

K-

Proton

Λ(1405) “Building block of kaonic nuclei”

… used to determine KbarN interaction

K-P P

K-pp

Most essential kaonic nucleus

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

…, 8BeK-,…

Nuclear many-body system with K-

Kaonic nuclei

at J-PARC

P PK-

Prototype system = K- pp

Experiments of K-pp searchFINUDA DISTO

J-PARC E15 SPring8/LEPSJ-PARC E27

• p + p → p + Λ + K+ at Tp=2.85 GeV

• K- pp???B. E. = 103 ±3 ±5 MeVΓ = 118 ±8 ±10 MeV

PRL104, 132502 (2010)

• Stopped K- on Li, C, Al, V targets

• K- pp???B. E.= 115 MeVΓ = 67 MeV

PRL 94, 212303 (2005)

• 3He(K-, n) X at PK = 1 GeV/c

• Attraction in K-pp subthreshold regionarXiv:1408.5637 [nucl-ex],

to appear in PTEP

• d(γ, π+K+) X at Eγ = 1.5 – 2.4 GeV

• No evidence of K-pp bound statePLB 728, 616 (2014)

• d(π+, K+) X at Pπ= 1.69 GeV/c

• ΣN cusp, Y* shiftPTEP 101D03 (2014)

• K-pp???

B. E. ~ 95 MeV

Γ ~ 162 MeV

PTEP 021D01 (2015)

Typical results of theoretical studies of K-pp

DHW AY BGL IS SGM

B(K-pp) 20±3 47 16 9 ~ 16 50 ~ 70

Width Γ 40 ~ 70 61 41 34 ~ 46 90 ~ 110

Method Variational

(Gauss)

Variational

(Gauss)

Variational

(HH)

Faddeev-AGS Faddeev-AGS

Potential Chiral

(E-dep.)

Pheno. Chiral

(E-dep.)

Chiral

(E-dep.)

Pheno.

Kinematics Non-rel. Non-rel. Non-rel. Non-rel. Non-rel.

Dependent on the type of KbarN interaction

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

-->> Shallow binding: B(K-pp) ~ 20 MeV

Phenomenological potential (Energy independent)

-->> Deep binding: B(K-pp) = 40~70 MeV

B(K-pp) < 100 MeV

B.E. of “K-pp”(=KbarNN-πYN with Jπ=0-, I=1/2) is less than 100 MeV.

2. Method

• Complex Scaling Method

• Feshbach projection on

coupled-channel Complex Scaling Method

“ccCSM+Feshbach method”

A. D., T. Inoue, T. Myo,

PTEP 2015, 043D02 (2015)

Resonant state

Coupled-channel

system

⇒ “coupled-channel

Complex Scaling Method”

• Λ(1405) = Resonant state & KbarN coupled with πΣ

• “K-pp” … Resonant state of

KbarNN-πYN coupled-channel systemDoté, 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)

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)

Complex rotation (Complex scaling) of coordinate

Resonance wave function → L2 integrable : ,i iU e e r r k k

Diagonalize Hθ = U(θ) H U-1(θ) with Gaussian base,

Continuum state appears on 2θ line. Resonance pole is off from 2θ line, and independent of θ. (ABC theorem)

tan-1 [Im E / Re E] = -2θ

Chiral SU(3) potential with a Gaussian form

Weinberg-Tomozawa term

of effective chiral Lagrangian

Gaussian form in r-space

Semi-rela. / Non-rela.

Based on Chiral SU(3) theory

→ Energy dependence

Constrained by KbarN scattering length

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

( 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

A non-relativistic potential (NRv2c)

: Gaussian form

ωi: meson energy

• Anti-kaon = Nambu-Goldstone boson

⇒ Chiral SU(3)-based KbarN potential

A. D., T. Inoue, T. Myo, Nucl. Phys. A 912, 66 (2013)

3. Hyperon resonance Λ(1405)

on ccCSM

Excited hyperon Λ（１４０５）

I=0 N-Kbar bound state

Not 3 quark state,

← A naive quark model fails.N. Isgar and G. Karl, Phys. Rev. D18, 4187 (1978)

But rather a molecular stateT. Hyodo and D. Jido,

Prog. Part. Nucl. Phys. 67, 55 (2012)

qbar

sqq q

qq

q <<

KbarN-πY coupled system with s-wave and isospin-0 state

Λ* on c.c. Complex Scaling Method

Schrödinger equation to be solved

… I=0 KbarN-πΣ coupled potentialV

: complex parameters to be determined

Wave function expanded with Gaussian base

Poles of I=0 KbarN-πΣ system found by ccCSM

KbarN potential:

a chiral SU(3) potential

(NRv2, fπ=110)

πΣ continuum

KbarN continuum

A. D., T. Inoue, T. Myo,

Nucl. Phys. A 912, 66 (2013)

θ=30°

Λ*

Lower pole

πΣ KbarN

A. D., T. Myo,

Nucl. Phys. A 930, 86 (2014)

“Complex-range

Gaussian basis”

-Γ/2

[Me

V]

M [MeV]

Double-pole structure of Λ(1405)

θ=40°

Higher pole

D. Jido, J.A. Oller, E. Oset, A. Ramos, U.-G. Meißner, Nucl. Phys. A 725 (2003) 181

ccCSM wfnc. of double pole

Norm (KbarN)

1.115+0.098i

Norm (πΣ)

-0.115-0.098i

Norm (KbarN)

0.097+0.154i

Norm (πΣ)

0.903-0.154i

KbarN dominant

πΣ dominant

Higher pole

Lower pole

A. D., T. Myo,

Nucl. Phys. A 930, 86 (2014)

Double-pole structure of Λ(1405)

1435

1331

KbarN

πΣ

fπ = 90 MeV 100 MeV 110 MeV 120 MeV

1349

Γ=134

1371

Γ=221

1395

Γ=275

1425

Γ=327

1420

Γ=46

1418

Γ=40

1418

Γ=33

1418Γ=28

Lower pole does exist in our potential.

It depends strongly on a parameter fπ.

A. D., T. Myo,

Nucl. Phys. A 930, 86 (2014)

P PK-

3. Three-body “K-pp” resonance

on ccCSM+Feshbach projection

“K-pp” =

KbarNN – πΣN – πΛN (Jπ = 0-, T=1/2)

ccCSM+Feshbach method• Λ(1405) = two-body system of KbarN-πΣ

→ Explicitly treat coupled-channel problem

• “K-pp” = three-body system of KbarNN-πYN

… High computational cost

For economical treatment of “K-pp”, we construct an effective KbarN

single-channel potential by means of Feshbach projection on CSM.

Formalism of ccCSM + Feshbach method

Schrödinger eq.

in model space “P” and out of model space “Q”

Eff

PPP PT U E E

Eff

P P PQ Q QPU E v V G E V

Schrödinger eq. in P-space :

Effective potential for P-space

P P PQ P P

QP Q Q Q Q

T v VE

V T v

Elimination of channels by Feshbash method

1

Q

QQ

G EE H

Q-space Green function:

1 1

Q n n

nQQ n

G EE H E

n

: expanded with Gaussian base.

Express the GQ(E) with Gaussian base using ECR

1

Q

Eff

P P PQ QPU E v V U UG VE

Q

G E

Extended Closure Relation in Complex Scaling Method

QQ n n nH 1n n

BC R

1n n

n

Diagonalize HθQQ with Gaussian base,

Well approximated

T. Myo, A. Ohnishi and K. Kato, PTP 99, 801 (1998)

R. Suzuki, T. Myo and K. Kato, PTP 113, 1273 (2005)

1

QQ QQH U H U

Applying this tequnique

to the two-body KbarN-πY system,

Effective single-channel KbarN potential

is constructed.

Using the UEffKN in “K-pp” three-body calculation,

the KbarNN-πYN coupled-channel problem is reduced

to the KbarNN single-channel problem.

VKN-KN VKN-πY

VπY-KN VπY-πY

UEffKN

“K-pp” … KbarNN - πΣN - πΛN (Jπ=0-, T=1/2) “K-pp” … KbarNN - πΣN - πΛN (Jπ=0-, T=1/2)

Apply ccCSM + Feshbach method to K-pp

• Schrödinger eq. for KbarNN channel :

; 0,1

' ; 0,1

barV K N Y I

V Y Y I

( 0,1)bar

Eff

K N IU E

For the two-body system, P = KbarN, Q = πY

( ,1) ( ,1) (3) (3) ( ,1) (3) (3)

2 2 1 1/ 2" " , , 0KNN KNN KNN

a a a NN Ta

K pp C G G S K NN

1 1x x x x

( ,2) ( ,2) (3) (3) ( ,2) (3) (3)

2 2 0 1/ 2, , 0KNN KNN KNN

a a a NNT

a

C G G S K NN

1 1x x x x

Ch. 1: KbarNN, NN:1E

Ch. 2: KbarNN, NN:1O

• Trial wave function

(3)

( , ) (3) (3) ( , ) (3) (3) ( , )

2 2 (3)

2

, exp ,KNN i KNN i KNN i

a a aG N A

1

1 1

xx x x x

x

• Basis function ＝ Correlated Gaussian

…including 3-types Jacobi-coordinates

Feshbach + ccCSM

(1,2

)bar barbar bar bi

arNNK NN K NN K NN

Eff

K N I K Ni

T EV EU

Self-consistency for complex KbarN energy

Kbar

NN

E(KN)Cal

( )

( )bari

Eff

K N II nE KNU

When E(KN)In= E(KN)Cal,

a self-consistent solution is obtained.

Effective KbarN potential has energy dependence…

• E(KN)In : assumed in the KbarN potential

• E(KN)Cal : calculated with the obtained K-pp

Self-consistency for complex KbarN energy

NNB K H H 1. Kaon’s binding energy: NNH : Hamiltonian of two nucleons

2

N K

KN N

N K

M m B KE M

M m B K

2. Define a KbarN-bond energy in two ways

How to determine the two-body energy in the three-body system? A. D., T. Hyodo, W. Weise,

PRC79, 014003 (2009)

: Field picture

: Particle picture

Particle pictureField picture

Self-consistent results

fπ=90~120MeV

NN pot. : Av18 (Central)

KbarN pot. : NRv2c potential

(fπ=90 - 120MeV)

fπ = 90

100

fπ = 90

100

110

120

Particle picture

(B, Γ/2) = (25~30, 15~32)

120110

100

fπ = 90×Unstable for scaling angle θ!

Field picture

(B, Γ/2) = (21~32, 9~16)

NN correlation density

Re ρNN

Im ρNN

NN repulsive core

NN pot. : Av18 (Central)

KbarN pot. : NRv2c potential

fπ=110, Particle pict.

Correlation density in Complex Scaling Method

,XNNN x x ,, NNNN x r x

,i

XN XN e r r 3 23 ,i ide e

R x R

N

N

Kbar

NN distance = 2.1 - i 0.3 fm

~ Mean distance of 2N in nuclear matter at normal density!

Energy composition of “K-pp” NN pot. : Av18 (Central)

KbarN pot. : NRv2c potential

(fπ=110MeV)

barKbar bar bar

NNNK NN K NN K N

EffV VH T V

Field pict. Particle pict.

TKNN 163.1 174.6

VNN -15.3 -15.9

VKN -153.3 -169.8

VKNEff -20.1 -16.1

-B.E. -25.6 -27.3

-5.5 -11.2

P PK-

4. Further analysis of “K-pp”

• SIDDHARTA constraint for K-p scattering length

• Another way of KbarN energy self-consistency

K-pp with SIDDHARTA data

M. Bazzi et al. (SIDDHARTA collaboration),

NPA 881, 88 (2012)

K-p scattering length (with improved Deser-Truman formula)

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

Precise measurement of 1s level shift of kaonic hydrogen

Strong constraint for the KbarN interaction!

U. -G. Meissner, U. Raha and A. Rusetsky, Eur. Phys. J. C 35, 349 (2004)

(Prediction)

“K-pp” with Martine value NN pot. : Av18 (Central)

KbarN pot. : NRv2c potential

(fπ=90 - 120MeV)

fπ = 100

110120

fπ = 90

100

110

120

aKN(I=0) = -1.7 + i0.68 fm

aKN(I=1) = (0.37) + i0.60 fm

“K-pp” with SIDDHARTA value NN pot. : Av18 (Central)

KbarN pot. : NRv2a-IHW pot.

(fπ=90 - 120MeV)

fπ = 100

110120

fπ = 90

100

110

120

Particle picture

(B, Γ/2) = (16~19, 14~25)

Field picture

(B, Γ/2) = (15~22, 10~18)

aKN(I=0) = -1.97 + i1.05 fm

aKN(I=1) = 0.57 + i0.73 fm

“K-pp” with SIDDHARTA value NN pot. : Av18 (Central)

KbarN pot. : NRv2a-IHW pot.

(fπ=90 - 120MeV)

E(KN) = -B(K)

E(KN) = -B(K)/2

E(KN) = -B(K)/2 - ΔBarnea, Gal, Livertz, PLB 712, 132 (2012)

Averaged KbarN energy in many-body system

fπ = 100

110120

fπ = 90

100

110

120

aKN(I=0) = -1.97 + i1.05 fm

aKN(I=1) = 0.57 + i0.73 fm

Summary of “K-pp”

NRv2c NRv2a-IHW DHW-HNJHfp=110 fp=110

aKN (I=0) -1.70 + i 0.68 -1.97 + i 1.04 -1.54 + i 1.14aKN (I=1) 0.68 + i 0.60 0.57 + i 0.73 0.39 + i 0.53

Λ * (High) -17.2 - i 16.6 -8.1 - i 13.3 -11.5 - i 21.9Λ * (Low) -39.8 - i 137.9 -45.1 - i 138.8

K-ppField -25.6 - i 11.6 -18.2 - i 12.3 -16.9 - i 23.5Particle -27.3 - i 18.9 -17.8 - i 17.2 -20.8 - i 29.2

†

† Dote, Hyodo, Weise, PRC 79, 014003 (2009)

Imaginary part is treated perturbatively.

5. Summary and future plans

5. SummaryWe have investigated

an excited hyperon Λ(1405) as a resonant state of KbarN-πΣ coupled system and

a prototype of Kbar nuclei “K-pp” as that of KbarNN-πYN coupled system,

based on a coupled-channel Complex Scaling Method (ccCSM)

using a chiral SU(3)-based KbarN-πY potential.

Λ(1405)

• Double pole structure is confirmed in our chiral SU(3) potential with a Gaussian form.

• Higher pole appears at 1420 MeV with a half decay witdth of ~20 MeV stably,

while Lower pole position strongly depends on a parameter of the potential.

• Main component of Higher pole is the KbarN, while that of Lower pole is πΣ.

“K-pp” with ccCSM+Feshbach projection

• πY’s are eliminated to reduce the coupled-channel problem to a single KbarNN channel

problem with Feshbach projection, which is well realized with Extended Closure Relation.

• With a constraint of Martine KbarN scattering length,

B(K-pp) = 20 ~ 30 MeV and Γ/2 = 10 ~ 30 MeV. NN distance ~ 2.2 fm → Normal density

• With a constraint of SIDDAHRTA K-p scattering length

and a prediction of K-n one by a chiral unitary model,

B(K-pp) = 15 ~ 22 MeV and Γ/2 = 10 ~ 25 MeV.

… Shallower binding compared with Martine’s case.

Cats in KEK

Thank you for your attention!

References:

1. A. D., T. Inoue, T. Myo,

NPA 912, 66 (2013)

2. A. D., T. Myo, NPA 930, 86 (2014)

3. A. D., T. Inoue, T. Myo,

PTEP 2015, 043D02 (2015)

5. Future plans

Full-coupled channel calculation of K-pp

… Deailed study for the double pole structure of K-pp

Application to resonances of other hadronic systems