The effective action approach applied to nuclear matter (II) (*)
TRANHUUPHAT(**) and NGUYENTUANANH(***)International Centre for Theoretical Physics - Trieste, Italy (ricevuto il 31 Luglio 1997; approvato il 21 Ottobre 1997)
Summary. — Within the framework of the effective action approach we present the numerical calculations based on the approximation, in which all interacting meson propagators are replaced by their free ones. This is the Hartree-Fock (HF) improved approximation since it contains both the quantum corrections to the mean-field theory and the higher-order effects of the HF traditional method.
PACS 21.65 – Nuclear matter.
PACS 21.40 – Hartree-Fock and random-phase approximations.
1. – Introduction
It is well known that the effective action approach provides a general formalism for the non-perturbative phenomena of physical systems; it is the only one that yields, at the same time, the system of coupled Schwinger-Dyson equations (SDE) and the ground-state energy density. In a previous paper [1] (referred to as I hereafter) we applied this approach to nuclear matter study and obtained the following main results:
1) Higher-order contributions are naturally included. 2) The SDE for meson mixings are absent.
3) The method works well even if the meson self-energies are very large as was pointed out by [2].
4) The contributions from the ring diagrams for scalar and vector mesons are explicitly involved in the energy density.
Since the effective action approach deals with the numerical calculations, much more complicated than those carried out in the nuclear SDE method [3], we will
(*) The authors of this paper have agreed to not receive the proofs for correction.
(**) Address after December, 1996: Vietnam Atomic Energy Commission, 59 Ly Thuong Kiet, Hanoi, Vietnam.
(***) Permanent address: High College of Physics, Institute of Physics, Hanoi, Vietnam.
restrict ourselves, in this paper, to the approximation, in which all the interacting meson propagators are replaced by their free ones and the vacuum polarization, furthermore, is not discussed. Therefore, the physical meaning of our numerical result is expressed in the fact that it provides the quantum corrections to the mean-field approximation (i.e. Hartree) or the higher-order effects of the HF traditional method.
As was suggested in [4-6], we confine our consideration to contributions from the density-dependent part of nucleon propagator, that is dominant at low density.
The present paper is organized as follows. In sect. 2 two free parameters gs and gv are defined so as to reproduce the nuclear saturation point. After fixing the coupling constants we calculate successively the saturation curve and the density dependence of the quantum part of nucleon self-energy. The conclusion and discussion are given in sect. 3.
2. – Numerical results
In this section we continue [3] to do the numerical calculations for the formulae obtained in [1]. First the condensed meson fields given by eqs. (2.1) and (2.2) are determined from the nucleon and scalar densities. By using these initial nucleon self-energies we calculate the nucleon propagator G(k) defined by eq. (2.4) and then the density dependent part SD(k) of the nucleon self-energies from eqs. (2.3) and (2.6). This iterating cycle is continued until the saturated values are attained. After the nucleon self-energy is fixed, the energy density of nuclear matter in
Fig. 1. – Contour map of binding energy per nucleon in the plane (gs, gv). The binding energy is
eqs. (3.6)-(3.9) of I is determined s 42l gs m2 s
d4 q ( 2 p)4 tr [G(q) ] 4i gs m2 s rS, (2.1) A04 2il gv mv2 d4q ( 2 p)4tr [g0G(q) ] 4 gv mv2 rB, (2.2) SD(k) 4SsD(k) 1SvD(k) , (2.3a) Ss D(k) 4igs2 d4q ( 2 p)4k
l ms2 tr GD(q) 1GD(q) C(q 2k)l
, (2.3b) Sv D(k) 42igv2 d4 q ( 2 p)4k
g4 l m2 v tr g4GD(q) 1gmGD(q) gnDmn(q 2k)l
, (2.3c) G21(k)4G21 0 (k) 2S(k)42igjkj[ 11Sv(k) ]2ig4[k41S4(k) ]2[M1Ss(k) ] , (2.4) k *i 4 ki[ 1 1Sv(k) ] , (2.5a) k *4 4 k41 S4(k) , (2.5b) M * (k) 4 [M1Ss(k) ] . (2.5c) GD(k) 4 [2igmk *m 1 M * (k) ] pi E * (k)d(
k *0 2 E * (k))
u(k0) u(kF2 NkN) 2 (2.6) 2 i 2g 2 v d4p ( 2 p)4 d4k ( 2 p)4tr [gmG(p) gnG(k) Dmn(p 2k) ] , E04 l p2 0 kF q2dq E * (q) , (2.7) rS4 l p2 0 kF q2dqM * (q) E * (q) , (2.8) rB4 lkF3 3 p2 . (2.9) Ebin4 E 2 M . (2.10)Two free parameters gs and gv are adjusted to reproduce the nuclear saturation point, the binding energy is 215.8 MeV at a density corresponding to a Fermi momentum kF4 1.42 fm21. The contour map of the binding energy per nucleon at the fixed density kF4 1.42 fm21 is shown in fig. 1. The line AB, corresponding to Ebin4 4 215.8 MeV, is approximately described by the equation gv4 1.26( gs2 7 ) 1 7.45.
Fig. 2. – Saturation curve, calculated by means of the values gs4 8.54, gv4 9.39.
The coupling constants gs and gvcan be defined uniquely by the condition that the binding energy 215.8 MeV is minimum at the normal density kF4 1.42 fm21. Their values are fixed to be gs4 8.54, gv4 9.39.
The saturation curve, calculated by means of the obtained values for gs and gv, is plotted in fig. 2. This result is similar to that of the nuclear SDE method [3] with slightly different values of coupling constants.
In order to have a comparison between different models, the values of coupling constants, effective nucleon mass and the energies calculated in four models: Hartree (H), HF, nuclear SDE method (NSD) and our method (EA), are listed in table I.
Next, the density dependence of the quantum part for three components Ss, So, Sv in the Dirac decomposition of the nucleon self-energies (see eqs. (3.1a)-(3.1c) in [1]) is considered for gsand gvgiven above.
TABLE I. – Coupling constants, effective nucleon mass, and the energies calculated in four models: H, HF, NSD and EA. The coupling constants are determined to reproduce the saturation point Ebin4 215.8 MeV at kF4 1.42 fm21. The energies are in units of MeV: nucleon
energy eN, two-tadpole energy from sigma condensed field esT, two-tadpole energy from omega
condensed field ev
T, correlation energy from sigma meson esC, and correlation energy from omega
meson ev C. gs gv M * eN esT evT esC evC H 9.57 11.670 524 895.5 191.9 2 164.0 0 0 HF 9.159 10.455 492 898.6 173.1 2 131.8 2 36.9 20.2 NSD 8.714 10.678 500 899.3 156.0 2 137.5 8.5 2 3.1 EA 8.54 9.39 685.4 779.5 150.5 2 106.3 2 32.1 17.2
Fig. 3. – a) Density dependence of Sscontributed from the sigma meson; b) density dependence
of Socontributed from the sigma meson.
The density dependence of Ss and Socontributed from the sigma meson are shown respectively in fig. 3.
The density dependence of Ssand Socontributed from the omega meson are shown respectively in fig. 4.
In both cases the contributions to Svare negligibly small.
The contributions from both sigma and omega mesons to Ss and So provide the density dependence as shown in fig. 5.
Fig. 4. – a) Density dependence of Sscontributed from the omega meson; b) density dependence
Fig. 5. – a) Density dependence of Sscontributed from the sigma and omega mesons; b) density
dependence of Socontributed from the sigma and omega mesons.
In comparison with the results quoted in [3] it follows from figs. 3-5 that our results are closer to those based on the HF traditional method than NSD but the numerical values for coupling constants, effective nucleon mass and the energies calculated in EA are in between the corresponding values obtained by HF and NSD as indicated in table I.
3. – Conclusion and discussion
In this paper we carried out the numerical calculations based on the bare-vertex approximation and the approximation, in which all interacting meson propagators are replaced by their free ones within the framework of the effective action. We calculated the coupling constants, the effective nucleon mass and the energies. Our results are good enough compared with those obtained using HF and NSD methods. However it is worth emphasizing that due to the largeness of the meson self-energies, as was shown in [2, 3], the free meson propagator approximation is not acceptable. Indeed, the ring diagram contribution to the energy density has the form
(
see eq. (3.5) in [1])
1 2
d4p ( 2 p)4tr ]ln [11Ps(p) C(p) ] 2Ps(p) C(p)( , 1 2 d4p ( 2 p)4tr ]ln [11Pma(p) Dan(p) ] 2Pma(p) Dan(p)( ,which cannot be neglected, here C(p) and Dan are, respectively, the propagators of
sigma and omega mesons, their expressions are given by eqs. (2.16) and (2.17) in [1]. Our next paper will be devoted to removing this shortcoming.
* * *
The authors would like to express their sincere gratitude to the International Centre for Theoretical Physics for the hospitality extended to them. The financial support of Vietnam National Program for Fundamental Research and of the World Laboratory is acknowledged with thanks.
R E F E R E N C E S
[1] TRANHUUPHATand NGUYENTUANANH, Nuovo Cimento A, 110 (1997) 475. [2] CELENZAL. S., PANTZINSA. and SHAKINC. M., Phys. Rev. C, 45 (1991) 205.
[3] NAKANOM., HASEGAWAA., KOUNOH. and KOIDEK., Phys. Rev. C, 49 (1994) 3061, 3076. [4] ALLENDESM. P. and SEROTB. D., Phys. Rev. C, 45 (1992) 2975.
[5] KREING., NIELSERM., PUFFR. D. and WILETSL., Phys. Rev. C, 47 (1993) 2485. [6] PRAKASHM., ELLISP. J. and KAPUSTAJ. I., Phys. Rev. C, 45 (1992) 2518.
