Effect of nucleon structure variation on the longitudinal response function

21 October 1999

Physics Letters B 465 Ž1999. 27–35

Effect of nucleon structure variation on the longitudinal response function K. Saito

1, 2

, K. Tsushima 3, A.W. Thomas

4

Special Research Center for the Subatomic Structure of Matter and Department of Physics and Mathematical Physics, The UniÕersity of Adelaide, SA 5005, Australia Received 20 April 1999; received in revised form 2 July 1999; accepted 10 September 1999 Editor: W. Haxton

Abstract Using the quark–meson coupling ŽQMC. model, we study the longitudinal response function for quasielastic electron scattering from nuclear matter. In QMC the coupling constant between the scalar Ž s . meson and the nucleon is expected to decrease with increasing nuclear density, because of the self-consistent modification of the structure of the nucleon. The reduction of the coupling constant then leads to a smaller contribution from relativistic RPA than that found in QHD-I. However, since the electromagnetic form factors of the in-medium nucleon are modified at the same time, the longitudinal response function and the Coulomb sum are reduced by a total of about 20% in comparison with the Hartree contribution. We find that the relativistic RPA and the nucleon structure variation both contribute about fifty–fifty to the reduction of the longitudinal response. q 1999 Published by Elsevier Science B.V. All rights reserved. PACS: 25.30.Fj; 21.60.-n; 24.10.Jv; 21.65.q f Keywords: Longitudinal response function; Nucleon structure effect; Quark-meson coupling model; Nuclear matter

There is still considerable interest in the longitudinal response for quasielastic electron scattering. Within the framework of nonrelativistic nuclear models and the impulse approximation, it is very difficult to reproduce the observed, quenched longitudinal response functions w1x. In the mid ’80s, several groups calculated the longitudinal response function using the classic version of Quantum Hadrodynamics ŽQHD-I. w2x Žor the Walecka model.. It was argued that the contribution of the relativistic random phase approximation ŽRRPA., which includes vacuum polarization, is very important in reducing the Coulomb sum rule w3,4x below the sum of the squares of the nucleon charges in the nucleus. There have been several other attempts to study the longitudinal response in nonrelativistic approaches w5x.

1

Permanent address: Tohoku College of Pharmacy, Sendai 981-8558, Japan. E-mail: [email protected] 3 E-mail: [email protected] 4 E-mail: [email protected] 2

0370-2693r99r$ - see front matter q 1999 Published by Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 2 6 9 3 Ž 9 9 . 0 1 0 8 7 - 4

K. Saito et al.r Physics Letters B 465 (1999) 27–35

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On the other hand, the nucleon has internal structure, and it is nowadays expected that this structure should be modified in a nuclear environment w6x. This is closely related to the issue of chiral restoration in QCD. In QHD-I w2x nuclear matter consists of point-like nucleons interacting through the exchange of point-like scalar Ž s . and vector Ž v . mesons. While more recent versions of QHD w7x, inspired by the ideas of effective field theory, are capable of incorporating the effects of hadron internal structure, they have not yet been applied to the problem of the longitudinal response function. It would clearly be very interesting to investigate the quenching of this response within such a framework, including, in addition, the structural changes of the nucleon in-medium. Recently, we have developed a relativistic quark model for nuclear matter, namely, the quark–meson coupling ŽQMC. model w8x, which could be viewed as an extension of the ‘‘original’’ QHD. However, in QMC the mesons couple to confined quarks Žnot to point-like nucleons. and the nucleon is described by the MIT bag model. This model yields an effective Lagrangian for a nuclear system w9x, which has the same form as that in QHD-I, but with a density dependent coupling constant between the s and the nucleon ŽN. – instead of a fixed value. Indeed, from the point of view of the energy of a nuclear system, the key difference between QHD-I and QMC lies in the density dependence of the s –N coupling, g s . ŽWe will revisit this later.. Although this difference may seem subtle, it leads to many attractive results w8,9x. We have already applied this model to various nuclear problems w10x and here we use it to study the effect of in-medium changes in the structure of the nucleon on the longitudinal response function of nuclear matter. First, let us briefly review the calculation of the longitudinal response function for quasielastic electron scattering from Žiso-symmetric. nuclear matter in QHD-I. The starting point is the lowest order polarization insertion, Pmn , for the v meson. This describes the coupling of a virtual vector meson or photon, of momentum q, to a particle-hole or nucleon–antinucleon excitation:

Pmn Ž q . s yig Õ2

d4 k

H Ž 2p .

4

Tr G Ž k . gm G Ž k q q . gn ,

Ž 1.

where GŽ k . is the self-consistent nucleon propagator Žwith momentum k . in relativistic Hartree approximation ŽRHA. given as G Ž k . s GF Ž k . q GD Ž k . s Ž g m km) q M ) .

ip

1 km) 2 y M ) 2 q i e

q Ek)

d Ž k 0) y Ek) . u Ž k F y < k < . .

Ž 2.

Here k ) m s Ž k 0 y g Õ V 0 , k . Ž V 0 is the mean value of the v field., Ek) s 'k 2 q M ) 2 Ž M ) is the effective nucleon mass in matter. and k F is the Fermi momentum. Using the nucleon propagator we can separate the polarization insertion into two pieces: one is the density dependent part, PmnD , which involves at least one power of GD , and the other is the vacuum polarization insertion, PmnF , which involves only GF . The former is finite, but the latter is divergent and must be renormalized. We choose to renormalize such that PmnF Ž q . vanishes at qm2 s mv2 and M ) s M Žwhere mv and M are respectively the free masses of the v meson and the nucleon.. We then find w11x

PmnF Ž q . s jmn P F Ž q . ,

Ž 3.

with jmn s ygmn q Ž qm qnrqm2 . and

P F Ž q. s

g Õ2 6p

q 2 2ln 2 m

M) M

y4

ž

M )2 qm2

M2 y

mv2

/ ž

q 1q2

M )2 qm2

/

ž

f Ž xq . y 1q2

M2 mv2

/

f Ž zÕ . ,

Ž 4.

K. Saito et al.r Physics Letters B 465 (1999) 27–35 )2

29

2

where x q s 1 y 4 Mqm2 , z Õ s 1 y 4mMv2 and

°

'y q 1 'y ln ' , y y1

~'y ln 1 q 'y y ip'y ,

f Ž y. s

¢

'

1y y

2 y y tany1

'

1

'y y

.

for 1 Fy - q`

for 0 - y - 1

.

Ž 5.

for y F0

We assume that the isospin degeneracy of the vacuum is 2. For PmnD , the explicit, analytical expressions can be found in Ref. w12x Žalso see Ref. w11x.. In the Hartree approximation, where only the lowest one nucleon ring is considered, the longitudinal response function, SLH , measured in electron scattering is simply given by SLH Ž q . s y

ž

ZGp2E Ž q . < q < 2 g Õ2pr B qm2

/

I m P LŽ q . .

Ž 6.

Here Z is the nuclear charge, r B the nuclear density, P LŽs P 33 y P 00 . the longitudinal component of the polarization insertion Žwe choose the direction of q as the z-axis. and Gp E is the proton electric form factor, which is usually parametrized by a dipole form in free space: 1 Gp E Ž Q 2 . s , Ž 7. 2 Ž 1 q Q 2r0.71 . with the space-like momentum transfer, Q 2 s yqm2 , in units of GeV 2 . For this initial investigation we omit a small Žand rather complicated. contribution from the anomalous moments w3x, in order to concentrate on the role of the variation of the structure of the nucleon. Since the vacuum polarization is real in the space-like region there is no modification of the Hartree response from this term. The RRPA for the longitudinal component of the polarization insertion, P LRPA , involves the sum of the ring diagrams to all orders. This summation has been discussed by many authors w3,4,11–13x. It involves s – v mixing in the nuclear medium, and is given by

P LRPA Ž q . s Ž 1 y D0 P s . P L q D0 P m2 re L ,

Ž 8.

where e L is the longitudinal dielectric function

e L s Ž 1 y d 0 P L . Ž 1 y D0 P s . y Ž qm2 rq 2 . D0 d 0 P m2 , with q s < q <, and the free meson propagators for the s and v mesons are respectively 1 1 D0 Ž q . s 2 and d 0 Ž q . s 2 , 2 qm y ms q i e qm y mv2 q i e

Ž 9.

Ž 10 .

where ms is the s meson mass. Here P s and P m are respectively the scalar and the time component of the mixed polarization insertions:

P s Ž q . s yig s2

d4 k

H Ž 2p .

P m Ž q . s ig s g Õ

4

d4 k

H Ž 2p .

4

Tr G Ž k . G Ž k q q . ,

Ž 11 .

Tr G Ž k . g 0 G Ž k q q . .

Ž 12 .

K. Saito et al.r Physics Letters B 465 (1999) 27–35

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The scalar polarization insertion can be again separated into two pieces. The density dependent part is finite and the explicit expression can be found in Ref. w12x. Because it does not involve GD , the vacuum component, P sF, is, of course, divergent and once again we need to renormalize it. First, we introduce the usual counter terms to the Lagrangian, which includes terms quadratic, cubic and quartic in the s field, as well as wavefunction renormalization w2x. To get the ‘‘physical’’ properties of the s meson in free space, we impose the following condition w11x:

E P sF Ž qm2 , M ) s M . s

E qm2

P sF Ž qm2 , M ) s M . s 0 at qm2 s ms2 .

Ž 13 .

Then, we find

P sF

Ž q. s

3 g s2 2p 2

q

qm2 3

1 6

ž

Ž

ms2 y qm2

M )2 qm2

.y

ž

M

)2

y

qm2 6

/ž

2ln

M) M

q f Ž xq . y f Ž zs .

M2

Ž f Ž x q . y 2 . y m2 Ž f Ž z s . y 2 . s

/

/ 2

y Ž M ) 2 y M 2 . Ž f Ž z s . y 2. q 2 M Ž M ) y M . q 3 Ž M ) y M . ,

Ž 14 .

where z s s 1 y 4 M 2rms2 . For the mixed polarization insertion there is no vacuum polarization and it vanishes at zero density. ŽThe explicit form can be also found in Ref. w12x.. As QHD-I involves only isoscalar mesons, the isovector RRPA response is the same as that obtained in Hartree approximation, Eq. Ž6.. This implies that the vacuum polarization only modifies the isoscalar response. It remains to study the effect of isovector mesons. ŽNote that the role of rho meson coupling Žwithout vacuum polarization. was studied in Ref. w14x, where it was found to reduce SL only slightly.. Since the longitudinal response is half isoscalar and half isovector, the longitudinal response function in RRPA is given by w3x SLRPA

Ž q. sy

ž

ZGp2E Ž q . < q < 2 g Õ2pr B qm2

/

Im

P LRPA Ž q . q P LŽ q . 2

.

Ž 15 .

Several authors w3,4x have calculated the longitudinal response function using this RRPA polarization, and reported that it is very important in reproducing the observed experimental data w1x. Now we are in a position to discuss the effect of changes in the internal structure of the nucleon in-medium. In order to do so, we consider the following modifications to QHD-I: Meson-nucleon vertex form factor. Since both the mesons and nucleons are composite they have finite size. In the region of space-like momentum transfer the finite-size effect will reduce the meson–N coupling. As the simplest example, we take a monopole form factor w15x at each vertex: FN Ž Q 2 . s

1 1 q Q 2rL2N

,

Ž 16 .

with a cut off parameter L N s 1.5 GeV. In principle, one could self-consistently calculate the form factor within QMC. However, as such changes are not expected to make a big difference, we use Eq. Ž16. in the following calculation.

K. Saito et al.r Physics Letters B 465 (1999) 27–35

Modification of the proton electric form factor.

31

Recently we have studied the electromagnetic form factors of the nucleon, not only in free space w16x but also in a nuclear medium, using the QMC model w17x Žsee also Ref. w18x.. Because the confined quark feels an attractive force due to the s , the quark wave function is modified in a nuclear medium. The ratio of the electric form factor of the proton in medium to that in free space, Gp E Ž r B ,Q 2 .rGp E Ž Q 2 ., is shown in Fig. 3 of Ref. w17x. The main result of that calculation is that the ratio decreases essentially linearly as a function of Q 2 , and that it is accurately parametrized at r B s r 0 Žs 0.15 fmy3 , the normal density of nuclear matter. as R p E Ž r 0 ,Q 2 . '

Gp E Ž r 0 ,Q 2 . Gp E Ž Q 2 .

, 1 y 0.26 = Q 2 .

Ž 17 .

This implies that the Želectric. rms radius of the proton at r 0 swells by about 5.5 % Žfor more details, see Ref. w17x.. Since the bag model reproduces the form factor measured in free space very well w16x and the latter is well described by Eq. Ž7., the in-medium proton form factor can be represented as Gp E Ž Q 2 . = R p E Ž r B ,Q 2 . 5. Density dependence of the In QMC the confined quark in the nucleon couples to the s field which coupling constants. gives rise to an attractive force. As a result the quark becomes more relativistic in a nuclear medium than in free space. This implies that the small component of the quark wave function, cq , is enhanced in medium w8,9x. The coupling between the s and nucleon is therefore expected to be reduced at finite density because it is given in terms of the quark scalar charge, HBag dVcq cq w9x. On the other hand, the coupling between the vector meson and nucleon remains constant, because it is related to the baryon number, which is conserved. To study the longitudinal response of nuclear matter, we first have to solve the nuclear ground state within RHA. In QHD-I the total energy density for nuclear matter is written as w11x 1 E s E0 q

2p

M 2 Ž MyM ) . 2

2

ms2 4M 2

q 32 f Ž z s . y 3 ,

Ž 18 .

where E0 has the usual form Žin RHA., given in Ref. w2x. Note that in Ref. w2x the renormalization condition on the nucleon loops is imposed at qm2 s 0. The second term on the r.h.s. of Eq. Ž18. w11x occurs because we chose the renormalization condition for the s at qm2 s ms2 Žsee Eq. Ž13... As measureable quantities cannot depend on this choice, our model gives the same physical quantities as those of Ref. w2x. To take into account the modifications Ž1. and Ž3., we replace the s- and v –N coupling constants in Eq. Ž18. by g s ™ g s Ž r B . = FN Ž Q 2 . , g Õ ™ g Õ = FN Ž Q

2

.,

Ž 19 . Ž 20 .

where the density dependence of g s Ž r B . is given by solving the nuclear matter problem self-consistently, using the MIT bag for the nucleon model Žsee Ref. w9x.. As in QHD-I, we have two adjustable parameters in the present calculation: g s Ž0. Žthe s –N coupling constant at r B s 0. and g Õ . 5

The nucleon–antinucleon excitation in Pmn contributes to the photon-nucleon vertex as an RRPA correction, which may in principle lead to double counting for the form factor. However, we ignore this correction because it is very small indeed w4x.

K. Saito et al.r Physics Letters B 465 (1999) 27–35

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Requiring the usual saturation condition for nuclear matter, namely Err B y M s y15.7 MeV at r 0 , we determine the coupling constants g s2 Ž0. and g Õ2 Ž g s2 Ž0. s 61.85 and g Õ2 s 62.61.. In the calculation we fix the quark mass to be 5 MeV, ms s 550 MeV and mv s 783 MeV, while the bag parameters are chosen so as to reproduce the free nucleon mass Ž M s 939 MeV. with the bag radius R 0 s 0.8 fm Ži.e., B 1r4 s 170.0 MeV and z s 3.295 w8,9x.. This yields the effective nucleon mass M ) rM s 0.81 at r 0 and the incompressibility K s 281 MeV. ŽWe do not consider the modification of the properties of the mesons themselves w9x in the present work.. Before presenting our main results we show, in Fig. 1, the density dependence of the scalar coupling constant. Notice that at r 0 g s has decreased by about 9%. The effective mass of the nucleon is also shown in the figure. Next, we show the longitudinal response function in QMC. Using the density dependent coupling constant, the meson–N form factors Žsee Eqs. Ž19. and Ž20.. and the in-medium proton electric form factor, we can calculate the longitudinal response of nuclear matter. The result is shown in Fig. 2. Because of the density dependent coupling, g s Ž r B ., the reduction of the response function from the Hartree result Žthe dotted curve., caused by the full RRPA Žthe dashed curve in the figure., becomes much smaller than that in QHD-I. On the other hand, the modification of the proton electric form factor is very significant, yielding a much bigger reduction in the response Žsee the upper solid curve.. We can see that the effect of the meson-nucleon form factor is to enhance the longitudinal response Žsee the dot-dashed curve., but the effect is relatively minor, as noted below Eq. Ž16.. It is also interesting to see the quark mass dependence of the longitudinal response. As an example, we consider the case of m q s 300 MeV, which is a typical constituent quark mass. For m q s 300 MeV and R 0 s 0.8 fm, the coupling constants required to fit the saturation properties of nuclear matter are: g s2 Ž0. s 68.69 and g Õ2 s 84.24, and the effective nucleon mass at r 0 and the incompressibility become 723 and 345 MeV, respectively. Using these parameters we show the result for the longitudinal response Žthe lower solid curve. in Fig. 2. In comparison with the case m q s 5 MeV, it is a little smaller and the peak position is shifted to the higher energy transfer side. This is related to the smaller effective nucleon mass in the case m q s 300 MeV than when m q s 5 MeV. The integrated strength of the longitudinal response Žor the Coulomb sum., C Ž q ., q

CŽ q. s

H0 dq S Ž q,q . , 0

L

0

Ž 21 .

is shown in Fig. 3 as a function of three-momentum transfer, q. For high q, the strength is about 20% lower in the full calculation than for the Hartree response. For low q, the full calculation with the constituent quark mass remains much lower than the Hartree result, while in case of the light quark mass it gradually approaches the Hartree one. This difference is caused by that the effective nucleon mass for m q s 5 MeV being larger in

Fig. 1. Density dependences of g s Ž r B .r g s Ž0. and M ) r M. The solid curve is for the ratio of the coupling constants, while the dotted curve is for the ratio of the nucleon masses.

K. Saito et al.r Physics Letters B 465 (1999) 27–35

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Fig. 2. Longitudinal response functions in QMC with m q s 5 MeV. We fix q s 550 MeV and r B s r 0 . The dotted curve is the result of the Hartree approximation Žsee Eq. Ž6.., where the effective nucleon mass is given by QMC and the proton electric form factor is the same as in free space. The dashed curve is the result of the full RRPA, without the modifications Ž1. and Ž2. Ži.e. FN s 1 and R p E s 1.. The dot-dashed curve shows the result of the full RRPA with the meson–N form factor but R p E s 1. The upper Žlower. solid curve shows the result of the full RRPA for m q s 5 Ž300. MeV, including all modifications.

matter than that for m q s 300 MeV. The 20% reduction found here is a little smaller than the value of approximately 30% found in QHD-I w3,4x. We would like to emphasize that these calculations are for nuclear matter and cannot be directly compared with the experimental data. Furthermore, there still remain discrepancies and uncertainties in the present experimental results w19,20x. We comment on the transverse response from nuclear matter. In Ref. w17x it was found that the in-medium modification of the nucleon magnetic form factor within QMC is very small: the calculated decrease in the proton Žneutron. magnetic form factor is about 1.5% Ž0.9%. at r 0 . Therefore, one would expect the total change in the transverse response caused by RRPA correlations and the effect of the variation of the structure of the nucleon to be much smaller than in the longitudinal response. This is certainly what one needs in order to fit the experimental data w1x. To complete our discussion it may be helpful to make some remarks on the relationship between QHD and the present work. Since we want to treat the nucleon relativistically and study how the longitudinal response in QMC deviates from that reported previously within QHD-I w3,4x, we have included antinucleon degrees of freedom explicitly. However, as is already well known, vacuum polarization diagrams involving antinucleons lead to unnaturally large effective interactions, such that the vacuum may not be stable w7,21x. Within mean-field approximation, QMC has been proven to be natural w22x. While the vacuum polarization diagrams introduced here do not strictly satisfy this criterion, the new features of QMC Žespecially the density dependent reduction of g s . mean that the situation is significantly better than that found in QHD-I. Recently, Serot and Walecka have proposed a new formulation of Quantum Hadrodynamics w7x, along the lines of modern effective field theory. Within that framework the effects of short-range interactions, vacuum polarization and internal structure are incorporated in a new Lagrangian involving non-linear couplings that are almost all natural. Many properties of nuclear systems have been fit using this approach. It is clear that on quite general grounds one could expect to incorporate the effects of internal structure, which are modelled within QMC, within this new formulation of QHD. For example, Muller ¨ and Jennings w23x have shown explicitly how

34

K. Saito et al.r Physics Letters B 465 (1999) 27–35

Fig. 3. Coulomb sum, C Ž q .rZ, at r 0 in the Hartree approximation with R p E s1 Žthe dotted curve. or the full RRPA with all modifications Žthe solid and dot-dashed curves are for m q s 5 and 300 MeV, respectively..

the effect of the density dependent reduction of the scalar coupling constant found within QMC, on the energy of nuclear matter, can be included within QHD by performing a re-definition of the scalar field. This naturally leads to new scalar self-couplings. In the future it will clearly be very interesting to calculate the response functions within such an approach. In the meantime, we feel that QMC, with its model of the in-medium changes of nucleon structure, provides useful insights into the physics that one can hope to learn by studying the longitudinal response. In summary, we have calculated the longitudinal response of nuclear matter using the QMC model. The reduction of the s –N coupling constant with density decreases the contribution of the RRPA, while the modification of the proton electric form factor in medium reduces the longitudinal response considerably. The longitudinal response, or the Coulomb sum, is reduced by about 20% in total, with RRPA correlations and the variation of the in-medium nucleon structure contributing about fifty-fifty. It will be interesting to extend this work to calculate both the longitudinal and transverse response functions for finite nuclei, in order to compare directly with the new experimental results which are anticipated soon w20x. Acknowledgements K.S. would like to acknowledge the warm hospitality at the CSSM, where this work was partly carried out. K.S. would especially like to thank M. Ericson, P.A.M. Guichon, W. Bentz and J. Morgenstern for valuable discussions on the longitudinal response function. This work was supported by the Australian Research Council and the Japan Society for the Promotion of Science. References w1x Z.E. Meziani et al., Phys. Rev. Lett. 52 Ž1984. 2130; Phys. Rev. Lett. 54 Ž1985. 1233; M. Deady et al., Phys. Rev. C 28 Ž1983. 631. w2x B.D. Serot, J.D. Walecka, Adv. Nucl. Phys. 16 Ž1986. 1.

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w3x C.J. Horowitz, Phys. Lett. B 208 Ž1988. 8; C.J. Horowitz, J. Piekarewicz, Phys. Rev. Lett. 62 Ž1989. 391. w4x H. Kurasawa, T. Suzuki, Nucl. Phys. A 445 Ž1985. 685; A 490 Ž1988. 571; K. Wehrberger, F. Beck, Phys. Rev. C 35 Ž1987. 298, 2337 ŽE.; Nucl. Phys. A 491 Ž1989. 587; X. Ji, Phys. Lett. B 219 Ž1989. 143; K. Tanaka, W. Bentz, A. Arima, Nucl. Phys. A 518 Ž1990. 229. w5x W.M. Alberico, M. Ericson, A. Molinari, Nucl. Phys. A 379 Ž1982. 429; W.M. Alberico, P. Czerski, M. Ericson, A. Molinari, Nucl. Phys. A 462 Ž1987. 269; W.M. Alberico, G. Chanfray, M. Ericson, A. Molinari, Nucl. Phys. A 475 Ž1987. 233; C.R. Chinn, A. Picklesimer, J.W. Van Orden, Phys. Rev. C 40 Ž1989. 790. w6x For an overview, see Quark Matter’97, Nucl. Phys. A 638 Ž1998.. w7x R.J. Furnstahl, B.D. Serot, H.B. Tang, Nucl. Phys. A 615 Ž1997. 441; B.D. Serot, J.D. Walecka, Int. J. Mod. Phys. E 6 Ž1997. 515. w8x P.A.M. Guichon, Phys. Lett. B 200 Ž1988. 235; K. Saito, A.W. Thomas, Phys. Lett. B 327 Ž1994. 9. w9x P.A.M. Guichon, K. Saito, E. Rodionov, A.W. Thomas, Nucl. Phys. A 601 Ž1996. 349; K. Saito, K. Tsushima, A.W. Thomas, Nucl. Phys. A 609 Ž1996. 339; Phys. Rev. C 55 Ž1997. 2637. w10x For example, see K. Saito, nucl-thr9811088, to appear in the proc. of the XIV international Seminar on High Energy Phyaics Problems, Dubna, 17–22 August, 1998. w11x K. Saito, K. Tsushima, A.W. Thomas, A.G. Williams, Phys. Lett. B 433 Ž1998. 243. w12x K. Lim, C.J. Horowitz, Nucl. Phys. A 501 Ž1989. 729. w13x S.A. Chin, Ann. Phys. 108 Ž1977. 301. w14x S. Nishizaki, H. Kurasawa, T. Suzuki, Phys. Lett. B 171 Ž1986. 1. w15x R. Machleidt, Adv. in Nucl. Phys. 16 Ž1989. 189. w16x D.H. Lu, A.W. Thomas, A.G. Williams, Phys. Rev. C 55 Ž1997. 3108. w17x D.H. Lu, A.W. Thomas, K. Tsushima, A.G. Williams, K. Saito, Phys. Lett. B 417 Ž1998. 217. w18x M. Bergmann, K. Goeke, S. Krewald, Phys. Lett. B 243 Ž1990. 185. w19x For example, see J. Jourdan, Phys. Lett. B 353 Ž1995. 189. w20x J. Morgenstern, private communication. Their group is now re-analysing quasielastic electron scattering data on 12 C and 208 Pb. w21x For example, see B. D Serot, H.B. Tang, Phys. Rev. C 51 Ž1995. 969. w22x K. Saito, K. Tsushima, A.W. Thomas, Phys. Lett. B 406 Ž1997. 287. w23x H. Muller, B.K. Jennings, Nucl. Phys. A 640 Ž1998. 55. ¨