using two-cluster RGM kernels

taupeselectionMechanics

Nov 14, 2013 (3 years and 7 months ago)

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2013.9.12 efb22

1.
Introduction

2. Three
-

and 4
-
cluster Faddeev
-
Yakubovsky equations using



2
-
cluster RGM kernels

3. Quark
-
model
baryon
-
baryon

interaction
fss2

4
.
3
N

and 4
N
bound
-
state problems by AV8’ and fss2

5.

Coulomb effect and charge dependence of the nuclear force

6. Application to the 4


獹獴敭

7
⸠†卵浭S特

Y
. Fujiwara: Kyoto University,
Japan

Four
-
body
Faddeev
-
Yakubovsky
equations


using two
-
cluster RGM kernels

--

Applications to
4
N
, 4
d

and
4


獹獴敭猠


2013.9.12 efb22

Introduction


3
-
baryon systems)


3
H (
n
+
n
+
p
),
3

H (
n
+
p
+



㨠扩湤楮朠敮敲杩敳e慮搠牭猠牡摩


Scattering observables of 3
-
nucleon systems

(
nd

and

pd

elastic
scattering, 3
-
body breakup processes)
with K.
Fukukawa

Comprehensive understanding of few
-
baryon systems in
terms of quark
-
model baryon
-
baryon interaction fss2


(
3
-
cluster systems



12
C (3

⤬
9
Be (
n
+2

⤬
9

Be (



)Ⱐ
6
He (2
n
+

⤬)
6

He (2

+

⤠

††
扯畮u
-
獴慴攠捡汣畬慴楯湳c畳楮朠


Ⱐ
n

,


則R步牮敬猠

General framework: 3
-
cluster and
4
-
cluster
Faddeev
-
Yakubovsky

equations using 2
-
cluster

RGM kernels

2013.9.12 efb22

Prerequisites of many
-
cluster

Faddeev
-
Yakubovsky

equations

1.
Equivalence to variational approach
using h.o. basis, SVM, Gaussian


expansion method, etc. for the bound
-
state problems

2.
2
-
cluster RGM kernels
are constructed from the 2
-
body force of
constituent particles (not the phenomenological 2
-
cluster potentials)





偡P汩l景牢楤摥渠獴慴攠
u

is naturally induced as the orthogonality



conditions for the relative motion of clusters

… i.e. ,
pairwise



orthogonality condition model

(OCM) introduced by H. Horiuchi

3.
For example, 2

,3

,4


獨潵汤s扥b捯湳楳瑥湴汹l摩獣畳獥搮



䥮摵捥搠3
-
扯摹b景牣攠
(
慮i
-
獹浭整物慴楯

敦e散琠慭潮朠3
-
捬畳瑥牳)



慮搠敦e散琠潦瑨攠潦f
-
獨敬氠s牡湳景牭慴楯渠⡬(k攠
ㄯ†1
N

effect from the



elimination of the energy
-
dependence) can be discussed.


... could be a hint
for
V
low
-
k

and

SRG transformation for the
NN

force.

4.
When existing Pauli forbidden state between 2
-
clusters,
Faddeev



redundant component

should be eliminated properly and the equation
becomes solvable.


(non
-
trivial in 4
-
body case and more)





2013.9.12 efb22

Removal of the energy dependence by the renormalized RGM


Matsumura, Orabi, Suzuki, Fujiwara, Baye, Descouvemont, Theeten

3
-
cluster semi
-
microscopic calculations using 2
-
cluster non
-
local RGM kernels
:


Phys. Lett. B659 (2008) 160; Phys. Rev. C76, 054003 (2007)

[


-

H
0
-

V
RGM
(

⤠崠


㴠=


w楴栠
V
RGM
(


V
D
+
G
+


K






[


-

H
0
-

V
RGM
]



= 0


with
V
RGM
=
V
D
+
G
+
W

1
1
0 D 0 D
[ ( ) ( )]
N
N
W H V G H V G
       
: Prog
. Theor. Phys.
107
(2002) 745; 993


Four
-
cluster Faddeev
-
Yakubovsky
formalism
using
two
-
cluster RGM kernels

N=1
-
K

Extra non
-
local kernel



   
 
 



  



 
 


 
 

  

  

     


 


RGM
( )
0
0
RGM
0 0
( ) ( )
0
0
0
[ ( ) ]
( ) (parameter)
( )
|[1 ( )
( )

(,) ( ) ( ) ( ) (,)
1
(,) ( )| |(

( )] [1 ( ) ( )]|
wit
0
)
h
new
new new
RGM RGM
V
T V V G T
T H u u H
V
v
H H
V
T
u G T T G u
(


=
E
-

E
int
)


 

 
int
{ } 0

( 1)
u
Ku u
Α
   
1 | |
u u
RGM
T
-
matrix

(also obtained by Kukulin's
method
)

2013.9.12 efb22

Projection
operator onto the
(pairwise) Pauli
-
allowed state

: 4
-
cluster Faddeev
-
Yakubovsky


equation
using
RGM
T
-
matrix

Cf
.
Non [4]
-
symmetric trivial
solutions in
the
4
α

system

慲攠敭潶慢汥.



i<j

|
u
i,j



u
i,j
|
ψ


=

|
ψ



in |
ψ





嬴]



㴠=:
偡畬椠汬潷敤

Þ

=


|
ψ




ψ


|




㸠>:
偡畬椠f牢楤摥


|
ψ


= (1/



i<j

|
u
i,j



u
i,j
|
ψ




4


捡獥

0 RGM,
0 (34)
0
0 (34)
0
(12) (23) (13) (23) (13) (24)
(34)
whe
[ ( ) ] 0

( ) ( ) [(1 ) ]
( ) ( ) [(1 ) ]
,
(1 ){[1 (1 )
Total wave functi
] (1 ) }
o

e
n

r

i j
i j
E H V
G E T E h P P
G E T E h P P
P P P P P P P P
P P P P
  
  
 

   

   
   
  
        
P P
P
,
| 0
i j
u
  
: 4
-
cluster
OCM using
V
RGM

(Faddeev redundant components)

Equivalent !

2013.9.12 efb22

NN

phase
shifts by
fss2

I



2

2013.9.12 efb22

2009.9.4 fb19


3
H

P
D
=
5.49
%

E
B
=
8.326 MeV

5.86
%

8.091 MeV

50 channels

(
I


6
)
with

np

force

PRC66 (2002) 021001(R),
PRC77 (2008) 027001

deuteron
D
-
state


probability

(triton)

Effect of

3
-
body force



㌵〠步k?

敦e散琠潦

charge
dependence



ㄹ〠步k

almost half of


0.5


1 MeV

for the meson

exchange models

2013.9.12 efb22

Faddeev
-
Yakubovsky equations for 4
-
identical fermions

0
Total wav
(3 body ca
e function
se)
(1 )
G tP
P
 


 
0 (34)
0 (34)
0
(12) (23) (13) (23) (13) (24)
(34)
Total wav
[(1 ) ]
[(1 ) ]
with ,
,
(1 ){[1 (1 )
e fu
] (1 }
n
)
ction
RGM RGM
G tP P
G tP P
t V V G t
P P P P P P P P
P P P P
  
  
 
  
  
 
  
      
p
3

q
3


12

(
12
s
12
)
I
12



I
max
= 6

by A. Nogga, Ph.D. thesis


3

12
+
3
+
4
,
12
+
34
+



(
sum
)
max

2013.9.12 efb22

4
He

fss2

10
-
10
-
5

AV8’

10
-
10
-
5



sum
max

E

(MeV)

KE (MeV)

R
c

(fm)

E

(MeV)

KE (MeV)

R
c

(fm)

2


24.73

76.52

1.498


㈱2㐶

㠳8〷

ㄮ㘰

4


27.32

85.84

1.443


㈴2㠸

㤷9㈹

ㄮ㔱5

6


27.75

88.05

1.433


㈵2㔲

㄰〮㤰

ㄮ㐹4

8


27.92

88.60

1.430


㈵2㠹

㄰㈮㌵

ㄮ㐸4

10


27.95


88.73

1.429


㈵2㤴

㄰㈮㘵

ㄮ㐸4


Stochastic Variational Method


25.92

102.35

1.486

deuteron

fss2

AV8’

(SVM)


AV18


d
(MeV)

2.2246

2.2436

2.242

2.2246

P
d

(%)

5.49

5.78

5.77

5.76

rms (fm)

1.960

1.961

1.961

1.967

Q
d

(fm2)

0.270

0.269

0.270



0.0252

0.0252

0.0250


d

(

0
)

0.849

0.847

0.847

KE (MeV)

(17.49)

19.89

19.881

19.814

Deuteron properties

H. Kamada et. al., Phys.

Rev. C64, 044001 (2001)

Benchmark test

R
c
exp
(
4
He)=
1.457(4
)

fm

3
H binding energy



350敖

missing

in
fss2:
almost half of
0.5


1 MeV for
meson exch. models

4

digits accuracy

2013.9.12 efb22


(3
q
)
-
(3
q
) folded cut
-
off Coulomb with
R
cou
= 10 fm





“smaller” than point Coulomb

800 keV


1
S
0

charge independence breaking (CIB) of fss2


㈰〠
×
2 keV

Scatt
. length

a
s

(fm)

F
BB

pp


17.80

0.9934

nn


18.0

0.9944

np


23.76

1

Approximate treatment in the isospin basis:


H. WitalaW.
Glöckle

and H. Kamada, Phys. Rev. C43, 1619 (1991)

reduction factor

only for
1
S
0

for isospin
I
=1

pairs

Coulomb effect and charge
dependence

of the nuclear force

3
1
3
1
4
1
2 1
H :
3
2 1
2
He : Coulomb
3 3
1
1
He : Coulomb
3 3
S
nn
pp
S
pp nn
S
F
f
F
f
F F
f



 

2013.9.12 efb22

3
H
,
3
He ,
4
He(



0 1 fm
-
1

5 fm
-
1

16 fm
-
1



for fss2

0 1.2 3 16 for AV8’

10 10 10
5

points

mesh

accyuracy

n
=4
-
4
-
2

2

digits

n
=6
-
6
-
3

3 digits

n
=10
-
10
-
5

4 digits

Coul (keV)

CIB (keV)

E

(MeV)

E
exp

(MeV)

diff (MeV)

3
H





ㄸ1


8⸱㐳


㠮㐸4

〮㌴

3
He

682

208


7⸴㌶


㜮㜱7

〮㈸

4
He

810

538


2⸶.


㈸⸳.

ㄮ㘶

(
4
He calculation is by
n
=6
-
6
-
3)

2013.9.12 efb22

Comparison with other calculations

model

P
d

(%)


3
H (MeV)

3
He (MeV)

4
He (MeV)

diff (MeV)

fss2


5.490


㠮ㄴ1


㜮㐳4


㈶2㘴

ㄮ㘶

CD
-
Bonn

4.833


㠮〱0


㜮㈸2


㈶2㈶

㈮〴

A嘱V

㔮㜶7


㜮㘲


㘮㤱9


㈴2㈵

㐮〲

乩橭⁉

㔮㘷


㜮㜴7


㜮〸0


㈴2㤸

㌮㌲

乩橭⁉

㔮㘵


㜮㘵


㜮〰0


㈴2㔶

㌮㜴

乩橭㤳

㔮㜵7


㜮㘶


㜮〱0


㈴2㔳

㌮㜷

䕸E.


㠮㐸4


㜮㜱7


㈸2㈹2


0.5


1 MeV 3


4 MeV missing in MEP’s

Ours:


㈸⸰䵥⬠〮㠠䵥⡃潵汯浢⤠⬠+⸵䵥⡃䥂⤠㴠


㈶⸷䵥

††
ㄮ㜠䵥浩獳楮朠


almost half of standard MEP’s

A. Nogga, H. Kamada, W. Glöckle, and B.R. Barrett, Phys. Rev. C
6
5, 054003 (2002)

2013.9.12 efb22

(
(
3
34)
(3
4
4
)
)
(1 ){[1 (1 )
| 0 , | (1 ) 0
| 0

, | (1
] (1 ) } 0
d
) 0
an
uf uF uf P uF uG
uu uG uu P uF u
P P P u u
G
F P G

       

      
      

(34)
(34)
(34
(34)
)
| [(1 ) ] |
| [(1 ) ] |


| (1 )
| (
with
1
1
)
uf P uF uG
uu P u
uF P P uF uG uf
u
F
G P P uF uG
uG
uu



     
   
   
     
 
Faddeev redundant components

1)
redundant component for the 3
-
body subsystem


in the Y
-
type Jacobi coordinates :
(1+
P
)|
uf

=0

2)
redundant component of the core
-
exchange type in
the H
-
type Jacobi coordinates :
(1+
P
)|
uu

=0

3) genuine 4
-
body

redundant
component :







we can prove

trivial

solution









4


(
and 4
d
’)
system


2013.9.12 efb22

0 (34)
0 (34)
(34)
(34) (34)
(34)
(34) (34)
[(1 ) ] |
|
[(1 ) ]
| (1 )
(1 ) | (1 )
| (1 )
(1 )
we can prove
|
|

|

(1
a
|
)
d
0
n
uf P
P uF u
G tP P uf
uF
G tP P uu
uG
G P
uu P
P uF uG P
u
u
 
 
 
  
 



    
 
 
   
     
   
   
  
 

  


(34)
(34)
| 0 , | (1 ) 0
| 0 , | (1 ) 0
f uf P
uu uu P
  
  
      
       
4) modified Faddeev
-
Yakubovsky equation :

for identical 4
-
boson systems









2013.9.12 efb22


sum

max

E
4


(MeV)

KE
(MeV)

R


(fm)

rms
(fm)

0


㐮㈱

ㄵ1㔱

㌮㘷

㌮㤵

2


4.16

15.20

3.69

3.96

4


6.53

20.71

3.27

3.57

6


㜮㈸

㈳2㄰

㌮〷

㌮㐰

8


1ㄮ㔶

㐳4〸

㈮㜱

㌮〷




ㄵ1㠲

㘶㌹

㈮ㄹ

㈮㘲




㌹3〶

ㄴ㈮㌳

ㄮ㔷

㈮ㄳ




㌹3ㄵ

ㄴㄮ㠰

ㄮ㔷

㈮ㄳ

N
tot

max

E
4


(MeV)

c
(00)

KE

(MeV)

R


(fm)

rms
(fm)

12


㌴3ㄴ

1

ㄸ㐮㤸

ㄮ㌸

㈮〰


ㄹ1㤹

1

ㄸ㐮㤸

ㄮ㌸

㈮〰




㌷3〴

〮㤶9

ㄶ〮㌴

ㄮ㐸

㈮〷


㈳2㐷

〮㤵9

ㄵ㠮㘰

ㄮ㐹

㈮〷




㌸3㈷

〮㤳9

ㄵ〮㠷

ㄮ㔳

㈮㄰


㈴2㤰

〮㤲9

ㄴ㠮〵

ㄮ㔴

㈮11




㌸3㜶

〮㤱9

ㄴ㔮㤵

ㄮ㔵

㈮ㄲ


25.50

0.901

142.43

1.57

2.13

20


㌸3㤶

〮㤰9

ㄴ㌮㌹

ㄮ㔷

㈮ㄳ


㈵2㜷

〮㠸8

ㄳ㤮㌷

ㄮ㔹

㈮ㄵ

4


敮敲杹e慮牭牡摩畳

Faddeev
-
Yakubovsky (4
-
4
-
2)

h.o. variation
(red: with Coulomb)

Volkov No.2
m
=0.605,
b
=1.36 fm

E
2

=

ㄮ㄰㔠
⠰⸲㔲⤠
MeV

E
3

=

㜮㌹ㄠ
(

㈮㌰㜩
景

N
tot
=60


largely overbound


change suddenly at ℓ
sum
max
=12 owing to
[(40)(40)](04)(40):
(00) channel


b

large,

rms radius


污牧攬e

扵琠獴楬氠
v敲
-
扩湤楮b

(rms)
exp
= 2.710

0⸰15晭

Cf.

S. Oryu, H. Kamada, H. Sekine, T. Nishino, and H. Sekiguchi, Nucl. Phys. A534 (1991)221

2013.9.12 efb22

Summary

We have studied the binding energy and charge rms radius of

-
灡牴楣汥
畳楮朠潵煵q牫
-
浯m敬扡特n
-
b慲祯渠楮瑥牡捴楯n

晳猲

周攠晲f浥m潲欠楳

“4
-
cluster
Faddeev
-
Yakubovsky equations using 2
-
cluster RGM kernel”
.

The results are


㈸⸰䵥
w楴桯畴䍯畬潭u
Ⱐ慮搠
牭

牡摩畳楳
ㄮ㐳


which is slightly too small.

If we incorporate the Coulomb force and the
effect of the charge
-
independence breaking of the
NN

force, we obtain

E


=

㈶⸶.
䵥



㈸⸰⬠+⸸⬠〮㔠†
vs
.
E

exp
=

㈸⸲㤶䵥

捨慲c攠牭猠牡摩畳㴠‱⸴㌹

††††††
癳⸠⁥硰.㨠ㄮ:㔷


〮〰㐠晭

The missing


ㄮ㘶䵥
晲潭數灥物浥湴n楳慬浯獴桡汦h潦瑨攠獴s湤慲搠
meson
-
exchange potentials, which is consistent with the 3
N

case.

Future problems


4

H,
4

He systems

(
under progress:
full inclusion
of

N
-

N

coupling )


4

He :

N
-

N

coupling and


-

N
-


捯異汩湧獨s畬搠扥b



simultaneously treated.

A method to eliminate the Faddeev redundant components in the 4
d
’ and

4


獹獴敭猠楳i灲潰潳敤e慮搠慰灬p敤e瑯瑨攠灲慣p楣慬捡汣畬慴楯湳⸠