K?????? -????????-

1
K?????? -????????-
???? (KEK)

• ???????
• ????????? ppK- ???
• Simple Correlated Model
• Test on two nucleons system
• Result of ppK-
• ???

2
K??????????
3
?????????
4
(1) ???????????????
Akaishi-san and Yamazaki-sans study
Phenomenological KN potential (AY KN potential)
Strongly attractive.

1. free KN scattering data
2. 1s level shift of kaonic hydrogen atom
3. binding energy and width of ?(1405)

K- proton
Y. Akaishi and T. Yamazaki, PRC 52 (2002) 044005
5
Collaboration with Akaishi-san and Yamazaki-san
According to the study with
Antisymmetrized Molecular Dynamics
G-matrix Phenomenological KN interaction
Kaonic nuclei has interesting properties
6
AMD G-matrix AY KN interaction studies
revealed

• E(K) gt 100 MeV for various light nuclei
• Drastic change of the structure of 8Be,
• isovector deformation in 8BeK-
• Highly dense state is formed in K nuclei.
• maximum density gt 4?0
• averaged density 24?0
• Proton satellite in pppK-

A. D., H. Horiuchi, Y. Akaishi and T. Yamazaki,
PLB 590 (2004) 51 PRC 70 (2004) 044313.
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(2) K-??????????????
Dense system
Lots of interesting phenomena!

• NN repulsive
• core

• Decay mode
• KN?pY
• KNN?YN

? Strongly attractive I0 KN interaction
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(3) ???????????????

• KN interaction??????????????????????

40Ca????????????? ???Kaon???????????????2?0????? ?
???????????
RMF???NL-SH??? J. Mares, E. Friedman and A. Gal,
Nucl. Phys. A770, 84 (2006)
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(3) ???????????????

• ppK- Prototype of K cluster???

ppK-????????? 50 70 MeV
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(4) ???????????????????
??????
KN???????????
??(??????)? Conventional???????G-matrix? ?
???????NN?????????????? ????????????????
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(4) ???????????????????
Decay width?
????????????????Sp?????? KNN?YN (Non-mesonic
decay, ?????) ???? ????? ?????????????
????????????????
??
BK gt100 MeV?? ???? G50 MeV
RMF???NL-SH??? (Pb?L-HS) J. Mares, E. Friedman
and A. Gal, Nucl. Phys. A770, 84 (2006)
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(4) ???????????????????
Effective KN potential ????
Strongly attractive
Strongly attractive
Weakly attractive
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(4) ???????????????????
Effective KN potential ????
??(???Weise????????)???
??????????????????kaon? ????????????????? ???
ppK- ????????????????? ???????proton???????????
Weakly attractive
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(4) ???????????????????
???????
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(4) ???????????????????
???????

• ??
• Final state interaction?
• K-pN??N????????N?
• ??????????????
• V. K. Magas, E. Oset, A. Ramos and H. Toki,
• PRC74, 025206 (2006)
• 6Li target??6Li??deuteron cluster?
• K-?????????
• M. Agnello et. al., NPA775, 35 (2006)

Very preliminary
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(5) ?????????????????

• ??????
• G-matrix???????????????
• ??????????AMD?????Unitary correlator?????

T. Neff and H. Feldmeier, Nucl. Phys. A713, 311
(2003)

• ?????????
• ??????????????????????????
• ???????????????????????????

ppK-
3fm
Nucleon
Kaon
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(5) ?????????????????

• KN potential????
• ????K?????????
• ?????kaon??????????????????
• ?????????????K?????????????????
• (????????????????)

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(6) ?????

• ???????????(1405)???
• ????(???????)?????(??)
• ?(1405)???????
• ????(??)
• ???????K?????????
• ???? ????????(???)
• ????????? ppK- ??? ????(???)?????(???)
• Kaonic 3,4Helium atom 2p???????
• ????(??)?????(??)?????(??)

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?????????ppK-???
Collaborating with W. Weise (TU Munich)
???????????
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1. Simple Correlated Model
Model wave function of ppK-
Normalization factor
Spin w. f. (NN)
Spatial part
Isospin w. f.
Detail of the spatial part
NN correlation function
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1. Simple Correlated Model
In this model, I assume only one configuration
total nucleons spin S0 and total nucleon
isospin TN1. Other configurations are ignored.
Therefore, this model is very simple. Single
particle motion of nucleons and kaon is described
with a single Gaussian, G(ri) and G(rK),
respectively. Two nucleons wave functions are
assumed to be the same G(ri). The NN
correlation is described with 1 minus
superposition of several Gaussians. We dont
introduce a correlation between a nucleon and a
kaon.
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1. Simple Correlated Model
Few remarks

• This model corresponds to the AMD case where all
  wave packets
• come together to the origin. But the NN
  correlation is respected.
• The angular momentum is very restricted.
• The orbital angular momentum of each particle
  measured from the center
• is zero and the relative one between any two
  particles are also zero.
• If we choose the variational parameters µ and ?
  independently,
• it is impossible to separate the wave function
  of the center-of-mass motion
• from the total wave function.
• The relation
• 
  should be held to separate the CM motion
  completely.

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1. Simple Correlated Model
Energy variation
This model wave function has the real variational
parameters,
which are included in the spatial part wave
function.
These real parameters are determined by the
Simplex method to minimize the total energy of
the system.
This time, The width parameters of the Gaussians
in the NN correlation are fixed to those of
Kamimura Gauss.
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2. Test on 2N system
First, I checked the reliability of this model in
case of pp system. The model wave function is as
follows.

• Variational parameters are
  determined by the Simplex method.
• are fixed to those of Kamimura Gauss.

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2. Test on 2N system
NN potential to test
Id like to know whether this model works
correctly under a potential such as Av18-like
which has a strong repulsive core or not. But
the Av18-like potential used in calculating ppK-
does not make two protons bound. So, I enhanced
the long-range attraction of this potential
slightly so that two protons are bound. The
test potential is shown as the pink line
(Dote_HC2) in the left panel. As can be seen,
the repulsive-core part of this potential is
almost the same as that of the Av18-like
potential shown as the blue line.
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2. Test on 2N system
Solve in two ways
I solve the same Hamiltonian by two methods.
Test potential (Dote_HC2)

• One way is the SCM model that will be applied to
  the calculation of ppK-.

• The other way is the Gaussian diagonalizing
  method. (GDM)

The relative wave function is
expanded by so-called Kamimura Gaussians.
We solve the Schroedinger equation
by the diagonalization
with the Gaussian base
.
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2. Test on 2N system
Result

• As for the GDM, I have confirmed that the
  solution is sufficiently
• converged up to the base number 25.
• This GDM solution can be regarded as the exact
  solution of this Schrodinger equation.
• The SCM method almost achieved to the exact
  solution when the base number is 9.

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2. Test on 2N system
Relative wave function
GDM N25
Test potential
MeV
SCM N9
fm
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3. Result of ppK-
Hamiltonian
This time, Coulomb force is neglected.
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KN potential
S-wave potential
P-wave potential
1, Gaussian shape
asapa
2, Energy dependent
Chiral SU(3) theory
KN scattering amplitude
KN scattering volume
3, P-wave potential including derivative operator.
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KN potential

1. The relation between T matrix and scattering
   amplitude

• Self energy at the low-density limit
• Klein-Gordon eq.

1. The optical potential from the self energy

Optical potential
Finite range (normalized Gaussian)
Two-body interaction
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KN potential
S-wave scattering amplitude
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KN potential
P-wave scattering volume
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Procedure of the present calculation

• Self-consistency of kaons energy is taken into
  account.

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Procedure of the present calculation
Remarks

• The imaginary parts are ignored in the current
  study.

• The kaons binding energy B(K)

B(K) -EK -(Etotal Enucl)
pp in ppK- K
Enucl
ppK
0
B(K)
ppK-
Etotal
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3. Result of ppK-
There doesnt exist any self-consistent solution
for the range parameter a lt 0.67 fm. This
result is the same as that obtained in the
previous AMD study reported in YKIS06 and so
on.
Kamimura Gauss, N10, r10.1 fm, rN9.0 fm P-wave
int. non-perturbative
a range parameter fm
Self consistency
a0.67 fm
a0.70 fm
a1.00 fm
a0.80 fm
a0.90 fm
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3. Result of ppK-
Property
fm
MeV
The total binding energy of ppK- is 42 76 MeV.
cf) It doesnt exceed 53 MeV in the previous
AMD study.
MeV
MeV
fm
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3. Result of ppK-
Property
fm
MeV
The relative distance between two nucleons is
larger than 1.0 fm. If the size of a nucleon
core is 0.5 fm, they dont touch. This result
is the same as that of the previous AMD study.
MeV
MeV
fm
39
???
40
????????ppK-???

• NN??????????????????(Av18-like)????ppK-?????

• KN?????????????????????
• s-wave??????p-wave????????

• ???????????????????
• ????LS0?T1??????
• ???????????????????????????????????

• ???AMD????????
• ???????????????Variation After Projection???????
• p-wave KN??????????????????

• ??
• ???????? 42 76 MeV
• (a1.00 0.67fm??????????????)
• ?????????1 fm???????
• ????????AMD?????????

41
??????

• KN???????????????????????????
• ppK-????????????????????????????5070MeV?

• KN?????????????????????????????????
• (???????????)
• ?????????????????????G????????
• ????????????????????

• ??????????????????????????????
• ?????????????????????
• ????????????????????????????????