Quark-model baryon-baryon interactions
and their applications
to few-body systems
Y. Fujiwara （Kyoto) Y. Suzuki （Niigata） C. Nakamoto (Suzuka)
M. Kohno（Kyushu Dental） K. Miyagawa（Okayama)
1. Introduction
2. B8 B8 interactions fss2 and FSS: spin-flavor SU6 symmetry
3. B8  interactions by quark-model G-matrix
4. Some applications
4.1. N interaction and 3H Faddeev calculation
4.2 effective  potential and 9Be Faddeev calculation
4.3.  s. p. potential and  , (3N) potentials
4.4. N total cross sections and  potential
5. Summary
2006.10.13 HYP2006 Mainz
Oka – Yazaki (1980) 
B8B8 interactions by fss2
Phys. Rev. C64 (2001) 054001
Phys. Rev. C65 (2002) 014001
A natural and accurate description of NN, YN,
YY interactions in terms of (3q)-(3q) RGM
• Short-range repulsion and LS by quarks
• Medium-attraction and long-rang tensor by S, PS and V
meson exchange potentials (fss2) (Cf. FSS without V)
Phys. Rev. C54 (1996) 2180
Model Hamiltonian
H =∑ 6i=1 (mi+pi2/2mi)
+∑ 6i<j (UijConf+UijFB+∑βUijSβ
+∑βUijPSβ + ∑βUijVβ)
f(3q)f(3q)|E-H|A {f(3q)f(3q)c(r)}=0
QMPACK homepage
PPNP in press
http://qmpack.homelinux.com/~qmpack/index.php
2006.10.13 HYP2006 Mainz
Arndt : SAID
Nijmegen : NN-OnLine
Lippmann-Schwinger (LS) RGM
P.T.P. 103
(2000) 755
Solve [ - H0 - VRGM() ] c=0 with VRGM()=VD+G+ K
in the mom. representation ( = E - Eint )
Born kernel qf|VRGM() |qi  T-matrix, G-matrix
1) non-local
2) energy-dependent
3) Pauli-forbidden states in N - N (I=1/2),  - N
-  (I=0),  -  (I=1/2) 1S0 : i.e. SU3 (11)s : Ku=u
3-cluster Faddeev formalism using VRGM()
   G 0 ( E )T  ( E  h ,   )(     )
 u | P     u |           0
self-consistency
equation for 
    P  | h  V
RGM
P.T.P. 107
(2002) 745;
993
(  ) | P  
B8 interaction by quark-model G-matrix
 : “(0s)4”
G (p, p’; K, , kF)
k’=p’- p , q’=(p+p’)/2
G (k’, q’; q1, q’)
k=k’
 - cluster folding
q1=q for direct and knock-on
V (k, q)
B8 relative q’
incident q1
=0.257 fm-2
in total c. m.
kF=1.35 fm-1
VW (R, q) : Wigner transform
k=pf - pi , q=(pf+pi)/2
V (pf , pi)
U(R)=VW(R, (h2/2)(E-U(R))
Lippmann - Schwinger equation
exact EB ,  (E)
2006.10.13 HYP2006 Mainz
Transcendental equation
Schrödinger equation
EBW , W(E)
“constant K , , kF”
n RGM by G-matrix of fss2
n sactt. phase shift
S1/2
P3/2
P1/2
2006.10.13 HYP2006 Mainz
exp
q1=0
q’=3/5 kF
kF=1.35 fm-1
B8B8 systems classified in the SU3 states with (l, )
S
B8B8(I)
1E, 3O (P =symmetric)
3E, 1O (P =antisymmetric)
0
NN(0)
NN(1)
―
(22)
(03)
―
‐1
N
N(1/2)
N(3/2)
‐2

N(0)
N(1)

(0)
(1)
(2)
‐3

(1/2)
(3/2)
‐4
(0)
(1)
(11)s
1
10
1
10
9
1
1
[(11)s+3(22)]
[3(11)s‐(22)]
(22)
2
1
2
1
(11)s+ 2 30(22)+ 2 2 (00)
5
3
1
1
(11)
‐
(22)+
(00)
s
10
2
5
3
(11)s+ 52 (22)
5
2
3
ｰ 5 (11)s+ 5
(22)
1
3
3
(11)
－
(22)－
(00)
2
10
8
s
5
―
(22)
1
10
1
10
2
1
2
―
(22)
complete Pauli forbidden
―
(11)a
1
[‐(11)a+ (30)+(03)]
3
1
[(30)‐(03)]
2
―
1
[2(11)a+ (30)+(03)]
6
―
1
[(11)s+3(22)]
[3(11)s‐（22）]
(22)
[‐(11)a+(03)]
[(11)a+(03)]
(30)
[‐(11)a+(30)]
[(11)a+(30)]
(03)
(30)
―
(30)
almost forbidden (=2/9)
Spin-flavor SU6 symmetry
1. Quark-model Hamiltonian is approximately SU3 scalar
・ no confinement contribution (assumption)
・ Fermi-Breit int. … quark-mass dependence only
・ EMEP … automatic SU3 relations for coupling constants
phenomenology Cf. OBEP: exp data  g, f,  … (integrated)
2. -on plays an important role through SU3 relations and FSB
3. effect of the flavor symm. breaking (FSB) by ms>mud , B, M masses
Characteristics of SU3 channels
1S, 3P (P-symmetric)
(22) attractive
pp
3S, 1P (P-antisymmetric)
(03) strongly attractive
np
(11)s strongly repulsive N(I=1/2) (30) strongly repulsive N(I=3/2)
(00) strongly attractive
H-particle channel
(11)a weakly attractive N(I=0)
“only this part is ambiguous”
1S
0
phase shifts for B8B8 interactions
with the pure (22) state (fss2)
S=‐2
S=0
1S
0
S=‐3
S=‐1
S=‐4
(22)
3S
1
phase
shifts
NN
3S
 (3/2)
fss2
1
(03)
NN (03) central only
(no  tensor)
(11)a
N (0) (11)a : weakly
 (0)
attractive
(30)
N (3/2)
(30) : Pauli
repulsion
+p differential cross sections
and +p, p asymmetries a()
Ahn et al. (KEK-PS E251, E289)
NP A648(1999)263, A761(2005)41
350 MeV/c  plab  750 MeV/c
Kurosawa et al. (KEK-PS E452B)
KEK preprint 2005-104 (2006)
 +p
+p elastic
aexp=0.44±0.2
at p=800±200 MeV/c
Kadowaki et al. (KEK-PS E452)
Euro. Phys. J. A15 (2002) 295
p elastic
reported by
K. Nakai
N interaction by fss2
Backward/Forward ratio
P-wave
N is weakly
attractive
fss2

FSS
N - N coupling : 3S1 + 3D1 by one- tensor
1P + 3P by FB LS (－)
1
1
2006.10.13 HYP2006 Mainz
from 3He
Faddeev
3
H
NN-NN CC Faddeev
(hypertriton)
Phys. Rev. C70, 024001 (2004)
N on-shell properties are directly reflected
1S / 3S relative
0
1
“ｄｅｕｔｅｒｏｎ” Λ(∑0 )
u
strength
s
d
exp’t
u u
d = 2.22 MeV
~5 fm
d
B=130 ±50 keV
p
close to NSC89 P (%)
~2 fm u
n
d
d
fss2 289 keV 0.80
FSS 878 keV 1.36
NN = 19.37 – 21.03 = －1.66
|ｄ |= 17.50 – 19.72 = －2.22 (MeV)
2006.10.13 HYP2006 Mainz
150 channel calculation
N １Ｓ0 ａｎｄ
model
３Ｓ
１
effective range parameters
FSS
as (fm) rs (fm)
－5.41 2.26
aｔ (fm) rt (fm)
－1.03 4.20
fss2
－2.59
2.83
－1.60
NSC89 －2.59
2.90
“fss2” －2.15
3.05
878
1.36
3.01
289
0.80
－1.38
3.17
143
0.5
－1.80
2.87
145
0.53
“fss2”: m c2 = 936 MeV  1,000 MeV
B (keV)
fss2
“fss2”
6 ch (S)
15 ch (SD)
102 ch (J4)
150 ch (J6)
137
198
288
289
44
85
145
145
2006.10.13 HYP2006 Mainz
B(keV) P(%)
favorable for 4H (1+)
Effect of the higher partial
waves is large
90 – 60 keV
vs. 20 – 30 keV in NSC89
BΛexp=130 ±50 keV
 effective local potentials
by G-matrix B8B8 interaction
effective potentials
quark-model N-N
ND
EB (exact)
－3.62 MeV
－3.18 MeV
EBexp=3.120.02
MeV
Cf. U(0)＝‐46 (FSS), ‐48 (fss2) MeV
in symmetric matter
(3.04 MeV) 2+
2 Faddeev for 9Be
Phys. Rev. C70, 024002, 0407002 (2004)
(0) 92 keV 0+
8Be
 + +
 RGM kernel (MN3R)
effective  pot. (SB u=0.98)
exp’t
-3.120.02 MeV
3067(3) keV 3/2+ 3026 keV
+5He
3024(3) keV 5/2+ 2828 keV
-6.620.04 MeV +
1/2
9

Be
calc.
l s splitting by N LS Born kernel
198 keV (fss2 quark+), 137 keV (FSS) : 3  5 times too large
Tamura et al. (BNL E930)
Eexp(3/2+ - 5/2+) = 43  5 keV Akikawa,
Phys. Rev. Let. 88, 082501 (2002)
2006.10.13 HYP2006 Mainz
ls splitting of 9Be by 2 Faddeev using
quark-model G-matrix  LS Born kernel

0.5 0
kF (fm-1)
1.07
‐10.5
-1.9
188
7
G-matrix fss2 (cont)
S (MeV fm5) FSS (cont)
Faddeev
E (keV)
fss2 (cont)
FSS (cont)
E exp (keV)
0.70
0
N Born
1.20
1.35
‐10.6 ‐10.7
-2.9
-3.6
194
198
34
59
43  5
-10.9
-7.8
198
137
FSS (cont) reproduces E exp at kF=1.25 fm-1 !
P-wave N-N coupling by LS(-) is important.
S-meson LS in fss2 is not favorable.
2006.10.13 HYP2006 Mainz
 potentials (VWC (R, 0)) by quark-model
G-matrix interaction
FSS
fss2
I=3/2
I=3/2
total
total
I=1/2
I=1/2
The Pauli repulsion of N(I=3/2) 3S1 is very strong.
2006.10.13 HYP2006 Mainz
 (3N) potentials by quark-model G-matrix
interaction ( 0+, T=1/2 channel)
3/2
V S ( 3 N ) ( S = 0, T = 1 / 2) = (4 / 3)V s
FSS
EB(exact)＝－3.79 MeV
+ (3 / 2)V t
1/ 2
+ (1 / 6)V s
fss2
EB(exact)＝－5.70 MeV
consistent with 4He (0+) resonance
2006.10.13 HYP2006 Mainz
1/ 2
(3N): (0s)3
=0.22 fm-2
q1=0
（－, K+) inclusive spectra on 28Si
exp: Noumi et al. PRL 89, 072301 (2002) ; 90, 049902 (E) (2003)
Saha et al. Phys. Rev. C70, 044613 (2004)
poster session
by M. Kohno
Repulsive U (q) in symmetric nuclear matter is experimentally confirmed.
 potentials (VWC (R, 0)) by quark-model
G-matrix interaction
fss2
FSS
I=1
total
total
I=0
I=0
Some attraction in the surface region.
I=1
+3.7
Tamagawa et al. (BNL-E906)
-N (in medium) = 30.7±6.7 -3.6 mb Nucl. Phys. A691 (2001) 234c
et al.
(eikonal approx.)= 20.9±4.5 +2.5 mb Yamamoto
Prog. Theor. Phys. 106 (2001)363
-2.4
-p /-n =1.1 +1.4+0.7
-0.7 -0.4 at plab=550 MeV/c

FSS
fss2

Ahn et al.
Phys. Lett. B
633 (2006) 214

More experiments are needed.

Ｓｕｍｍａｒｙ
Quark-model description for the baryon-baryon
interaction is very successful to reproduce many
experimental data. In particular, the extension of
the (3q)-(3q) RGM study for the NN and YN
interactions to the strangeness S=-2, -3, -4 sectors
has clarified characteristic features of the B8B8
interactions. The results seem to be reasonable if
we consider
1) spin-flavor SU6 symmetry
2) weak π-on effect in the strangeness sector
3) effect of the flavor symmetry breaking
We have analyzed B8, B8(3N) interactions based on
the G-matrix calculations of fss2 and FSS.
2006.10.13 HYP2006 Mainz
Characteristics of fss2 and FSS
S=0 ・ triton binding energy … fss2: +150 keV (3 body force?)
S＝‐1 p and +p interactions are progressively known.
・ + p total and differential cross sections and
polarization … fss2, FSS
・ N 1S0 and 3S1 attraction (relative strength)
( 3H Faddeev calculation: 289 keV for fss2)
・ small ls splitting in 9Be excited states (FSS)
・ N (I=1/2 1S0), N (I=3/2 3S1) repulsion
 repulsive s. p. and  potentials … fss2, FSS
S＝‐2  interaction is not much attractive !
・  interaction |V|<|VN|<|VNN|
B  1 MeV (Nagara event 6He) … fss2
・ N in-medium total cross section (fss2, FSS)
… strong isospin dependence of  s.p. potential
・ N (I=0 3S1): (11)a  0 or weakly attractive (fss2, FSS)
vs. ESC04(d): strongly attractive