BLED WORKSHOPS
IN PHYSICS
VOL. 1, NO. 1
Proceedins of the Mini-Workshop
Few-Quark Problems (p. 29)
Bled, Slovenia, July 8-15, 2000
A Realistic Description of Nucleon-Nucleon and
Hyperon-Nucleon Interactions in the SU6 Quark
Model?
Yoshikazu Fujiwara1??, M. Kohno2, C. Nakamoto3 and Y. Suzuki41Department of Physics, Kyoto University, Kyoto 606-8502, Japan2Physics Division, Kyushu Dental College, Kitakyushu 803-8580, Japan3Suzuka National College of Technology, Suzuka 510-0294, Japan4Department of Physics, Niigata University, Niigata 950-2181, Japan
Abstract. The SU6 quarkmodel for theNN and YN interactions, developed by the Kyoto-
Niigata group, is upgraded to incorporate some effects of scalar and vector mesons ex-
changed between quarks. The phase-shift agreement in theNN sector at the non-relativistic
energies up to Tlab = 350 MeV is greatly improved. The essential feature of the N-N
coupling is qualitatively similar to that obtained from the previous models. The G-matrix
calculation of the N-N coupled-channel system shows that the  single-particle poten-
tial is repulsive in ordinary nuclear matter. The single-particle spin-orbit strength for the particle is found to be very small, in comparison with that of the nucleons.
The quark-model study of the nucleon-nucleon (NN) and hyperon-nucleon (YN)
interactions is motivated to gain a natural and accurate understanding of the fun-
damental strong interaction, in which the quark-gluon degree of freedom is be-
lieved to be the most economical ingredient to describe the short-range part of
the interaction, while the medium- and long-range parts are dominated by the
meson-exchange processes. We have recently achieved a simultaneous and real-
istic description of the NN and YN interactions in the resonating-group (RGM)
formalism of the spin-flavor SU6 quark model. [1–3] In this approach the effec-
tive quark-quark interaction is built by combining a phenomenological quark-
confining potential and the colored version of the Fermi-Breit (FB) interaction
withminimum effectivemeson-exchange potentials (EMEP) of scalar and pseudo-
scalar meson nonets directly coupled to quarks. The flavor symmetry breaking
for the YN system is explicitly introduced through the quark-mass dependence
of the Hamiltonian. An advantage of introducing the EMEP at the quark level
lies in the stringent relationship of the flavor dependence appearing in the vari-
ousNN and YN interaction pieces. In this way we can utilize our rich knowledge
of the NN interaction to minimize the ambiguity of model parameters, which is
crucial since the present experimental data for the YN interaction are still very
scarce.? Talk delivered by Yoshikazu Fujiwara?? E-mail: fujiwara@ruby.scphys.kyoto-u.ac.jp
30 Y. Fujiwara
In this report we upgrade our model to incorporate vector mesons and the
momentum-dependent Bryan-Scott terms, and compare the NN and YN observ-
ables with the existing experimental data. This model is dubbed fss2 since it is
based on our previous model FSS [2,3] The agreement to the phase-shift parame-
ters in the NN sector is greatly improved. The model fss2 shares the good repro-
duction of the YN scattering data and the essential features of the N-N cou-
pling with our previous models [1–3]. The single-particle (s.p.) potentials ofN, 
and  are predicted through the G-matrix calculation, which employs the quark-
exchange kernel explicitly. [4] The strength of the s.p. spin-orbit potential is also
examined by using these G-matrices. [5] These applications of fss2 are discussed
by Kohno in the present symposium.
A new version of our quark model is generated from the Hamiltonian which
consists of some extra pieces of interaction generated from the scalar (S), pseu-
doscalar (PS) and vector (V) meson-exchange potentials acting between quarks:H = 6Xi=1mi
2 + p2i2mi + 6Xi<j 0UCfij +UFBij +X USij +X UPSij +X UVij 1A :
It is important to include the momentum-dependent Bryan-Scott term in the S-
and V-meson contributions, in order to ensure the correct asymptotic behavior of
the s.p. potentials in high-momentum region. [6] We have calculated UN(q1) by
using the so-called teff prescription and made sure that it turns into repulsive
beyond the incident momentum q1  6 fm-1, while our old model FSS is too at-
tractive, having the minimum value  -70 MeV around q1  10 fm-1. Another
important feature of fss2 is the introduction of vector mesons for improving the
fit to the phase-shift parameters in theNN sector. Since the dominant effect of the!-meson repulsion and the LS components of ,! and K mesons are already ac-
counted for by the FB interaction, only the quadratic LS (QLS) component of the
octet mesons is expected to play an important role to cancel partially the strong
one-pion tensor force. Further details of the model fss2 is given in [7]. The model
parameters are determined by fitting the most recent result of the phase shift anal-
ysis, SP99 [8], for the np scattering with the partial waves J  2 and the incident
energies Tlab  350MeV, under the constraint of the deuteron binding energy and
the 1S0 NN scattering length, as well as the low-energy YN total cross sections.
The deuteron D-state probability of fss2 is 5.5 %, which is slightly smaller than
5.88 % in FSS [2].
Figure 1 shows some important low-partial wave NN phase-shift param-
eters, compared with the experiment SP99. The previous result by FSS is also
shown with the dotted curves. The 3D2 phase shift is greatly improved by theQLS component. The good accuracy of the NN phase-shift parameters continues
up to Tlab  600MeV, where the inelasticity of the S-matrix becomes appreciable.
The total cross sections of the NN and YN scattering predicted by fss2 are
comparedwith the available experimental data in Fig. 2. The “total” cross sections
for the scattering of charged particles (i.e., pp, +p and -p systems) are calcu-
lated by integrating the differential cross sections over cosmin = 0:5  cosmax =-0:5. The solid curves indicate the result in the particle basis, while the dashed
A Realistic Description of Nucleon-Nucleon... 31
− 20
0
20
40
60
80
0 100 200 300
δ 
(d
eg
)
Tlab (MeV)
np
3S1
3D1
ε 1
(a)
fss2
FSS
SP99
− 20
0
20
40
60
80
0 100 200 300
δ 
(d
eg
)
Tlab (MeV)
np
1S0
3D2
1D2
(b)
fss2
FSS
SP99
− 40
− 20
0
20
40
0 100 200 300
δ 
(d
eg
)
Tlab (MeV)
np
3P0
3P2
3P1(c)
fss2
FSS
SP99
− 40
− 30
− 20
− 10
0
0 100 200 300
δ 
(d
eg
)
Tlab (MeV)
np
1P1
(d)
fss2
FSS
SP99
Fig. 1. Comparison of np phase shifts with the phase-shift analysis SP99 by Arndt et al.
The dotted curves are by FSS.
curves in the isospin basis. In the latter case, the effects of the charge symmetry
breaking, such as the Coulomb effect and the small difference of the threshold en-
ergies for -p and 0n channels, are neglected. The empirical total cross sections
for the np and pp systems include the inelastic cross sections. This is the reason
why our result with no inelasticity underestimate these total cross sections above
the pion threshold. If we properly compare the total elastic cross sections with
an appropriate angular range, we find that the agreement with the experiment is
excellently good up to Tlab = 800 MeV. We find that the cusp structure of the p
total cross sections at the N threshold is enhanced by the effect of the P-waveN-N coupling due to the antisymmetric spin-orbit force (LS(-) force), which
is a new feature of the YN interaction as the scattering of non-identical particles.
32 Y. Fujiwara
10
100
1000
0 200 400 600 800
σ t
ot
 (
m
b)
Tlab (MeV)
np
fss2
10
100
1000
0 200 400 600 800
σ t
ot
 (
m
b)
Tlab (MeV)
pp
fss2
isospin
0
50
100
150
0 200 400 600 800 1000
σ t
ot
 (
m
b)
pΣ  (MeV/c)
Σ +p elastic
fss2
isospin
0
50
100
150
0 200 400 600 800 1000
σ t
ot
 (
m
b)
pΣ  (MeV/c)
Σ − p elastic
fss2
isospin
0
50
100
150
0 200 400 600 800 1000
σ t
ot
 (
m
b)
pΛ  (MeV/c)
Λ  p elastic
fss2
isospin
0
50
100
150
0 200 400 600 800 1000
σ t
ot
 (
m
b)
pΣ  (MeV/c)
Σ −  p →  Σ 0 n
fss2
isospin
0
5
10
15
0 200 400 600 800 1000
σ t
ot
 (
m
b)
pΛ  (MeV/c)
Λ  p →  Σ 0 p
fss2
isospin
0
50
100
150
0 200 400 600 800 1000
σ t
ot
 (
m
b)
pΣ  (MeV/c)
Σ −  p →  Λ  n
fss2
isospin
Fig. 2. CalculatedNN and YN total cross sections compared with the available experimen-
tal data.
A Realistic Description of Nucleon-Nucleon... 33
In the present model fss2, the N(I = 1=2) 3P1 resonance still stays at theN channel, which is similar to the situation in RGM-H [2]. As to the N-N
coupling in the positive-parity states, the present 3S1-3D1 tensor coupling by the
one-pion tensor force is rather close to that in RGM-F [1]. The -p inelastic cap-
ture ratio at rest is rR = 0:442 in fss2, which is slightly smaller than the recent
empirical values rRexp = 0:474  0:016 and 0:465  0:011 [9,10].
In summary, we have upgraded our quark model FSS to fss2, by incorporat-
ing the momentum-dependent Bryan-Scott term and the vector-meson exchange
potential acting between quarks. With a few phenomenological ingredients, the
accuracy of the model in the NN sector has now become almost comparable to
that of the OBEP models. The existing data for the YN scattering are well repro-
duced and the essential feature of theN-N coupling is almost unchanged from
our previous models. [1,2] We also calculatedNN and YNG-matrices in ordinary
nuclear matter, by solving the Bethe-Goldstone equations for the quark-exchange
kernel. Similar to our previous models, fss2 predicts a repulsive  s.p. potential,
the strength of which is about 10 MeV. The repulsive isoscalar part of the  s.p.
potential is reported to be compatible with recent BNL (K-; ) data [11]. The
relative ratio of the Scheerbaum factors for  toN is about 1/5 in fss2, due to the
strong effect of LS(-) force and the effect of the flavor symmetry breaking. It can
be further reduced by many-body effects such as the starting-energy dependence
and the density dependence of the G-matrix. This small spin-orbit potential for
the might be confirmed in future experiments.
References
1. Y. Fujiwara, C. Nakamoto and Y. Suzuki, Prog. Theor. Phys. 94 (1995) 215, 353.
2. Y. Fujiwara, C. Nakamoto and Y. Suzuki, Phys. Rev. Lett. 76 (1996) 2242; Phys. Rev.
C54 (1996) 2180.
3. T. Fujita, Y. Fujiwara, C. Nakamoto and Y. Suzuki, Prog. Theor. Phys. 100 (1998), 931.
4. M. Kohno, Y. Fujiwara, T. Fujita, C. Nakamoto and Y. Suzuki, Nucl. Phys.A674 (2000),
229.
5. Y. Fujiwara, M. Kohno, T. Fujita, C. Nakamoto and Y. Suzuki, Nucl. Phys.A674 (2000),
493.
6. Y. Fujiwara, M. Kohno, C. Nakamoto and Y. Suzuki, Theor. Phys. 103 (2000), 755.
7. Y. Fujiwara, M. Kohno, C. Nakamoto and Y. Suzuki, The electric proceedings of this
workshop. http://www-f1.ijs.si/Bled2000.
8. Scattering Analysis Interactive Dial-up (SAID), Virginia Polytechnic Institute, Blacks-
burg, Virginia R. A. Arndt: Private Communication.
9. V. Hepp and H. Schleich, Z. Phys., 214 (1968), 71.
10. D. Stephen, Ph.D. thesis, Univ. of Massachusetts, 1970 (unpublished)
11. J. Dabrowski: Phys. Rev. C60 (1999), 025205.