Nuclear matter properties in the relativistic random phase approximation

Volume 208, number 1

PHYSICS LETTERS B

7 July 1988

NUCLEAR MATTER P R O P E R T I E S IN THE RELATIVISTIC R A N D O M PHASE APPROXIMATION Xiangdong JI H i K. Kellogg Radiation Laboratoo', California Institute o f Technology. Pasadena, CA 91125, USA Received 28 February 1988

Nuclear matter properties at zero temperature are studied in terms of the relativistic a-o) model, including the random phase approximation (RPA) contribution. At normal density, the medium polarization adds about - 35 MeV of binding energy to the mean field result. The scalar and vector effective interaction strengths are fitted to the nuclear matter saturation conditions under various approximations for the energy functional including the RPA term. The effective mass and bulk modulus are calculated with these parameters. The relative importance of different contributions to the binding energy is analyzed.

Relativistic models for nuclear matter and finite nuclei have been studied extensively in recent years [ 1 ]. Incorporating the power of relativistic quantum field theory, such as relativistic kinematics and explicit meson degrees of freedom, these models are very successful in explaining certain physical phenomena, such as the observed spin-orbit force yielding the right order of single-particle orbits, which has no natural explanations within the traditional non-relativistic potential models. The simplest of relativistic models, on which this paper will concentrate exclusively, is the a-o) model, in which nucleons interact with each other via exchange of the neutral scalar a meson and the neutral vector o) meson [2,3]. The properties of nuclear matter in the a-co model were first studied by Walecka in the mean field approximation more than ten years ago [2]. He derived the energy density of nuclear matter under the approximation ~eMFT ~

( g v2 / 2 m v2 )PB2 + ( m~/2g~) ( M - M * ) 2

stants between mesons and the nucleon; they are fitted to the experimental nuclear matter saturation conditions (kv= 1.42 fro-~, EB= -- 15.75 MeV). PB is the nucleon density and is related to the Fermi mo3 2 mentum, kv, via pB=2k~/3rt . The results of the minimization and the fitting are listed in the first row of table 1. The mean-field approximation is based on the observation that at very high nucleon density quantum fluctuations in the Heisenberg equations of motion for the meson field operators are negligible and thus Table 1 Vector and scalar interaction strengths, gv and gs, obtained in the fits described in the text. The effective mass, M*, in each case is obtained by minimizing the energy functional. Kv ~ (MeV) is the compressibility defined in ref. [8]. The first four lines are taken from ref. [4]. MFA is the mean-field approximation, VAC is the vacuum polarization, EXC denotes the exchange correction, and RPA is the random phase approximation contribution. Contributions

gv

gs

M* / M

KV1

MFA MFA+VAC MFA+EXC M F A + V A C + EXC

11.67 8.91 10.55 8.32

9.57 7.92 9.21 7.97

0.57 0.72 0.60 0.72

550 470 600 490

MFA+RPA MFA+RPA+VAC MFA+RPA+EXC MFA+RPA+VAC+EXC

10.92 9.27 10.82 7.79

7.35 0.74 7 . 7 1 0.73 7.50 0.74 7.08 0.77

350 340 410 360

kv

+ (~-~5~)2 d 3 k ~

z,

(1)

o

where mv, ms and Mare the masses of vector meson, scalar meson and free nucleon, respectively. M* is the effective nucleon mass in the nuclear medium which is obtained by minimizing this mean-field energy functional, gv and gs are the Yukawa coupling con0370-2693/88/$ 03.50 © Elsevier Science Publishers B.V. ( North-Holland Physics Publishing Division )

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PHYSICS LETTERS B

a classical treatment of the meson field is appropriate. In fact, Chin has speculated that all quantum corrections to the energy density behave like k~ for large kv [ 4 ], whereas the vector meson repulsion in the first term of the mean-field energy density ( 1 ) clearly behaves like k 6. But for the nuclear densities of greatest interest, such as in neutron stars or in the ground state of nuclear matter, quantum corrections are certainly important. Studies of quantum corrections to the mean-field energy density of eq. ( 1 ) have been undertaken in previous works. The first such correction comes from the mass shift of anti-nucleons in the nuclear Dirac sea within the Hartree approximation and contains a divergent integral. After performing a standard renormalization, the finite vacuum correction can be expressed as ?VAC= -- ( 1/4g 2) [34*4 In (M*/M) + M 3( M - M * )

- 7M2 (M-M'*)2 + ~ - M ( M - M * )3 25 ( M - M * ) 4 ] - Z~

.

(2)

When this correction is added to the mean-field energy density ( 1 ) and the total density is again minimized with respect to M*, one finds that the effective nucleon mass at saturation changes from 0.56 M in the mean-field approximation to 0.72 M [3,4 ]. The interaction strengths obtained from the refitting are listed in the second row of table 1. It should be noted that the vacuum correction adds a repulsive contribution to the nuclear energy density. The mean-field approximation plus the vacuum correction discussed above is the Hartree approximation. The next correction is naturally the Fock exchange contribution, which was studied by Chin in a second-order approximation [4] and by Horowitz and Serot in a self-consistent Hartree-Fock approximation with no vacuum contribution [5]. The energy density of the second-order exchange from the scalar meson has the form

~×c

d4k d4p =g~ f (2~z) 4 (2~z)4

× Tr[GD(p+k)GD(p) ]Zlo(k),

(3)

where GD(P) is the density dependent part of the Hartree nucleon propagator in nuclear matter 20

7 July 1988

GD(P) = ( p ' 7 + M * ) i6(po - E p ) 0( [Pl --kF),

2E.

(4)

with Ep = x/P 2+ M.2, and Ao(k) is the free scalar meson propagator. A similar expression is obtained for the vector meson contribution. Taking the total energy density as the sum of the mean field ( 1 ) and the second-order exchange scalar and vector energies, minimizing with respect to M* and refitting the coupling constants, Chin obtained essentially the same results as the later more elaborate calculation of ref. [5], in which several coupled nonlinear integral equations were solved. The parameters from Chin's variational calculation are listed in the third row of table 1. Unlike in weak coupling theories, higher-order quantum corrections to the classical result are not necessarily small in any model of strongly interacting systems. Thus, even if in one approximation the experimental nuclear matter saturation conditions are satisfied, adding another correction can totally destroy the agreement. Therefore, the following renormalization philosophy is implicit in the above calculations. Whenever a higher-order term is included in energy density, the parameters of the theory, such as coupling constants, are refitted to the saturation conditions. The other physical observables then must be calculated in the same approximation with the refitted parameters. This approach is very similar to the traditional renormalization scheme for overcoming divergence difficulties. In this way, physical quantities, such as the nucleon effective mass and the compressibility, do not vary drastically when higher-order processes are included. However, whenever possible, higher-order processes must be still studied to verify the essential physics of lower-order calculations and the convergence of physical quantities calculated in different approximation schemes. In this letter the effects on the energy density of the medium polarization corrections to the meson propagation are studied in this context. The RPA contribution to the energy density of nuclear matter was first studied by Chin [ 4 ], following similar work in the relativistic electron gas system [ 6 ]. Perturbatively, it amounts to summing all Feynman diagrams for the closed meson propagators with arbitrary numbers of nucleon polarizations. To obtain a finite result for the energy-momentum tensor,

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PHYSICS LETTERS B

the wave functions and masses of the nucleon and of the mesons at finite nucleon density must be renormalized in a manner similar to what is done at zero nucleon density. The final expression for the RPA energy density is similar to the results of a non-relativistic RPA calculation [7], except that the Fourier transform of the static potential is replaced by the RPA meson propagator and the definitions of the polarization propagators are different. To be more specific, the relativistic RPA energy can be divided into four different parts: the scalar meson contribution (,~s), the longitudinal contribution from the vector meson (dE), the transverse contribution from the vector meson (~T), and finally the scalar-vector meson mixing contribution (ESL). They can be expressed collectively in the equation

E~p~= ~s + EL + & + EsL --

2i f &~k {

[ln(1-3'Hs)+3'Hs]

+ [In( 1 - D ' H L ) + D ' H L ] + 2 [ln(1 - D ' H T ) + D ' H T ] + In 1 - ( A , _ ~ _ H s ) ( D ,

,

where A' and D' are the scalar and vector meson propagators in the RPA approximation. For the sake of simplicity, we will replace them by free meson propagators Ao and Do and thus ignore the vacuum fluctuation contribution (which gives rise to the problem of tachyon poles and will be discussed in a separate paper [8]. Hs is the scalar polarization propagator

d4p Tr[ GD(P+k)GD(p)]. H s ( k ) = - i g ~ j I (--~)4

(6)

Note that GD(P) is a matrix in both Dirac spin and isospin spaces. HL and HT are the longitudinal and transverse polarization propagators of the vector meson. They are defined in terms of the vector-polarization propagator d4p /-/z,,(k) = -ig2v f (27~) 4 X Tr[GD as

(p+k)7~GD(p)7~],

(7)

7 July 1988

HL =//33 -Hoo =

HT=H,I

(k2/k2)Hoo,

=H22.

(8) (9)

HSL is the scalar-vector mixing polarization "> =Ms3 2 -H20 = HgL

(k2/k2)H20.

(10)

where

Hs.(k)=igsgv f

d4p (2Z() 4

× Tr[GD(p+k)7;,GD(p) ]

( 11 )

(S refers to scalar, not to a running index). In these equations, the notation follows that of ref. [ 9 ]. The spatial integral of eq. (5) can be reduced to one for the radial variable only, because all the polarization propagators are independent of the direction of the three-momentum transfer k. A Wick rotation can be performed on the fourth component, o9, of the integration momentum so that the space metric becomes euclidian. In doing so, the imaginary parts of the polarization propagators disappear (the pole is removed from the real axis) and the integrand ofeq. (5) becomes an even function of o). Thus, a two-dimensional integration over the polarization propagators is to be performed to calculate the RPA contribution to the nuclear energy density. Analytical expressions for the polarization propagators have been given already in refs. [10,11] and will not be presented here, because of their length. Care must be taken when evaluating the inverse tangent function stemming from a logarithmic function of a complex argument after the m--,ico substitution is performed. We evaluate the remaining two-dimensional integral numerically. In going to polar coordinates, the integration over the radial part is done by using a fourth-order Bode formula, while the angular part is done by the Gauss-Legendre quadrature method [ 12 ]. The integrand in eq. (5) has logarithmic singularities for some densities and coupling constants. Thus, a "hole" is cut around the singularity and the integration across the hole is done by an analytical approximation. The final result of the integration should be independent of the size of the hole. By varying the size of the holes and the number of grid points used for the numerical integration, we calculate the RPA energy with an error of less than one percent (less than 0.5 MeV at saturation). 21

V o l u m e 208, n u m b e r 1

PHYSICS LETTERS B

We have taken the mass of the nucleon as 939 MeV, the mass of the scalar meson to be 550 MeV and that of the vector meson as 738 MeV. For our first calculation, we assume that the effective coupling constants are the same as the mean field results, gv = 11.67 and gs = 9.57. The nucleon effective mass is taken to be the mass that minimizes the mean-field energy functional only. The resulting R P A contribution to the nuclear-matter energy density is attractive at almost all values of the density. For kv between 0.6 and 2.5 f m - J it is represented by the dashed curve in fig. 1. At the saturation density (kv= 1.42 f m - ~), the RPA energy is large ( - 3 5 MeV). The total energy density with both the mean field and RPA contributions (shown as the solid curve in fig. 1 ) has m o v e d the saturation density to about 2.0 f m - 1 at an energy of - 107 MeV. At high density, the RPA contribution to the energy density behaves as k 4 (from the dimensional analysis) and is thus negligible in comparison to the mean-field energy density. Because of the large coupling constants, the RPA energy is not small at normal density. The effect of the RPA correction can also be studied by refitting the effective coupling constants. We have made four different fits by taking different approximations for the energy density functional: (i) the sum of mean field (1) and RPA (5); (ii) the sum of mean field 200

. , , ....

,

7 J u l y 1988

( 1 ), RPA (5), and vacuum polarization (2); (iii) the sum of mean field ( 1 ), RPA ( 5 ), and second-order exchange contribution (3); and (iv) the sum o f mean field ( 1 ), RPA (5), vacuum polarization (2), and second-order exchange contribution (3). The scalar and vector coupling constants are fitted to the condition that nuclear matter saturates at an energy of - 15.75 MeV per nucleon for a Fermi m o m e n t u m of kv = 1.42 f m - '. The resulting parameters are displayed in rows 58 of table 1. The coupling constants are systematically reduced as compared to the mean-field fit. Thus, one might speculate that they would approach some asymptotic values if more corrections were added in. The effective mass at saturation density is calculated in every case by minimizing the energy functional with the corresponding fitted parameters. We observe that the values obtained in the different approximations are very close to each other, and thus M * = 0 . 7 5 M is the typical value for the model. In every one of the four cases under consideration, the compressibility is reduced by about 30% when RPA corrections are included. This is probably due to the attractive nature of the RPA energy contribution. With the new coupling constants, we have calculated different contributions to the total energy density. In fig. 2, we display the calculation for the energy density in approximation (i) (the mean-field plus

....

/

40

/ /

100

//

MFA /

2O v

/

o

/

/ MFA

/ /

/

v

MFA R I

_1oo

0

2o

MFA+RPA RPA

- - 2 0 0

. . . .

~

,

I

1

I

I

I

~

1.5

I

~1

2

1

1

40

W ~

L

2.5 1

kv(f

m

Fig. 1. M e a n field a n d R P A c o n t r i b u t i o n s to the e n e r g y d e n s i t y u s i n g the m e a n - f i e l d p a r a m e t e r s a n d effective m a s s e s f r o m ref. [2].

22

1.5

1)

k v ( f m 1) Fig. 2. M e a n field energy, R P A energy a n d t h e i r s u m , c a l c u l a t e d w i t h p a r a m e t e r s fitted to s a t u r a t i o n .

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PHYSICS LETTERS B

RPA contributions only). The mean-field contribution is repulsive for all densities, so that the cancellation of scalar a n d vector energies in the mean-field a p p r o x i m a t i o n ( 1 ) is not responsible for binding. Rather, saturation is due to a balance between the mean-field repulsion and the R P A attraction. This shows how the b i n d i n g m e c h a n i s m can differ, even in different levels o f a p p r o x i m a t i o n in the same model. In fig. 3, the energy curves o f the mean-field app r o x i m a t i o n , the v a c u u m polarization, the exchange correction, and the RPA in a p p r o x i m a t i o n ( i v ) are displayed, together with the sum o f the four. In this case, the R P A correction almost cancels the sum o f the v a c u u m polarization and the exchange correction. The mean field energy density alone has a mini m u m o f - 2 0 M e V at 1.7 fm -~ which is not too far from the e x p e r i m e n t a l saturation point. This points out that although the different high-order terms are large, they can p r o d u c e a small net effect on the nuclear equation o f state, so that one must be careful in c o m b i n i n g different contributions. We also calculate the effective masses for different nuclear densities in a p p r o x i m a t i o n ( i v ) and compare t h e m with the mean-field results (fig. 4). At low densities, the effective mass in the two a p p r o x i m a tions differs little. But for large densities, the effec20

/

/ / /

10

EXC

~

~

//

VAC

~ o < "~'-\\

/

/

I

~

-10

•

\

RPA

/

x, ........... \ T OO T ATL A L ~ \ \

20

.

/ //

\

1.5

1.0 \\ +VAC+EXC

0.8

"~ 0.6 MFA

\\

0.4 \ \ \

0.2

0.0

\ \

.........

I .........

1

I .

2

.

.

.

.

3

kr (fro -1) Fig. 4. Comparison of the nucleon effective masses as calculated in the mean-field approximation and the approximation (iv) described in the text. rive mass in approximation (iv) is considerably larger than that o f the m e a n field alone. Thus one has to be careful when using the effective mass to calculate other physical quantities, such as the response functions for electron scattering; the same a p p r o x i m a t i o n must be used in calculating both the energy density and the d y n a m i c a l observables. In conclusion, we have studied nuclear m a t t e r properties (the nucleon effective mass, the bulk modulus, and the energy densities) by including the RPA contribution to the energy functional. The RPA term is a large negative energy at n o r m a l densities. W h e n the interaction strengths are refitted to the ground state properties of nuclear matter, we find that the effective nucleon mass becomes 0.77 M a n d the bulk modulus is reduced to 300 MeV. Finally, the saturation m e c h a n i s m is quite sensitive to cancellations a m o n g the different terms.

/ MFA

1

7 July 1988

2

kF(frn -1 ) Fig. 3. Similar to fig. 2, except that the energy density is approximated as the sum of the mean field, the RPA, the vacuum polarization, and the second-order exchange.

The a u t h o r thanks S.E. K o o n i n and D.A. Wasson for m a n y useful discussions and W. Bauer and B.H. Wildenthal for careful reading o f the manuscript. The work was supported by N S F grants PHY-85-5682 and PHY86-04197.

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PHYSICS LETTERS B

References [11B.D. Serot and J.D.Walecka, Adv. Nucl. Phys. 16 (1986) 1. [2] J.D. Walecka, Ann. Phys. 83 (1974) 491. [3] J.D. Walecka, Phys. Lett. B 59 (1975) 109. [4] S.A. Chin, Ann. Phys. 108 (1977) 301. [5] C.J. Horowitz and B.D. Serot, Nucl. Phys. A 399 (1983) 529. [6] J.A. Akhiezer and S.V. Peletminski, Sov. Phys. JETP 11 (1960) 1316.

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[7] A.L. Fetter and J.D. Walecka, Quantum theory of manyparticle systems (McGraw-Hill, New York, 1971 ). [8] X. Ji, in preparation. [ 9 ] J.D. Bjorken and S.D. Drell, Relativistic quantum mechanics (McGraw-Hill, New York, 1964). [10] H. Kurasawa and T. Suzuki, Phys. Lett. B 154 (1985) 16. [ 11 ] H. Kurasawa and T.Suzuki, Nucl. Phys. A 455 ( 1985 ) 685. [ 12] M. Abramowitz and I.A. Stegun, Handbook of mathematical functions (Dover, New York, 1965 ).