fulltext

The effective action approach applied to nuclear chiral sigma model (*)

TRANHUUPHAT(1)(**) and PHANHONGLIEN(2)

(1) International Centre for Theoretical Physics - Trieste 34100, Italy (2) Department of Physics, Hanoi National University - Hanoi, Vietnam (ricevuto il 19 Agosto 1997; approvato il 21 Ottobre 1997)

Summary. — The nuclear chiral sigma model of nuclear matter is considered by

means of the Cornwall-Jackiw-Tomboulis (CJT) effective action. The method provides a very general framework for investigating many important problems: chiral symmetry in nuclear medium, energy density of nuclear ground state, nuclear Schwinger-Dyson (SD) equations, etc. It is shown that the SD equations for sigma-omega mixing are actually not present in this formalism. For numerical computation purposes the Hartree-Fock (HF) approximation for ground-state energy density is also presented.

PACS 21.65 – Nuclear matter.

PACS 21.60 – Nuclear-structure models and methods.

1. – Introduction

The description of nuclear properties within the framework of relativistic quantum field theory turns out to be more and more crucial for nuclear theoretical study. The relativistic mean field theory of Walecka and its HF formalism extension [1] have proven to be very powerful for considering the ground-state properties of both the nuclei (spherical and deformed) over the entire range of the periodic table [2] and the nuclei far from the stability line [3]. However, the recent considerations [4, 5] indicate that it is not possible to construct a reliable approximation scheme for these theories. In this respect, it is worth to mention that the CJT effective action method [6-8], which obviously includes the nuclear SD equations approach [9], may hopefully provide a promised approximation beyond two-loop calculations. Its priority is expressed by the fact that the vacuum expectation values of field operators and the propagators are treated on the same footing; therefore it takes into account all the possible correlation effects—those that are extremely important for nuclear processes. In addition to the preceding trend,

(*) The authors of this paper have agreed to not receive the proofs for correction.

(**) Permanent address: Vietnam Atomic Energy Commission, 59 Ly Thuong Kiet, Hanoi, Vietnam.

one has made great attempts to investigate the role of chiral symmetry in nuclear phenomena. Much interest is focused on chiral models of nuclear matter and nuclei [10-18], because they not only give insights into the mechanism of saturation in nuclear matter [13, 16], but also important constraints in nuclear medium [16, 19]. The chiral s model with omega meson inclusion was adopted to be the starting point for nuclear chiral dynamics consideration in two elegant publications of Arima and collaborators [20, 21]. With regard to the above-mentioned result of [7, 8], the present paper also deals with the application of composite operators effective action to this nuclear chiral s model. In this connection, it is possible to consider our work as being complementary to [20, 21].

This paper is organized as follows. In sect. 2 the Slavnov-Taylor identities of CJT effective action are derived for a quantum system of fermion-boson interation. Section 3 is devoted to considering the CJT effective action and SD equations for nuclear chiral s model. Section 4 deals with the ground-state energy density and its HF approximation that is necessary for numerical computation purposes. The conclusion and discussion are given in sect. 5.

2. – Slavnov-Taylor identities

In this section the derivation of the Slavnov-Taylor identities for CJT effective action is carried out analogously to that of [22].

Let S(C , C , Fn) be the action of fermion-boson interacting system, where C are

operator of fermion field and Fn (n 41, 2, R , N) operators of N boson fields. The

generating functional W[h, h , J , H , K ] for connected Green’s functions reads exp

[

iW[h, h , J , H , K ]

]

4 Z[h, h , J , H , K ] 4 4 1 Z[ 0 ]



_{DC DC}

_»

n DFnexp

k

i

m

S(C , C , Fn) 1 1



dx [C(x) h(x) 1h(x) C(x)1Jn(x) Fn(x) ] 1 11 2



_{dx dy[ 2 C(x) H(x , y) C(y) 1Fm(x) Kmn(x , y) Fn(y) ]}

nl

_,

which leads to the mean values of field operators C , C, Fn and the propagator

S(x , y)

(

Dmn(x , y)

)

of fermion (boson) field in the presence of external sources:

dW d h(x) 4 c(x) , (2.1) dW dh(x) 4 c(x) , (2.2) dW dJn(x) 4 fn(x) , (2.3)

and d2W d h(x) dh(y) 4 S(x , y) , (2.4) d2_W dJn(y) dJm(x) 4 Dmn(x , y) . (2.5)

The CJT effective action G[c, c , f , S , D] is then defined as the double Legendre transform of W: (2.6) G_{[c, c , f , S , D] 4W[h, h, J, H, K]2} 2



dx [c(x) h(x) 1h(x) c(x)1Jn(x) fn(x) ] 2 21 2



_{dx dy [ 2 c(x) H(x , y) c(y) 1fm(x) Kmn(x , y) fn(y) 2} 21 2



_{dx dy [ 2 S(x , y) H(x , y) 1Dmn(x , y) Kmn(x , y) ] .}

Starting from (2.6) one obtains

dG dc(x) 4 2h(x) 2



_{dy c(y) H(y , x) ,} (2.7) dG d c(x) 4 2 h(x) 2



_{dy H(x , y) c(y) ,} (2.8) dG dfm(x) 4 2 Jm(x) 2



dy Kmn(x , y) fn(y) , (2.9) dG dG(x , y) 4 2 H(x , y) , (2.10) dG dDmn(x , y) 4 21 2Kmn(x , y) . (2.11)

Suppose now that the action S is invariant under the transformations of field operators C KC1eL(x) , (2.12) C K C1eL*(x) , (2.13) FnK Fn1 eFn(x) . (2.14)

The invariance of Z requires [22] that

which implies that

(2.15)



_{dx [aL* (x)b h(x) 1h(x)aL(x)b1 aF}n(x)b Jn(x) ] 1

1



dx dy

k

aL*a(x) cb(y) 1Ca(x) Lb(y)b Hab(x , y) 1

11

2aFm(x) fn(y) 1Fm(x) Fn(y)b Kmn(x , y) ] 40 , where a b denotes the quantum average in the presence of external sources.

Substituting (2.7)-(2.11) into (2.14) we ultimately arrive at the Slavnov-Taylor identities for CJT effective action

(2.16)



dx

{

aFn(x)b dG dfn(x) 1 aL* (x)b dG d c(x) 1 dG dc(x) aL(x)b

}

2 22



dx dy

{

2aFn(x)b fm(y) dG dDmn(x , y) 1 aL* (x)b dG dS(x , y)c(y) 1 1 c(y) dG dS(x , y)aL(y)b

n

1



_{dx dy}

_{

₂

_»

_C(x) dG dS(x , y)L(y) 1L*(x) dG dS(x , y)C(y)

«

1

1aFm(x) Fn(y) 1Fm(x) Fn(y)b

dG dDmn(x , y)

}

4 0 .

3. – Effective action for nuclear chiral s model

Let us consider the chiral s-v model of nuclear matter, whose Lagrangian reads (3.1) _{L 4 C[ig}m ¯m2 g(s 1 i p K t K g5) 2gvgmVm] C 1 1 2[ (¯ms) 2 1 (¯mp K )2 ] 2 2m 2 2 (s 2 1 pK2) 2 l 2 4 (s 2 1 pK2)2 2 1 4FmnF mn 1 m 2 v 2 VmV m_,

where C, s, pK, and Vmare the field operators of nucleon, sigma meson, pion, and meson,

respectively, and Fmn 4¯mVn2 ¯nVm. The model has three coupling constants g, gvand

l. For convenience, the notation in this paper is the same as in ref. [23]. It is known that

(3.1) is invariant under the chiral transformation C KC84C1i(aK2 bKg5) t K 2 C, (3.2a) C K C84 C2iC(aK1 bKg5) t K 2 , (3.2b)

s Ks84s2 bKKp, (3.3a) p K K pK8 4 pK2 aK3 pK1 b K s. (3.3b)

It is evident that (3.1) is a special case of the one we presented in sect. 2, in which Fn

(n 41, 2, 3) are specified to be s, p, v meson fields and the chiral transformation (3.2), (3.3) correspond to (2.12), (2.13), respectively.

We denote the propogators and external currents corresponding to s , p and v fields as follows: For s: Dmn(x , y) KC(x, y) , Jn(x) KJ(x) , Kmn(x , y) KK(x, y) . For p: Dmn(x , y) KDij(x , y) (i , j 41, 2, 3) , Jn(x) KJi(x) , Kmn(x , y) KLij(x , y) . For v: Dmn(x , y) KDm , n(x , y) (m , n 40, 1, 2, 3) , Jn(x) KJm(x) , Kmn(x , y) KMmn(x , y) .

Accordingly, the generating functional for connected Green function W and the CJT effective action G are specified as W[h, h , J , Ji, Jmn] and G[c, c , s , pi,

Vm, S , C , Dij, Dmn], respectively. Then we have, of course, equations like (2.1)-(2.5) and

(2.7)-(2.11). To proceed further let us emphasize that when all external sources vanish we have

c 4c40 ,

and s , pi, Vmtend to their ground-state mean values

s Kv4 aFNsNFb , piK pi4 aFNpiNFb ,

VmK Vm4 aFNVmNFb ,

For a finite density system v and Vmdo not vanish, and moreover, the symmetry of the

ground state yields

pi4 0 , Vm4 dm0v .

v and v are independent of space-time coordinates owing to the homogeneity of nuclear

matter.

In free space v Kv0and v K0.

The nuclear ground state is defined to be the configuration that fulfills the system of equations like (2.7)-(2.11) for vanishing external sources, namely,

g

dG dc

h

0 4 0 ,

g

dG d c

h

0 4 0 , dG dv 4 0 ,

g

dG dpi

h

4 0 , (3.4) dG dv 4 0 , (3.5) dG dG 4 0 , G 4 ]S, C, Dij, Dmn( . (3.6)

Equations (3.6) produce the system of SD equations.

Next, inserting L and Fngiven by (3.2) and (3.3) according to (2.15) we arrive at the

Slavnov-Taylor identities for chiral symmetry: (3.7)



dx

{

pi(x) dG ds(x)2s(x) dG dpi(x) 2i 2

g

c(x) g5t i dG d c(x)1 dG dc(x)g5t i c(x)

h

}

₂ 2i



dx dy

g

c(x) g5ti dG dS(x , y)c(y) 1c(y) dG dS(x , y) g5t i_c(x)

h

2 2i



dx dy

{

»

C(x) dG dS(x , y)g5t i_C(y)

«

1

»

C(x) g5ti dG dS(x , y)C(y)

«

}

2 22



dx dy

{

s(x) dG dDij(x , y) pj(y) 2pi(x) dG dC(x , y)s (y)

}

1 1



dx dy

{

apj(x) s(y) 1s(x)pj(y)b dG dDij(x , y) 2

2as(x) pi(y) 1pi(x) s(y)b

dG

dC(x , y)

}

4 0 .

Differentiating (3.7) with respect to pi and letting all external sources equal zero we

obtain dik dG dv 4 v

g

d2G dpidpj

h

0 4 2 v

g

d 2_W dJidJk

h

21 0 4 vDik21( 0 ) . (3.8)

If we add to (3.1) a term, which explicitly breaks the chiral symmetry, for example

L K L 4L1cs ,

(3.9)

with c being constant, then, instead of (3.4), we get immediately

dG

dv 4 2 c .

(3.10)

Combining (3.8) and (3.10) provides

dG dv 4 vD

21_{( 0 ) 42 c .}

(3.11)

Equation (3.11) expresses the Goldstone theorem in nuclear medium: for exact chiral symmetry _{(c 40) in the nuclear ground state we have either D}21_{( 0 ) 40}

(Nambu-Goldstone phase), or v 40

(

Wigner phase; the in-medium mass of nucleon is zero due to (3.21)

)

. We will return to (3.11) again later when the explicit form for CJT effective action will be found. To this end, first let us shift the field s and Vm

s 4v1s , Vm4 d0 mv 1Wm.

Then the expression for G can be derived directly basing on [1]: (3.12) _{G 4S(v, v)1i Tr [ln S}21 0 S] 2S021(v , v) S 11]2 2 i 2Tr [ln C 21 0 C 2C021(v) C 11]2 i 2Tr [ln D 21 0 , ijDij2 D0 , ij21(v) Dij1 1 ] 2 2 i 2Tr [ln D 21 0 , mnDmn2 D210 , mnDmn1 1 ] 1 G2,

where the trace, the logarithm and the product C21

0 C , D0 , ij21Dij, R are taken in the

functional sense; C0, D0 , ij, D0 , mnand S0are, respectively, the propagators of free sigma,

pion, omega and nucleon, their momentum representation reads

C21 0 (p) 4p22 ms22 ie ; ms24 m21 3 l2v02, D21 0 , ij(p) 4dij(p22 mp22 ie); mp24 m21 l2v0, D21 0 , ij(p) 4p22 mv22 ie , S021(p) 4 p× 2M2ie; M 4gv0,

and the momentum representations of C0(v), D0 , ij(v) and S0(v , v) are as follows:

C21 0 (p , v) 4p22 ms22 3 l2(v22 v02) , D21 0 , ij(p , v) 4dij

(

p22 mp22 l2(v22 v02)

)

, S21 0 (p ; v , v) 4 p× 2gv2gvg0v .

Fig. 1. – The bold solid line represents the nucleon propagator S, the solid line-pion propagator

Dij, the dashed line-sigma propagator C and the dashed-dotted line-omega propagator Dmn. The

mesonic T matrices are given in [23]. G , Gj_{and G}n _{are the NN-meson irreducible vertices (s , p}

and v).

G2is given by all those two-particle irreducible vacuum graphs which, upon cutting off

one line, yield proper self-energy graphs.

Then it is easily verified that, corresponding to Lagrangian (3.1), only the diagrams shown in fig. 1 are under discussion.

and consequently all propagators turn out to be functions of the difference of two space-time coordinates.

Starting from (3.12) and fig. 1 it is easy to write down the expression for the effective potential V in momentum space:

(3.13) _{V [v , v , S , C , D , d] 4} l 2 4 (v 2 2 v02)21 m2 p 2 (v 2 2 v02) 2c(v2v0) 2 2m 2 v 2 v 2 2 i



d 4_p ( 2 p)4Tr ]ln [S 21 0 (p) S(p) ] 2S021(p ; v , v) S(p) 11(1 1 i 2



d4_p ( 2 p)4]ln [C 21 0 (p) C(p) ] 2C021(p ; v) C(p) 11(1 1 i 2



d4_p ( 2 p)4]ln [D 21 0 , ij(p) Dij(p) ] 2D0 , ij21(p , v) Dij(p) 11(1 1 i 2



d4_p ( 2 p)4]ln [D 21 0 , mn(p) Dmn(p) 2D210 mn(p) Dmn(p) 11(1 11 2g



d4_p ( 2 p)4 d4_k ( 2 p)4Tr ]S(p) G(p, k) S(k) C(p2k)(2 2i 2g



d4 p ( 2 p)4 d4 k ( 2 p)4Tr ]g5t i_{S(p) G}j_{(p , k) S(k) D} ij(p 2k)(1 11 2gv



d4p ( 2 p)4 d4k ( 2 p)4Tr ]gmS(p) Gn(p , k) S(k) Dmn(p 2k)(1 13 2l 2 v



d 4 p ( 2 p)4 d4k ( 2 p)4D(p) D(p 1k) C(k) T(k; p, 2p2k)1 1l 2_v 2



d4_p ( 2 p)4 d4_k ( 2 p)4C(p) C(p 1k) C(k) T(p, k, 2p2k; )1 115 4 l 2



d 4 p ( 2 p)4 d4 k ( 2 p)4D(p) D(k) 1 3 2l 2



d 4 p ( 2 p)4 d4 k ( 2 p)4D(p) C(k) 1 13 4l 2



d 4 p ( 2 p)4 d4k ( 2 p)4C(p) C(k) 1 l2 4



d4p ( 2 p)4 d4q ( 2 p)4 d4k ( 2 p)4 3 3

m

3 C(p) C(q) D(p 1q2k) D(k) T(p, q; 2p2q1k, 2k)1

11 6D(p) D(q) D(p 1q2k) D(k)(dijdkl1 dikdjl1 dildjk) 3 3Tijkl(; p , q , 2p2q1k, 2k)

n

1 1 8l 2



d 4_{p d}4_{q d}4_k ( 2 p)12 3 3C(p) C(q) C(p 1 q 2 k) C(k) T(p , q , 2p 2 q 1 k , 2k ; ) . Substituting (3.13) into (3.6) we arrive at the system of SD equations

S21_{(p) 4S}21 0 (p) 2G(p) , (3.14) C21_{(p) 4C}21 0 (p) 2P(p) , (3.15) D21_{(p) 4D}21 0 (p) 2S(p) , (3.16) D21 mn (p) 4D210 , mn(p) 2Pmn(p) , (3.17)

in which G, P, S and Pmn are, respectively, the self-energy of nucleon, sigma, pion and

omega,

.

`

`

/

`

`

´

G(p) 4g(v2v0) 1G – (p) , G(p) 4gvg0v 1ig



d4_k ( 2 p)4S(k) G(k , p) 3 3C(k 2 p) 2 g



d 4 k ( 2 p)4[g5t i S(k) Gj(k , p) Dij(k 2p) ]1 1igv



d4_k ( 2p)4[gmS(k) Gn(k , p) Dmn(k 2p) ] . (3.18)

.

`

`

`

`

/

`

`

`

`

´

P(p) 43l2_(v2 2 v02) 1P(p) , P(p) 42ig



d 4_k ( 2 p)4Tr [S(k) G(k , k 1p) S(k1p) ]2 23 il2v



d 4_k ( 2 p)4[D(k) D(k 1p) T(p, k2p2k)1 1C(k) C(k 1 p) T(p , k 2 p 2 k ; )] 2 23 il2



d 4_k ( 2 p)4[D(k) 1C(k) ]2 2il2



d 4 q ( 2 p)4 . d4 k ( 2 p)4vv[ 3 C(q) D(q 1p2k) D(k) T(p, q; 2p2q1k, 2k)1 1C(q) C(q 1 p 2 k) C(k) T(p , q , 2p 2 q 1 k , 2k ; )] , (3.19)

.

`

`

`

`

/

`

`

`

`

´

Sik (p)4 l2(v22 v02) dik1 sik, Sij (p) 4g



d 4 k ( 2 p)4Tr [g5t i_{S(k) G}j (k , k 1p) S(k1p) ]2 2idijl2



d4k ( 2 p)4[C(k) 15D(k)12vC(k) D(k1p) T(2k); k1p, 2p) ]2 2il2



d 4_q ( 2 p)4 d4_k ( 2 p)4[dijC(q) C(k) D(p 1k2p)3 3T(q , k ; 2q 2 k 1 p , 2p) 1 1 3D(q) D(k) D(q 1k2p)3 3(diadbc1 dibdac1 dicdab) Tabcj(; q , k , 2q2k1p, 2p) ] , S_{(p) 4} 1 3dikS ik_, (3.20)

Pmn(p) is given by eqs. (2.15b) and (2.18) of ref. [7].

Introducing now the in-medium mass of nucleon, sigma and pion, M 4M1g(v2v0) 4gv4M v V0 , (3.21) m2 s4 ms21 3 l2(v22 v02) , (3.22) m2 p4 mp21 l2(v22 v02) , (3.23)

and taking into account (3.18)-(3.20) the SD equations (3.14)-(3.16) provide

S(p) 4 1 p × 2M2G–(p) , (3.24) C(p) 4 1 p2 2 m2s2 P(p) , (3.25) D(p) 4 1 p2 2 m2p2 S(p) . (3.26)

Differentiating (3.13) with respect to v we get (3.27) ¯V ¯v 4 2 [c 2 m 2 pv] 2ig



d4p ( 2 p)4tr S(p) 13il 2_v



d 4 p ( 2 p)4[C(p) 1D(p) ]1 1il2



d 4 p ( 2 p)4 d4k ( 2 p)4[ 3 D(p) D(p 1k) C(k) T(k; p, 2p2k)1 1C(p) C(p 1 k) C(k) T(p , k , 2p 2 k ; )] 4 0 , where m2 pis defined by (3.23).

On the other hand, (3.11) and (3.26) give ¯V ¯v 4 2 vD 21_{( 0 ) 4v}

₍

_m2 p1 S( 0 )

)

4 c (3.28)

(3.27) and (3.28) altogether simply lead to

Ss v 4 S( 0 ) , (3.29) where Ss4 2ig



d4p ( 2 p)4Tr S(p) 13il 2 v



d 4 p ( 2 p)4[C(p) 1D(p) ]1 1il2



d 4_p ( 2 p)4 d4_k ( 2 p)4[ 3 D(p) D(p 1k) C(k) T(k; p, 2p2k)1 1C(p) C(p 1 k) C(k) T(p , k , 2p 2 k ; )] . (3.29) was proven in [20] by another method. Here it is necessary to emphasize that (3.29) holds only in the ground state.

Next the Goldstone theorem is considered in detail. Let us first rewite (3.28) for vanishing c,

¯V

¯v 4 v

(

m

2

p1 S( 0 )

)

4 0 ,

which implies that for exact chiral symmetry, in the nuclear ground state either the pion propagator has a pole at k 40 (Nambu-Goldstone phase) or the expectation value v4 aFNsNFb vanishes (Wigner phase).

The stability condition ¯2V ¯v2 4

g

d2_W dj dj

h

0 4 2 C21( 0 ) , m2s1 p( 0 ) D 0 (3.30)

decides which of the two phases is actually realized in the nuclear ground state. For completeness let us write down the equations for omega meson:

¯V ¯v 4 2 m 2 vv 2ilgv



d4_p ( 2 p)4Tr [v 0 S(p) ] 40 (3.31)

from which one obtains v 42il gv m2 v



d4_p ( 2 p)4Tr [g 0 S(p) ] 42 il gv m2 v rB,

where l is an isospin multiplicity and rBis the nucleon density of nuclear matter.

Finally it is indispensable to emphasize that the SD equations for sigma-omega mixing no longer exist in the present formalism. This is because there is no 2PI vacuum graph which encompasses those ring diagrams, shown in fig. 2, as its subgraphs.

4. – Energy density of nuclear ground state

It is known that the energy density of nuclear ground state corresponds exactly to the effective potential V for the propagators fulfilling the SD equations (3.14)-(3.17), and the condensed fields v and v satisfying (3.27), (3.31).

After substituting (3.14)-(3.17) into (3.13) we obtain

(4.1) _{E 4} 1 4l 2_(v2 2 v02)21 mp2 2 (v 2 2 v02) 2c(v2v0) 2 mv2 2 v 2 2 2i



d 4 p ( 2 p)4Tr ]ln [11G(p) S(p) ]2G(p) S(p)1gvg0vS(p)(1 1 i 2



d4_p ( 2 p)4]ln [ 1 1 P(p) C(p) ] 2 P(p) C(p)( 1 1 i 2



d4 p ( 2 p)4]ln [ 1 1 Sik(p) Dik(p) ] 2Sik(p) Dik(p)(1 1 i 2



d4p ( 2 p)4]ln [ 1 1 Pmn(p) Dmn(p) ] 2Pmn(p) Dmn(p)(1 11 2g



d4_p ( 2 p)4 d4_k ( 2 p)4Tr ]S(p) G(p, k) S(k) C(p2k)(2 1i 2g



d4_k ( 2 p)4 d4_k ( 2 p)4Tr ]g5t i_{S(p) G}j_{(p , k) S(k) D} ij(p 2k)(1 11 2gv



d4 p ( 2 p)4 d4 k ( 2 p)4Tr ]gmS(p) Gn(p , k) S(k) Dmn(p 2k)(1 13 2l 2_v



d 4_p ( 2 p)4 d4_k ( 2 p)4D(p) D(p 1k) C(k) T(k; p, 2p2k)1 11 2l 2_v



d 4_p ( 2 p)4 d4_k ( 2 p)4C(p) C(p 1k) C(k) T(p, k, 2p2k; )1

115 4 l 2



d 4_p ( 2 p)4 d4_k ( 2 p)4D(p) D(k) 1 3 4l 2



d 4_p ( 2 p)4 d4_k ( 2 p)4C(p) C(k) 1 13 2il 2



d 4_p ( 2 p)4 d4_k ( 2 p)4D(p) C(k) 1 1 4l 2



d 4_p ( 2 p)4 d4_k ( 2 p)4 d4_q ( 2 p)4 3 3

m

3 C(p) C(q) D(p 1q2k) D(k) T(p, q; 2p2q1k, 2k)1 11 6D(p) D(q) D(p 1q2k) D(k)(dijdkl1 dikdjl1 dildkj) 3 3Tijkl(; p , q , 2p2q1k, 2k)

n

1 1 8l 2



d 4_p ( 2 p)4 d4_q ( 2 p)4 d4_k ( 2 p)4 3 3C(p) C(q) C(p 1 q 2 k) C(k) T(p , q , 2p 2 q 1 k , 2k ; ) . All the terms appearing in (4.1) are physically meaningful, namely, the first four terms represent the mean-field part. The fifth term consists of two parts, the first one

2i



d

4

p

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

represents the quantum correction and the second one,

i



d

4_p

( 2 p)4Tr G(p) S(p)

represents the ground-state energy shift [24]. The sixth and seventh terms are the contributions from the ring diagrams for sigma and omega mesons [25, 26], respectively.

The remaining terms are exactly the high-order correlation energy.

For numerical calculation, that is the subject of our next paper, it is worth dealing with the energy density in the HF approximation having the graphical representation shown in fig. 1a)-h) with the bare vertices. Then we get

(4.2) EHF4 1 4l 2_(v2 2 v02)21 m2 p 2 (v 2 2 v02) 2C(v2v0) 2 m2 v 2 v 2 2 2i



d 4 p ( 2 p)4Tr ]ln [11G(p) S(p) ]2G – (p) S(p) ] 1gvg0vS(p)(1 1i 2



d4_p ( 2 p)4]ln [ 1 1 P(p) C(p) ] 2 P – (p) C(p) ](1 1i 2



d4_p ( 2 p)4]ln [ 1 1 Sik(p) Dik(p) ] 2Sik(p) Dik(p)(1

1i 2



d4_p ( 2 p)4]ln [ 1 1 Pmn(p) Dmn(p) ] 2Pmn(p) Dmn(p)(2 2i 2 g 2



d 4_p ( 2 p)4 d4_k ( 2 p)4Tr ]S(p) S(k) C(p2k)(2 23 i 2 g 2



d 4_p ( 2 p)4 d4_k ( 2 p)4Tr ]g5S(p) g5S(k) D(p 2k)(2 2i 2 g 2 v



d4 p ( 2 p)4 d4 k ( 2 p)4Tr ]gmS(p) gnS(p) Dmn(p 2k)(1 23 i(l2v)2



d 4_p ( 2 p)4 d4_k ( 2 p)4[D(p) D(p 1k) C(k)1C(p) C(p1k) C(k) ]1 13 4l 2



d 4_p ( 2 p)4 d4_k ( 2 p)4[C(p) C(k) 12C(p) D(k)15D(p) D(k) ] . The HF self-energy parts of nucleon, sigma, pion and omega are given, respectively, by (4.3) _{G(p) 4g(v2v}0) 1gvg0v 1g2



d4k ( 2 p)4S(k) C(k 2p)2 23 g2



d 4_k ( 2 p)4g5S(k) g5D(k 2p)1g 2 v



d4_k ( 2 p)4gmS(k) gnDmn(k 2p) , (4.4) _{P(p) 43l}2_(v2 2 v02) 2g2



d4 k ( 2 p)4Tr ]S(k) S(k1p)(2 26(l2v)2



d 4_k ( 2 p)4[D(k) D(k 1p)13C(k1p) ]23il 2



d 4_k ( 2 p)4[D(k) 1C(k) ] , (4.5) Sik (p) 4l2_(v2 2 v02) dik1 g2



d4k ( 2 p)4Tr ]g5t i_{S(k) g} 5tkS(k 1p)(2 2idijl2



d 4_k ( 2 p)4[C(k) 15D(k) ]24(l 2 v)2dik



d 4_k ( 2 p)4C(k) D(k 1p) , (4.6) Pmn(p) 4gv2



d4_k ( 2 p)4Tr ]gmS(k) gnS(k 1p)( .

Inserting (4.3)-(4.6) into (4.2) we arrive at the final expression for EHF.

To end this section, it is interesting to remark that ¯V

¯v 4

¯E

which, together with (3.28), provide the Goldstone theorem obtained in [20]: for exact chiral symmetry (c 40), the minimization of the energy density with respect to v leads either to a pole in the pion propagator at q 40 (Goldstone mode) or to a vanishing expectation value v 40 (Wigner mode).

5. – Conclusion and discussion

In the preceding sections the CJT effective action was used to study systematically the chiral symmetry in the nuclear medium modelled by the chiral s-v model. We did not discuss those that were carried out in [20] such as the renormalization procedure, the constraints imposed by the baryon current conservation on omega-nucleon interaction in the medium, etc. This is because our main aim is to present in detail a general formalism that enlightens its priority in connection with two fundamental problems concerned: the energy density and the Goldstone theorem in nuclear medium. In a transparent way the CJT effective action approach simultaneously provides general expression for the nuclear energy density involving explicity contributions from all parts: mean fields, ring diagrams and high-order correlation energy, and the Goldstone theorem which was stated more precisely. For exact chiral symmetry, in the nuclear ground state either the pion propagator has a pole at zero momentum (Nambu-Goldstone phase) or the expectation value v 4 aFNsNFb vanishes (Wigner phase). Due to the fact that our next paper is intended to be devoted to numerical calculations of nuclear properties, we restricted the energy density in HF approximation.

In concluding we would like to stress that the CJT effective action approach is able to clear up the physical insight into chiral symmetry in nuclear medium and at the same time, hopefully to produce an adequate framework for treating the calculations beyond the two-loop approximation.

* * *

One of the author (THP) would like thank the International Centre for Theoretical Physics, Trieste, for the hospitality extended to him. The support of Vietnam Foundation for Scientific Research is acknowledged.

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