- Email: [email protected]
Quark model approach to the charge dependent NN interaction
NUCLEAR PHYSICS ELSEVIER
A
Nuclear PhysicsA631 (1998)462c-466c
Quark Model Approach to the Charge Dependent N-N Interaction Fan Wang ~, Pu Xu b, Jia-Lun Ping c, Di Qing a aCenter for Theoretical physics, Nanjing University, Nanjing 210093 bPhysics Department, Nanjing University of Science and Technology, Nanjing, 210094 CPhysics Department, Nanjing Normal University, Nanjing 210097
The charge symmetry breaking and charge independence breaking N-N 1S0 scattering length difference A a c s s and AacIB are calculated by a resonating group method with a quark cluster model.By adding the QED-QCD interference effect to the quark mass difference and the electromagnetic interaction,the AaCSB and AacxB can be reproduced with model parameters constrained by the hadron isomultiplet mass splitting.
There are mainly two approaches to the hadron interactions. One is the meson exchange model, the other is the quark cluster model. The meson exchange model[l] fits the hadron interactions quantitatively with the aid of many phenomenological meson-baryon coupling constants and form factors (or short range phenomenologies). The quark cluster model[2] gives rise to a deeper explanation of the short range repulsion but most of them can not get the N-N intermediate range attraction and invoke the effective meson exchange again. Wang et al[3] develop a model taking into account the quark delocalization and color screening, which is able to yield the N-N short range repulsion and the intermediate range attraction simultaneously. It also gives an explanation why nuclear force and molecular force are similar. For the charge dependent N-N interaction, the meson exchange model attributes the charge independence breaking mainly to the 7r mass difference and the charge symmetry breaking mainly to the p - w mixing[4]. Even though the primary origin of the charge dependence of the hadron interaction is the quark mass difference and the electromagnetic interaction, but these two effects play role indirectly in the meson exchange model. The electromagnetic interaction affects the 7vmasses and the quark mass difference induces the p - w mixing. However, as pointed out by Yang and Hwang[5], that the meson exchange model fit of the A a c s s and Aacls is controversial. Another question remains is that the quark mass difference is not an observable one, because the quarks are permanently confined. It is mainly determined through the hadron isomultiplet masses. Such determined quark mass difference is in turn dependent on the assumption of the primary origin of the charge dependence. Goldman, Maltman and Stephenson (GMS)[6] proposed that 0375-9474/98/$19.00 © 1998 ElsevierScienceB.V. All rights reserved. PII S0375-9474(98)00048-7
463c
E Wang et al./Nuclear Physics A631 (1998) 462c-466c
the QED-QCD interference induces a new charge dependent q-q interaction. If this effect is taking into account in the hadron isomultiplet mass fitting, GMS can fit the existed hadron isomultiplet masses (with large experimental errors) with a quark mass difference -4.96 MeV _< A m ( = m d - - m u ) ~_ 6.28 MeV and their most favorable parameter set having a Am=0.66 MeV which is quite different from the A m ~ 3MeV recommended in Ref. [4]. If the QED-QCD interference effect is really as important as proposed by GMS, one has to add this QED-QCD interference effect to the quark mass difference and electromagnetic interaction to reanalyze the charge symmetry breaking in nuclear and particle physics[4]. There are also few direct quark model approaches to the charge dependent N-N interaction[7,8]. It seems impossible to fit the charge symmetry and charge independence breaking simultaneously in this approach, because the quark mass difference does not affect the n-p scattering and the electromagnetic effect is not large enough[7] to reproduce the charge independence breaking. The QED-QCD interference effect contributes both to the charge symmetry and charge independence breaking. It is interesting to check if one can deduce the charge symmetry and charge independence breaking appeared in the N-N interaction simultaneously in a quark model approach by taking three sources of charge dependence the quark mass difference, the electromagnetic interaction, the QED-QCD interference-into account together. Motivated by the above mentioned reasons we did a quark cluster model calculation 1 of the N-N 1S0 scattering length differences / k a c s B -..= app - ann and A a c i B ~-- 5(app ÷ a n n ) - - anp~ where app(nn,np) is the pp(nn, np) 1S0 scattering length. The u,d quark mass difference A m = rn d - - mu, the q-q electromagnetic interaction Vq~ and the QCD-QED interference induced q-q interaction Vqlq are all taking into account in this calculation. A resonating group method based on the quark cluster model is used to do the N-N scattering length calculation[7]. The Hamiltonian of the N-N system is assumed to be H ( 1 . . . 6) = ~ ( m i + ~m~) -- T~ + ~ ( V i C + Vi~ + Vi~j + ViIj) p?
= E(m + ~ ) - T~ + E(V~[ + ~ ) + 2(Smi + 5K~) + ~ ( 5 ~
+ ~ +
~), (~)
where mi is the quark mass, Tc is the total center of mass kinetic energy, Vzj with superscript C, G, ~, and I represents the confinement, gluon exchange,photon exchange and the QED-QCD interference q-q interaction respectively, m is the average u,d quark mass ?Tt z
ra~+mg 2
(~Trti z
SKi
Am
---- m d - m u
ll~i - - ~
--__ - - ~
( 1
1~ 2
= X~mi--~m)Pi
(2)
Tiz '
P~Am. r
=- 2 m - ~ m
~z,
(3) (4)
v~J + ~iJ = ( ~ :~ . As + ~Q~Qj)(~ - ~ ( ~ + ~ + ~,~,',~jjo~,j~ + . . . ) ,
(5)
E Wang et al. /Nuclear Physics A631 (1998) 462c-466c
464c
=
-
=
+
+
(6)
~Ai " Aj(AQiQjf~l(rij) + BQiQ~£ . Sjfm~g(rij) + C(Q~ + Q~)f~(ri3) + D(Q~ +Q~)£.Sjfm~9(rij) ), 1
fob(r) =
r
~
(7) 87r
fmo
(r)-- < 3~--~(~>~o~
here c~s((~) is the quark-gluon (electromagnetic) coupling constant, ac is the confinement strength, Qi, Si, T~z and A~ are the quark charge, spin, third component of isospin and color SUn generator. A,B,C and D are the parameters of the QED-QCD interference induced quark interactions. In principle, we can directly use the Hamiltonian (1) and the resonating group method to get both the charge independent and the charge dependent N-N interactions[8]. In fact the quark model approach has not been able to get a charge independent N-N interaction as good as the phenomenological or the meson exchange model ones[2,3]. To focus this study on the charge dependent N-N interaction, we assume the charge independent part can be well described by the Reid soft core phenomenological potential[9]. The resonating group equation for the N-N scattering is reduced to[7] [V ~ + k 2 - ~(VNN + V~m)]F(r) = ~ f K(r, r')F(r')dr', K ( r , r ' ) = --9([¢N~(1,2,3)¢N~(4,5,6)]ST [ ~ 0 ~
[ P34[¢N~(1,2,3)¢N~(4,5,6)]ST),
(8)
where the VNN is the Reid soft core potential, V~m is the direct electromagnetic interaction between nucleons, p is the reduced mass p - M n +4M p , CN is the quark model nucleon wave function (color singlet, SU~ ~ symmetric, Gaussian orbital with a size parameter b), Pa4 is an quark exchange operator, [ ]ST means coupling the nucleon spin-isospin to channel spin S and isospin T. O~ including the 6mi, SKi, 5Vi~, Vi~ and Vi} five charge dependent terms. The first four terms had been studied in [7]. Here we concentrate on the last QED-QCD interference term Vi~. We take two approaches to study the effect of V~. One is to follow GMS[6], i.e., assume that the QED-QCD interference induces a q-q interaction as shown in Eq.(7), and fix all the model parameters (the average u,d quark mass m, the u,d quark mass difference Am, quark gluon coupling constant as, the nucleon size b and the A,B,C,D) from an overall fit to the baryon isomultiplet mass splitting. We call this a phenomenological model (MP). Due to the big error involved in the experimental hadron masses used in the overall fitting, especially those of E* and Ec, GMS tested 135 sets of parameters. The results listed in table I labeled by MP correspond to their most favorable set. Other parameter sets have been tested too except Am < 0 cases, because those are unacceptable from a general view point on the quark mass difference [4]. In table I, only two additional results labeled as MP I and MP II are listed to show the sensitivity of the N-N scattering lengths
E Wang et al./Nuclear Physics A631 (1998) 462c-466c Table 1. Calculated AaCSB~aCIB and the model parameters. m(MeV) A m(MeV) as b(fm) A(MeV) exp. MP 330 0.6609 1.624 0.617 -1.666 MPI 330 3.7724 1.624 0.617 -1.278 MPII 330 5.8927 1.624 0.617 -1.104 MQI 330 4.37 1.624 0.617 1.182 MQII 336 1.91 1.624 0.80 0.911 MQP 330 0.6609 1.624 0.617 2.804
B(MeV) 5.894 5.543 4.844 -4.592 -2.032 -2.444
C(MeV) 5.340 1.551 -0.953 0 0 2.824
465c
D(MeV)
AacsB
AaCIB
-6.258 0.296 4.861 -0.301 -0.133 -1.040
~ 1.5 2.06 2.21 3.45 3.36 2.16 1.50
~ 5.7 -1.52 1.12 1.02 4.26 7.33 5.73
to the variation of the phenomenological parameters. Another approach is to take the A,B,C and D parameters from a perturbative QED-QCD calculation[10], where the vertex electromagnetic penguin, photon and gluon box and crossing diagrams are included. The calculated interference interaction is
v/
~-
m2 4 2 2 ~ ~(Qi + Qj) ~ ~(r,j) +...}.
('di(~jOi • OjO(?'ij ) -
(9)
There are many other terms in the perturbative QED-QCD interference calculated result. Only three terms as shown in Eq.(9), which have the same form as assumed by GMS[6], have been included in our scattering calculation. We call this a QCD model (MQ). The parameters other than those appeared in Eq.(9) are determined by re-fitting the n-p mass difference with two choices of the nucleon size b but fixed as. They are labeled as MQ I and MQ II in table I. For the GMS phenomenological models, the calculated AacsB(,,~2-3 fm) is a little too large in comparison with the value 1.5 fm recommended in Ref. [4]. The AactB is much too small (,-~1 fm in comparison with 5.7 fm recommended in Ref. [4]) and for the GMS most favorable parameter set it is -1.5 fm which is in the wrong direction. The QCD model is plausible. The MQ II gives rise to AacsB=2.16 fm, AacIB=7.33 fm, these are not too far from the recommended value 1.5 and 5.7 fm respectively. To show the plausibility further, in table I we show another result labeled as MQP, where the parameters are fixed by an overall fit with both the n - p, A ++ - A ° mass differences and the scattering length differences. Our conclusion is that taking into account the quark mass difference, the electromagnetic interaction and the QED-QCD interference effect together, it is possible to deduce the charge dependent effect appeared in the N-N interaction based on a quark cluster model. The GMS phenomenological model with their present model parameters can not explain the charge asymmetry appeared in the N-N 1S0 scattering lengths. The perturbative QCD model appears plausible, but quantitatively need further work, especially to check if the perturbative QCD-QED calculation can derive an interference interaction to fit the isomultiplet mass splittings and the N-N scattering length differences directly. This work is supported by the NSF(19675018), SEDC and SSTC of China.
466c
E Wang et al./Nuclear Physics A631 (1998) 462c-466c
REFERENCES 1. M.M. Nagels, T.A. Rijken and J.J. de Swart, Phys. Rev., D12 (1975) 744; M. Lacombe, B. Loiseau, J.M. Richard and R. Vink Man, Phys. Rev., D12 (1975) 1495; R. Machleidt, K. Holinde and C. Elster, Phys. Rep., 149 (1987) 1. 2. C.W. Wong, Phys. Rep., 136 (1986) 1; F. Myhrer and J. Wroldsen, Rev. Mod. Phys., 60 (1988) 629; K. Shimizu, Rep. Prog. Phys., 52 (1989) 1. . F. Wang, G.H. Wu, L.J. Teng and T. Goldman, Phys. Rev. Lett., 69 (1992) 2901; G.H. Wu, L.J. Teng, J.L. Ping, F. Wang and T. Goldman, Phys. Rev. C53 (1995) 1161; Mod. Phys. Lett. A10 (1995) 1895. G.A. Miller, B.M.K. Nefkens and I. Slaus, Phys. Rep., 194 (1990) 1.
4.
5. S.N. Yang, this proceedings; Pauchy W-Y Hwang, this proceedings. 6. T. Goldman, K. Maltman and G.J. Stephenson Jr., Phys. Lett., B228 (1989) 396; G.J. Stephenson Jr., K. Maltman and T. Goldman, Phys. Rev., D43 (1991) 860; and private communication. ,
Ming-Zong Wang, Fan Wang and Chun-Wa Wong, Nucl. Phys., A483 (1988) 161.
8. M. Chemtob and S.N. Yang, Nucl. Phys., A420 (1984) 461; K. Brauer, A. Faessler and E.M. Henley, Phys. Lett., B163 (1985) 46. ,
R.V. Reid, Ann. Phys., 50 (1968) 411.
10. Hong-shi Zong, Wei-ning Su, Fan Wang and Han-xin He, Chin. J. Nucl. Phys., 15 (1993) 316; Particles and Nuclei, PANIC XIII Book of Abstract, Vol. 1 (Perugia, Italy, June 1993) p.15.
A
Nuclear PhysicsA631 (1998)462c-466c
Quark Model Approach to the Charge Dependent N-N Interaction Fan Wang ~, Pu Xu b, Jia-Lun Ping c, Di Qing a aCenter for Theoretical physics, Nanjing University, Nanjing 210093 bPhysics Department, Nanjing University of Science and Technology, Nanjing, 210094 CPhysics Department, Nanjing Normal University, Nanjing 210097
The charge symmetry breaking and charge independence breaking N-N 1S0 scattering length difference A a c s s and AacIB are calculated by a resonating group method with a quark cluster model.By adding the QED-QCD interference effect to the quark mass difference and the electromagnetic interaction,the AaCSB and AacxB can be reproduced with model parameters constrained by the hadron isomultiplet mass splitting.
There are mainly two approaches to the hadron interactions. One is the meson exchange model, the other is the quark cluster model. The meson exchange model[l] fits the hadron interactions quantitatively with the aid of many phenomenological meson-baryon coupling constants and form factors (or short range phenomenologies). The quark cluster model[2] gives rise to a deeper explanation of the short range repulsion but most of them can not get the N-N intermediate range attraction and invoke the effective meson exchange again. Wang et al[3] develop a model taking into account the quark delocalization and color screening, which is able to yield the N-N short range repulsion and the intermediate range attraction simultaneously. It also gives an explanation why nuclear force and molecular force are similar. For the charge dependent N-N interaction, the meson exchange model attributes the charge independence breaking mainly to the 7r mass difference and the charge symmetry breaking mainly to the p - w mixing[4]. Even though the primary origin of the charge dependence of the hadron interaction is the quark mass difference and the electromagnetic interaction, but these two effects play role indirectly in the meson exchange model. The electromagnetic interaction affects the 7vmasses and the quark mass difference induces the p - w mixing. However, as pointed out by Yang and Hwang[5], that the meson exchange model fit of the A a c s s and Aacls is controversial. Another question remains is that the quark mass difference is not an observable one, because the quarks are permanently confined. It is mainly determined through the hadron isomultiplet masses. Such determined quark mass difference is in turn dependent on the assumption of the primary origin of the charge dependence. Goldman, Maltman and Stephenson (GMS)[6] proposed that 0375-9474/98/$19.00 © 1998 ElsevierScienceB.V. All rights reserved. PII S0375-9474(98)00048-7
463c
E Wang et al./Nuclear Physics A631 (1998) 462c-466c
the QED-QCD interference induces a new charge dependent q-q interaction. If this effect is taking into account in the hadron isomultiplet mass fitting, GMS can fit the existed hadron isomultiplet masses (with large experimental errors) with a quark mass difference -4.96 MeV _< A m ( = m d - - m u ) ~_ 6.28 MeV and their most favorable parameter set having a Am=0.66 MeV which is quite different from the A m ~ 3MeV recommended in Ref. [4]. If the QED-QCD interference effect is really as important as proposed by GMS, one has to add this QED-QCD interference effect to the quark mass difference and electromagnetic interaction to reanalyze the charge symmetry breaking in nuclear and particle physics[4]. There are also few direct quark model approaches to the charge dependent N-N interaction[7,8]. It seems impossible to fit the charge symmetry and charge independence breaking simultaneously in this approach, because the quark mass difference does not affect the n-p scattering and the electromagnetic effect is not large enough[7] to reproduce the charge independence breaking. The QED-QCD interference effect contributes both to the charge symmetry and charge independence breaking. It is interesting to check if one can deduce the charge symmetry and charge independence breaking appeared in the N-N interaction simultaneously in a quark model approach by taking three sources of charge dependence the quark mass difference, the electromagnetic interaction, the QED-QCD interference-into account together. Motivated by the above mentioned reasons we did a quark cluster model calculation 1 of the N-N 1S0 scattering length differences / k a c s B -..= app - ann and A a c i B ~-- 5(app ÷ a n n ) - - anp~ where app(nn,np) is the pp(nn, np) 1S0 scattering length. The u,d quark mass difference A m = rn d - - mu, the q-q electromagnetic interaction Vq~ and the QCD-QED interference induced q-q interaction Vqlq are all taking into account in this calculation. A resonating group method based on the quark cluster model is used to do the N-N scattering length calculation[7]. The Hamiltonian of the N-N system is assumed to be H ( 1 . . . 6) = ~ ( m i + ~m~) -- T~ + ~ ( V i C + Vi~ + Vi~j + ViIj) p?
= E(m + ~ ) - T~ + E(V~[ + ~ ) + 2(Smi + 5K~) + ~ ( 5 ~
+ ~ +
~), (~)
where mi is the quark mass, Tc is the total center of mass kinetic energy, Vzj with superscript C, G, ~, and I represents the confinement, gluon exchange,photon exchange and the QED-QCD interference q-q interaction respectively, m is the average u,d quark mass ?Tt z
ra~+mg 2
(~Trti z
SKi
Am
---- m d - m u
ll~i - - ~
--__ - - ~
( 1
1~ 2
= X~mi--~m)Pi
(2)
Tiz '
P~Am. r
=- 2 m - ~ m
~z,
(3) (4)
v~J + ~iJ = ( ~ :~ . As + ~Q~Qj)(~ - ~ ( ~ + ~ + ~,~,',~jjo~,j~ + . . . ) ,
(5)
E Wang et al. /Nuclear Physics A631 (1998) 462c-466c
464c
=
-
=
+
+
(6)
~Ai " Aj(AQiQjf~l(rij) + BQiQ~£ . Sjfm~g(rij) + C(Q~ + Q~)f~(ri3) + D(Q~ +Q~)£.Sjfm~9(rij) ), 1
fob(r) =
r
~
(7) 87r
fmo
(r)-- < 3~--~(~>~o~
here c~s((~) is the quark-gluon (electromagnetic) coupling constant, ac is the confinement strength, Qi, Si, T~z and A~ are the quark charge, spin, third component of isospin and color SUn generator. A,B,C and D are the parameters of the QED-QCD interference induced quark interactions. In principle, we can directly use the Hamiltonian (1) and the resonating group method to get both the charge independent and the charge dependent N-N interactions[8]. In fact the quark model approach has not been able to get a charge independent N-N interaction as good as the phenomenological or the meson exchange model ones[2,3]. To focus this study on the charge dependent N-N interaction, we assume the charge independent part can be well described by the Reid soft core phenomenological potential[9]. The resonating group equation for the N-N scattering is reduced to[7] [V ~ + k 2 - ~(VNN + V~m)]F(r) = ~ f K(r, r')F(r')dr', K ( r , r ' ) = --9([¢N~(1,2,3)¢N~(4,5,6)]ST [ ~ 0 ~
[ P34[¢N~(1,2,3)¢N~(4,5,6)]ST),
(8)
where the VNN is the Reid soft core potential, V~m is the direct electromagnetic interaction between nucleons, p is the reduced mass p - M n +4M p , CN is the quark model nucleon wave function (color singlet, SU~ ~ symmetric, Gaussian orbital with a size parameter b), Pa4 is an quark exchange operator, [ ]ST means coupling the nucleon spin-isospin to channel spin S and isospin T. O~ including the 6mi, SKi, 5Vi~, Vi~ and Vi} five charge dependent terms. The first four terms had been studied in [7]. Here we concentrate on the last QED-QCD interference term Vi~. We take two approaches to study the effect of V~. One is to follow GMS[6], i.e., assume that the QED-QCD interference induces a q-q interaction as shown in Eq.(7), and fix all the model parameters (the average u,d quark mass m, the u,d quark mass difference Am, quark gluon coupling constant as, the nucleon size b and the A,B,C,D) from an overall fit to the baryon isomultiplet mass splitting. We call this a phenomenological model (MP). Due to the big error involved in the experimental hadron masses used in the overall fitting, especially those of E* and Ec, GMS tested 135 sets of parameters. The results listed in table I labeled by MP correspond to their most favorable set. Other parameter sets have been tested too except Am < 0 cases, because those are unacceptable from a general view point on the quark mass difference [4]. In table I, only two additional results labeled as MP I and MP II are listed to show the sensitivity of the N-N scattering lengths
E Wang et al./Nuclear Physics A631 (1998) 462c-466c Table 1. Calculated AaCSB~aCIB and the model parameters. m(MeV) A m(MeV) as b(fm) A(MeV) exp. MP 330 0.6609 1.624 0.617 -1.666 MPI 330 3.7724 1.624 0.617 -1.278 MPII 330 5.8927 1.624 0.617 -1.104 MQI 330 4.37 1.624 0.617 1.182 MQII 336 1.91 1.624 0.80 0.911 MQP 330 0.6609 1.624 0.617 2.804
B(MeV) 5.894 5.543 4.844 -4.592 -2.032 -2.444
C(MeV) 5.340 1.551 -0.953 0 0 2.824
465c
D(MeV)
AacsB
AaCIB
-6.258 0.296 4.861 -0.301 -0.133 -1.040
~ 1.5 2.06 2.21 3.45 3.36 2.16 1.50
~ 5.7 -1.52 1.12 1.02 4.26 7.33 5.73
to the variation of the phenomenological parameters. Another approach is to take the A,B,C and D parameters from a perturbative QED-QCD calculation[10], where the vertex electromagnetic penguin, photon and gluon box and crossing diagrams are included. The calculated interference interaction is
v/
~-
m2 4 2 2 ~ ~(Qi + Qj) ~ ~(r,j) +...}.
('di(~jOi • OjO(?'ij ) -
(9)
There are many other terms in the perturbative QED-QCD interference calculated result. Only three terms as shown in Eq.(9), which have the same form as assumed by GMS[6], have been included in our scattering calculation. We call this a QCD model (MQ). The parameters other than those appeared in Eq.(9) are determined by re-fitting the n-p mass difference with two choices of the nucleon size b but fixed as. They are labeled as MQ I and MQ II in table I. For the GMS phenomenological models, the calculated AacsB(,,~2-3 fm) is a little too large in comparison with the value 1.5 fm recommended in Ref. [4]. The AactB is much too small (,-~1 fm in comparison with 5.7 fm recommended in Ref. [4]) and for the GMS most favorable parameter set it is -1.5 fm which is in the wrong direction. The QCD model is plausible. The MQ II gives rise to AacsB=2.16 fm, AacIB=7.33 fm, these are not too far from the recommended value 1.5 and 5.7 fm respectively. To show the plausibility further, in table I we show another result labeled as MQP, where the parameters are fixed by an overall fit with both the n - p, A ++ - A ° mass differences and the scattering length differences. Our conclusion is that taking into account the quark mass difference, the electromagnetic interaction and the QED-QCD interference effect together, it is possible to deduce the charge dependent effect appeared in the N-N interaction based on a quark cluster model. The GMS phenomenological model with their present model parameters can not explain the charge asymmetry appeared in the N-N 1S0 scattering lengths. The perturbative QCD model appears plausible, but quantitatively need further work, especially to check if the perturbative QCD-QED calculation can derive an interference interaction to fit the isomultiplet mass splittings and the N-N scattering length differences directly. This work is supported by the NSF(19675018), SEDC and SSTC of China.
466c
E Wang et al./Nuclear Physics A631 (1998) 462c-466c
REFERENCES 1. M.M. Nagels, T.A. Rijken and J.J. de Swart, Phys. Rev., D12 (1975) 744; M. Lacombe, B. Loiseau, J.M. Richard and R. Vink Man, Phys. Rev., D12 (1975) 1495; R. Machleidt, K. Holinde and C. Elster, Phys. Rep., 149 (1987) 1. 2. C.W. Wong, Phys. Rep., 136 (1986) 1; F. Myhrer and J. Wroldsen, Rev. Mod. Phys., 60 (1988) 629; K. Shimizu, Rep. Prog. Phys., 52 (1989) 1. . F. Wang, G.H. Wu, L.J. Teng and T. Goldman, Phys. Rev. Lett., 69 (1992) 2901; G.H. Wu, L.J. Teng, J.L. Ping, F. Wang and T. Goldman, Phys. Rev. C53 (1995) 1161; Mod. Phys. Lett. A10 (1995) 1895. G.A. Miller, B.M.K. Nefkens and I. Slaus, Phys. Rep., 194 (1990) 1.
4.
5. S.N. Yang, this proceedings; Pauchy W-Y Hwang, this proceedings. 6. T. Goldman, K. Maltman and G.J. Stephenson Jr., Phys. Lett., B228 (1989) 396; G.J. Stephenson Jr., K. Maltman and T. Goldman, Phys. Rev., D43 (1991) 860; and private communication. ,
Ming-Zong Wang, Fan Wang and Chun-Wa Wong, Nucl. Phys., A483 (1988) 161.
8. M. Chemtob and S.N. Yang, Nucl. Phys., A420 (1984) 461; K. Brauer, A. Faessler and E.M. Henley, Phys. Lett., B163 (1985) 46. ,
R.V. Reid, Ann. Phys., 50 (1968) 411.
10. Hong-shi Zong, Wei-ning Su, Fan Wang and Han-xin He, Chin. J. Nucl. Phys., 15 (1993) 316; Particles and Nuclei, PANIC XIII Book of Abstract, Vol. 1 (Perugia, Italy, June 1993) p.15.