The nucleon–nucleon interaction and properties of the nucleon in a πρω soliton model including a dilaton field

3 February 2000

Physics Letters B 473 Ž2000. 200–208

The nucleon–nucleon interaction and properties of the nucleon in a prv soliton model including a dilaton field Ulf-G. Meißner a

a,1

, A. Rakhimov

a,2

, U. Yakhshiev a

b,3

Forschungszentrum Julich, Institut fur Germany ¨ ¨ Kernphysik (Theorie), D-52425 Julich, ¨ b International Centre for Theoretical Physics (ICTP), 34014, Trieste, Italy

Received 24 March 1999; received in revised form 8 October 1999; accepted 14 December 1999 Editor: J.-P. Blaizot

Abstract We investigate an extended chiral soliton model which includes p , r , v and s mesons as explicit degrees of freedom. The Lagrangian incorporates chiral symmetry and broken scale invariance. A scalar–isoscalar s-meson is associated with a quarkonium dilaton field with a mass ms f 550 . . . 720 MeV. We show that the inclusion of the scalar field slightly changes the static and electromagnetic properties of the nucleon. In contrast, it plays a significant role in nucleon–nucleon dynamics and gives an opportunity to describe well the two-nucleon interaction. q 2000 Published by Elsevier Science B.V. All rights reserved. Keywords: Skyrme model; Sigma meson; Nucleon properties; Meson-nucleon form factors; Nucleon-nucleon interaction

1. Introduction In Ref. w1x Furnstahl, Tang and Serot ŽFTS. proposed a new model for nuclear matter and finite nuclei that realizes QCD symmetries such as chiral symmetry, broken scale invariance and the phenomenology of vector meson dominance. An important feature of this approach is the inclusion of light scalar degrees of freedom, which are given an anomalous scale dimension. The vacuum dynamics 1

E-mail: [email protected] E-mail: [email protected]; permanent address: institute of nuclear physics, academy of sciences, usbekistan Žcis. 3 E-mail: [email protected]; permanent address: institute of applied physics, tashkent state university, tashkent-174, usbekistan Žcis. 2

of QCD is constrained by the trace anomaly and related low-energy theorems of QCD. The scalar– isoscalar sector of the theory is divided into a low mass part that is adequately described by a scalar meson Žquarkonium . and a high mass part Žgluonium., that can be ‘‘integrated out’’, leading to various couplings among the remaining fields. The application of the model to the properties of nuclear matter as well as finite nuclei gave a satisfactory description. Further developments of the model w2,3x showed that the light scalar related to the trace anomaly can play a significant role not only in the description of bound nucleons but also in the description of heavy-ion collisions. Here a natural question arises: What is the role of this light quarkonium in the description of the properties of a single nucleon, when it is taken into

0370-2693r00r$ - see front matter q 2000 Published by Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 2 6 9 3 Ž 9 9 . 0 1 4 9 5 - 1

U.-G. Meißner et al.r Physics Letters B 473 (2000) 200–208

account in topological nonlinear chiral soliton models, which are similar to the FTS effective Lagrangian on the single nucleon level? In the present paper we introduce a dilaton field into the prv-model w4x and investigate some properties of single nucleons which emerge as solitons in the sector with baryon number one Ž B s 1. as well as some features of the nucleon–nucleon interaction. It is well known that a scalar–isoscalar meson, 4 the sigma, plays an important role in the nucleon– nucleon ŽNN. interaction especially within one-boson-exchange ŽOBE. models w6x. We remark that the missing medium range attraction was a long standing puzzle in Skyrme like models. Lately it has been shown that w7x explicit inclusion of a scalar meson into the Skyrme model produces in a natural way the desired attraction. But these studies have two shortcomings. Firstly, the scalar meson used in such models w7x has nothing to do with OBE phenomenology, as it has a large mass and is identified with a gluonium state. Secondly, the Lagrangians used in Refs. w7x do not include explicit omega mesons at all, which may ‘‘spoil’’ the mechanism of the attraction due to its strong repulsion. 5 Therefore, it would be quite interesting to investigate the central part of the NN interaction when the light scalar Ž s . and v-mesons are both taken into account explicitly. This is what is done here. In particular, it is important that if one is able to properly describe the intermediate range attraction in the central NN interaction, the successful description of the single nucleon properties within the prv model should not be destroyed. We also note that in an soliton approach with explicit regulated two-pion loop graphs one is able to get the proper intermediate range attraction. In that case, however, one does not stay within a simple OBE approach any more Žas done here. and also needs to calculate the modifications of the isovector two-pion exchange to the r and so on. For compari-

4

Here and in what follows, we call it sigma meson for simplicity. Although we assume it to be some kind of quarkonium state, its precise dynamical nature is of no direct relevance for the following arguments. 5 Of course, the stabilization of the soliton in Refs. w7x via higher derivative terms also leads to repulsion in the central NN interaction, but this is less easily be interpreted as single meson exchange.

201

son, we mention that recent developments of the original Bonn OBE potential performed at Julich ¨ also include multi-meson exchanges leading to a renormalization of various interactions, couplings and cut-off parameters w9x. The model we investigate is closely related to the OBE approximation of the NN force. 2. The p rvs model Including a s-meson by means of the scale invariance and trace anomaly of QCD into the prvmodel w8x can be done in terms of the following chiral Lagrangian of the coupled prvs system, Ls

fs2 ey2 s 2 y

m

Em sE s y

fp2 ey2 s 2

fp2 ey2 s 4

TrLm L m

Tr lm q rm q igtrm q ig vm

2

3 1 q g vm B m y Ž vmn v mn q rmn r mn . 2 4 q

fp2 mp2 ey3 s 2 fs2 ms2

Tr Ž U y 1 .

1 y ey4s Ž 4s q 1 . , Ž 1. 16 where the pion fields are parameterized in terms of U s expŽ it P prfp . and j s 'U , the leftrrighthanded currents are given by Lm s UqEm U, lm s jqEm j , rm s jEm jq, and the pertinent vector meson Ž r , v . field strength tensors are rmn s Em rn y En rm q g w rm = rn x and vmn s Em vn y En vm . Furthermore, the topological baryon number current is given by B m s ´ m a bg TrLa Lb LgrŽ24p 2 .. In Eq. Ž1., fp is the pion decay constant Ž fp s 93 MeV., g s g rp p is determined through the KSFR relation g s mrŽ'2 fp . and fs is a parameter of the model. The model assumes the masses of r and v mesons to be equal, mr s mv s m. The mass of the s is related to the gluon condensate in the usual way w1,10x, ms s 2 C g rfs . Being ‘‘mapped’’ onto the states of a nucleon, the Lagrangian Eq. Ž1. will be similar to the FTS effective Lagrangian. To see that, one simply considers the linear approximation in the fluctuation variable f Ž r . s S0 y SŽ r . of the scalar field SŽ r . s expŽys Ž r ... y

(

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202

Nucleons arise as soliton solutions from the Lagrangian Eq. Ž1. in the sector with baryon number B s 1. To construct them one goes through a two step procedure. First, one finds the classical soliton which has neither good spin nor good isospin. Then an adiabatic rotation of the soliton is performed and it is quantized collectively. 6 The classical soliton follows from Eq. Ž1. by virtue of spherical symmetrical ansatze ¨ for the meson fields: U Ž r . s exp Ž it rˆQ Ž r . . ,

vm Ž r . s v Ž r . dm 0 ,

r ia s ´ i a k rˆk

GŽ r . gr

s Ž r. ss Ž r . .

,

Ž 2.

In what follows, we call Q Ž r ., GŽ r ., v Ž r ., and s Ž r . the pion-, r-, v-, and s-meson profile functions, respectively. The pertinent boundary conditions to ensure baryon number one and finite energy are Q Ž0. s p ,GŽ0. s y2, v X Ž0. s s X Ž0. s 0,Q Ž`. s GŽ`. s v Ž`. s s Ž`. s 0. To project out baryonic states of good spin and isospin, we perform a time-independent SUŽ2. rotation U Ž r ,t . s A Ž t . U Ž r . Aq Ž t . ,

j Ž r ,t . s A Ž t . j Ž r . Aq Ž t . , s Ž r ,t . s s Ž r . , v i Ž r ,t . s

fŽ r.

t P r 0 Ž r ,t . s

r 2 g

Ž K = rˆ . i ,

AŽ t . t P Ž K j 1Ž r .

qrK ˆ P rˆj 2 Ž r . . Aq Ž t . ,

t P r i Ž r ,t . s A Ž t . t P r i Ž r . Aq Ž t . ,

Ž 3.

with 2 K the angular frequency of the spinning soli˙ This leads to the time-dependent ton, it P K s Aq A. Lagrange function L Ž t . s dr Ls y MH Ž Q ,G, v , s .

H

˙ ˙q . . q L Ž Q ,G, v , s , f , j 1 , j 2 . Tr Ž AA

Ž 4.

Minimizing the classical mass MH ŽQ ,G, v , s . leads to coupled differential equations for Q ,G, v and s subject to the aforementioned boundary con-

6

We again refer the reader to Ref. w8x for details.

ditions. In the spirit of the large Nc-expansion, one then extremizes the moment of inertia LŽQ ,G, v , s , f , j 1 , j 2 . which gives the coupled differential equations for j 1 , j 2 and f in the presence of the background profiles Q ,G, v and s . The pertinent boundary conditions are f Ž0. s f Ž`. s 0, j 1X Ž0. s j 1Ž`. s 0, j 2X Ž0. s j 2 Ž`. s 0, 2 j 1Ž0. q j 2 Ž0. s 2. The masses of the nucleon, MN , and of the D, MD , are then given by MN s MH q 3r8 L and MD s MH q 3r15 L. The electromagnetic form factors are obtained in the usual way w8x. To show the modification due to the s field, we only display the isoscalar electric form factor, it reads: GES Ž q 2 . s y

4p m 2 3g

`

H0

j0 Ž qr . v Ž r . ey2 s r 2 dr . Ž 5 .

The normalization is GES Ž0. s GEV Ž0. s 1r2. Similarly, meson–nucleon vertex form factors may be calculated w11x. Of course, such strong interaction form factors are model-dependent quantities. In the soliton approach, however, they arise naturally as Fourier transforms of the meson distribution within the extended nucleon. Stated differently, the soliton acts as a source of an extended meson cloud, which leads to a meson–nucleon interaction region of finite extension. In momentum space, this extension can be interpreted as a corresponding form factor. We mention that such a picture underlies the inclusion of strong form factors say in OBE models. Coming back to our approach, these form factors are most easily evaluated in the Breit-frame. For the explicit expressions, we refer to Ref. w8x. The ‘‘electric’’ and ‘‘magnetic’’ vector meson–nucleon form factors, called GEr , v Ž t . and GMr , v Ž t . are connected to the Dirac, F1Ž t ., and Pauli, F2 Ž t ., form factors through the following relations: GEi Ž t . s F1i Ž t . q tF2i Ž t .r 4 MN2 , GMi Ž t . s F1i Ž t . q F2i Ž t . Ž i s r , v ..

3. Results and discussion 3.1. Static and electromagnetic properties of the nucleon Using the formulas given above we have calculated static and electromagnetic properties of the

U.-G. Meißner et al.r Physics Letters B 473 (2000) 200–208

nucleon. As can be seen from Eq. Ž1., the Lagrangian has no free parameters in the prv sector. In the actual calculations, the parameters mp ,m, fp are fixed at their empirical values, mp s 138 MeV, m s mr s mv s 770 MeV, fp s 93 MeV, g s mrŽ'2 fp . s 5.85. In the s-meson sector there are two free parameters: fs and ms . The former is poorly known. For this purpose we may use the analogy between the the Lagrangian Eq. Ž1. and that of FTS. Actually, as it has been pointed above, by the substitution expŽys Ž r .. s Ž SŽ r .rS0 .1r d, where S0 is the vacuum expectation value of the scalar field SŽ r . in free space, and d is its anomalous scale dimension Žin the sense of an effective field theory., the kinetic term of the scalar field can be rewritten as that of the FTS Lagrangian: Lskin s fs2 ey2 s Em sE msr2 s S 2rS02

Ž1yd .rd

Em SE m Sr2 .

Ž 6.

From the last equation it is clear that fs s S0 d. The ranges for the parameters S0 and d have been studied in nuclear matter calculations w1,2x. In particular, it was shown that for d G 2 the much debated Brown–Rho ŽBR. scaling may be recovered. Therefore, assuming that there is no dependence of d on the density, we shall use the best value d s 2.6 found in Refs. w1,2x. The values for S0 were found to

203

be S0 s 90.6 % 95.6 MeV w1x. Consequently, we put S0 s fp s 93 MeV and hence fs s 242 MeV. The mass of the s or equivalently the gluon condensate C g s ms2 d 2 S02r4 s ms2 d 2 fp2r4 is uncertain. We thus consider two cases: ms s 550 MeV and ms s 720 MeV, in accordance with recent pp phase shift analyses w5x and with OBE values. From here on, the scalar sector is entirely fixed by the values for fs and ms and we do not need to discuss the parameter d any more. We stress again that the precise nature of such a scalar–isoscalar field is not relevant here, only that it should not be a pure gluonium state. A summary of static nucleon properties obtained in both cases, i.e. with ms s 550 MeV and ms s 720 MeV, is given in Table 1. One immediately observes that the nucleon mass is again overestimated. This may not be regarded as a deficiency, since it is known that quantum fluctuations tend to decrease the mass substantially w12x. To estimate the influence of the s-meson, we also show the results given by the minimal version of the prv model. As can be seen from Table 1, the inclusion of a light sigma meson into the basic prv model just slightly changes the nucleon mass and its electromagnetic properties. This may be explained by the fact that the role of intermediate scalar–isoscalar meson in gamma–nucleon interactions is negligible. In contrast, the presence of the sigma-meson

Table 1 Nucleon properties in the prv and prvs models.

prv

prvs

prvs

ms wMeVx C g1r4 wMeVx

y y

550 258

720 295

300 % 400

B 1r4 wMeVx MN wMeVx L wfmx MD y MN wMeVx rH s ² rB2 :1r2 wfmx ² rE2 :1r2 wfmx p ² rE2 :n wfm2 x ² rM2 :1r2 p wfmx

y 1560 0.88 344 0.5 0.92 y0.20 0.84

119 1492 0.88 350 0.5 0.94 y0.16 0.85

121 1511 0.88 350 0.5 0.94 y0.16 0.85

y 939 y 293 ; 0.5 0.86 " 0.01 y0.119 " 0.004 0.86 " 0.06

² rM2 :1r2 wfmx n m p wn.m.x m n wn.m.x < m p rm n < gA ² rA2 :1r2 wfmx p

0.85 3.34 y2.58 1.29 0.88 0.63

0.85 3.33 y2.53 1.30 0.95 0.66

0.85 3.33 y2.53 1.30 0.95 0.66

0.88 " 0.07 2.79 y1.91 1.46 1.26 " 0.006 0.65 " 0.07

Exp.

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leads to an enhancement of the axial-vector coupling constant as it was first observed in Ref. w13x. Although the physical mechanism of this change is not clear, it may be understood as mainly due to modifications of certain meson profiles. This is in marked contrast to the inclusion of pion loop effects, which tend to lower the axial coupling even further w12x. One finds g A s 0.88 for the prv and g A s 0.95 for the prvs model, respectively. One may hope that a more refined inclusion of the s meson might give the desired value g A s 1.26. 3.2. Meson–nucleon form factors and NN interaction One of the usual ways to calculate the meson– nucleon interaction potential within topological soliton models is the so-called product ansatz w7x. Within this approximation, the two-Skyrmion potential as a function of the relative angles of orientation between the Skyrmions has a compact form, and the extraction of the NN potential by projection onto asymptotic two-nucleon states is straightforward. This procedure gives only three nonvanishing channels: the central, spin–spin and tensor potential. At large and intermediate distances, the latter two compare well with e.g. the phenomenological Paris potential w14x. The major inadequacy found in such type of calculations is the lack of an intermediate range attraction in the central potential. Although many remedies have been proposed, this result may not be genuine for Skyrme like models. In fact, the product ansatz, which is not a solution of the equations of motion, can only be considered accurate at large distances, and the failure of these calculations to reproduce the central range attraction may simply be the failure of the product ansatz to provide an adequate approximation to the exact solution. 7 Although the lack of central attraction may be recovered by the inclusion of a scalar–isoscalar meson, the inherent ansatz dependence of the trial configuration remains a major shortcoming of the product approximation w16x. There is another method of using the soliton model for calculating the nuclear forces as first done by Holzwarth and Machleidt w17x. They proposed to

7

An early study giving credit to this line of reasoning can be found in Ref. w15x.

calculate VN N within the OBE model taking coupling constants and meson–nucleon form factors from a microscopical model such as the Cloudy Bag model or the Skyrme model. It was shown that the Skyrme form factor is a soft pion form factor that is compatible with the p N and NN systems. We shall use this strategy to investigate the NN potential within the present model. The meson–nucleon form factors are given by a well known procedure, proposed first by Cohen w11x. Although they were derived in a microscopic and consistent way, these form factors should not be directly used in standard OBE schemes. The reason is that the OBE schemes w6x in momentum space use form factors defined for fields propagating on a flat metric, whereas the definition of form factors used here involves a nontrivial metric. Hence, before using the soliton model form factors in an OBE scheme, one should modify the procedure to calculate these by redefining the meson fields. The modification for the pion-nucleon form factor in a prv model is clearly outlined in Refs. w17x. Now, applying this procedure to the Lagrangian in Eq. Ž1., we get the following p NN form factor: Gp Ž yq 2 . s

8p MN fp 3q `

=

H0

s

Ž q 2 q mp2 .

j1 Ž qr . MT Ž r . sin Ž Q . r 2 dr

(

8p MN fp 3

` j1

H0

yrF XX Ž r . q

Ž qr . qr

y2 F X Ž r .

2FŽ r. r

qrmp2 F Ž r . r 2 dr ,

Ž 7.

where MT s 1 q 2tan2 Ž Qr2 . ey2 s˜ ,

(

F Ž r . s 3 q cos 2 Ž Q Ž r . . y 4cos Ž Q Ž r . . ey s˜ ,

Ž 8. with s˜ a properly rescaled s field, see below. The influence of the metric factor MT on the pion-nucleon form factor is illustrated in Fig. 1. It is seen that

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205

properties of the nucleon given in Table 1. Note also that, using the above rescaling in the last term of Eq. Ž1., it follows that ms˜ s ms . The sigma–nucleon form factor is then given by G s Ž yq 2 . s y4p

`

H0

j0 Ž qr . s˜ XX q

2 s˜ X r

yms2 s˜ r 2 dr ,

Fig. 1. The normalized p NN form factor in the prvs model Ž ms s 720 MeV.. The solid line represents the form factor when the metric factor is included ŽEq. Ž7.., while the dashed line gives the result with no metric factor. The dotted line is a monopole form factor with Lp s1300 MeV.

without the inclusion of MT , the form factor is softer than in OBE models Žthe dashed line in Fig. 1., while its inclusion via Eqs. Ž7. and Ž8. gives a behavior closer to OBE models. In fact, a monopole approximation at small q 2 s t of the normalized form factor Gp Ž t .rGp Ž0. f Lp2 rŽ Lp2 y t . gives Lp s 860 MeV and Lp s 1100 MeV for MT s 1 and MT / 1 respectively, compared to its empirical fit, LpOB E s 1300 MeV Ždotted line in Fig. 1.. Note, however, that our results for Lp are in line with recent coupled-channel calculations of the Julich ¨ group w18x. There, a monopole form factor with Lp , 800 MeV is obtained. We do not want to stress here any qualitative comparison but rather like to point out that our approach also leads to cut-off values well below the ones obtained in OBE approaches. Introducing a flat metric requires a canonical form for the kinetic part of the Lagrangian, which determines the dynamics of the field fluctuations. The kinetic term of the scalar meson in Eq. Ž1., Lskin s fs2 ey2 s Em sE msr2, can be easily rewritten in the canonical way, Lskin s Em sE ˜ ms˜r2, by an appropriate redefinition of the basic sigma field. This new field s˜ may then be identified with the real sigma field. Clearly this redefinition does not change the static

Ž 9.

and may be used in OBE models. We have not introduced any metric factors in the form factors of the heavier mesons since these should play a lesser role than in the case of the pion. For small values of the squared four-momentum transfer t, each form factor can be parameterized by a monopole form: Gi Ž t . s g i Ž L2i y m2i .rŽ L2i y m 2i . Ž i s p , r , v , s .. We present in Table 2 the range parameters Žcut-offs. and the coupling constants of the resulting meson–nucleon dynamics. One can see that the inclusion of the s-meson does not significantly affect meson–nucleon form factors as given by the prv model. As can be seen from Table 2, the values for meson–nucleon coupling constants are close to their empirical values Žin some cases obtained by OBE model fits.. This is one of the main advantages of the inclusion of a scalar–isoscalar meson as done in the present approach. The corollary of the present model is that it gives significant information on the s-nucleon interaction. As it is seen from the Table 2, the value for gs N N and the cut-off parameter of sigma–nucleon vertex Ls are smaller than their OBE prediction LsOB E f 1300 % 2000 MeV. This contrast is evidently seen from the Fig. 2, where G s Ž t .rG s Ž0. for two cases Ž ms s 720 MeV and ms s 550 MeV. is displayed by the solid and dashed lines, respectively. The band enclosed by the dotted lines refers to the OBE monopole form factor with LsOBE s 1300 . . . 2000 MeV. One can conclude that the present model gives a softer s NN form factor than one obtains in OBE approaches. As it had been noticed before, the t-plane for each form factor has a cut along the positive real axis extending from t s t 0 to `. The cut for the s-nucleon vertex function starts at t 0 s 4 mp2 , reflecting the kinematical threshold for the s pp channel. More precisely, this result follows from the

™

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Table 2 Meson–nucleon coupling constants and cut-off parameters of meson–nucleon form factors. The L i Ž i s p , r , v , s . are cutoff parameters in equivalent monopole fits 1rŽ1 y trL2i . to the normalized form factors Gi N N Ž t .rGi N N Ž0. around t s 0. The empirical values are from an OBE potential fit w6x

prv Gp N N Ž0. Gs N N Ž0. F1r Ž0. F2r Ž0. F2r Ž0.rF1r Ž0. F1v Ž0. F2v Ž0. F2v Ž0.rF1v Ž0. Lp ŽGeV. Ls ŽGeV. L1r ŽGeV. L2r ŽGeV. L1v ŽGeV. L2v ŽGeV.

14.74 y 2.55 14.33 5.6 8.78 y2.15 y0.24 1.2 y 0.62 0.92 0.95 1.12

prvs Ž550.

prvs Ž720.

OBErEmp.

13.97 6.2 2.76 15.01 5.43 10.73 y2.78 y0.25 1.1 0.59 0.63 0.92 0.89 0.86

14.17 6.19 2.68 14.67 5.47 10.15 y2.65 y0.26 1.1 0.60 0.63 0.92 0.91 0.89

13.53 9.1Ž12.41. 2.24 13.7 6.1 11.7 0 0 1.3 % 2.0 1.3 % 2.0 1.3 1.3 1.5 y

™

asymptotic behavior of the meson profiles: For r `, we have Q Ž r . ; expŽymp r .rmp r and s Ž r . ; Q 2 Ž r . Žas a consequence of the equations of motion.. Once the vertex function of the corresponding meson–nucleon interaction has been found, its appropriate contribution to the NN interaction may be

easily calculated by using well known techniques from OBE. The detailed formulas are given elsewhere w6,19x. In particular, the contribution of the s-meson exchange to the central potential is given by Vsc Ž r . s

Fig. 2. The sigma–nucleon form factor Gs N N Ž q 2 .r Gs N N Ž0.. The dashed and solid lines are for ms s 550 MeV and ms s 720 MeV, respectively. Typical OBE monopole fits with Ls s1.3 . . . 2 GeV are shown by the band enclosed by the dotted lines.

`

H0

k 2 dk Gs2 N N Ž k 2 . 2p 2

k 2 q ms2

j0 Ž kr . .

Ž 10 .

The central NN potential in the T s 0,S s 1 state Žthe deuteron state. is presented in Fig. 3 in comparison with Paris potential. Our prediction is in good agreement with the empirical one. Note that the desired attraction in the central VN N has been obtained before in the prv model by means of twomeson exchange w19x. In conclusion, it should be noted that we do not intend to describe all NN phase shifts staying solely in the framework of the present model. Besides other mesons, which are usually included in an OBE picture, the full model should also take in account e.g. NDr couplings. In addition the 2p exchange and its strong mixing with the s meson exchange Žsee e.g. Ref. w20x. should be considered. Another reason which limits the accuracy of NN phase shifts in the present model is that the s NN coupling is not sensitive to the mass of the sigma Žsee Table 2., whereas that is the case in the OBE phenomenology. In fact, even when the 2p exchange is disregarded,

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tor coupling is somewhat improved. In the twonucleon sector, this extended prvs Lagrangian leads to the correct intermediate range attraction in the central potential and a soft s NN formfactor for both values of the sigma meson mass, ms s 550 MeV and ms s 720 MeV, considered here.

Acknowledgements We would like to thank R. Machleidt for providing us with computer codes and M. Musakhanov for useful discussions. This work was supported in part by Deutscher Akademischer Austauschdienst ŽDAAD.. Fig. 3. The central potential in the Ss1, T s 0 state for prv and prvs models Ždashed and solid lines, respectively.. No contribution from two-meson exchange has been taken into account. The dotted line corresponds to the Paris potential w14x

References

the pure OBE model has to consider two types of sigmas with nearly the same masses but with quite different coupling constants. So, we refrain from performing direct calculations of NN phase shifts in the present model. Instead, we point out that the meson–nucleon form factors found in the present model could be useful in a wider context of calculations of nucleon–nucleon observables Žalso at finite temperatures or densities. and may give more information on meson–nucleon and nucleon–nucleon dynamics. To summarize, we have developed a topological chiral soliton model with an explicit light scalar–isoscalar meson field, which plays a central role in nuclear physics, based on the chiral symmetry and broken scale invariance of QCD. Although we used the parameters given by FTS, we do not stress that the dilaton field used in our model should have an anomalous scale dimension. On the contrary, we expect that it can be identified with the sigma obtained by pp rescattering or by analyzing pion-pion scattering phase shifts. We have shown that for the single nucleon properties, the successful description of the electromagnetic observables of the prv model is not modified and even the value for the axial-vec-

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