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On the effective σ-boson exchange in the relativistic Dirac-Brueckner approach to nuclear matter
Volume 160B, number 6
PHYSICS LETTERS
17 October 1985
O N T H E EFFECTIVE o-BOSON EXCHANGE IN T H E liELATIVISTIC DIRAC-BRUECKNER A P P R O A C H T O NUCLEAR MATIqER R. M A C H L E I D T 1 TRII.n~ 4004 Wesbrook Mall, Vancouver, B.C., Canada V6T 2A3 and Institut fiir Theoretische Kernphysik der Universiti~t Bonn, Nussallee 14-16, D-5300 Bonn, West Germany
and R. B R O C K M A N N Institute of Theoretical Physics, University of Regensbur~ D-8400 Regensburg West Germany Received 30 May 1985 A relativistic form of the Brueckner theory of nuclear matter is applied to an extended meson-exchange model for the NN-interaction which contains explicit 2*r-and *r0-exchange. This model avoids the effective o-boson which is characteristic of the simplified meson exchange, as e.g. the one-boson-exchange(OBE) potential. It turns out that the relativistic saturation effects found earlier within the OBE model are confirmed by the extended and more realistic model. In particular it is found that the relativistic effects caused by the explicit 2*r- and *r0-exchange are well simulated by the effective o-boson of the OBE model.
It is well known that the saturation properties of nuclear matter cannot be explained in the framework of conventional non-relativistic nuclear physics [1]. T o overcome this problem, a relativistic extension of the non-relativistic B r u e c k n e r - H a r t r e e - F o c k approach was first pursued by the Brooklyn group [2]. The main difference of these relativistic calculations compared to non-relativistic ones is the explicit treatment of the lower components of the nucleon spinor wave functions. It turns out that the ratio of the upper to the lower component for a free nucleon is different from that for a nucleon in the medium. The lower component of the nucleon spinor in the medium is determined by the D i r a c - H a r t r e e - F o c k equations. Shakin and coworkers [2] started with one of the earlier relativistic OBEPs of the Bonn group; they worked out the binding energy per nucleon as a function of density for nuclear matter and found large
1 Supported in part by Deutsche Forschungsgemeinschaft.
364:
relativistic corrections. However, they treated the medium modifications of the small components of the nucleon spinors only in first order perturbation theory. Horowitz and Serot [3] solved the relativistic Bethe-Goldstone equation, treating the small components fully self-consistently. However, they did not start with a quantitative free NN-interaction but rather with Walecka's effective interaction, consisting of an isoscalar scalar and vector boson exchange o n l y . Finally, in ref. [4], the Dirac-BruecknerH a r t r e e - F o c k (DBHF) equations were solved correctly, starting from modern OBEPs which fit recent N N phase shifts accurately and which are based on the present state of the art of meson theory. The lower components of the nucleon wave functions were treated fully self-consistently. It turned out that these calculations yield results for the binding energy and saturation density which are located on a curve (a new Coester line) which passes through the empirically determined saturation point (E/A = - 1 5 . 6 MeV and K F = 1.36
0370-2693/85/$ 03.30 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
Volume 160B, number 6
PHYSICS LETTERS 0
fm-1). Ref. [4] presents an OBEP which, for the first time, fits phase shifts of free NN-scattering on the one hand and describes nuclear matter in DBHF quantitatively on the other hand. The OBEP consists essentially of ¢r-, o-, o~- and p-exchange. The o-boson provides the intermediate range attraction between the two nucleons and contributes a major part to the relativistic effects in nuclear matter. However, one must raise the question whether the relativistic effects in nuclear matter survive, when the o-boson is replaced by the explicit two-meson-exchange mechanisms (2¢r- and ¢rp-exchanges) which it simulates [5]. It is the aim of this paper to answer this question. The basic quantity of Brueckner theory [6] is the reaction matrix, G, satisfying an integral equation, which reads in operator notation
. . . .
r
where V denotes the NN-potential (or more general: the kernel of irreducible contributions to the NN-interaction), Q the Pauli projector and 1 / e the two-nucleon propagator in the medium. G is the infinite sum of ladder diagrams obtained by the successive application of V in strict analogy with the definition of the T-matrix in scattering. Within Brueckner theory the energy per nucleon in nuclear matter as a function of the density is in lowest order in G: +
½(G)/A,
(2)
where A is the number of particles, K F the Fermi momentum and T the kinetic energy operator. In conventional Brueckner calculations the operators in eq. (1) act on free nucleon plane wave spinors. Characteristic results obtained in this scheme are displayed in fig. 1. The squares and circles denote the saturation minima of conventional Brueckner calculations, using different NN-interactions. The numbers 1-5 denote different NN-interactions, which are described in refs. [4,7-9], e.g., number I corresponding to ref. [7] ,i, number 2 to ref. [8] ,2. The squares in fig. 1 ,t Ref. [7], ignoring mesonic effects. ,2 Number 3 by the OBEP A of ref. [4]; number 4: ref. [9], including mesonic effects; number 5: ref. [9], ignoring mesonic effects.
.
.
.
.
i
-5
/ OBEP t 2 ~,FULL °~//.~//MODEL
Eli I I0
~_
D3 Z
w
-20
\xx, \\ \'\ \,
-25
0 [] 3 4 04
\\,
05
\k \~\.
OBEP
-3o
")~,\
-35
MODEL
05
~,'/,
FULL ~ " ( ~ ' I
I
I
I
I
I
I
I
1.5
(])
G = V - V(Q/e)G,
E / A ( KF) = ( T ) / A
17 October 1985
/
~
I
I
-
I
2.0
KF(fm -I) Fig. 1. Energy per nucleon, E/N, in nuclear matter versus the Fermi momentum, K p Comparison of the Dirac-Brueckner (full and long dash-dot curve) and the conventional Brueckner approach (dash and small dash-dot curve) for the full meson-exchange model of fig. 2 and an OBEP, respectively. More details are described in the text.
are used when free nucleon energies were applied for the particle spectrum above the Fermi surface. A circle with the same number as a square refers to a corresponding calculation, using a continuous single particle spectrum. These results form a "Coester band" which does not meet the empirical range of nuclear matter saturation denoted by the shaded rectangle. The essential idea of the relativistic extension of Brueckner theory is to realize that the nucleons in nuclear matter are exposed to a strong average scalar and vector field. They should therefore not be treated as free particles. The nucleon spinors in nuclear matter satisfy the following Dirac equation (¢~ - M -
~.)~(k,
s ) = 0,
(3)
with
E = A(k) + yoB(k),
(4)
the self-energy operator, and A and B the 365
Volume 160B, number 6
PHYSICS LETTERS
attractive scalar and repulsive vector potential in nuclear matter, respectively. With )~r - M + A, the solutions are
fi ( k , s ) = ( E + )(t I1/2 ( E° '+* 2P~ /
) x,
(5)
Table 1 OBEP in time-ordered perturbation theory approximatingthe Bonn meson model for the NN-interaction[5]. A form-factor (A~ - m2)/(A2 +p2), where p is the three-momentumof the exchanged meson, is applied at each meson-nucleonvertex. For formalism and explicit formulae see ref. [9]. Meson
Meson mass m, (MeV)
g2(t = m2) (f/g)
A,~(GeV)
~r r/ o 8 to #
138.0 548.8 550.0 983.0 782.6 769.0
14.4 5.0 8.88 1.05 20.0 0.9 (6.1)
2.0 2.0 2.0 2.0 2.0 2.0
with J~ -= (k 2 + 1~7/2)1/2 and Xs a Pauli spinor. The normalization is
fi+(k,s)fi(k,s)= l.
(6)
In contrast to NN-scattering, where free Dirac spinors are used, the meson-exchange contributions to the NN-interaction in nuclear matter are now evaluated by spinors of the type eq. (5), where A is determined self-consistently. (For more details see ref. [4]). This leads to the before mentioned change of the ratio between upper and lower components of the Dirac spinors as compared to the free ones. Since A is density dependent, the interaction between two nucleons in nuclear matter becomes density dependent, too. Results for OBEPs, applied in the framework of this extended Brueckner theory, were given in ref. [4]. In this paper, we replace the OBEP by an extended model for the NN-interaction which contains explicit 2~r- and ~rp-exchange (see fig. 2) and which replaces the effective o-boson of the OBE model. It is based on the Bonn model [5] which, for these nuclear matter calculations, was slightly simplified to keep the computing time within reasonable limits. The simplifications take advantage of the fact that part of the crossed two-boson-exchange diagrams cancel each other
.
(al
.
.
.
.
17 October 1985
to a large extent. The crossed 2rr-exchange diagrams are effectively included by an increase of the NA~r coupling from f 2 a , = 0.23-0.36. These modifications are done such that the fit of the N N phase shifts is retained. Furthermore, the absolute size of the uncorrelated and correlated 27r-exchange is kept in agreement with the original model [5]. The results obtained with our extended mesonexchange model for the NN-interaction in the Dirac-Brueckner approach to nuclear matter (fig. 1, full line) show that the empirical nuclear matter properties can be described quantitatively. We repeated the calculations with an OBEP defined in table 1, which represents the OBE approximation to the latest Bonn model [5] and obtained
I: :1 + I: :l + I ...... I
(c)
Vv-1 (b) Fig. 2. Meson-exchangecontribution to the NN-interaction: (a) one-meson-exchange,(b) 2*r-exchange;the double line stands for intermediate A(1232)-isobars; the o' denotes the *r~rs-waveinteraction; (c) ~rp-exchange.The sum of these three contributions defines the kernel of the interaction denoted by "full model" in this work.
366
Volume 160B, number 6
PHYSICS LETTERS
-I0
-20
ALL:M >~
-50
BQ.-40 W
-50
-60
J
J
u
. i
,.2
1.3
J
i
~.4
1.5
"h A L L : M ~.6
K F (fm -I) o10
b ALL:M
-20
-30 v _40 W
-50
-60
I.I
i
l
,
,
\)ALL
1.2
1.3
1.4
1.5
1.6
M
K v (fm H)
Fig. 3. "Potential energy", Er,ot, versus Fermi momentum(a) shows the results for the full modeldefinedin fig. 2; in (b) the OBE-approximation is used. Contributions from different meson exchangesare displayed. More details are describedin the text. quite similar results (fig. 1, long dash-dot line). The other two curves in fig. 1 were obtained with a G-matrix, which was evaluated using free plane wave spinors instead of the self-consistently determined spinors of eq. (5). In fig. 3, the potential energy per nucleon, Epot (that is the second term on the fight-hand side of
17 October 1985
eq. (2)) is displayed as a function of Fermi momentum. Fig. 3a shows the results for the full model defined in fig. 2, while in fig. 3b the OBE approximation is used. The consideration starts with the lower full line labelled "All : M", which displays the results for the conventional Brueckner calculation. On top of this potential energy_ denoted by "All M " we add e.g. in case of "to : M " the difference in the potential energy from to-meson-exchange evaluated with Dirac spinors using 37/and M, respectively. When comparing fig. 3a and fig. 3b, two interesting points arise. First the Dirac effect on the (2~r + ~rp) contribution to the full model is quite similar to that of the OOBE of the OBEP, whereas the effect on the 2 ~r-exchange alone is rather large (dash-dot line, fig. 3a). The effects on the 2~r- and ~rp-exchange are essentially caused by the correct normalization, eq. (6), of the Dirac spinors, eq. (5), involved in the evaluation of those diagrams. In case of the incorrect normalization, ~(k, s) × ~(k, s) = 1, there would be no relativistic medium effect on the 2~r-exchange. Secondly, the potential energy per nucleon is the same for the extended mesonexchange model, called full model, and the OBEP. Note that in this work we apply time-ordered perturbation theory like in ref. [5]. Therefore, for reasons of consistency, the OBEP used here is also constructed in time-ordered perturbation theory. This leads to an energy-dependent potential, which differs off-shell from the prescriptions obtained for the three-dimensional relativistic reductions suggested by Blancenbecler and Sugar [10] or Thompson [11]. In our earlier work [4], the latter approach was used which results in an energy-independent potential, which is more convenient in applications to nuclear structure. Due to the off-shell differences the size of the relativistic effects differs for the two OBEPs. However, the quality of the effect is the same, leading to the correct nuclear matter saturation in both cases. In conclusion it is found that the relativistic effects caused by the explicit 2~r- and ~rp-exchanges are well simulated by the effective o-boson of the OBE model. Furthermore, the relativistic saturation effect of the Dirac-Brueckner approach, which we found eaFlier within the OBE model [4], 367
Volume 160B, number 6
PHYSICS LETTERS
m a i n t a i n s w h e n a realistic a n d e x t e n d e d m e s o n e x c h a n g e m o d e l is applied.
O n e o f t h e a u t h o r s (R.B.) w o u l d like to t h a n k W. Weise for many helpful comments.
References [1] B.D. Day, Phys. Rev. C24 (1981) 1203; Phys. Rev. Lett. 47 (1981) 226. [2] M.R. Anastasio, L.S. Celenza and C.M. Shakin, Phys. Rev. C23 (1981) 2273; M.R. Anastasio, L.S. Celenza, W.S. Long and C.M. Shakin, Phys. Rep. 100 (1983) 327. [3] C.J. Horowitz and B.D. Serot, Phys. Lett. 137B (1984) 287.
368
17 October 1985
[4] R. Brockmann and R. Machleidt, Phys. Lett. 149B (1984) 283; R. Machleidt and R. Brockmann, contrib. LAMPF Workshop on Dirac approaches to nuclear physics (Los Alamos, January 1985); R. Brockmann and R. Machleidt, to be published. [5] R. Machleidt, in: Quarks and nuclear structure, ed. K. Bleuler, Lecture Notes in Physics, Vol. 197 (Springer, Berlin, 1984), p. 352; R. Machleidt, K. Holinde and C. Elster, to be published. [6] B.D. Day, Rev. Mod. Phys. 39 (1967) 719; 50 (1978) 495. [7] R. Machleidt and K. Holinde, Nucl. Phys. A350 (1980) 396. [8] K. Holinde and R. Machleidt, Nucl. Phys. A247 (1975) 495. [9] K. Kotthoff, K. Holinde, R. Machleidt and D. Schi~tte, Nud. Phys. A242 (1975) 429. [10] R. Blancenbecler and R. Sugar, Phys. Rev. 142 (1966) 1051. [11] R.H. Thompson, Phys. Rev. D1 (1970) 1738.
PHYSICS LETTERS
17 October 1985
O N T H E EFFECTIVE o-BOSON EXCHANGE IN T H E liELATIVISTIC DIRAC-BRUECKNER A P P R O A C H T O NUCLEAR MATIqER R. M A C H L E I D T 1 TRII.n~ 4004 Wesbrook Mall, Vancouver, B.C., Canada V6T 2A3 and Institut fiir Theoretische Kernphysik der Universiti~t Bonn, Nussallee 14-16, D-5300 Bonn, West Germany
and R. B R O C K M A N N Institute of Theoretical Physics, University of Regensbur~ D-8400 Regensburg West Germany Received 30 May 1985 A relativistic form of the Brueckner theory of nuclear matter is applied to an extended meson-exchange model for the NN-interaction which contains explicit 2*r-and *r0-exchange. This model avoids the effective o-boson which is characteristic of the simplified meson exchange, as e.g. the one-boson-exchange(OBE) potential. It turns out that the relativistic saturation effects found earlier within the OBE model are confirmed by the extended and more realistic model. In particular it is found that the relativistic effects caused by the explicit 2*r- and *r0-exchange are well simulated by the effective o-boson of the OBE model.
It is well known that the saturation properties of nuclear matter cannot be explained in the framework of conventional non-relativistic nuclear physics [1]. T o overcome this problem, a relativistic extension of the non-relativistic B r u e c k n e r - H a r t r e e - F o c k approach was first pursued by the Brooklyn group [2]. The main difference of these relativistic calculations compared to non-relativistic ones is the explicit treatment of the lower components of the nucleon spinor wave functions. It turns out that the ratio of the upper to the lower component for a free nucleon is different from that for a nucleon in the medium. The lower component of the nucleon spinor in the medium is determined by the D i r a c - H a r t r e e - F o c k equations. Shakin and coworkers [2] started with one of the earlier relativistic OBEPs of the Bonn group; they worked out the binding energy per nucleon as a function of density for nuclear matter and found large
1 Supported in part by Deutsche Forschungsgemeinschaft.
364:
relativistic corrections. However, they treated the medium modifications of the small components of the nucleon spinors only in first order perturbation theory. Horowitz and Serot [3] solved the relativistic Bethe-Goldstone equation, treating the small components fully self-consistently. However, they did not start with a quantitative free NN-interaction but rather with Walecka's effective interaction, consisting of an isoscalar scalar and vector boson exchange o n l y . Finally, in ref. [4], the Dirac-BruecknerH a r t r e e - F o c k (DBHF) equations were solved correctly, starting from modern OBEPs which fit recent N N phase shifts accurately and which are based on the present state of the art of meson theory. The lower components of the nucleon wave functions were treated fully self-consistently. It turned out that these calculations yield results for the binding energy and saturation density which are located on a curve (a new Coester line) which passes through the empirically determined saturation point (E/A = - 1 5 . 6 MeV and K F = 1.36
0370-2693/85/$ 03.30 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
Volume 160B, number 6
PHYSICS LETTERS 0
fm-1). Ref. [4] presents an OBEP which, for the first time, fits phase shifts of free NN-scattering on the one hand and describes nuclear matter in DBHF quantitatively on the other hand. The OBEP consists essentially of ¢r-, o-, o~- and p-exchange. The o-boson provides the intermediate range attraction between the two nucleons and contributes a major part to the relativistic effects in nuclear matter. However, one must raise the question whether the relativistic effects in nuclear matter survive, when the o-boson is replaced by the explicit two-meson-exchange mechanisms (2¢r- and ¢rp-exchanges) which it simulates [5]. It is the aim of this paper to answer this question. The basic quantity of Brueckner theory [6] is the reaction matrix, G, satisfying an integral equation, which reads in operator notation
. . . .
r
where V denotes the NN-potential (or more general: the kernel of irreducible contributions to the NN-interaction), Q the Pauli projector and 1 / e the two-nucleon propagator in the medium. G is the infinite sum of ladder diagrams obtained by the successive application of V in strict analogy with the definition of the T-matrix in scattering. Within Brueckner theory the energy per nucleon in nuclear matter as a function of the density is in lowest order in G: +
½(G)/A,
(2)
where A is the number of particles, K F the Fermi momentum and T the kinetic energy operator. In conventional Brueckner calculations the operators in eq. (1) act on free nucleon plane wave spinors. Characteristic results obtained in this scheme are displayed in fig. 1. The squares and circles denote the saturation minima of conventional Brueckner calculations, using different NN-interactions. The numbers 1-5 denote different NN-interactions, which are described in refs. [4,7-9], e.g., number I corresponding to ref. [7] ,i, number 2 to ref. [8] ,2. The squares in fig. 1 ,t Ref. [7], ignoring mesonic effects. ,2 Number 3 by the OBEP A of ref. [4]; number 4: ref. [9], including mesonic effects; number 5: ref. [9], ignoring mesonic effects.
.
.
.
.
i
-5
/ OBEP t 2 ~,FULL °~//.~//MODEL
Eli I I0
~_
D3 Z
w
-20
\xx, \\ \'\ \,
-25
0 [] 3 4 04
\\,
05
\k \~\.
OBEP
-3o
")~,\
-35
MODEL
05
~,'/,
FULL ~ " ( ~ ' I
I
I
I
I
I
I
I
1.5
(])
G = V - V(Q/e)G,
E / A ( KF) = ( T ) / A
17 October 1985
/
~
I
I
-
I
2.0
KF(fm -I) Fig. 1. Energy per nucleon, E/N, in nuclear matter versus the Fermi momentum, K p Comparison of the Dirac-Brueckner (full and long dash-dot curve) and the conventional Brueckner approach (dash and small dash-dot curve) for the full meson-exchange model of fig. 2 and an OBEP, respectively. More details are described in the text.
are used when free nucleon energies were applied for the particle spectrum above the Fermi surface. A circle with the same number as a square refers to a corresponding calculation, using a continuous single particle spectrum. These results form a "Coester band" which does not meet the empirical range of nuclear matter saturation denoted by the shaded rectangle. The essential idea of the relativistic extension of Brueckner theory is to realize that the nucleons in nuclear matter are exposed to a strong average scalar and vector field. They should therefore not be treated as free particles. The nucleon spinors in nuclear matter satisfy the following Dirac equation (¢~ - M -
~.)~(k,
s ) = 0,
(3)
with
E = A(k) + yoB(k),
(4)
the self-energy operator, and A and B the 365
Volume 160B, number 6
PHYSICS LETTERS
attractive scalar and repulsive vector potential in nuclear matter, respectively. With )~r - M + A, the solutions are
fi ( k , s ) = ( E + )(t I1/2 ( E° '+* 2P~ /
) x,
(5)
Table 1 OBEP in time-ordered perturbation theory approximatingthe Bonn meson model for the NN-interaction[5]. A form-factor (A~ - m2)/(A2 +p2), where p is the three-momentumof the exchanged meson, is applied at each meson-nucleonvertex. For formalism and explicit formulae see ref. [9]. Meson
Meson mass m, (MeV)
g2(t = m2) (f/g)
A,~(GeV)
~r r/ o 8 to #
138.0 548.8 550.0 983.0 782.6 769.0
14.4 5.0 8.88 1.05 20.0 0.9 (6.1)
2.0 2.0 2.0 2.0 2.0 2.0
with J~ -= (k 2 + 1~7/2)1/2 and Xs a Pauli spinor. The normalization is
fi+(k,s)fi(k,s)= l.
(6)
In contrast to NN-scattering, where free Dirac spinors are used, the meson-exchange contributions to the NN-interaction in nuclear matter are now evaluated by spinors of the type eq. (5), where A is determined self-consistently. (For more details see ref. [4]). This leads to the before mentioned change of the ratio between upper and lower components of the Dirac spinors as compared to the free ones. Since A is density dependent, the interaction between two nucleons in nuclear matter becomes density dependent, too. Results for OBEPs, applied in the framework of this extended Brueckner theory, were given in ref. [4]. In this paper, we replace the OBEP by an extended model for the NN-interaction which contains explicit 2~r- and ~rp-exchange (see fig. 2) and which replaces the effective o-boson of the OBE model. It is based on the Bonn model [5] which, for these nuclear matter calculations, was slightly simplified to keep the computing time within reasonable limits. The simplifications take advantage of the fact that part of the crossed two-boson-exchange diagrams cancel each other
.
(al
.
.
.
.
17 October 1985
to a large extent. The crossed 2rr-exchange diagrams are effectively included by an increase of the NA~r coupling from f 2 a , = 0.23-0.36. These modifications are done such that the fit of the N N phase shifts is retained. Furthermore, the absolute size of the uncorrelated and correlated 27r-exchange is kept in agreement with the original model [5]. The results obtained with our extended mesonexchange model for the NN-interaction in the Dirac-Brueckner approach to nuclear matter (fig. 1, full line) show that the empirical nuclear matter properties can be described quantitatively. We repeated the calculations with an OBEP defined in table 1, which represents the OBE approximation to the latest Bonn model [5] and obtained
I: :1 + I: :l + I ...... I
(c)
Vv-1 (b) Fig. 2. Meson-exchangecontribution to the NN-interaction: (a) one-meson-exchange,(b) 2*r-exchange;the double line stands for intermediate A(1232)-isobars; the o' denotes the *r~rs-waveinteraction; (c) ~rp-exchange.The sum of these three contributions defines the kernel of the interaction denoted by "full model" in this work.
366
Volume 160B, number 6
PHYSICS LETTERS
-I0
-20
ALL:M >~
-50
BQ.-40 W
-50
-60
J
J
u
. i
,.2
1.3
J
i
~.4
1.5
"h A L L : M ~.6
K F (fm -I) o10
b ALL:M
-20
-30 v _40 W
-50
-60
I.I
i
l
,
,
\)ALL
1.2
1.3
1.4
1.5
1.6
M
K v (fm H)
Fig. 3. "Potential energy", Er,ot, versus Fermi momentum(a) shows the results for the full modeldefinedin fig. 2; in (b) the OBE-approximation is used. Contributions from different meson exchangesare displayed. More details are describedin the text. quite similar results (fig. 1, long dash-dot line). The other two curves in fig. 1 were obtained with a G-matrix, which was evaluated using free plane wave spinors instead of the self-consistently determined spinors of eq. (5). In fig. 3, the potential energy per nucleon, Epot (that is the second term on the fight-hand side of
17 October 1985
eq. (2)) is displayed as a function of Fermi momentum. Fig. 3a shows the results for the full model defined in fig. 2, while in fig. 3b the OBE approximation is used. The consideration starts with the lower full line labelled "All : M", which displays the results for the conventional Brueckner calculation. On top of this potential energy_ denoted by "All M " we add e.g. in case of "to : M " the difference in the potential energy from to-meson-exchange evaluated with Dirac spinors using 37/and M, respectively. When comparing fig. 3a and fig. 3b, two interesting points arise. First the Dirac effect on the (2~r + ~rp) contribution to the full model is quite similar to that of the OOBE of the OBEP, whereas the effect on the 2 ~r-exchange alone is rather large (dash-dot line, fig. 3a). The effects on the 2~r- and ~rp-exchange are essentially caused by the correct normalization, eq. (6), of the Dirac spinors, eq. (5), involved in the evaluation of those diagrams. In case of the incorrect normalization, ~(k, s) × ~(k, s) = 1, there would be no relativistic medium effect on the 2~r-exchange. Secondly, the potential energy per nucleon is the same for the extended mesonexchange model, called full model, and the OBEP. Note that in this work we apply time-ordered perturbation theory like in ref. [5]. Therefore, for reasons of consistency, the OBEP used here is also constructed in time-ordered perturbation theory. This leads to an energy-dependent potential, which differs off-shell from the prescriptions obtained for the three-dimensional relativistic reductions suggested by Blancenbecler and Sugar [10] or Thompson [11]. In our earlier work [4], the latter approach was used which results in an energy-independent potential, which is more convenient in applications to nuclear structure. Due to the off-shell differences the size of the relativistic effects differs for the two OBEPs. However, the quality of the effect is the same, leading to the correct nuclear matter saturation in both cases. In conclusion it is found that the relativistic effects caused by the explicit 2~r- and ~rp-exchanges are well simulated by the effective o-boson of the OBE model. Furthermore, the relativistic saturation effect of the Dirac-Brueckner approach, which we found eaFlier within the OBE model [4], 367
Volume 160B, number 6
PHYSICS LETTERS
m a i n t a i n s w h e n a realistic a n d e x t e n d e d m e s o n e x c h a n g e m o d e l is applied.
O n e o f t h e a u t h o r s (R.B.) w o u l d like to t h a n k W. Weise for many helpful comments.
References [1] B.D. Day, Phys. Rev. C24 (1981) 1203; Phys. Rev. Lett. 47 (1981) 226. [2] M.R. Anastasio, L.S. Celenza and C.M. Shakin, Phys. Rev. C23 (1981) 2273; M.R. Anastasio, L.S. Celenza, W.S. Long and C.M. Shakin, Phys. Rep. 100 (1983) 327. [3] C.J. Horowitz and B.D. Serot, Phys. Lett. 137B (1984) 287.
368
17 October 1985
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