Application of coupled-channel Complex Scaling Method to the K bar N -πY system

Application of coupled-channel Complex Scaling Method to the KbarN-πY system

1. Introduction

2. coupled-channel Complex Scaling Method• For resonance states• For scattering problem

3. Set up for the calculation of KbarN-πY system• “KSW-type potential”• Kinematics / Non-rela. Approximation

4. Results• I=0 channel: Scattering amplitude / Resonance pole• I=1 channel: Scattering amplitude

5. Summary and future plans

A. Doté (KEK Theory center / IPNS/ J-PARC branch)T. Inoue (Nihon univ.)

T. Myo (Osaka Inst. Tech. univ.)

International workshop “Inelastic Reactions in Light Nuclei”, 09.Oct.’13 @ IIAS, The Hebrew univ., Jerusalem, Israel

A. Dote, T. Inoue, T. Myo,Nucl. Phys. A912, 66-101 (2013)

1. Introduction

Kaonic nuclei = Nuclear system with Anti-kaon “Kbar”

Excited hyperon Λ(1405) = a quasi-bound state of K- and protonAttractive KbarN interaction!

Difficult to explain by naive 3-quark model Consistent with “repulsive nature” indicated by

KbarN scattering length and 1s level shift of kaonic hydron atom

T. Hyodo and D. Jido, Prog. Part. Nucl. Phys. 67, 55 (2012)

KbarN threshold = 1435MeV

Antisymmetrized Molecular Dynamics with a phenomenological KbarN potential

Anti-kaon can be deeply bound and/or form dense state?

Anti-kaon bound in light nuclei with ~ 100MeV binding energy Dense matter (average density = 2 ～ 4 ρ0)

Relativistic Mean Field applied to medium to heavy nuclei with anti-kaons

• AMD studyA. D., H. Horiuchi, Y. Akaishi and T. Yamazaki, PLB 590 (2004) 51; PRC 70 (2004) 044313.

• RMF studyD. Gazda, E. Friedman, A. Gal, J. Mares, PRC 76 (2007) 055204; PRC 77 (2008) 045206;A. Cieply, E. Friedman, A. Gal, D. Gazda, J. Mares, PRC 84 (2011) 045206.T. Muto, T. Maruyama, T. Tatsumi, PRC 79 (2009) 035207.

Antisymmetrized Molecular Dynamics with a phenomenological KbarN potential

Anti-kaon can be deeply bound and/or form dense state?

Anti-kaon bound in light nuclei with ~ 100MeV binding energy Dense matter (average density = 2 ～ 4 ρ0)

Relativistic Mean Field applied to medium to heavy nuclei with anti-kaons

Prototype of kaonic nuclei = “K-pp” Variational calculation with Gaussian base Av18 NN potential Chiral SU(3)-based KbarN potential

B. E. ~ 20MeV, Γ = 40 ~70MeVpp distance ~ 2.1 fm

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

PRC79, 014003 (2009)

Theoretical studies of K-pp

• Doté, Hyodo, Weise PRC79, 014003(2009)

Variational with a chiral SU(3)-based KbarN potential• Akaishi, Yamazaki PRC76, 045201(2007)

ATMS with a phenomenological KbarN potential• Ikeda, Sato PRC76, 035203(2007)

Faddeev with a chiral SU(3)-derived KbarN potential• Shevchenko, Gal, Mares PRC76, 044004(2007)

Faddeev with a phenomenological KbarN potential• Barnea, Gal, Liverts PLB712, 132(2012)

Hyperspherical harmonics with a chiral SU(3)-based KbarN potential • Wycech, Green PRC79, 014001(2009)

Variational with a phenomenological KbarN potential (with p-wave)

• Arai, Yasui, Oka/ Uchino, Hyodo, Oka PTP119, 103(2008)

Λ* nuclei model /PTPS 186, 240(2010)• Nishikawa, Kondo PRC77, 055202(2008)

Skyrme model

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

All studies predict that K-pp can be bound!

Theoretical studies of K-pp

• Doté, Hyodo, Weise PRC79, 014003(2009)

Variational with a chiral SU(3)-based KbarN potential• Akaishi, Yamazaki PRC76, 045201(2007)

ATMS with a phenomenological KbarN potential• Ikeda, Sato PRC76, 035203(2007)

Faddeev with a chiral SU(3)-derived KbarN potential• Shevchenko, Gal, Mares PRC76, 044004(2007)

Faddeev with a phenomenological KbarN potential• Barnea, Gal, Liverts PLB712, 132(2012)

Hyperspherical harmonics with a chiral SU(3)-based KbarN potential • Wycech, Green PRC79, 014001(2009)

Variational with a phenomenological KbarN potential (with p-wave)

• Arai, Yasui, Oka/ Uchino, Hyodo, Oka PTP119, 103(2008)

Λ* nuclei model /PTPS 186, 240(2010)• Nishikawa, Kondo PRC77, 055202(2008)

Skyrme model

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

All studies predict that K-pp can be bound!

50 60 70 80 90 100 110 120 130

Width (KbarNN→πYN) [MeV]

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

Akaishi, Yamazaki [2](Variational, Phenomenological)

Exp. : FINUDA [6]if K-pp bound state.

Exp. : DISTO [7]if K-pp bound state.

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, Mares [3](Faddeev, Phenomenological)

[1] PRC79, 014003 (2009) [5] PLB94, 712 (2012)[2] PRC76, 045201 (2007)[3] PRC76, 044004 (2007)[4] PRC76, 035203 (2007)

[6] PRL94, 212303 (2005)[7] PRL104, 132502 (2010)

Typical results of theoretical studies of K-pp

Barnea, Gal, Liverts [5](HH, Chiral SU(3))

-100 MeV

50 60 70 80 90 100 110 120 130

Width (KbarNN→πYN) [MeV]

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

Akaishi, Yamazaki [2](Variational, Phenomenological)

Exp. : FINUDA [6]if K-pp bound state.

Exp. : DISTO [7]if K-pp bound state.

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, Mares [3](Faddeev, Phenomenological)

[1] PRC79, 014003 (2009) [5] PLB94, 712 (2012)[2] PRC76, 045201 (2007)[3] PRC76, 044004 (2007)[4] PRC76, 035203 (2007)

[6] PRL94, 212303 (2005)[7] PRL104, 132502 (2010)

Typical results of theoretical studies of K-pp

Barnea, Gal, Liverts [5](HH, Chiral SU(3))

Kbar+N+N threshold

π+Σ+N threshold

From theoretical viewpoint, K-pp exists between Kbar-N-N and π-Σ-N thresholds!

Key points to study kaonic nuclei

Similar to the bound-state calculation

1. Consider a coupled-channel problem

Kbar + N + N

“Kbar N N”

π + Σ + N

2. Treat resonant states adequately.

Coupling of KbarN and πY

Bound below KbarN threshold, but a resonant state above πY threshold.

Their nature?

3. Obtain the wave function to help the analysis of the state.

“coupled-channel Complex Scaling Method”

KbarN (-πY) interaction??

4. Confirmed that CSM works well on many-body systems.

“Chiral SU(3)-based potential”… Anti-kaon = Nambu-Goldstone boson

2. Complex Scaling Method

Λ(1405) = a building block of kaonic nuclei

Proton

• For Resonance states

• For Scattering problem

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

Λ(1405) with c.c. Complex Scaling Method

(Jπ=0-, T=1/2)

N (Jπ=1/2+, T=1/2)

π(Jπ=0-, T=1)

Σ(Jπ=1/2+, T=1)

Kbar + N

Λ(1405)

π + Σ

1332 [MeV]

Complex Scaling Method for Resonance

By Complex scaling, …

Complex rotation of coordinate (Complex scaling) : ,i iU e e r r k k

1 ,H U HU U

• Resonant state

exp expR Rik r i i r

κ, γ : real, >0Divergent

Damping under some condition

exp sin cos exp cos sin

iRU i i re

Complex Scaling Method for ResonanceComplex rotation of coordinate (Complex scaling)

: ,i iU e e r r k k

1 ,H U HU U

By Complex scaling,

Resonance wave function: divergent function damping function⇒Boundary condition is the same as that for a bound state.

Resonance energy (pole position) doesn’t change.

ABC theorem“The energy of bound and resonant states is independent of scaling angle θ.”

† J. Aguilar and J. M. Combes, Commun. Math. Phys. 22 (1971),269. E. Balslev and J. M. Combes, Commun. Math. Phys. 22 (1971),280

Diagonalize Hθ with Gaussian base,

we can obtain resonant states, in the same way as bound states!

Complex Scaling Method for Resonance

πΣ KbarN

pS continuum KbarN continuum

=30 deg.

B. E. (KbarN) = 28.2 MeVΓ = 40.0 MeV… Nominal position of Λ(1405)

Akaishi-Yamazaki potential†

• Local Gaussian form• Energy independent

2( 0), 0 exp 0.66 fmI

AY IV V r

KbarN πΣ

( 0) 436 412MeV

412 0IV

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

Test calculation with a phenomenological potential

2. Complex Scaling Method

• For Resonance states

• For Scattering problem

Calculation of KbarN scattering amplitude

Calc. of scattering amplitude with CSM

With help of the CSM, all problems for bound, resonant and scattering states can be treated with Gaussian base!

Calc. of scattering amplitude with CSMA. T. Kruppa, R. Suzuki and K. Katō, PRC 75, 044602 (2007)

1. Separate incoming wave

• Unknown• Non square-integrable

llπh x i kr -2

square-integrable for 0 < θ < π

2. Complex scaling r → reiθ

3. Calculate scattering amplitude with help of Cauchy theorem

Born term is OK!

Scattered part is unknown.

Scattered wave function along reiθ is known!By Cauchy theorem, we can obtain as

Expanding with square-integrable basis function (ex: Gaussian basis)

r eiθ

, 0SCl kdz j kz V z z

Cauchy theorem

can be expanded with Gaussian basis

Scattering problem can be solved as bound-state problem by matrix calculation!

Equation to be solved:

Linear equation to be solved with matrix calculation!

3. Set up for the calculation of

KbarN-πY system

• “KSW-type potential” … Chiral SU(3)-based• Kinematics• Non-rela. approximation of KSW-type potential

“KSW-type potential” … chiral-SU(3) based

N. Kaiser, P. B. Siegel and W. Weise, NPA 594, 325 (1995)

Effective Chiral Lagrangian

Pseudopotential • Delta-function type (→ Yukawa / separable type)• Up to order q2

• Local Gaussian form in r-space

• Weinberg-Tomozawa term

• Energy dependence

“KSW-type potential”

C M Mr

f sg rV

3/ 2 ( )3( )1 expI

jjg r r a

a(I)ij : range parameter [fm]

→ Easy to handle in many-body calculation with Gaussian base

→ 　 Relative strength between channels determined by the SU(3) algebra

← Chiral SU(3) theory

( 0)332

KbarN πΣ

Constrained by KbarN scattering length (Martin’s value)

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

Kinematics1. Non-relativistic

2. Semi-relativistic

2 2 2 2

c cc c

c c c c

k kE M mM m

M mM m

k E M m

2 2 2 2 2 2

1/ 22 2

c c c c c c

E M k m k

2 22 2

SR c cc K N Y

H m M c c

, 2bar

NR c cc K N Y c

H m M c c

Reduced mass

Reduced energy

Non-rela. approximation of KSW potential- Original

( ) ( )28

IijI I

ij ii j

i jj ij

CV r g r

2( )( )3/ 2 ( ) 3

1 exp IIij ijI

g r r ddp

( ) ( ) ( ) 2I I Iij ii jjd d d

• Normalized Gaussian

• Range parameter for coupling potential

→ (NRv1) Non-rela. approx. version 1

( ) ( )2

IijI I

ij ii j

Comparison of the flux factor for differential cross sectionbetween non-rela. and rela.

→ (NRv2) Non-rela. approx. version 2

( ) ( )2

IijI I

ij i j ijjm m

CV r g r

, 1i i i im M @ non-rela. limit (small p2)

Kinematics

- Original

(NRv1) Non-rela. approx. ver. 1

(NRv2) Non-rela. approx. ver. 2

( ) ( )28

IijI I

ij ii j

i jj ij

CV r g r

( ) ( )28

1IijI I

ij i j iji j

CV r g r

( ) ( )2

IijI I

ij i j ijjm m

CV r g r

Non-rela.

Semi-rela.

Potential

4. Result

• I=0 channel• I=0 KbarN scattering length

Range parameters of KSW-type potential• Scattering amplitude• Resonance pole … wave function and size

Using Chiral SU(3) potential “KSW-type”… r-space, Gaussian form, Energy-dependent

I=0 KbarN scattering lengthRange parameters of KSW-type potential

2( 0) ( 0)33/ 2

1 expI Iij ij

ijijdV r V s r

,babar bar rK Nij

K N K Nd d

Range parameters: ,, ,

2bar bar barK N K N K Nd d dp pp S SS

Pion decay constant “fπ” as a parameter

fπ = 90 ~ 120 MeV

Found sets of range parameters (dKN,KN, dπΣ,πΣ) to reproduce the Martin’s value.

Two sets found in semi-rela. case. (SR-A, SR-B)

Kinematics MartinKSW- type pot. NR- v1 NR- v2 SR- A SR- B

Range 0.44 0.44 0.50 0.37parameters [fm] 0.61 0.64 0.71 0.35

KbarN scatt. Re - 1.701 - 1.700 - 1.700 - 1.696 - 1.70length [fm] Im 0.681 0.681 0.681 0.681 0.68

Non- rela. Semi- rela.

dKN,KN 　　　　　

dπΣ,πΣ

Re aI=0KN 　　　　　

Im aI=0KN

fπ=110 MeV

Scattering amplitude … Non-rela. / KSW-type NRv1

fπ=110 MeV

Iij i jI

i j ij

C M MV r g r

πΣ KbarN πΣ KbarN

Resonance structure appears in KbarN and πΣ channels.

KbarN → KbarN πΣ → πΣ

Resonance structure at 1413 MeV

Scattering amplitude … Non-rela. / KSW-type NRv2

fπ=110 MeV

ij i j ijj

CV r g r

f m mp

Essentially same as NRv1 case.

Scattering amplitude … Semi-rela. / KSW-type SR-A

fπ=110 MeV

Qualitatively similar to non-rela. amplitudes.

( 0)28

Iij i jI

ij i j iji j

C M MV r g r

Scattering amplitude … Semi-rela. / KSW-type SR-B

fπ=110 MeV

Very different behavior of πΣ amplitude from other cases.

( 0)28

Iij i jI

ij i j iji j

C M MV r g r

(Another set of SR)

Pole position of the resonance fπ=90 - 120MeV

fpi NRv1 NRv2 SR- A SR- BM - Γ / 2 M - Γ / 2 M - Γ / 2 M - Γ / 2

90 1419.8 - 26.0 1419.9 - 23.1 1423.6 - 26.4 1419.0 - 14.4100 1417.3 - 23.1 1418.0 - 19.8 1421.5 - 26.3 1419.6 - 13.2110 1416.6 - 19.5 1417.8 - 16.6 1419.5 - 25.0 1420.0 - 12.8120 1416.9 - 16.7 1418.3 - 14.0 1417.5 - 22.9 1418.9 - 11.7

- Γ /

M [MeV]

- Γ /

Complex energy plane

Pole position of the resonance

- Γ /

fπ=90 - 120MeV

90 1419.8 - 26.0 1419.9 - 23.1 1423.6 - 26.4 1419.0 - 14.4100 1417.3 - 23.1 1418.0 - 19.8 1421.5 - 26.3 1419.6 - 13.2110 1416.6 - 19.5 1417.8 - 16.6 1419.5 - 25.0 1420.0 - 12.8120 1416.9 - 16.7 1418.3 - 14.0 1417.5 - 22.9 1418.9 - 11.7

M [MeV]

Non-rela.

(M, Γ/2) = (1418.2 ± 1.6, 21.4 ± 4.7) … NRv1 (1418.9 ± 1.1, 18.6 ± 4.6) … NRv2

- Γ /

fπ=90 - 120MeV

90 1419.8 - 26.0 1419.9 - 23.1 1423.6 - 26.4 1419.0 - 14.4100 1417.3 - 23.1 1418.0 - 19.8 1421.5 - 26.3 1419.6 - 13.2110 1416.6 - 19.5 1417.8 - 16.6 1419.5 - 25.0 1420.0 - 12.8120 1416.9 - 16.7 1418.3 - 14.0 1417.5 - 22.9 1418.9 - 11.7

M [MeV]

Semi-rela.

(M, Γ/2) = (1420.5 ± 3, 24.5 ± 2) … SR-A

- Γ /

fπ=90 - 120MeV

90 1419.8 - 26.0 1419.9 - 23.1 1423.6 - 26.4 1419.0 - 14.4100 1417.3 - 23.1 1418.0 - 19.8 1421.5 - 26.3 1419.6 - 13.2110 1416.6 - 19.5 1417.8 - 16.6 1419.5 - 25.0 1420.0 - 12.8120 1416.9 - 16.7 1418.3 - 14.0 1417.5 - 22.9 1418.9 - 11.7

M [MeV]

Semi-rela.(Another set)

(M, Γ/2) = (1419.5 ± 0.5, 13.0 ± 1.4) … SR-B

Different behavior from other cases

- Γ /

“Wave function” of the resonance pole fπ=110 MeVθ=30°

Non-rela. (NRv2)

πΣ component also localizeddue to Complex scaling.

Mean distance between meson and baryon

2MBr fm

mNR: ～ 1.3 - 0.3 i [fm]SR: ～ 1.2 - 0.5 i [fm]

? Somehow small. cf) 1.9 fm @ M = 1423 MeV†

(B = 12 MeV)

- Difference of binding energy? - Different definition of pole? Gamow state or bound state

† A. D., T. Hyodo, W. Weise, PRC79, 014003 (2009)

4. Result

• I=1 channel

• Range parameters of the KSW-type potential• Scattering amplitude

Using Chiral SU(3) potential (KSW-type)… r-space, Gaussian form, Energy-dependent

I=1 channel … KbarN - πΣ - πΛ

Data : aI=1KbarN = 0.37 + i 0.60 fm

(A. D. Martin)† 1 1

.barbar barI I

K N thrK N K Na f s

bar bar bar barK N K N K N K N

d d dd

p p p p

Range parameters:

† A. D. Martin, Nucl. Phys. B 179, 33 (1981)

2( 1)32 3/ 2

1 exp8

ij i jij

CV r flux factor r

Potential … Weinberg-Tomozawa term only

1 1.barbar bar

I IK N thrK N K N

bar bar bar barK N K N K N K N

d d dd

p p p p

Range parameters:

1 1 3 200

SU(3) C.G. coefficients for I=1 channel

2( 1)32 3/ 2

1 exp8

ij i jij

CV r flux factor r

(A. D. Martin)†

,, ,2bar bar barK N K N K N

d d dp pp S SS

1 1.barbar bar

I IK N thrK N K N

bbar ba bar ar rK NK N K N

Range parameters:

1 1 3 200

d d dp pp S SS

2( 1)32 3/ 2

1 exp8

ij i jij

CV r flux factor r

(A. D. Martin)†

1 1.barbar bar

I IK N thrK N K N

babar bar ba rr K NI IK N K N K N

dd p p

Range parameters:

1 1 3 200

“Isospin symmetric choice”

Ref.) chiral unitary model

2( 1)32 3/ 2

1 exp8

ij i jij

CV r flux factor r

(A. D. Martin)†

d d dp pp S SS

A. D. Martin : aI=1KbarN = 0.37 + i 0.60 fm

• “Iso-symmetric choice” : {dKN,KN, dπΣ,πΣ} fixed to I=0 channel ones• Search dKN,πΣ to reproduce Re or Im aI=1

KbarN of Matin’s value.

Difficult to reproduce Re aI=1KbarN Martin’s value within our model.

When Re aI=1KbarN is reproduced, Im aI=1

KbarN deviates largely from Martin’s value.

(fit)(fit)

(I=0 one)

(Search)

NRv2 (c): aI=1KbarN = 0.657 + i 0.599 fm

fπ=110 MeV

Search dKN,πΣ to reproduce Im aI=1KbarN of Matin’s value.

KbarN πΣ πΛ

SR-A (c): aI=1KbarN = 0.659 + i 0.600 fm

I=1 channel … KbarN - πΣ - πΛ• Only dKN,KN fixed to I=0 channel ones

In a study with separable potential, the cutoff parameter for KbarN is not so different between I=0 and 1 channels.†

• Search {dπΣ,πΣ, dKN,πΣ} to reproduce Re and Im aI=1KbarN of Matin’s value.

When the only dKN,KN is fixed iso-symmetrically, we can find a set of {dπΣ,πΣ, dKN,πΣ}

to reproduce simultaneously Re and Im of Martin’s value

(fit)(fit)(fit)

† Y. Ikeda and T. Sato, PRC 76, 035203 (2007)

(I=0 one)

(Search)

NRv2 (a): aI=1KbarN = 0.376 + i 0.606 fm

fπ=110 MeV

Search {dπΣ,πΣ, dKN,πΣ} to reproduce Re and Im aI=1KbarN of Matin’s value.

SR-A (a): aI=1KbarN = 0.375 + i 0.605 fm

KbarN πΣ πΛ

A narrow resonance exists at a few MeV below πΣ threshold

πΣ repulsive ???

5. Summary and

Future plan

5. Summary

Scattering and resonant states of KbarN-πY system is studied with a coupled-channel Complex Scaling Method using a chiral SU(3) potential

• A Chiral SU(3) potential “KSW-type” … r-space, Gaussian form, energy dependence• Calculated scattering amplitude with help of CSM

Scattering states as well as resonant and bound states are treated with Gaussian base.

Non-rela. / Semi-rela. kinematics and two types of non-rela. approximation of KSW-type potential are tried.

Determined by the KbarN scattering length obtained by Martin’s analysis of old data. fπ dependence (fπ = 90 ～ 120MeV) Found two sets of range parameters in SR case.

KbarN-πY system is essential for the study of Kbar nuclear system which is expected to be an exotic nuclear system with strangeness.

5. Summary and future plans

Future plansThree-body system (KbarNN-πYN); Updated data of K-p scattering length by SHIDDARTA

• Pole position • “Size” of the I=0 pole stateNR: ～ 1.3 – 0.3i fmSR: ～ 1.2 – 0.5i fm

(M, Γ/2) NRv1 : (1418.2 ± 1.6, 21.4 ± 4.7) NRv2 : (1418.9 ± 1.1, 18.6 ± 4.6) SR-A : (1420.5 ± 0.5, 24.5 ± 2) SR-B : (1419 ± 1, 13.0 ± 2)

I=0 channel (KbarN-πΣ)

I=1 channel (KbarN-πΣ-πΛ)• Difficult to reproduce Re aI=1

KbarN of Martin’s value within our model, in case of “iso-symmetric choice” of range parameter.

Another self-consistent solution NR: (~1360, 40~90), SR: (1350~1390, 30~100) … Lower pole of double pole?

Thank you very much!

A. D. is thankful to Prof. Katō for his advice on the scattering-amplitude calculation in CSM, and to Dr. Hyodo for useful discussion.

Backup slides

Rough estimation of charge radiusK-

Distance

Mr Distancem M

mr Distancem M

2 2 2c p CM K CM

† Calculated from electric form factor with chiral unitary model T. Sekihara, T. Hyodo and D. Jido, PLB 669, 133 (2008)

Pole position [MeV] <rc2> [fm2]

NRv2 1417.8 – i 16.6 -0.49 + i 0.24

SR-A 1419.5 – i 25.0 -0.37 + i 0.37

(Sekihara et al)† 1426.11 – i 16.54 -0.131 + i 0.303

Charge radius

Chiral (HW-HNJH): B ～ 12 MeV, Distance = 1.86 fm → <r2c> = -1.07 fm2

Charge radius derived from electric form factor

T. Sekihara, T. Hyodo and D. Jido, PLB 669, 133 (2008)

Rough estimation

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…

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)

Relate to various interesting physics such as … Restoration of chiral symmetry in dense matter Interesting structure Neutron star

Excited hyperon 　 Λ （１４０５）

I=0 Proton-K- bound state with 30MeV binding energy

　 Not 3 quark state,　　←　 can’t be explained with 　　　　　 a simple quark model…

　 But rather a molecular state

T. Hyodo and D. Jido, Prog. Part. Nucl. Phys. 67, 55 (2012)

Today, K-pp has been focused in theor. and exp. studies!As the first trial of our study with ccCSM, consider Λ(1405).

Prototype of Kaonic nuclei

3HeK-, pppK-, 4HeK-, pppnK-, …8BeK-,…

Complicated nuclear system with K-

Proton

Λ(1405)

Building block of Kaonic nuclei

Asymptotic behavior of the wave function before/after complex scaling

exp cos exp sin

exp cos sin

exp sin cos exp cos sin

iRU i i re

i i r i

i ikU i ke re

• Bound state

• Resonant state

• Continuum state

expB Br

expR Rik r

expk ikr

κB : real, >0

κ, γ : real, >0

k : real, >0

Damping Damping under some condition

Oscillating Oscillating

Diverging Damping under some condition

Complex Scaling Method for Resonance Great success in nuclear physics … in particular, unstable nuclei.

Well applied to usual calculation of bound states… diagonalize with Gaussian base (Gaussian Expansion Method, Correlated Gaussian)

Tiny modification of matrix elements 22 2 2

02 , exp2

i idT V V r am dr

Possible to study many-body systems … up to 5-body system (4He+n+n+n+n)

“Cluster-Orbital Shell Model”

T. Myo, R. Ando, K. Kato, PLB691, 150 (2010)

Complex scaling

r → r eiΘ

• Unknown• Non square-integrable

llπh x i kr -2

Separate the incoming wave ! lj krPoint 1

Point 2

Expanding with square-integrable basis function (ex: Gaussian basis)

• Calculation of scattering amplitude

We don’t have which is a solution along r,

but we have which is a solution along reiθ.

,SCl k r

,,SCl k r θ

r eiθ

Cauchy theoremPoint 3

Calc. of scattering amplitude with CSM

is independent of θ. SClf k

expressed with Gaussian base

A. T. Kruppa, R. Suzuki and K. Katō, PRC 75, 044602 (2007)

Test calculation with AY potentialResonance pole

πΣ KbarN

pS continuum KbarN continuum

=30 deg.

B. E. (KbarN) = 28.2 MeVΓ = 40.0 MeV

… Nominal position of Λ(1405)

Akaishi-Yamazaki potential†

• Phenomenological potential• Local Gaussian form• Energy independent

2( 0), 0 exp 0.66 fmI

AY IV V r

KbarN πΣ

( 0) 436 412MeV

412 0IV

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

Complex Scaling Method for ResonanceStudy of unstable nuclei … Resonance of 8He

“Cluster-Orbital Shell Model”

T. Myo, R. Ando, K. Kato, PLB691, 150 (2010)

Bound state

Resonance

Continuumon 2θ line

Test calculation with AY potentialUnitarity violation of S-matrix

Phase shift sum

| |Det S| -1 |

EKbarN [MeV]

Checked by Continuum Level Density method (R. Suzuki, A. T. Kruppa, B. G. Giraud, and K. Katō, PTP119, 949(2008))

δKbarN + δπΣ [deg.]

Test calculation with AY potential

KbarN (I=0) scattering amplitude

Scattering length (Scatt. amp. @ EKbarN=0) = -1.77 + i 0.47 fm

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

Scattering length = -1.76 + i 0.46 fm

0IKN KNf fm

EKbarN [MeV]

Scattering amplitude … Non-rela. / KSW-type SR

fπ=90 MeV

( 0)28

Iij i jI

ij i j iji j

C M MV r g r

Kinematics-potential mismatched case

… A virtual state exists. (aπΣ, re)=(61,-6.3) fm satisfies -aπΣ /2 < re

†. † Y. Ikeda et al, PTP 125, 1205 (2011)

Singular behavior at πΣ threshold

I=0 KbarN scattering lengthRange parameters of KSW-type potential

fπ=110 MeV

Data : aI=0KbarN = -1.70 + i 0.68 fm (A. D. Martin)

Found sets of range parameters (dKN,KN, dπΣ,πΣ) to reproduce the Martin’s value.

Two sets found in semi-rela. case. (SR-A, SR-B)

Kinematics MartinKSW- type pot. NR- v1 NR- v2 SR- A SR- B

Range 0.44 0.44 0.50 0.37parameters [fm] 0.61 0.64 0.71 0.35

KbarN scatt. Re - 1.701 - 1.700 - 1.700 - 1.696 - 1.70length [fm] Im 0.681 0.681 0.681 0.681 0.68

Non- rela. Semi- rela.

dKN,KN 　　　　　

dπΣ,πΣ

Re aI=0KN 　　　　　

Im aI=0KN

Resonant structure in the scattering amplitude

Energy @ Re fKbarN → KbarN = 0 Energy @ Re fπΣ → πΣ = 0

fπ=90 - 120MeV

KSW - Semi rela.KbarN → KbarNfπ = 90MeV

Resonant structure in the scattering amplitude

Energy @ Re fKbarN → KbarN = 0 Energy @ Re fπΣ → πΣ = 0

fπ=90 - 120MeV

Strange behavior?

“Wave function” of the resonance pole fπ=110 MeVθ=30°

πΣ component also localizeddue to Complex scaling.

* The wave functions shown above are multiplied by a phase factor so that the KbarN wfn. becomes real at r=0.

Non-rela. (NRv2) Semi rela. (SR-A)

Semi rela. (SR-B)

“Size” of the resonance pole state

2MBr fm

Mean distance between meson and baryon

fπ=90 - 120MeV

NR: ～ 1.3 - 0.3 i [fm]SR: ～ 1.2 - 0.5 i [fm]

? Somehow small. cf) 1.9 fm @ M = 1423 MeV†

(B = 12 MeV)

- Difference of binding energy? - Different definition of pole? Gamow state or bound state

† A. D., T. Hyodo, W. Weise, PRC79, 014003 (2009)

Strange nuclear physics

Nucleus

Hypernuclei…u

d sHyperon

baryon = qqq

Strangeness is introduced through baryons.

Strange nuclear physics

Nucleus

Kaonic nuclei !

sK- meson(anti-kaon, Kbar)

Strangeness is introduced through mesons …

meson = qqbar

Key points to study kaonic nuclei

Similar to the bound-state calculation

1. Consider a coupled-channel problem

Kbar + N + N

“Kbar N N”

π + Σ + N

2. Treat resonant states adequately.

KbarN couples with πY.

Bound state for KbarN channelbut a resonant state for πY channel

Their nature?

3. Obtain the wave function to help the analysis of the state.

“coupled-channel Complex Scaling Method”

KbarN (-πY) interaction??

“Chiral SU(3)-based potential”

Excited hyperon 　 Λ （１４０５）

939 p,n

1115 Λ

1190 Σ

p + K-1435

Λ(1405)1405

1250 Λ + π

Σ + π1325

Mysterious state; Λ(1405)Quark model prediction … calculated as 3-quark state

N. Isgar and G. Karl, Phys. Rev. D18, 4187 (1978)

Λ(1405) can’t be well reproduced as a 3-quark state!

observed Λ(1405)calculated Λ(1405)

( 0)28

CV r r

赤字のファクターは、微分断面積で適切な相対論的 flux factor を得るために導入されている。

KSW の（１４）式、 Born 近似での微分断面積の表式　（相対論的な場合）

ki: meson momentum 、　 ωi: reduced energy

1 exp2

ii j j i ij

k d i Vk

r k k r r

３． KSW 非相対論版の考察

非相対論的な場合の Born 近似での微分断面積

31 exp2ij j i ijf d i Vp

r k k r r

2 2ij i i iij ij

i j j j

d v kf fd v k

fij : 散乱振幅vi : 速度

kij : 運動量μi : 換算質量

1 exp2

1 1 exp2

ij i ij i ij

ij j i ij

d k d i Vd k

k d i Vk

r k k r r

相対論・非相対論での Born 近似微分断面積の表式の比較

1 1 exp2

ij ij j i ij

d k d i Vd k

p r k k r r

1 exp2

ij ij j i ij

d k d i Vd k

p r k k r r相対論的：

非相対論的：

は１に取っている。

微分断面積の表式は、相対論的なものと非相対論的なもので、明らかに対応している。

非相対論的な表式の換算質量（ μi ）が、相対論的表式では換算エネルギー（ ωi ）に対応。

推測 KSW の pseudo-potential 中のファクター

が微分断面積の表式とつじつまが合うように導入されているのならば、このポテンシャルの非相対論版を考える際には、前ページの対応関係からこのファクター中の換算エネルギーを換算質量に置き換えればいいのでは？

つまり

( 0)( 0)

C M MV r g r

カイラル理論を尊重するので、メソンのエネルギー ωi は残しておく。　（続きは次ページ）

• 1 / √s はどうするか？そのまま、系の全エネルギー（ ETot ）として残しておく？？

Comparison with other studies of I=0 channel

Comparison with other studies of I=0 KbarN-πΣ

Kaiser-Siegel-Weise

Hyodo-Weise Ikeda-Hyodo-Jido-Kamano-Sato-Yazaki

Dote-Inoue-Myo

Base Charge Isospin Isospin Isopin

Terms All terms up to q2 WT term WT term WT term

Regularization Yukawa / Separable form

Dimensional regularization

/ Separable formGaussian form

Constraint

Low energy scattering data,

Branching ratio at

KbarN threshold

Low energy scattering data,

Branching ratio at KbarN threshold

KbarN scattering length

/ Λ(1405) pole

KbarN scattering length

Ref. NPA 594, 325 (1995) PRC 77, 035204 (2008) PTP 125, 1205 (2011) NPA 912, 66 (2013)

IHJKSY, HW:

Different energy dependence of interaction kernel

Proportional to just meson energy

Not proportional to meson energy… including relativistic q2 correction and non-static effect of baryons

IHJKSY

Comparison with other studies of I=0 KbarN-πΣUse the same interaction kernel as that of IHJKSY and HW in our ansatz (Flux factor and Gaussian form factor)

Suppressed the enhancement near the πΣ threshold, but still different from the amplitudes of IHJKSY.

Different interaction kernel Use of flux factor Different regularization scheme (Gaussian form factor vs Dimensional reg.)

* Gaussian form factor enhances the amplitude far below the threshold as |E| exp [c|E|] with E → -∞. (c>0)

1. IntroductionNeutron star in universe

Radius ～ 10kmMass ～　 Msun

High dense at core ～ > 2 ρ0

Maybe …

Involving strangeness

Kbar nuclear system on the Earth

• Self-bound Kbar-nuclear system

… light nucleus with K- mesons Involving strangeness via K- meson

1. IntroductionNeutron star in universe

Radius ～ 10kmMass ～　 Msun

High dense at core ～ > 2 ρ0

Maybe …

Involving strangeness

Kbar nuclear system on the Earth

Density [fm-3]0.00 0.14

Density [fm-3]0.0 0.75 1.5

3HeK- 4×4 fm2… light nucleus with K- mesons

Involving strangeness via K- meson

Strong attraction by K- meson ⇒ Dense nuclear system

† A. D., H. Horiuchi, Y. Akaishi and T. Yamazaki, PLB 590 (2004) 51; PRC 70 (2004) 044313.

A phenomenological KbarN potential† ⇒ 　 B(K) ～ 100MeV, 2 ～ 4ρ0

• FINUDA collaboration (DAΦNE, Frascati)

• K- absorption at rest on various nuclei (6Li, 7Li, 12C, 27Al, 51V)• Invariant-mass method

Strong correlation between emitted p and Λ (back-to-back)

Invariant mass of p and Λ

ΛIf it is K-pp, …Total binding energy = 115 MeVDecay width = 67 MeV

PRL 94, 212303 (2005)

6 35 4

14 211 3

Experimental studies of K-pp

• Re-analysis of KEK-PS E549

• DISTO collaboration

- p + p -> K+ + Λ + p @ 2.85GeV- Λp invariant mass- Comparison with simulation data

T. Yamazaki et al. (DISTIO collaboration), PRL104, 132502 (2010)

- K- stopped on 4He target- Λp invariant mass

T. Suzuki et al (KEK-PS E549 collaboration), arXiv:0711.4943v1[nucl-ex]

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

Strong Λp back-to-back correlation is confirmed.Unknown strength is there in the same energy region as FINUDA.

Experimental studies of K-pp

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

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

E27: Search for a nuclear Kbar bound state K-pp in the d(π+, K+) reaction (T. Nagae (Kyoto))

E31: Spectroscopic study of hyperon resonances below KN threshold via the (K-, n) reaction on deuteron (H. Noumi (Osaka))

as of 2011

Experiments of kaonic nuclei at J-PARC

as of 2011

Dr. Fujioka’s talk (KEK workshop, 7-9. Aug. 08)

at K1.8BR beam line

1.8GeV/c

as of 2011

Missing mass spectroscopy

at K1.8BR beam line

1.8GeV/c

Invariant mass spectroscopy

Dr. Fujioka’s talk (KEK workshop, 7-9. Aug. 08)

All emitted particles will be measured.　　「完全実験」

as of 2011

π+ K+

Λ(1405)np p

Preceding E15, E27 experiment was performed in June. Analysis is now undergoing!

Use lots of pion Missing mass

as of 2011

KbarN interaction

Λ(1405)

These days, K-pp has been focused in both of theoretical and experimental studies!

Proton

Λ(1405)

3HeK-, pppK-, 4HeK-, pppnK-, …8BeK-,…

Complicated nuclear system with K-

Most essential kaonic nucleus!

Coupled Channel Chiral Dynamics (Chiral Unitary model)

We are thinking about KbarN interaction.S=-1 meson-baryon system is constrained by

The leading couplings between

Low mass pseudo-scalar meson octet (Nambu-Goldston bosons)

and Baryon octet

are determined by

Spontaneous breaking ofSU(3)×SU(3) Chiral symmetry

Chiral SU(3) dynamics !

Chiral low-energy theorem tells us …

KbarN πΣ ηΛ KΞ

For I=0 channel,

Remark

• There are no free parameters as for coupling.• There is an attractive interaction in πΣ-πΣ channel, while AY potential doesn’t have it.

Energy and mass of baryon in channel i

Weinberg-Tomozawa term derived from Chiral SU(3) effective Lagrangian

f : Psuedo-scalar meson decay constant

3. Solve coupled channel Bethe-Salpeter equation.

T-matrix of coupled channel scattering

+ … + + … …

2. Using the WT-term as a building block,

1. Weinberg-Tomozawa term derived from Chiral SU(3) effective Lagrangian:

• K-p scattering length • Threshold branching ratio• Total cross section of K-p scattering

• Threshold branching ratio • Total cross section of K-p scattering

T. Hyodo, S. I. Nam, D. Jido, and A. Hosaka, Phys. Rev. C68, 018201 (2003)

3. Solve coupled channel Bethe-Salpeter equation.

T-matrix of coupled channel scattering

+ … + + … …

2. Using the WT-term as a building block,

Λ(1405) is dynamically generated as meson-baryon system.

1. Weinberg-Tomozawa term derived from Chiral SU(3) effective Lagrangian:

• K-p scattering length • Threshold branching ratio• Total cross section of K-p scattering

I=0 πΣ mass distribution

T. Hyodo, S. I. Nam, D. Jido, and A. Hosaka, Phys. Rev. C68, 018201 (2003)

Dynamical generation of Λ(1405) !

Remark:Calculated with πΣ-πΣ scattering amplitude.

Chiral SU(3) potential (KSW)N. Kaiser, P. B. Siegel and W. Weise, NPA 594, 325 (1995)

a: range parameter [fm]Mi , mi : Baryon, Meson mass in channel iEi : Baryon energy, ωi : Meson energy

2 22 2 2 2

i i i i

Reduced energy: i ii

( 0)332

KbarN πΣ

Energy dependence of Vij is controlled by CM energy √s.

Flavor SU(3) symmetry

Original: δ-function type

Present: Normalized Gaussian type

C M Mr

f sg rV

3/ 2 ( )3( )1 expI

jjg r r a

a: range parameter [fm]

Energy dependence is determined by Chiral low energy theorem. ← Kaon: Nambu-Goldstone boson

i ji j

C M MV r

Chiral SU(3) potential (KSW)Energy dependence

( 0) 0, @ 0.5 , 100ij

IV r s a fm f MeVp

√s [MeV]

KbarN-KbarN

πΣ-πΣ

KbarN-πΣ

KbarN thresholdπΣ threshold

Chiral SU(3) potential = Energy-dependent potential

Calculational procedure

Perform the Complex Scaling method.

Then, find a pole of resonance or bound state. Calculateds

Calculated Assumeds sCheck Finished !If Yes

Self consistency for the energy!

Assume the values of the CM energy √s.

MBH T V s M Assumeds

4. Experiments related to Kbar nuclear

physics

Ｋ原子核に関係する実験

• LEPS / SPring-8

πΣ invariant mass 測定

Ｋ原子 (Kaonic atom)

Λ(1405)

• CLAS / JLab

• Kaonic 4He atom, 2p レベルシフト (3d→2p X 線測定 ) @ KEK (E570) S. Okada et. al., Phys. Lett. B653, 387 (2007)

• Kaonic hydrogen atom, 1s レベルシフト @ DEAR group, DAΦNE, G. Beer et al., Phys. Rev. Lett. 94, 212302 (2005) Frascati National Laboratories

• Kaonic hydrogen, deuterium @ SIDDHARTA group

• Kaonic 3He atom, 2p レベルシフト (3d→2p X 線測定 ) @ J-PARC (E17, DAY-1)

γ + p → K+ + Λ(1405), Λ(1405) → π Σ

K. Moriya and R. Schumacher, Nucl. Phys. A835, 325 (2010)J. K. Ahn, Nucl. Phys. A835, 329 (2010)

π-Σ+, π0Σ0, π+Σ- が全て押さえられた

M. Bazzi et al., Phys. Lett. B704, 113 (2011)

• LEPS / SPring-8

Λ(1405)

• CLAS / JLab

γ + p → K+ + Λ(1405), Λ(1405) → π Σ

DEAR exp. for kaonic hydrogen atom

シフトの符号は同じだが、ＫＥＫの前回の実験（ＫｐＸ）と重ならない！

Kaonic hydrogen atom, 1s のレベルシフト @ DEAR Collaboration, DAΦNE, Frascati National Laboratories

G. Beer et al., Phys. Rev. Lett. 94, 212302 (2005)

cf) KEK exp. M.Iwasaki et al., Phys. Rev. Lett. 78, 3067 (1997)

B. Borasoy et al., Phys. Rev. Lett. 94, 213401 (2005)

KEK exp.

DEARCoupled channel chiral dynamics (Chiral unitary model) でDEAR の結果を合わすのには苦労する。かろうじてギリギリ合わせられる程度。。。

SHIDDARTA exp. for kaonic hydrogen atom

KEK 実験（ＫｐＸ）とコンシステントな結果

Kaonic hydrogen atom, 1s のレベルシフト @ SHIDDARTA Collaboration, DAΦNE, Frascati National Laboratories

K-p 散乱長が精密に決定

理論計算にとって重要なインプットに強い拘束条件

KbarN subthreshold での散乱振幅の振る舞い、Λ(1405) のポールの位置、が制限される。

Y. Ikeda, T. Hyodo and W. Weise, Phys. Lett. B706, 63 (2011)

• LEPS / SPring-8

Λ(1405)

• CLAS / JLab

γ + p → K+ + Λ(1405), Λ(1405) → π Σ

KEK E570 for kaonic 4He atom

シフトは 0 eV と consisitent

Kaonic 4He atom, 2p のレベルシフト 3d→2p X 線測定 @ KEK, E570

理論の予言がほぼ 0eV に対して、過去の実験ではシフトは平均 -43 eV

“Kaonic helium puzzle”

S. Okada et. al., Phys. Lett. B653, 387 (2007)

パズルは解けた！S. Hirenzaki et al., Phys. Rev. C61, 055205 (2000)

J-PARC for kaonic 3He atomKaonic 3He atom, 2p のレベルシフト 3d→2p X 線測定 @ J-PARC, E17 DAY-1

S. Okada et. al., Phys. Lett. B653, 387 (2007)

Kaonic 4He atom, 2p のレベルシフト … ほぼ 0 eV と確定

赤石氏の計算

KbarN potential の強度が二つの領域に絞られた。

さらに 3He でシフトが測定されることでKbarN potential の強度に絞りを掛けることが期待できる。

Y. Akaishi, Proceedings of EXA’05, Austrian Academy of Sciences press, Vienna, 2005, p.45

※ 特定領域研究「ストレンジネスで探るクォーク多体系」研究会 2007 での岡田氏のスライドより引用

• LEPS / SPring-8

Λ(1405)

• CLAS / JLab

γ + p → K+ + Λ(1405), Λ(1405) → π Σ

Λ(1405) - πΣ invariant mass 測定 -

三つの異なる電荷状態が抑えられた

• LEPS / Spring-8K. Moriya and R. Schumacher, Nucl. Phys. A835, 325 (2010)

• CLAS / Jefferson Laboratory

γ + p → K+ + Λ(1405), Λ(1405) → π Σ

ピークの順番が理論(chiral unitary) と違う？

Highest peak実験：　 Σ+ π-

理論：　 Σ- π+

理論

J. K. Ahn, Nucl. Phys. A835, 329 (2010)

p (γ, K+ π) Σ at Eγ = 1.5-2.4GeVCharged πΣ を測定

What is the object observed experimentally?

A bound state of K-pp, or another object such as πΣN ???

Only what we can say from only this spectrum is that“There is some object with B=2, S=-1, charge=+1”…

• DISTO collaboration

Self-consistent solutions for complex energy

0 0KN I Ki N In outZ ZH p p S S 2Z E i : complex energy

Self-consistent solution … in outZ Z

Ex) KSW ; fπ = 100 MeV, a= 0.47 fm ( θ=35°)

Self-consistency for complex energy should be considered!

Chiral potential … Energy-dependent potential

( 0)28

Iij i jI

i ji j

C M MV r g r

Self-consistency achieved after a few times iteration.

A. D., T. Inoue and T. Myo, Proc. of BARYONS’ 10, AIP conf. proc. 1388, 549 (2010)

•fπ = 90 MeV

Scattering amplitude

Scattering amplitude … KSW org. – Non. rela.

• Phase shift (πΣ)

KbarN → KbarN

πΣ → πΣ

KbarN → πΣ

δ （ πΣ ）

πΣ→ KbarN

fπ=90 MeV

( 0)28

Iij i jI

ij i j iji j

C M MV r g r

πΣ KbarN

Scattering amplitude … KSW NRv1 – Non. rela.

fπ=90 MeV

KbarN → KbarN

KbarN → πΣ

πΣ→ KbarN

πΣ → πΣ

δ （ πΣ ）

Iij i jI

C M MV

Er g r

πΣ KbarN

fπ=90 MeV

KbarN → KbarN

KbarN → πΣ

πΣ→ KbarN

πΣ → πΣ

δ （ πΣ ）

ij i j ijj

CV r g r

f m mp

πΣ KbarN

Scattering amplitude … KSW org. – Semi rela.

fπ=90 MeV

KbarN → KbarN

KbarN → πΣ

πΣ→ KbarN

πΣ → πΣ

δ （ πΣ ）

( 0)28

Iij i jI

ij i j iji j

C M MV r g r

πΣ KbarN

Succeeded in nuclear physics … Especially used in the study of unstable nuclei.

Resonant state of 6He

S. Aoyama, T. Myo, K. Kato and K. Ikeda, Prog. Theor. Phys. 116, 1 (2006)

Continuum

Resonance

Complex Scaling Method

Result• Scattering amplitude

• Position of resonant structure appeared in the scattering amplitude

• “Wave function” of the resonance pole• “Size” of the resonance pole state• θ dependence of “Size” and “Wave function”• Comparison with Hyodo-Weise’s result

•Potential: KSW original

•Kinematics: Non. rela.

fπ=90 MeV

δ （ πΣ ）

KbarN → πΣ

πΣ→ KbarN

δ （ πΣ ）

fπ=100 MeV

δ （ πΣ ）

KbarN → πΣ

πΣ→ KbarN

δ （ πΣ ）

fπ=110 MeV

δ （ πΣ ）

KbarN → πΣ

πΣ→ KbarN

δ （ πΣ ）

fπ=120 MeV

δ （ πΣ ）

KbarN → πΣ

πΣ→ KbarN

δ （ πΣ ）

•Potential: KSW NRv1

fπ=90 MeV

δ （ πΣ ）

KbarN → πΣ

πΣ→ KbarN

δ （ πΣ ）

fπ=100 MeV

δ （ πΣ ）

πΣ → πΣ

KbarN → πΣ

πΣ→ KbarN

δ （ πΣ ）

KbarN → KbarN

δ （ πΣ ）

fπ=110 MeV

πΣ → πΣ

KbarN → πΣ

πΣ→ KbarN

δ （ πΣ ）

KbarN → KbarN

πΣ → πΣ

δ （ πΣ ）

fπ=120 MeV

δ （ πΣ ）

πΣ → πΣ

KbarN → πΣ

πΣ→ KbarN

δ （ πΣ ）

KbarN → KbarN

•Potential: KSW NRv2

πΣ → πΣ

δ （ πΣ ）

fπ=90 MeV

δ （ πΣ ）

KbarN → πΣ

πΣ→ KbarN

δ （ πΣ ）

fπ=100 MeV

δ （ πΣ ）

KbarN → πΣ

πΣ→ KbarN

δ （ πΣ ）

πΣ → πΣ

δ （ πΣ ）

fπ=110 MeV

δ （ πΣ ）

KbarN → πΣ

πΣ→ KbarN

δ （ πΣ ）

πΣ → πΣ

δ （ πΣ ）

fπ=120 MeV

δ （ πΣ ）

πΣ → πΣ

KbarN → πΣ

πΣ→ KbarN

δ （ πΣ ）

KbarN → KbarN

•Potential: KSW original

•Kinematics: Semi rela.

πΣ → πΣ

δ （ πΣ ）

fπ=90 MeV

δ （ πΣ ）

KbarN → πΣ

πΣ→ KbarN

δ （ πΣ ）

πΣ → πΣ

δ （ πΣ ）

fπ=100 MeV

δ （ πΣ ）

KbarN → πΣ

πΣ→ KbarN

δ （ πΣ ）

πΣ → πΣ

δ （ πΣ ）

fπ=110 MeV

δ （ πΣ ）

KbarN → πΣ

πΣ→ KbarN

δ （ πΣ ）

πΣ → πΣ

δ （ πΣ ）

fπ=120 MeV

δ （ πΣ ）

KbarN → πΣ

πΣ→ KbarN

δ （ πΣ ）

Position of resonant structure

appeared in the scattering amplitude

Position of resonant structure appeared in the scattering amplitude

fπ=90 MeV

Kinematics Non- rela. Semi- rela.KSW potential Original Non- rela. v1 Non- rela. v2 Original

Range a(KbarN) 0.593 0.576 0.574 0.487parameter [fm] a(π Σ ) 0.541 0.725 0.751 0.457

KbarN Scatt. Re - 1.701 - 1.700 - 1.700 - 1.703leng. [fm] Im 0.679 0.677 0.687 0.677

Resonance position [MeV]f11=0 E 1402.5 1408.4 1411.1 1417.0

E(KbarN) - 32.5 - 26.6 - 23.9 - 18.0

f22=0 E 1354.6 1388.8 1393.0 1422.0E(KbarN) - 80.4 - 46.2 - 42.0 - 12.9

δ (π Σ )=0 E 1354.6 1388.8 1393.0 1422.1E(KbarN) - 80.4 - 46.2 - 42.0 - 12.9

KbarN → KbarN

πΣ → πΣ

fπ=100 MeV

E(KbarN) - 30.2 - 23.8 - 21.4 - 17.0

f22=0 E 1384.7 1399.3 1402.6 1421.5E(KbarN) - 50.3 - 35.7 - 32.4 - 13.5

δ (π Σ )=0 E 1384.7 1399.3 1402.6 1421.5E(KbarN) - 50.3 - 35.7 - 32.4 - 13.5

KbarN → KbarN

πΣ → πΣ

fπ=110 MeV

E(KbarN) - 27.8 - 21.6 - 19.5 - 16.1

f22=0 E 1394.7 1405.3 1408.1 1421.4E(KbarN) - 40.3 - 29.7 - 26.9 - 13.6

δ (π Σ )=0 E 1394.7 1405.3 1408.1 1421.4E(KbarN) - 40.3 - 29.7 - 26.9 - 13.6

KbarN → KbarN

πΣ → πΣ

fπ=120 MeV

E(KbarN) - 26.4 - 20.0 - 18.0 - 15.7

f22=0 E 1399.7 1409.1 1411.7 1420.9E(KbarN) - 35.3 - 25.9 - 23.3 - 14.1

δ (π Σ )=0 E 1399.7 1409.1 1411.7 1420.9E(KbarN) - 35.3 - 25.9 - 23.3 - 14.1

KbarN → KbarN

πΣ → πΣ

Resonance position in KbarN → KbarN

Resonance position in πΣ → πΣ

“Wave function” of

the resonance pole

“Wave function” of the resonance pole

Org. – Non.rela.

NRv1 – Non.rela. NRv2 – Non.rela.

Org. – Semi rela.

fπ=90 MeVθ=30°

Org. – Non.rela. Org. – Semi rela.

fπ=100 MeVθ=30°

Org. – Non.rela.

Org. – Semi rela.

fπ=110 MeVθ=30°

Org. – Non.rela.

Org. – Semi rela.

fπ=120 MeVθ=30°

“Size” of the resonance

pole state

Pole position E 1423.3 1419.8 1419.9 1419.0[MeV] B(KbarN) 11.7 15.2 15.1 16.0

Γ / 2 28.5 26.0 23.1 14.4

Meson- Baryon distance[fm]KbarN ch. Re 1.19 1.20 1.25 1.21

Im - 0.45 - 0.35 - 0.35 - 0.49

π Σ ch. Re 0.29 0.33 0.34 0.12Im 0.21 0.24 0.23 - 0.09

KbarN+π Σ Re 1.19 1.20 1.25 1.21Im - 0.39 - 0.28 - 0.29 - 0.49

“Size” of the resonance pole state fπ=90 MeVθ=30°

Γ / 2 28.0 23.1 19.8 13.2

Im - 0.45 - 0.34 - 0.34 - 0.49

π Σ ch Re 0.34 0.39 0.38 0.11Im 0.16 0.15 0.13 - 0.08

KbarN+π Σ Re 1.22 1.31 1.36 1.21Im - 0.40 - 0.29 - 0.30 - 0.49

Γ / 2 26.6 19.5 16.6 12.8

Im - 0.48 - 0.37 - 0.37 - 0.49

π Σ ch Re 0.36 0.39 0.37 0.11Im 0.11 0.05 0.04 - 0.06

KbarN+π Σ Re 1.23 1.36 1.42 1.18Im - 0.43 - 0.34 - 0.34 - 0.49

Γ / 2 25.6 16.7 14.0 11.7

Im - 0.49 - 0.39 - 0.39 - 0.42

π Σ ch Re 0.37 0.36 0.34 0.14Im 0.06 - 0.01 - 0.02 - 0.04

KbarN+π Σ Re 1.22 1.38 1.44 1.16Im - 0.45 - 0.38 - 0.38 - 0.42

“Size” of the resonance pole state

θ=30°

2MBr fmMean distance between meson and baryon

Pole position of the resonancefpi Org- NR NRv1- NR NRv2- NR Org- SR

M - Γ / 2 M - Γ / 2 M - Γ / 2 M - Γ / 290 1423.3 - 28.5 1419.8 - 26.0 1419.9 - 23.1 1419.0 - 14.4100 1421.7 - 28.0 1417.3 - 23.1 1418.0 - 19.8 1419.6 - 13.2110 1420.5 - 26.6 1416.6 - 19.5 1417.8 - 16.6 1420.0 - 12.8120 1419.5 - 25.6 1416.9 - 16.7 1418.3 - 14.0 1418.9 - 11.7

- Γ /

M [MeV]

90100110

120 90

100110

θ dependence of “Size” and

“Wave function”

θ dependence of “Size” NRv2-NRfπ=90 MeV

θ [deg]

θ 10 15 20 25 30 35

Pole positionE 1416.8 1419.2 1419.8 1419.9 1419.9 1419.8[MeV] Γ / 2 18.5 22.9 23.1 23.1 23.1 23.1

Meson- Baryon distance[fm]KbarN ch. (Re) KbarN 0.95 1.24 1.25 1.25 1.25 1.24

(Im) KbarN - 0.43 - 0.41 - 0.35 - 0.35 - 0.35 - 0.35

π Σ ch. (Re) π Σ 12.36 3.61 0.71 0.35 0.34 0.34(Im) π Σ 4.27 4.81 1.17 0.33 0.23 0.23

KbarN+π Σ(Re) KbarN+π Σ12.38 3.61 0.87 1.23 1.25 1.25(Im) KbarN+π Σ 4.23 4.67 0.44 - 0.26 - 0.29 - 0.29

θ dependence of “Size” Org-SRfπ=90 MeV

θ [deg]

θ 10 15 20 25 30 35

Pole positionE 1416.7 1418.9 1419.0 1419.0 1419.0 1418.9[MeV] Γ / 2 14.5 14.5 14.4 14.4 14.4 14.4

Meson- Baryon distance[fm]KbarN ch. (Re) KbarN 0.98 1.21 1.21 1.21 1.21 1.21

(Im) KbarN - 0.69 - 0.50 - 0.49 - 0.49 - 0.49 - 0.49

π Σ ch. (Re) π Σ 8.62 0.73 0.13 0.06 0.12 0.12(Im) π Σ 7.05 2.33 0.51 - 0.11 - 0.09 - 0.09

KbarN+π Σ(Re) KbarN+π Σ 8.60 0.54 1.10 1.21 1.21 1.22(Im) KbarN+π Σ 6.99 2.00 - 0.48 - 0.49 - 0.49 - 0.49

θ dependence of “Wave function” NRv2-NRfπ=90 MeV

KbarN bound state πΣ resonant state

r [fm]

θ =10 deg.

r [fm]

θ =15 deg.

r [fm]

θ =20 deg.

r [fm]

θ =25 deg.

r [fm]

θ =30 deg.

r [fm]

θ =35 deg.

Comparison with Hyodo-Weise’s result

Scattering amplitudes

HW: PRC 77, 035204 (2008)

Fig. 4 I=0 KbarN/πΣ scattering amplitude in two channel model

Comparison with Hyodo-Weise’s resultPole position HW: PRC 77, 035204 (2008)

Fig. 5 Pole positions of I=0 scattering amplitude

(Upper pole in two channel model)

z = 1419 – 14.4i MeV

Schrödinger equation to be solved

: complex parameters to be determined

Wave function expanded with Gaussian base

c.c. CSM for resonant states

Phenomenological potentialY. Akaishi and T. Yamazaki,

PRC 52 (2002) 044005= Energy independent potential

Chiral SU(3) potential N. Kaiser, P. B. Siegel and W. Weise, NPA 594, 325 (1995)= Energy dependent potential

Complex-rotate , then diagonalize with Gaussian base.

c.c. CSM for resonant states

Complex scaling of coordinate

ABC theorem

The energy of bound and resonant states is independent of scaling angle θ.

J. Aguilar and J. M. Combes, Commun. Math. Phys. 22 (1971),269E. Balslev and J. M. Combes, Commun. Math. Phys. 22 (1971),280

Chiral SU(3) potential (KSW)N. Kaiser, P. B. Siegel and W. Weise, NPA 594, 325 (1995)

Original: δ-function type

Present: Normalized Gaussian type

( 0)28

i ji j

MV r r

3/ 2 3

1 expg r ra

a: range parameter [fm]Mi , mi : Baryon, Meson mass in channel iEi : Baryon energy, ωi : Meson energy

2 22 2 2 2

i i i i

Reduced energy: i ii

( 0)332

KbarN πΣ

Energy dependence of Vij is controlled by CM energy √s.

( 0)28

i ji j

C M Mr rV

Energy dependence is determined by Chiral low energy theorem. ← Kaon: Nambu-Goldstone boson

Flavor SU(3) symmetry

Chiral SU(3) potential (KSW)Energy dependence

( 0) 0, @ 0.5 , 100ij

IV r s a fm f MeVp

√s [MeV]

KbarN-KbarN

πΣ-πΣ

KbarN-πΣ

KbarN thresholdπΣ threshold

Chiral SU(3) potential = Energy-dependent potential

Calculational procedure

Perform the Complex Scaling method.

Then, find a pole of resonance or bound state. Calculateds

Calculated Assumeds sCheck Finished !If Yes

Self consistency for the energy!

Assume the values of the CM energy √s.

MBH T V s M Assumeds

やったこと1. Chiral SU(3) potential で、 r-space ガウス型を仮定

レンジパラメータ a 、相互作用の強さ fπ の決定が必要

2. A.D. Martin の解析による I=0 KbarN 散乱帳 aKbarN(I=0) を再現するような (a, fπ) を探す。

散乱振幅の計算は、 Kruppa 流に複素スケーリングを使って計算

3. 見つけた (a, fπ) に対して、これまでの結合チャネル複素スケーリングで、 Λ(1405) に相当する KbarN-πΣ 系の共鳴エネルギーを求める。

Chiral SU(3) potential はエネルギー依存性をもつが、複素エネルギーに対して self consistency を課して計算。

( 0)28

i ji j

C M MV r

3/ 2 3

1 expg r ra

( 0) 1.70 0.07 0.68 0.04KbarN

Ia i fm

A. T. Kruppa, R. Suzuki and K. Kato, PRC 75, 044602 (2007)

結果のまとめ1. この形のポテンシャルでは Martin の散乱長の実部・虚部を同時に　　合わせることはできない。 fπ=90 ～ 120MeV の範囲では。

　　　 fπ=75.5MeV というカイラルリミットより小さい値を使えば、一応再現できる。

2. 散乱長の実部・虚部、一方だけなら再現できるポテンシャルが存在。　　　各々の場合で、複素エネルギーに対して self-consistent な KbarN-πΣ 系の　　　共鳴エネルギーが求まる。

散乱長実部を尊重した場合： (EKbarN, Γ/2) = (-19, 27) ～ (-5,8) MeV 虚部を尊重した場合 : (EKbarN, Γ/2) = (-32 ～ -24, 28 ～ 66) MeV

3. 散乱長の再現に関わらず、このポテンシャルでは PDG に載っている　　　 Λ （ 1405 ）のエネルギー及び幅 (EKbarN, Γ/2) = (-29±4, 25±1) MeV は　　　再現できない。

※ 　 fπ=75.5 で無理やり散乱長を再現した場合 : (EKbarN, Γ/2) = (-23, 34) MeV

Mysterious state; Λ(1405)Quark model prediction … calculated as 3-quark state

N. Isgar and G. Karl, Phys. Rev. D18, 4187 (1978)

Λ(1405) can’t be well reproduced as a 3-quark state!

observed Λ(1405)calculated Λ(1405)

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

Kbar + N

Λ(1405)

π + Σ

1332 [MeV]

B. E. (KbarN) = 27 MeVΓ (πΣ) ～ 50 MeV

Jπ = 1/2-

(Jπ=0-, T=1/2)

N (Jπ=1/2+, T=1/2)

π(Jπ=0-, T=1)

Σ(Jπ=1/2+, T=1)

Complex-rotate , then diagonalize with Gaussian base.

Complex scaling of coordinate

ABC theorem

The energy of bound and resonant states is independent of scaling angle θ.

J. Aguilar and J. M. Combes, Commun. Math. Phys. 22 (1971),269E. Balslev and J. M. Combes, Commun. Math. Phys. 22 (1971),280

Calc. of scattering amplitude with CSM “Kruppa method” A. T. Kruppa, R. Suzuki and K. Katō,

PRC 75, 044602 (2007)

• Radial Schroedinger eq.

• Asymptotic behavior of wave func.

• Equation to be solved

V(r) : short-range potential

• Asymptotic behavior of scattering part of wave func.

llh x i kr p

, llj kr h kr

: Ricatti-Bessel, Hankel func.

is not square-integrable ,SCl k r

Search the solution in the form

Separate the incoming wave ! lj kr

Point 1

PRC 75, 044602 (2007)

• Complex scaling

Equation to be solved

2,i i ir r e r r e r e

• Expanding with square-integrable basis function (ex: Gaussian basis)

Point 2

i r : square integrable function

Linear equation:

PRC 75, 044602 (2007)

• Calculation of scattering amplitude

We don’t have which is a solution along r,

but we have which is a solution along reiθ.

,SCl k r

,,SCl k r

r eiθ

Cauchy theoremPoint 3

2. Application of CSM to Λ(1405)

• Coupled-channel Complex Scaling Method (ccCSM)

• Energy-independent KbarN potential

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

Energy-independent potential

436 412exp 0.66 fm MeV

412 0AY IV r

KbarN πΣ

1. free KbarN scattering data2. 1s level shift of kaonic hydrogen atom3. Binding energy and width of Λ(1405)

= K- + proton

The result that I show hereafter is not new, because the same calculation was done by Akaishi-san, when he made AY potential.

Remark !

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

= 0 deg.

= 5 deg.

=10 deg.

=15 deg.

=20 deg.

=25 deg.

=30 deg.

=35 deg.

=40 deg.

trajectory

=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… (1405) !

3. ccCSM with an energy-dependent potential

for Λ(1405)

Result

Range parameter (a) and pion-decay constant fπ are ambiguous in this model. Various combinations (a,fπ) are tried.

fπ = 95 ～ 105 MeV

Self-consistency for complex energy

ReCalculateds E

Search for such a solution that both of real and imaginary parts of energy are identical to assumed ones.

(B.E., Γ)Calculated = (B.E., Γ)Assumed

Calculated complex en ys E erg

More reasonable?

0T V V G ZTZ Z Z Z

Pole search of T-matrix is done on complex-energy plane.

Z: complex energy

Self consistency for complex energy KSWfπ = 100 MeV

a=0.47, θ=35°

-B [MeV]

1 step-Γ/2 [MeV]

obtained by the self-consistency for the real energy

a=0.47, θ=35°

-B [MeV]

2 steps-Γ/2 [MeV]

a=0.47, θ=35°

-B [MeV]

3 steps-Γ/2 [MeV]

a=0.47, θ=35°

-B [MeV]

4 steps-Γ/2 [MeV]

a=0.47, θ=35°

5 steps-Γ/2 [MeV]

-B [MeV]

Self consistent!

Assumed

-B [MeV]

a=0.60

a=0.50

a=0.47

a=0.45

S.C. for real energyS.C. for complex energy

Repulsively shifted!

Mean distance between Kbar (π) and N (Σ)

Kbar (π)

N (Σ)

Distance

Chiral (HW-HNJH): B ～ 12 MeV, Distance = 1.86 fm

Λ （１４０５）、他の計算との比較

Rough estimation of charge radiusK-

Distance

Mr Distancem M

mr Distancem M

2 2 2c p CM K CM

Charge radius

Charge radius derived from electric form factor

T. Sekihara, T. Hyodo and D. Jido, PLB 669, 133 (2008)

Rough estimation

Application of coupled-channel Complex Scaling Method to the K bar N -πY system

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