Field theoretical description of exchange terms in nuclear systems

ELSEVIER

Nuclear

Field theoretical M. Serra”,

Physics

description

T. Otsukaabc,

A722

(2003)

of exchange

Y. Akaishid,

502c-506~

terms in nuclear

systems

P. Ringe and S. Hirose”

“Department of Physics, University of Tokyo, Hongo, Bunkyo-ku, Tokyo, 113-0033, Japan bCenter for Nuclear Hongo, Bunkyo-ku,

Study, Tokyo,

‘RIKEN,

Wako-shi,

dKEK, ePhysics

Hirosawa,

Oho, Tsukuba-shi, Department,

University 113-0033, Saitama

Ibaraki

Technical

of Tokyo, Japan 351-0198,

305-0801, University

Japan

Japan Munich,

85747 Garching,

Germany

In this work we suggest a possible origin of the a-meson in nuclear systems in terms of the bare Nucleon-Nucleon interaction. Starting from an effective interaction determined by a G-matrix calculation, we show that the intermediate range of this effective interaction can be described to a good approximation by an exchange of scalar mesons, whose fitted masses and coupling constants are in good agreement with the values of standard relativistic mean field parametrization. 1. INTRODUCTION Relativistic Mean Field (RMF) models are phenomenological approaches to the nuclear many-body problem. In this framework, nucleons are treated as Dirac spinors and they can interact amongst themselves only-by the exchange of effective particles, named mesons [I]. In spite of their very good predictive power for the description of ground state properties of stable and exotic nuclei [2], their success as well as their foundation have not been completely understood yet. In RMF the most important meson for the attraction amongst the nucleons is the isoscalar scalar a-meson whose mass and coupling constant are determined through a fit to nuclear data. Although the a-meson is usually interpreted as a resonance in pion-pion scattering, up to now the connection between this resonance and the 0 meson of RMF has not been shown. Furthermore, ab-initio calculations have shown that the nuclear binding mainly originates from the tensor force of the bare NucleonNucleon (NN) interaction [3], whereas in RMF it is only due to the a-meson. In this work we propose a novel interpretation of a-meson of RMF. Starting from a bare NN interaction and by employing a G-matrix formalism, we show that the o-meson of RMF can be connected to the tensor force of the bare NN interaction. In addition, with this analysis we are able to investigate the density dependence and some isospin properties of the resulting effective G-matrix in its intermediate attractive range. This manuscript is organized as follows. In Section 2 the properties of the effective potentials obtained by the 0375-9474/03/$ - see front matter doi:lO.l016/S0375-9474(03)01416-7

0 2003 Elsevier

Science

B.V.

All rights

reserved.

M. Serra

et al. /Nuclear

Physics

A722

(2003)

502c-506~

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G-matrix calculations are briefly presented, and, in particular their density dependence is discussed. In Section 3 the effective G-matrix interactions are fitted to one-mesonexchange potentials and the results are discussed. A brief summary and some conclusions are given in Section 4.

2. G-MATRIX

EFFECTIVE

POTENTIALS

In this study we consider the G-matrix interaction for symmetric nuclear matter derived from the NN interaction of Ref. [4] by employing the method of Nagata, Bando and Akaishi [5]. Using this method the G-matrix interaction may be written as a two-body potential, function of the inter-nucleon distance and of the spin-isospin degrees of freedom. The two-nucleon state on which the NN interaction is acting on, is labeled by its total spin, singlet (s) or triplet (t), and by its reflection symmetry in the coordinate (even (e) or odd (0)). A two-body potential with spin-isospin dependences may be written as V(l,

2) = L(T)IqrI;

+ Ke(7-)n:rI;

+ v&(?-)II;n~

f v,o(7-)II;II;

(1)

where II: (u = s, t, e, o and u = a,p) is the projection operator of the spin (0) and parity (p) properties mentioned above. In this study, Vacu’s are central potentials given a as function of the inter-nucleon distance T. Because in the present investigation we are interested in the intermediate attractive range of the effective interaction, the odd potentials may be neglected. Furthermore, it is important to remark that the tensor term of the NN force gives a very large contribution to the attraction in the 3E state [6]. The potentials V,, and V,, are shown respectively in the left and in the right panels of Figure 1. Minima of about -115 MeV for V,, and -90 MeV for V,, at normal density are found at T M 1 fm. For T 2 1.5 fm these potentials decrease to zero with a Yukawa-function behaviour. The short-range (T < 0.8 fm) repulsion is due to short-range correlations. Let us now consider the density dependence of these two potentials in the intermediate-long range T > 1 fm. For both the states the depth of the potential increases at lower densities. For each /CF, V,, is more attractive than V,,, and, in addition, shows a stronger density dependence. This result, well known from the Brueckner theory, implies that a large part of the density dependence of the effective interaction originates from the tensor term of the NN interaction. In particular, at the minimum of attraction T s 1 fm, the difference between the potentials calculated at the densities 0.6 and 1.36 fm-’ is about 50 MeV for 3E and only about 20 MeV for iE.

3. FIT TO A YUKAWA

POTENTIAL

In this section the attractive intermediate is fitted to a one-meson exchange potential I&(?-)

= hc$~

range of the effective

potentials

V,,

and V,,

(4

in which p and g are the mass and the coupling constant of an efIective meson. In the oneboson-exchange model of the NN interaction, the attractive parts of V,, and V,, originate from the exchange of a isoscalar scalar and of a isovector scalar effective field, i.e. a aand a &meson respectively. Assuming, as a first approximation, that these fields have

M. Serra et al. /Nuclear

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100

3E with tensor

50 w2

0

g

-50

s -k>

-100 -150

0

1

- -

k,=0.6 fm-’

2

3

r (fm>

I 40

‘-\/ / 1

- -

k,=O.6 fm-’

2

3

4

r (fm>

Figure 1. Lejl: Singlet-even (‘E) potential obtained by the G-matrix calculation for densities denoted by the Fermi momenta k~ = 1.36 (solid), 1.0 (dashed), 0.6 (longdashed) fm-‘. Right: Triplet-even (3E) potential with effects of the renormalized tensor interaction included. See also previous caption.

the same masses and coupling constants, the projection operators for the isospin in the ‘E and in the 3E states

in which Ti and fZ are the the two fields to the total (scalar term) contributes weight in the 3E state. If

isospin operators, can give an estimation of the contribution of attractivelpotential. From Eq. (3) it follows that the o-meson mostly to ‘E *state, while the S-meson (Ti . &) has the largest the assumption of a one-meson-exchange potential (Eq. (2)) for

M. Sevra et al. /Nuclear Physics A722 (2003) 502c-506~

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Figure 2. Function M/(r) (defined in Eq. (4)) as a function of T for V,, at normal density (filled diamond). The solid and dashed lines result from a fit of Vt, to a superposition of two Yukawa-functions.

the effective interactions V,, is correct, then the following function

should be, approximately, a straight line with a tangent -p for this range. In Figure 2 W(r) is shown for V,, for normal density. We find that, although W(T) has a linear behavior, we need a superposition of two Yukawa-functions with different masses and coupling constants to fit the 3E potential. An analogous result is found for the ‘E state. In Table 1 the numerical values for masses ~1, pz, and coupling constants gr, and g2 of the isoscalar scalar mesons are given for the triplet-even and for the singlet-even states at normal density. The parameter p1 is about 590 MeV in both the cases, while g1 is larger for 3E than for ‘E. This is a consequence of the larger attraction produced by I/t,. On the other side, pz and g2 are found to be the same in both the states. In particular, m2 is about 200 MeV and it may represent more complicated processes than one-meson-exchange. In order to make a comparison with the o-meson of RMF, we consider the parameters of the first interval. Concerning the mass, the value obtained by the fit is 590 MeV, comparable with the values of standard non-linear relativistic mean field models (500 - 550 MeV). Concerning the coupling constant, combining the g1 of the triplet-even and singlet-state with the corresponding isospin factors, we find the value 10.4 for the coupling constant of the scalar meson. This is also in very good agreement with the standard parameters of RMF models.

506~

M. Serra et al. /Nuclear

Physics A722 (2003) 502c-506~

Table 1 Masses and coupling constants pi, ,LL~,gi, and ~2 resulting from the fit of the 3E and of the iE potentials in terms of the superposition of two Yukawa potentials (see also discussion in the text).

4. CONCLUSIONS

AND

OUTLOOK

In this work the attractive parts of G-matrix interactions for infinite nuclear matter have been investigated and fitted to one-meson-exchange potentials. In such a way we have been able to suggest a possible origin of the a-meson in nuclear systems. In particular we have shown that it can be connected to the tensor force of the bare NN interaction, and therefore RMF calculation become more consistent with ab-initio approaches to the nuclear many-body problem. Furthermore, the values of the mass and coupling of the scalar meson constant determined by fitting the effective G-matrix interactions are very close to the values used in successful parametrizion of RMF. Another interesting point resulting from this analisys is the strong density dependence connected to the 3E state. This could mean that in a RMF model the S-meson could play a role for exotic nuclei at low energy regime.

REFERENCES 1. 2. 3. 4. 5. 6.

J.D. Walecka, Ann. Phys. (N.Y.) 83, 491 (1974); B.D. Serot and J.D. Walecka, Adv. Nucl. Phys. 16, 1 (1986). P. Ring, Progr. Part. Nucl. Phys. 37, 193 (1996) B.S. Pudliner, V.R. Pandharipande, J. Carlson, S.C. Pieper and R.B. Wiringa, Phys. Rev. C56, 1720 (1997) R. Tamagaki and T. Takatsuka:Prog. Theor. Phys. 105, 1059 (2001). S. Nagata, H. Bando and Y. Akaishi, Prog. Theor. Phys. Suppl. 65, 10 (1979). Y. Akaishi, Int. Rev. Nucl. Phys. Vo1.4, 259 (1986)