Theoretical study of a prototype systemof kaonic nuclei "K-pp"
Akinobu Doté (KEK Theory Center, IPNS / J-PARC branch)
Takashi Inoue (Nihon university)
Takayuki Myo (Osaka Institute of Technology)

1.

Introduction2.

Effective single-channel potential by “Feshbach method withcoupled-channel Complex Scaling Method”

3.

Result of “K-pp” calculated with ccCSM+Feshbach method

4.

Other quantum number case – Jπ=1- … SNN=1

5.

Summary and future plan
Workshop on Progress on J-PARC hadron physics in 2014
30. Nov. ’14 @ IQBRC, Tokai, Ibaraki, JapanKaonic nuclei
1. Introduction
PK-

P

Prototype system = K pp

at J-PARC

Kaonic nuclei
K-ProtonKbarN two-body system

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

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

Strongly attractive KbarN potential

PK-

P

Prototype system = K pp
 Doorway to dense matter†→ Chiral symmetry restoration in dense matter Interesting structure† Neutron star

Nuclear many-body system with K3HeK-, pppK-,
4HeK-, pppnK-,…, 8BeK-,…
† A. D., H. Horiuchi, Y. Akaishi and T. Yamazaki, PRC70, 044313 (2004)

at J-PARC

Kaonic nuclei
PK-P

Prototype system = K pp

at J-PARC

Kaonic nuclei

Experiments of K-pp search
FINUDA

DISTO

K- pp???
B. E.= 115 MeV
Γ = 67 MeV
PRL 94, 212303 (2005)

P
J-PARC E15

K-

P

J-PARC E27

K- pp???
B. E. = 103 ±3 ±5 MeV
Γ= 118 ±8 ±10 MeV
PRL104, 132502 (2010)

SPring8/LEPS

Prototype system = K pp

Attraction in K-pp subthreshold region
arXiv:1408.5637 [nucl-ex]

ΣN cusp, Y* shift
PTEP 101D03 (2014)

at J-PARC
No evidence of K-pp bound state
PLB 728, 616 (2014)

Kaonic nuclei

Theoretical studies of K-pp
Dote-HyodoWeise

Barnea-GalLiverts

PRC79, 014003(2009)

PLB712, 132(2012)

B(K-pp)20±3ΓIkedaKamano-Sato

ShevchenkoGal-Mares

PRC76, 045201(2007)

PTP124, 533(2010)

PRC76, 044004(2007)

1647

9 ～ 1650 ～ 70

40 ～ 70

41

61

34 ～ 46

90 ～ 110

MethodVariational(Gauss)Variational(H. H.)

Variational(Gauss)Faddeev-AGS

Faddeev-AGS

Potential

Chiral(E-dep.)

Chiral(E-dep.)

Pheno.

Chiral(E-dep.)

Pheno.

P

AkaishiYamazaki

K-P

Prototype system = K pp

• Chiral pot. (E-dep.)→ Small binding• Phenomenological pot. (E-indep.) → Large binding

B(K-pp) < 100 MeV
K-pp should be bound, but exist as
a resonance between KbarNN and πΣN thresholds.

at J-PARC

• Resonant nuclei
stateKaonic• Coupled-channelsystem⇒ coupled-channel
KP
P

Complex Scaling Method
Prototype system = K pp

• Anti-kaon = Nambu-Goldstone boson

atKbarJ-PARC⇒ Chiral SU(3)-based
N potential

2. Effective single-channel potential by“Feshbach projection with ccCSM”
πΣ,πΛKbarN

QP

Reduce the coupled-channel problem
to a single channel problem

Complex Scaling Method for ResonanceComplex rotation of coordinate (Complex scaling)
U   : r  r ei , k  k eiH  U   HU 1   ,  U   

Diagonalize Hθ with Gaussian base,
we can obtain resonant states, in the same way as bound states!

S. Aoyama, T. Myo, K. Katoand K. Ikeda, PTP116, 1 (2006)† J. Aguilar and J. M. Combes, Commun. Math. Phys. 22 (1971),269.

 Continuum state appears on 2θ line.
E. Balslev and J. M. Combes, Commun. Math. Phys. 22 (1971),280
 Resonance pole is off from 2θ line, and independent of θ. (ABC theorem†)

Formalism of ccCSM + Feshbach method
Elimination of channels by Feshbash method
 TP  vP VQP

Schrödinger eq.
in model space “P” and out of model space “Q”

 T  U E   E 

Schrödinger eq. in P-space :PEffPP

VPQ   P  P E
TQ  vQ   Q  Q 

PEffective potential for P-space

UEffP E   vP  VPQ GQ  E VQP

Q-space Green function:

GQ  E  1E  H QQ

Extended Closure Relation in Complex Scaling Method
H QQ
 n   n  nH QQ
 U   H QQU 1    

CR B

n

 n  1

Diagonalize HθQQ with Gaussian base, n n  1

Well approximatednT. Myo, A. Ohnishi and K. Kato, PTP99, 801 (1998)

Express the GQ(E) with Gaussian base using ECRGQ  E  11   n nE  H QQE  nn n: expanded with Gaussian base.

U PEff  E   vP  VPQ U 1   GQ  E U   VQPGQ  E 

3. Result of K-pp calculated withccCSM + Feshbach method

Apply ccCSM + Feshbach method to K-pp
(Jπ=0-, T=1/2)“K-pp” … KbarNN - πΣN - πΛN (JFor the two-body system, P = KbarN, Q = πYV  K bar N   Y ; I  0,1V  Y   Y '; I  0,1Feshbach + ccCSM

U KEffbar N ( I 0,1)  E • Schrödinger eq. for KbarNN channel :Eff
 TK bar NN  VNN   U K bar Ni ( I )  EK bar N    K bar NN  E  K bar NNi 1,2


• Trial wave function( KNN ,1)
" K  pp "   Ca( KNN ,1) Ga( KNN ,1)  x1(3) , x(3) x1(3) , x(3)2  S NN  0  K  NN 1 T 1/ 2
2   Gaa

( KNN ,2)
  Ca( KNN ,2) Ga( KNN ,2)  x1(3) , x(3) x1(3) , x(3)2  S NN  0  K  NN 0 T 1/ 22   Gaa• Basis function ＝ Correlated Gaussian…including 3-types Jacobi-coordinates

Ch. 1: KbarNN, NN:1E
Ch. 2: KbarNN, NN:1O(3)
( KNN ,i )(3)
(3)
( KNN ,i )  x1 Ga( KNN ,i )  x1(3) , x(3)
Nexp

x,xA
  1 2  a
 (3)  2 a x 2  

Chiral SU(3) potential with a Gaussian form
 Weinberg-Tomozawa termof effective chiral LagrangianA non-relativistic potential (NRv2c)

 Gaussian form in r-space Semi-rela. / Non-rela.
 Based on Chiral SU(3) theory→ Energy dependence( I  0,1)ij

Vr   gij  r  Cij( I 0,1)
8 f2



i   j  r d 2 
expij


 3 / 2 dij3
11g ij  r 
mi m j: Gaussian form

ωi: meson energy

Constrained by KbarN scattering lengthaKN(I=0) = -1.70+i0.67fm, aKN(I=1) = 0.37+i0.60fm

A. D. Martin, NPB179, 33(1979)A resonance pole corresponding to Λ(1405) at(M, -Γ/2) ~ (1420, -20) MeV
BKN ~ 15MeV
A. D., T. Inoue, T. Myo, Nucl. Phys. A 912, 66 (2013)

Double-pole structure is confirmed.
Lower pole ~ (1395, -138) MeV at fπ=110 MeV
A. D., T. Myo, Nucl. Phys. A 930, 86 (2014)

Self-consistency for complex KbarN energy
Kbar

NU KEffbar N ( I )  EK bar N EKNiN

How to determine the two-body energy in the three-body system?
1. Kaon’s binding energy:

BK   H  H 
NNH NN : Hamiltonian of two nucleons

2. Define a KbarN-bond energy in two ways

 M  m  B  K 
EKN  M N     N K
 M N  mK  B  K  2

Field picture

A. D., T. Hyodo, W. Weise,
PRC79, 014003 (2009): Field picture: Particle picture

Particle picture

Result
Fix the KbarN energy at Λ*… self-consistent for Λ* in free space
Kbar-N-N“K-pp”

NN pot. : Av18 (Central)
KbarN pot. : NRv2c potential(fπ=110MeV)EKNN [MeV]Λ*(-17, -18) MeV(-28.6, -21.6)
MeV

Kbar-N-Ncontinuum

Λ*-N continuum-Γ/2 [MeV]θ = 30 deg.

NN pot. : Av18 (Central)
KbarN pot. : NRv2c potential(fπ=90 - 120MeV)

Self-consistent results
fπ=90~120MeV

120

110
100

×fπ = 90

Unstable for scaling angle θ!

Field picture(B, Γ/2) = (21~32, 9~16)

120
110
100
fπ = 90
Particle picture(B, Γ/2) = (25~30, 15~32)

NN correlation density

NN pot. : Av18 (Central)
KbarN pot. : NRv2c potential
fπ=110, Particle pict.

Correlation density in Complex Scaling Method NN ,  x     r NN ,  x NN  x     XN ,  x  
 e 3i  d 3 R 2  xe  i , R r XN ,  r XN ei

NN repulsive core

Kbar

NRe ρNNNIm ρNNNN distance = 2.1 - i 0.3 fm~ Mean distance of 2N in nuclear matter at normal density!

KbarN correlation density
Kbar I=0,1

KbarN (I=0) [Re]NN pot. : Av18 (Central)
KbarN pot. : NRv2c potential
fπ=110, Particle pict.

Λ* [Re]NKbar

Λ*I=0 N

NKbarN (I=0) [Im](I=1)
Λ* [Im]

I=0 KbarN compacter than I=1 oneStrong KbarN attraction in I=0

I=0 KbarN seems similar to Λ*Λ* survives in K-pp

4. Other quantum-number case
Jπ=1- state … SNN=1

J-PARC E27 experimentK-pp search by d (π+, K+) reaction at 1.69 GeV/c
Inclusive spectrum: Ichikawa et al., PTEP 101D03 (2014)

2p-coincidence
analysis

Ichikawa’s talkat EXA2104

If the observed state is really the K-pp and if spin-flip is not so strong in the reaction,
spin of K-pp formed from deuteron should be 1 because the deuteron has the spin 1.
(Prof. T. Harada, Osaka E.C. university )How is Jπ = 1- state … SNN=1?
" K  pp " " K d " 

LKNN  0, NN : s  wave

LKNN  0, NN : s  wave

S NN  0  K  NN 1 T 1/ 2

Jπ=0-, T=1/2

 K  NN 0 T 1/ 2

Jπ=1-, T=1/2

S NN = 1“K- + Deuteron”-like channel“K- d” studied simply with …
• NN potential:

Av4’ (3E, 3O) (fitted with 5-range Gaussian functions)… Tensor force is incorporated into central potential.

• KbarN potential: A phenomenological potential
Y. Akaishi and T. Yamazaki,… Energy independent
PRC 52 (2002) 044005• Λ*-fixed ansatz… E(KbarN) in effective KbarN potential is fixed to the Λ* energy.

How is Jπ = 1- state … SNN=1?

NN pot. : Av4’ (5 Gauss)
KbarN pot. : AY potential
Λ* fixedbar
K NN threshold
EKNN [MeV](-28, -21) MeV… Λ* resonanceKbar-N-N
continuum~ (-2, 0 ) MeV
N-N: 4.3 fm
Kbar-[NN] : 12 fm… Deuteron with distant Kbar

Kbar-d continuum
Λ*-N continuum
θ=25°-Γ/2[MeV]

No resonance of KbarNN (SNN=1)

5. Summaryand future plans

5. Summary and future plans
A prototype of Kbar nuclei “K-pp” = Resonance state of KbarNN-πYN coupled system“coupled-channel Complex Scaling Method + Feshbach projection”… Represent the Q-space Green function with the Extended Complete Set
well approximated by Gaussian base

⇒ Eliminate πＹ channels to reduce the problem to a KbarNN single channel problem.

K-pp studied with ccCSM+Feshbch method• Used a Chiral SU(3)-based potential(Gaussian form in r-space)• Self-consistency for kaon’s complex energy

NRv2c potential case(B, Γ/2) = (21~31, 9~16) MeV : “Field picture”(25~30, 15~32) MeV : “Particle pict.”

Mean NN distance ~ 2.2 fm → Normal density• Correlation density in CSM shows
effect of NN repulsive core and Λ* survival in K-pp resonance.
• Jπ=1- state (“Deuteron+K-”-like channel) seems not to exist as a resonance state.

Future plans Full-coupled channel calculation of K-pp
 Consider with the KbarN potential constrained by the latest data of SIDDAHRTA
 Application to resonances of other hadronic systems

Kaonic nuclei
Thank you very much!
PK-P

Prototype system = K pp
Reference)
A. D., T. Inoue, T. Myo,at[nucl-th]
J-PARCarXiv: 1411.0348