Λ(1405) and “Kpp” - J-PARC Theory top...

35
Λ(1405 ) and “ K - pp” on a coupled - channel scheme 1. Introduction 2. Method Complex Scaling Method (CSM) Chiral SU(3) - based K bar N interaction 3. Results Λ(1405) as a K bar N - πΣ system with coupled - channel CSM “K - pp” as a K bar NN - 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 1 st building), Tokai., Ibaraki, Japan

Transcript of Λ(1405) and “Kpp” - J-PARC Theory top...

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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

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1. Introduction

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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:

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“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-

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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)

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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.

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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)

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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)

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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

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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)

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3. Hyperon resonance Λ(1405)

on ccCSM

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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 <<

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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

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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

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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)

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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)

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P PK-

3. Three-body “K-pp” resonance

on ccCSM+Feshbach projection

“K-pp” =

KbarNN – πΣN – πΛN (Jπ = 0-, T=1/2)

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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.

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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

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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

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“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

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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

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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

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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)

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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!

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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

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P PK-

4. Further analysis of “K-pp”

• SIDDHARTA constraint for K-p scattering length

• Another way of KbarN energy self-consistency

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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)

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“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

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“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

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“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

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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.

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5. Summary and future plans

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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.

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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