fulltext

The effective action approach applied to nuclear matter - I (*)

TRANHUUPHAT(**) and NGUYENTUANANH(***)

International Centre for Theoretical Physics - 34100 Trieste, Italy (ricevuto il 9 Dicembre 1996; approvato il 13 Maggio 1997)

Summary. — Within the framework of the Walecka model (QHD-I) the application of the Cornwall-Jackiw-Tomboulis (CJT) effective action to nuclear matter is presented. The main feature is the treatment of the meson condensates for the system of finite nuclear density. The system of coupled Schwinger-Dyson (SD) equations is derived. It is shown that the SD equations for sigma-omega mixings are absent in this formalism. Instead, the energy density of the nuclear ground state does explicitly contain the contributions from the ring diagrams, amongst others. In the bare-vertex approximation, the expression for energy density is written down for numerical computation in the next paper.

PACS 21.60 – Nuclear-structure models and methods.

1. – Introduction

In recent years, many investigations have been devoted to a relativistic description of nuclear properties. The relativistic mean field (RMF) theory, pioneered by Walecka [1], with nonlinear self-interaction has turned out to be very powerful for considering the ground-state properties of both spherical and deformed nuclei over the entire range of the periodic table [2]. It is further demonstrated that the RMF model is able to describe successfully the properties of nuclei far from the stability line [3]. The relativistic Hartree-Fock formalism is an important extension of RMF. However, it has not been possible to construct a reliable approximation scheme for these approaches. The results of Furnstahl et al. [4] indicate that the loop expansion is not useful to QHD at least up to two-loop order. Moreover, Celenza et al. [5] pointed out that the perturbative approach and other approximation schemes, such as loop expansion and the random phase approximation, are invalidated due to the extremely large self-energies of sigma and omega mesons. Recently, Nakano et al. [6] proposed the

(*) 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. E-mail: phatHictp.trieste.it

(***) Address: Hanoi National University; High College of Physics, Institute of Physics, Hanoi, Vietnam. E-mail: anhHictp.trieste.it

nuclear Schwinger-Dyson formalism which may hopefully provide an alternative approximation to QHD beyond two-loop calculations.

In this paper, adopting the Lagrangian of QHD-I, we consider the effective action for composite operators formulated by Cornwall, Jackiw and Tomboulis [7]. We then derive a set of SD equations together with the explicit form of self-energies, propagators and energy density. We show that the SD equations for sigma-omega meson mixings are, in fact, not present in this formalism.

Our work is organised as follows. In sect. 2, we derive the formula for effective action (and potential) and therefrom we arrive at the SD equations which fully coincide with those of [6]. Section 3 deals with the explicit expressions for self-energies and for the energy density in the bare-vertex approximation, they will be used for numerical calculation in the next paper. The discussion and conclusion are given in sect. 4.

2. – Effective action and SD equations

We start with the following Lagrangian density [1]: L 42C[gm¯m1 M] C 2 1 2[¯ms¯ms1 ms 2_s2 ] 2 1 4FmnFmn2 1 2 mv 2_A m 2 1 (2.1) 1gsCsC 1igvC ACC,

where Fmn4 ¯mAn2 ¯nAm, C and s, Am represent the field operators of nucleon and mesons, respectively, gs and gv are coupling constants, M and ms, mv are physical

masses of nucleon and mesons. For convenience, the notation in this paper is the same as in ref. [6]. The gamma matrices satisfy

gmgn1 gngm4 2 dmn. (2.2)

A four-vector is defined as Am4 (Ai, A4) 4 (Ai, iAo) and a scalar product of four-vectors is AmBm4 AiBi1 A4B44 AiBi2 AoBo, where repeated Greek indices range between 1 and 4, repeated Latin indices range between 1 and 3 and A 4 (A1, A2, A3).

The generating functional Z[h, h , J , Jmn, H , K , Lmn] corresponding to (2.1) reads

Z[h, h , J , Jm, H , K , Lmn] 4N



D C DC DsDAm3 (2.3)

3exp

k

i

k

_{S 1}



_{dx [h(x) C(x) 1C(x) h(x)1J(x)}s_{(x) 1J}m(x) Am(x) ] 1 1



dx dy [C(x) H(x , y) C(y) 1s(x) K(x , y)s_{(y) 1A}m(x) Lmn(x , y) An(y) ]

l

l

, where N is the normalization constant guaranteeing that Z[ 0 , 0 , 0 , 0 , 0 , 0 , 0 ] 41 and

S is the classical action,

S 4



dx L .

Corresponding to (2.3), the connected generating functional W 4W[h, h, J, Jm,

The mean values of the field operators C , C,sand Amare defined by dW d h 4 C , dW dh 4 C , dW dJ 4 s , dW dJm 4 Am.

It is evident that when all the external sources vanish we have C 4 C 40

and

s K s 4 aFNs_{NFb ,}

AmK Am4 aFNAmNFb , where NFb is the ground state of nuclear matter.

It is worth emphasizing that s and Am do not vanish for the system of finite nuclear density, in which there exists the meson condensation. Moreover, the symmetry of the ground state yields

Am4 dmoAo.

Next the propagators of nucleon, G(x , y), and of mesons, C(x , y) and Dmn(x , y), are introduced, respectively, as follows:

dW dH(x , y) 4 1 2 [C(x) C(y) 1ˇG(x, y) ] , (2.4a) dW dK(x , y) 4 1 2[s(x) s(y) 1ˇC(x, y) ] , (2.4b) dW dLmn(x , y) 4 1 2[Am(x) An(y) 1ˇDmn(x , y) ] . (2.4c)

In view of (2.4) the CJT effective action G 4G[C, C, s, Am, G , C , Dmn] is defined by means of the Legendre transform

G 4W2



dx

[

_{h(x) C(x) 1C(x) h(x)1J(x) s(x)1J}m(x) Am(x)

]

2 (2.5)

21

2



dx dy

[

C(x) H(x , y) C(y) 1s(x) K(x, y) s(y)1Am(x) Lmn(x , y) An(y)

]

2

21

2

Since the physical processes correspond to vanishing external sources, the configuration of the ground state is given by

dG ds(x) 4 0 , (2.6a) dG dAm(x) 4 0 , (2.6b) dG dG(x , y) 4 0 , (2.7a) dG dC(x , y) 4 0 , (2.7b) dG dDmn(x , y) 4 0 . (2.7c)

(2.6) are the equations of motion for condensed meson fields s and Am, and (2.7) is the set of Schwinger-Dyson equations for propagators G, C and Dmn.

Now the explicit form for G can be derived immediately basing on [7],

G 4S1 ˇ 2Tr ln CoC 21 1 ˇ 2Tr CoC 21 1 ˇ 2Tr ln Do . mnDmn 21 1 ˇ 2Tr Do . mnDmn 21 2 (2.8) 2ˇ Tr ln GoG212 Tr S21(s , A) G 1G2,

where the trace, the logarithm and the product CoC21, Do . mnDmn21, R are taken in the functional sense, Co, Do . mn and Go are, respectively, the propagators of free sigma, omega and nucleon,

S(s , A) 4 d

2_S

d C(x) dC(y) ,

G2is given by all those two-particle irreducible vacuum graphs which, upon opening one

line, yield proper self-energy graphs.

Then it is easily verified that, corresponding to Lagrangian (2.1), only the two diagrams shown in fig. 1 are those under discussion.

G24 i 2gs 2 Tr GGGC 2 i 2gv 2_{Tr g} mGGnGDmn. (2.9)

When all the external sources vanish, C, C vanish and the condensed meson fields

s, Ao become independent of space-time coordinates owing to the homogeneity of nuclear matter. In that case, the effective action G manifests the translational invariance and consequently all propagators turn out to be functions of the difference of two space-time coordinates.

Fig. 1. – Solid line represents the nucleon propagator G, wavy line denotes sigma propagator C, dotted line stands for omega propagator Dmn. The nucleon-sigma and nucleon-omega vertices are

denoted by gsGand igvGm, respectively.

Starting from (2.8) and (2.9) we arrive at the expression for effective potential

V[s, Ao, G , C , Dmn] in momentum space, V[s, Ao, G , C , Dmn] 4 (2.10) 4 2ms 2 2 s 2 2 mv 2 2 Ao 2 2



d 4_p ( 2 p)4 tr [ ln Go 21_{(p) G(p) 2S}21_{(s, A} o; p) G(p) 11]1 11 2



d4_p ( 2 p)4 tr [ ln Co 21_{(p) C(p) 2C} o21(p) C(p) 11]1 11 2



d4_p ( 2 p)4 tr [ ln Do . mn 21 _{(p) D} mn(p) 2Do.mn21(p) Dmn(p) 11]1 1 i 2gs 2



d 4 p ( 2 p)4 d4 k ( 2 p)4 tr G(p) G(p , k) G(k) C(p 2k)2 2 i 2gv 2



d 4_p ( 2 p)4 d4_k ( 2 p)4 tr gmG(p) Gn(p , k) G(k) Dmn(p 2k) , where the trace tr is taken over Lorentz indices and

Co(k) 42(k21 ms22 ie)21, Do . mn(k) 42(k21 mv22 ie)21

g

dmn1 kmkn mv2

h

, Go(k) 42(igmkm1 M 2 ie)21, S21_{(s, A} o; k) 42igmkm2 M 2 igss 1gvgoAo. Inserting (2.10) into (2.6) yields the equations for s and Ao,

s 42l gs ms2



d4q ( 2 p)4 tr [G(q) ] 4i gs ms2 rS, (2.11) Ao4 2il gv mv2



d4q ( 2 p)4 tr [goG(q) ] 4 gv mv2 rB, (2.12)

in which l is an isospin multiplicity, rSand rBare scalar and nucleon density of nuclear matter, respectively.

The SD equations are derived by substituting (2.10) into (2.7),

G21_{(k) 4G} o21(k) 2S(k) , (2.13a) S_{(k) 42ig}ss 1igs2



d4 q ( 2 p)4G(q) G(2q, k) C(k2q)1 (2.13b) 1gvgoAo2 igv2



d4_q ( 2 p)4gmG(q) Gn(2q, k) Dmn(k 2q) , C21_{(k) 4C} o21(k) 2Ps(k) , (2.14a) Ps(k) 42ilgs2



d4_q ( 2 p)4 Tr [G(q) G(2q, k1q) G(k1q) ] , (2.14b) Dmn21(k) 4Do . mn21 (k) 2Pmn(k) , (2.15a) Pmn(k) 4ilgv2



d4q ( 2 p)4 Tr [gmG(q) Gn(2q, k1q) G(k1q) ] . (2.15b)

It is clear that (2.13), (2.14) and (2.15) exactly coincide with those obtained in [6], S, Ps and Pmn given by (2.13b), (2.14b) and (2.15b), are the self-energies for nucleon, sigma meson and omega meson, respectively.

Concerning the SD equations for sigma-omega mixings it is simple to check that they are not present in this formalism. This is because there is no 2PI vacuum graph which contains the ring diagrams, shown in fig. 2, as its subgraphs.

This kind of graph can be the ingredient of a certain 1PI graph like the one indicated in fig. 3, which contributes to the ordinary effective action G[s , A].

Finally, let us investigate the self-energies of scalar meson and vector meson, they characterize the in-medium effect.

The scalar meson propagator is easily deduced from (2.14),

C(k) 4 21

k2_{1 m}s21 Ps(k) 2ie

. (2.16)

The solution to eq. (2.15) is more complicated because Pmn(k) has a matrix structure. We adopt the projection method presented in [6] to derive the vector meson

Fig. 3.

propagator as the solution to (2.15),

Dmn(k) 4dmnA(k) 1 kmkn k2 B(k) 1dmidni

g

dij2 kikj k2

h

F(k) , (2.17a) where A(k) 4Dl_(k) (2.17b) B(k) 4

g

1 1 k 2 mv2

h

Do(k) 2Dl(k) , (2.17c) F(k) 4Dt (k) 2Dl_{(k) ,} (2.17d) Dt(k) 42(k2 1 mv21 Pt(k) 2ie)21, (2.17e) Dl (k) 42(k2 1 mv21 Pl(k) 2ie)21, (2.17f )

Pt(k) and Pl_{(k) are transverse and longitudinal components of P}

mn(k), respectively, Pmn4

.

`

`

`

`

`

´

Pt₂ k1 2 k2W 2k2k1 k2 W 2k3k1 k2 W 2k4k1 k2 P l 2k1k2 k2 W Pt 2 k2 2 k2W 2k3k2 k2 W 2k4k2 k2 P l 2k1k3 k2 W 2k2k3 k2 W Pt 2 k3 2 k2W 2k4k3 k2 P l 2k1k4 k2 P l 2k2k4 k2 P l 2k3k4 k2 P l k2 k2P l

ˆ

`

`

`

`

`

˜

(2.18) with W 4Pt2 (k42Ok2) Pl.

The nucleon propagator G(k) will be discussed in the next section.

3. – Bare-vertex approximation

Our main purpose in this section is to derive the formula for energy density in the bare-vertex approximation, where G(p , q) 41, Gm(p , q) 4gm. However, for

completeness, let us first discuss the contributions given by the density-dependent part of the nucleon propagator, which is dominant at lower density as was suggested in [8-10]. To this end, let us follow [6] starting with the nucleon self-energy S(k), (2.13b), whose density-dependent part SD(k) reads

SD(k) 4SDs(k) 1SDv(k) , (3.1a) SDs(k) 4igs2



d4_q ( 2 p)4

k

l ms2 tr GD(q) 1GD(q) C(q 2k)

l

, (3.1b) SDv(k) 42igv2



d4 q ( 2 p)4

k

g4 l mv2 tr g4GD(q) 1gmGD(q) gnDmn(q 2k)

l

, (3.1c)

where GD(k) is the density part of G(k), which is defined below by eq. (3.4).

In terms of the nucleon self-energy SD(k), (3.1), the nucleon propagator G(k) is

given by the SD equation (2.13), which is rewritten as follows:

G21_{(k) 4G}

o21(k) 2S(k) 42igjkj[ 1 1Sv(k) ] 2ig4[k41 S4(k) ] 2 [M1Ss(k) ] ,

(3.2)

where Sv_{, S}4_{and S}s _{are three components of Dirac decomposition of S} D(k).

Equation (3.2) leads to the definition of effective quantities:

ki* 4ki[ 1 1Sv(k) ] , (3.3a) k4* 4k41 S4(k) , (3.3b) M * (k) 4 [M1Ss_{(k) ] .} (3.3c)

By means of these effective quantities we obtain

G(k) 4GF(k) 1GD(k) (3.4a) GF(k) 4 [2igmkm* 1M *(k) ] 21 k *2_{1 M * (k)}2_{2 ie} , (3.4b) GD(k) 4 [2igmkm* 1M *(k) ] pi E * (k)d

(

ko* 2E *(k)

)

u(ko) u(kF2 NkN) . (3.4c)

Now the energy density of the nuclear ground state is considered. As is well known, it corresponds exactly to the effective potential for the propagators fulfilling the SD equations (2.13)-(2.15) and for condensed fields satisfying equations of motion (2.11), (2.12). This is the major advantage of this approach.

After substituting (2.11)-(2.15) into (2.10) we get V 4 1 2 gs2 ms2 r2S2 1 2 gv2 mv2 rB22



d4_p ( 2 p)4 tr ]ln [11S(p) G(p) ]2S(p) G(p)(1 (3.5) 11 2



d4_p ( 2 p)4 tr ]ln [11Ps(p) C(p) ] 2Ps(p) C(p)(1 11 2



d4_p ( 2 p)4 tr ]ln [11Pma(p) Dan(p) ] 2Pma(p) Dan(p)(1 1i 2 gs 2



d 4 p ( 2 p)4 d4 k ( 2 p)4 tr [G(p) G(p , k) G(k) C(p 2k) ]2 2 i 2gv 2



d 4 p ( 2 p)4 d4k ( 2 p)4 tr [gmG(p) Gn(p , k) G(k) Dmn(p 2k) ] .

All terms of (3.8) are physically meaningful, namely, the first and second terms come from disconnected two-tadpole diagrams and they cancel half of the Hartree energy. The third term consists of two parts, the first one,

2



d

4

p

( 2 p)4 tr ln [ 1 1S(p)G(p) ]

represents the quantum correction and the second one,

2



d

4

p

( 2 p)4 tr S(p) G(p)

represents the ground-state energy shift [11]. The fourth and fifth terms express the contributions from the ring diagrams for sigma and omega mesons [12, 13], respectively. And the last two terms are exactly the high-order correlation energy.

It is interesting to remark that the terms corresponding to the contributions from the ring diagrams do not occur in the SD formalism developed by [6], these contributions are implicitly involved in the propagators only.

Bearing all this in mind we obtain the explicit form for energy density in the bare-vertex approximation: E 4Eo1 1 2 gs2 ms2 rS22 1 2 gv2 mv2 r2B2



d4_p ( 2 p)4 tr ln [ 1 1S(p) G(p) ]1 (3.6) 11 2



d4_p ( 2 p)4 tr ]ln [11Ps(p) C(p) ] 2Ps(p) C(p)(1 11 2



d4_p ( 2 p)4 tr ]ln [11Pma(p) Dan(p) ] 2Pma(p) Dan(p)(1

1 i 2gs 2



d 4_p ( 2 p)4 d4_k ( 2 p)4 tr [G(p) G(k) C(p 2k) ]2 2 i 2gv 2



d 4_p ( 2 p)4 d4_k ( 2 p)4 tr [gmG(p) gnG(k) Dmn(p 2k) ] , where Eo4 l p2



0 kF q2_{dq E * (q) ,} (3.7) rS4 l p2



0 kF q2dqM * (q) E * (q) , (3.8) rB4 lkF3 3 p2 . (3.9)

(3.6) proves that the effective potential approach works well even if the meson self-energies are very large as was indicated by [5].

Therefore, the binding energy per one nucleon Ebinis usually given as

Ebin4 E 2 M ,

(3.10)

which may be computed by means of (2.6)-(2.18) and (3.1)-(3.9).

The numerical results will be reported in a forthcoming paper [14], in which two free parameters gs and gv are adjusted to reproduce the nuclear saturation point; the

binding energy Ebin4 215.8 MeV at a density corresponding to a Fermi momentum

kF4 1.42 fm21.

4. – Conclusion and discussion

This is the first publication of a series of papers dealing with the application of the CJT effective action in nuclear matter modelled by QHD-I.

The approach used proves very powerful since it is the only one that allows us to determine consistently both the propagators of the fields concerned through solving the SD equations and the energy density of the nuclear ground state. Furthermore, the theory naturally accounts for all high-order corrections as well as contributions from the ring diagrams for scalar and vector mesons, which are claimed to be important for nuclear matter study [12, 13]. Here it is necessary to pay special attention to the fact that the SD equations for sigma-omega mixings are no longer derived within the CJT effective action method, which provides the very general formalism in quantum field theory. This possibly leads to the conclusion that the SD equations for meson mixings are actually an artifact and should not be taken into consideration.

* * *

The authors would like to thank Profs. M. A. VIRASORO, L. BERTOCCHI, S. RANDJBAR-DEAMI and the International Centre for Theoretical Physics, Trieste, for hospitality. The financial support of ICSC-World Laboratory is acknowledged with thanks.

R E F E R E N C E S

[1] WALECKAJ. D., Ann. Phys., 83 (1974) 491; SEROTB. D. and WALECKAJ. D., Adv. Nucl. Phys., 16 (1986) 1.

[2] GAMBHIRY. K., RINGP. and THIMETA., Ann. Phys., 198 (1990) 132.

[3] TOKIH., SUGAHARAY., HIRATAD., TANIHATAI. and CARLSONB., Nucl. Phys. A, 524 (1991) 633.

[4] FURNSTAHLR. J., PERRYR. J. and SEROTB. D., Phys. Rev. C, 40 (1989) 321. [5] CELENZAL. S., PANTZIRISA. and SHAKINC. M., Phys. Rev. C, 45 (1991) 205.

[6] NAKANOM., HASAGAWAA., KOUNOH. and KOIDEK., Phys. Rev. C, 49 (1994) 3061, 3076. [7] CORNWALLJ., JACKIWR. and TOMBOULISE., Phys. Rev. D, 10 (1974) 2428.

[8] ALLENDESM. P. and SEROTB. D., Phys. Rev. C, 45 (1992) 2975.

[9] KREING., NIELSENM., PUFFR. D. and WILETSL., Phys. Rev. C, 47 (1992) 2485. [10] PRAKASHM., ELLISP. J. and KAPUSTAJ. I., Phys. Rev. C, 45 (1992) 2518.

[11] FETTER A. L. and WALECKA J. D., Quantum Theory of Many-Particle Systems (McGraw-Hill Book Company) 1971.

[12] CHINS. A., Ann. Phys., 108 (1977) 301.

[13] MCNEILJ. A., PRICEC. E. and SHEPARDJ. R., Phys. Rev. C, 47 (1992) 1534.

[14] TRANHUUPHATand NGUYENTUANANH, The effective action approach applied to nuclear matter - II (in preparation).