quark condensate in the nuclear matter

z. Phys. A 353,455-466 (1996) ZEITSCHRIFT FOR PHYSIK A �9 Springer-Verlag 1996

Quark condensate in the nuclear matter E.G.Drukarev, M.G.Ryskin, V.A.Sadovnikova

Petersburg Nuclear Physics Institute, Gatchina, St.Petersburg 188350, Russia

Received: 13 June 1995

Communicated by V.V. Anisovich

Abstract. We calculate the quark condensate in the nuclear matter, taking into account the single-pion and two-pion exchanges between nucleons. We find the dependence of the averaged value of the quark operator qq on the density of the matter p. At very low density the nonlinear terms are proportional to p2 and increase the tendency to restoration of the chiral symmetry. At larger values of density the account of interaction inside the matter slower down the restoration of chiral symmetry compared to the gas approximation law. The leading nonlinear term in Fermi momentum power expansion becomes of the order p4/3. The value of the condensate at the saturation value of density is obtained. The role of multinucleon effects is analyzed.

PACS: 14.80.Dq, 21.65.+f.

1. Introduction

The expectation value of the quark operator

qq = ~u +dd (1)

is known to describe the breakdown of chiral symmetry. Thus the chiral properties of the nuclear matter are connected with the expectation value

t~(p) = (NMlq(O)q(O)INM) (2)

with I N M } - the ground state of the matter while p stands for the density.

The value of ~ in vacuum is [1]

2 2 t~(0) = (0[~q]0} - -2f~zm~ - 0.029 GeV 3. (3)

TFb u + T/~ d

Here mq;~r are the masses of u and d quarks and that of pion while f~ ~ 92 MeV.

The gas approximation formula

n(p) = t~(O) + p(X[glqIN} + . . . (4)

with IN} being the free nucleon was used firstly by Drukarev and Levin [2]. Equation (4) is quite trivial. Indeed, it is obvious that if the density p is small enough, the second

term in right-hand side of Eq. (4) dominates in the expansion of t~ in powers of Fermi momentum pp. This approximation corresponds to the neglection of the interaction in the system. The nonlinear in density contributions to ~(p) are caused by the interactions between the nucleons of the matter. They will be the subject of the investigation.

It is important to find the value of t~ at the saturation point p = P0. It is also interesting to obtain the function ~(p). The nonlinear contributions, neglected in Eq(4), determine the character of restoration of chiral symmetry while the value of p increases. This problem was analyzed by M.Ericson [3]. Also the QCD sum rules ap- proach to the investigation of nuclear matter [2, 4, 5] ties the nucleon-nucleon interaction in vector and scalar chan- nels with the values of condensates ( N M I q T o q ] N M ) and (NMICtq[NM) = t~(p). The former condensate is linear in p due to the baryon number conservation. The form of dependence of ec on p becomes important for the analysis of the problem of the saturation of nuclear matter.

The nonlinear contributions to ~ were considered in a number of papers. Drukarev and Levin [2, 4] and Drukarev and Ryskin [5] showed that in the consistent expansion of

in powers of Fermi momentum the leading nonlinear contribution comes from the account of the one-pion exchange Fock term. Chanfray and M.Ericson [6] expressed the contribution of the pion field through the pion excess number [7]. Also Chanfray et al. [8] considered t~ as the result of gen- eralization of Eq. (3) for the case of finite densities. Cohen et al. [9] and Lutz et al. [10] analyzed the nonlinear contributions to n in framework of Nambu-Jona-Lazinio model. The limits of validity of the gas approximation equation (4) were investigated in the papers of Celenza et al. [11]. The subject was considered also by Saito and Thomas [12].

In the present paper we calculate the function ~(p) taking into account the two-nucleon configurations in the nuclear matter. The consistent treatment of the Pauli principle yields the account of certain three-nucleon configurations. We con- sider n as the result of the interaction of the operator ~q with the pions which are exchanged by the nucleons. We single out the one-pion and two-pion exchanges. The account of the exchange by larger number of pions between the two nucleons just makes the wave function of initial state more

456

Fig. 1. The interaction of the operator (lq (the dark blob) with the pion field emerging in the single-pion exchange. The solid lines denote the nucleons; wavy lines denote the pions

Pl

P2

Pt Pl

p2 P2

a b

p2 Pl ~ / Pl

Pt P2 P2

p~ p~ P . ~ _ ~ . . ~ t

d e

Fig. 2a-e. The interaction of the operator ~q with the pion field created by the two-pion exchange between the two nucleons. The solid line stands for baryons (N or A). The other notations are the same as in Fig. l

precise (the latter is the plane wave in the case of nuclear matter).

Hence, the function t~(p) is treated as the amplitude of the interaction of the operator ~q with the virtual pions of the diagrams shown in Figs. 1,2. The amplitude contains the condensate

r / = (Tr[qqlTr) - 2m~ (5) ?nu + TY~d

as a factor. We take into account N N , N A and A A configurations

as the two-baryon intermediate states. This approximation is often used in the calculations which involve the two-pion exchange [7, 13, 14]. It is based on the small values of the coupling constants for the heavier baryons and on the larger values of their masses.

If the value of p is small enough, the function t~(p) can be presented as power series in Fermi momentum PF. For PF --> m~r (p >_ 2 . 10 .2 Fm -3) the single-pion exchange contributes the terms of the order p4/3 to t~ [2]. To estimate the contribution of the two-pion exchange note that the integration of the diagrams of Fig. 2 over momenta of the initial nucleons Pl,2 give the factor of the order p2. In the case of N N intermediate states the integral over pion momenta k is saturated by k ~ PF leading to the contributions of the order p5/3 to ~. In the case of N A and A A intermediate states the integral over k is dominated by larger values and these states contribute the terms of the order p2. The region k ~ p f provides the terms containing the powers of the ratio p F / A with

A = mA -- m ~ 300 MeV (6)

being the N A mass splitting.

If the density is very small, i.e. PF << m~ (p << 2 �9 10 . 2 Fm -3) the one-pion exchange term is quenched since k < 2pF. In this case the nonlinear contribution to t~(p) is determined by two-pion exchange. The leading nonlinear term is of the order p2.

We single-out the terms linear in p:

to(p) = ~(0) + p(NtCtq[N ) + S(p);

S(p) =-- pF(p). (7)

Thus the dependence of F on p determines all the nonlinear effects. Recall also the values of the saturation density P0 and that of corresponding Fermi momentum PF0

P0 = 0.17 Fm -3 = 1.3- 10 -3 GeV3;

PF0 = 268 MeV/c . (8)

Introduce also

z = PF/PFO;

We present

S(p) = pt(z);

P/Po = z 3- (9)

t (z) = ~ A n z ~ (10) n=l

with An being dependent on the value of the pion mass m~. At n = 1,2 the function t is determined by the values of pion momenta k ~ PF. Hence,

A~ = ~ n = 1,2. (11)

For larger n m~ is compared to other parameters of the same dimension which do not depend on density. Thus

t (Z ) = (D1 z + ~P2 + A n ( m ~ ) z ~. (12) ~=3

It is instructive to compare the function t (z) to that with the last term considered in the chiral limit

r ( z ) = ~1 z + ~2 z 2 + anz ~ (12a) n=3

a n = A ~ ( 0 ) if n > _ 4 .

At n = 3 the coefficient A3 contains Inm, , ; we determine a3 as the sum of this term and the chiral limit of the contribution regular in m r . Note also that the contribution of the order p2 to t~ contains the terms proportional to In p. They are included into A3.

The meaning of the function t (z) is obvious. Equation (7) can be presented in the form

~(p) = ~(0) + p ( ( N l q q l N ) + t(z)). (#)

Thus the sum in the brackets presents the "effective value" of (N lqq lN) if we use the gas approximation equation (4).

If z = PF/PFo is small enough we can expect the convergence of the power expansions of Eqs. (12) and (12a). However at z ~ 1 all the terms with n > 3 can be of the same order of magnitude. Indeed, the contributions with n _> 4 contain the powers of the ratio p F / A . The latter is close to unity at z ~ 1. Hence, at z ~ 1 we present

( P F ) pz (13) ~(p) = P2(z)p + U

457

with P2 - a certain polynom of the second power in z, U is a certain function. Our aim is the calculation of/:)2 and U.

The interactions involving larger number of nucleons can be accounted for by modification of the propagators of nucleons and pions. In the first case kinetic energies of nucleons depend on effective mass m*. The pion propagators obtain self-energy insertions [15]. The latter include the functions gNN(P) and gNA(P)~ which describe the correlations between the baryons. The values of these functions are available at P = P0 only. Besides, the accuracy of the values is not sufficient for making an unique conclusion about the role of multinucleon effects.

Note that the interactions of the operator qq with vector mesons contribute to the terms of the order p2 in expansion of ~; and to the higher ones. The interaction occurs at the distances of the order of the inversed masses of the mesons. Thus the consistent calculation of the contribution should be based on the quark structure of the nucleons. We expect to present the analysis of the point elsewhere.

The contributions of the diagrams of Figs. 1 and 2 are determined by various intervals of the values of pion momenta k. Some of the integrals are divergent in the limit of the point nucleons. The account of finite size of the latter effectively cuts off the integrals at certain k < ra. This enables us to use the nonrelativistic approximation in description of the baryons and the vertexes of these interactions with pions.

2. The contribution of the single-pion exchange

It was shown in the papers [2, 4, 5] that the leading nonlinear contribution to ~ comes from the interaction of the operator qq with the pion field of the single-pion interaction if p is small enough. The contribution is illustrated by Fig. 1. It is

= - 4 Sp (270 3 ( i JToD~(t%, k)GForl (14)

with the trace over the spin variables of nucleon. In Eq. (14) Do (j~o 2 k2 2 = - - mTr + ie) -I is the propagator of the free pion, while

1 i Fo = ~ - ~ ' 7 5 ~ V " = v ~ f = X*((rk)x (15)

is the vertex of 7cNN interaction following from the chiral Lagrangian - see, e.g. [15]. In Eq. (15) g)(X) denotes the (non)relativistic spin function of the nucleon.

The nonrelativistic nucleon propagator in medium is [16]

G(p, E) = Go(p, E) + Gp(p, E) (16)

with

1 G0(p, E) - (17)

E - ~(p) + i~

while e(p) denotes the eigen value of the energy of the nucleon with momentum p while

Up(p, E) = 27ri 6( E - e(p))O(pF - p). (18)

Since the potential energies of all the nucleons in nuclear matter are the same, Go coincides with the free propagator in vacuum. The function Gp accounts the antisymmetriza- tion of the propagating nucleon with those belonging to the

occupied states of the Fermi sea. Th terms caused by Gp are known as the Pauli-blocking terms.

While the first term of Eq. (16) leads to the correction which is already included in the free nucleon, the second term gives [5]

99i = 87r2 f~ r! 1 - --PF arctan--m~ (19)

( 2 ~ ) 4 2 2 2 4p~ + m,~ m~ _ _ 4pF + rr~ ] ra~ In + - - In .

The function 991 is shown in Fig. 5. At m~ = 0 we find ~1 = -4 .16 ; at the physical value of ra~ we obtain 991 = - 1 . 1 3 if PF = p~o.

Note that at PF ~ PF0 the chiral limit of the equation

~(p) = ~(0) + p N lqq[N ) + Pe #l (20) PFO

leads to the values of ~(p) very close to those provided by physical value of rn~ ~ 138 MeV. The reason is that the value of (N[qqlN) is connected with 7rNcr-term by the relation

2(7 (N lqq lN) - (21)

/TL u + ~t~ d

The value of cr can be obtained by extrapolation of the data on low-energy %N scattering to Cheng-Dashen point [18]. The nowadays experimental value is [19]

~ 60 MeV (22)

while

cr = Z - cr~ (23)

with the numerical value (71 ~ 15 MeV [20]. In the chiral limit when ra 2 ~, m~,d ~ 0 while m 2 / ( m ~ + rod) -+const

2. cr/(ra~ + m~) --+const and ~rl/or we obtain also (7 ~ m~, m~ -+ 0 [21]. In the chiral limit we must put cr = ~ and 991 = 991(0). Putting rrz~ + raa = 11 MeV [22] we find the value of t~ very close to that provided by values m~r 138 MeV and cr ~ 45 MeV.

3. The estimation of the contributions of the two-pion exchange graphs

Here we obtain the dependence of tile graphs shown in Figs. 2,3 on the density p. Note firstly that we shall average the operator qq over the pion states only, but not over the intermediate baryon states. Indeed, in the case of N N intermediate states all the contributions containing IN]qqlN) are already accounted for by Eq. (7). In the case of 2VA and A A intermediate states we meet the contributions proportional to d -= (Atqqlz3) - (NlqqIN) . However at least in the additive quark model [23] we find d = 0 [24]. We carry out the calculations in this limit. We discuss the possible influence of finite value of d in Sect. 10.

Hence, we shall average the operator qq over the pion states only. From the technical point of view this move is supported by the following argument. The value of the condensate (NIqqlN)~ for the nucleon in the matter can be presented as

Pl

P

458

b

p2

d

c

e

Fig. 3a-e. The diagrams of Fig. 2 in the particle-hole picture

( O Z N x (Nlqq[N),~ = (N,~]CtqlN,~) + (NlqqlN) \ - ~ - - ~ - j . (24)

Here IN(,~)) is the state of the nucleon (in the matter) with the energy E, SN being the self-energy. Using the mul- tiplicative character of renormalization IN,~) = ZU2{N); OZN/OE = Z - 1 we find (NIgtqlN),~ = (NlqqIN), prov- ing the earlier statement.

Now we start to estimate the contributions of the diagrams of Fig. 2. Each of them contributes

3 d3pl d p~ S(p) = (27l.)3 (2rc)3 05(Pl, P2) (25)

with Pl,2 -< PF are the momenta of initial state nucleons while ~ denotes the sum over the spin and isospin states. The function 05 is the result of integration over pion momenta

d4 k 05([01, P2) = ~ WQgl , P2, k) (26)

or, after integration over k0

f d3 k @Q;Ol, ]92) -- ~ ~)(~01, j02,/C). (27)

The integration over Pl and P2 in Eq. (25) leads to the factor p2. It is the integral over k which is the subject of estimation. It is instructive to present the integral over/Co as the sum of the residues in the baryon and pion poles.

In the case of N N intermediate states the leading contribution comes from the nucleon pole. In the limit of point nucleons

/C4 1 ~ h6(k) [/C2 _ (/cq)]/ra " C N N .T] (28)

with

( f N N r c ~ 4 . CNN = \ ~ - - - ~ / , f x x ~ = 1.0, (29)

while h(/c) = (/C2 + m~)l/2 and q = Pl - - / )2 . The condensate rl is determined by Eq. (5). In the chiral limit ra~r = 0 the integral over /C is determined by /C ~ q ~ PF. Thus & 1/pF and

s ~ p 2 ~ ~ p~. (30) PF

The finite value of m~ provides another scale of momentum k. At PF > m~ the region k ~ m r appears to be strongly quenched. The region gives the contributions to ~ containing the powers of m~/pF. This is the way how the function ~pl(ra~/pF) introduced in Eq. (11) is formed.

The pion pole in k0 in Eq. (26) gives

/C4 ~ ~ CNN~ (31)

and the integral in Eq. (27) obtains logarithmic divergence at large k in the limit of the point nucleons. The exchange terms depend on q and obtain the contributions containing In q. Thus ~c contains the terms

S ~ p2; S N p21np. (32)

The logarithmic divergence at the upper limit is removed by the account of the finite size of the nucleons. The formfactor of ~rNN vertex can be presented as [14]

F = F0 exp - ~ -~ (33)

with A = A1 = 0.67 GeV [14] corresponding to the size of the nucleon (@2})1/2 ~ 0.8 Fro.

Note that although the integral presented by Eq. (27) converges for the contribution of the nucleon pole, the account of finite value of A does modify its numerical value. Hence, the contribution contains also the higher terms of expansion in powers of PF.

For the N A intermediate states we find the contribution of the nucleon pole in the chiral limit to be

(34) 1 Cevza /C2 5' ~ [/C2 _ (/cq)]/m + A - -

with CNA = CNN(fNA~/fNN~) 2. The integral determined by Eq. (27) is saturated by k ~ (raA) 1/2 ~ 500 MeV in the limit of the point baryons. The finite sizes of the baryons can be taken into account by Eq. (33) with d = A 2 (generally speaking, A2 r A1) modifying the numerical value of S ,0 2 .

The contribution of the pion pole does not contain the terms ~ In rn~ this time due to the finite value of A. If A2 >> A the integrals over h are determined by the region A << k << A 2. However the realistic values are A2 ~ A1 2A and the integral is dominated by k ~ B, leading to S ~-, p2. In the exchange terms the region k ~ q ~ PF provides the contributions depending on the ratio pF/A, i.e. S ~ p2~_f (e~_) with f - a certain function with a finite value f(0). At PJv ~ PF0 we find p f / A ~ 1. However the physical meaning of the two parameters is quite different.

The Pauli blocking terms contribute S ~ p2 m /py ,-, p5/3 and S ~ p2pF/A ~ p7/3 in the case of N N and N A states. The structure of the contributions of A A states is the same as that in N A case.

To summarize, in the case of N N intermediate states the contribution to t~ is determined by pion momenta k ~ PF leading to the terms of the order pp~. If the intermediate states N A and z~A are included, the regions k ~ (mA) 1/2 and k ~ A provide the contributions of the order p2, while

the region k ~ PF causes the higher order terms in expansion in powers of PF.

Note that the chiral Lagrangian contains also the scalar ~-N interaction. It can be taken into account by replacement of the product (o -k )G(ak) by 7rN a-term. This provides the contribution of the order p2. It vanishes in the chiral limit being numerically very small at observable value of m~.

In the concrete calculations we use the value fzz~ ~ = 4f2N~v [13]. To analyze the possible influence of the corrections to this relation which may be of the order of 15% we shall investigate the dependence of our results on

f~vza~ C N A (35) 11'- 4f2VN. ~ - C N N "

We also put A2 -- A1 in Eq. (33), following ref.[14]. We discuss the latter assumption in Sect. 10.

To avoid too cumbersome formulas we shall present the equations for the functions ~ and r - Eqs. (26),(27) for the special case of the exchange by two neutral pions between neutron and proton (between two neutrons in exchange graphs). We call them 7r ~ graphs. Thus the total contribution will be expressed "in units" of these contributions - see Eqs. (44)-(47), (60),(66), etc. below.

4. T h e N N i n t e r m e d i a t e s t a t e s

Note firstly that we are looking for the real contribution to ~. Imaginary part corresponds to the iteration of the single-pion exchange. It is already accounted for in the wave function. Thus considering the product of the nucleon propagators

G(pl - k)G(p2 + k) = Go(p1 - k)Go(p2 + k)

+(Go(PI - k)Gp(p2 + k) + (1 --+ 2)) (36)

we must include only the pole contribution while calculating the contribution of the first term.

1. The contribution of the nucleon pole

It can be presented as

S = S f .-k Sp (37)

with the two terms corresponding to those in Eq. (36). To obtain S f we calculate for 7r ~ graph (see the definition in the end of the latest section) of Fig. 2a

4k 4 1 ~2f = h 6 ( k ) [k 2 _ ( k q ) ] / m C N N ~ (q = Pl -- P2). (38)

Here we used the possibility to change the signs of momenta Pl,2. Eq. (38) leads to

21m f ~ f -- d k ~ In C N N ~ . (39)

For the exchange graph of Fig. 2b

29(k, q)CN N~I ~ i ~ = h4(k)[(q _ k)2 + m 2 ] [ , ~ 2 _ ( k q ) ] / m (40)

with 9(k , q) = 1/2 ( - k 2 ( k - q)2 + 2(k, k - q)2). This leads to the rather complicated expression for e f t .

+ p e r m u t a t i o n s

459

5 Fig. 4. An example of the interaction of the operator ~q with the pion field created by the two-pion exchange between three nucleons

The diagrams of Fig. 2 c,d vanish in nonrelativistic approximation since the poles of both nucleon propagators are in the same half-plane of complex variable k0. The graph of Fig. 2e also vanishes in the nonrelativistic approximation after the renormalization is carried out - see similar situation in ref.[25].

The Pauli blocking contribution of all the graphs of Fig. 2a-e is finite. Note, however, that the diagram of Fig. 2d contributes to the imaginary part only. There are certain re- lations between the other contributions. Since both external nucleons as well as one of intermediate ones belong to the Fermi sea, the integrand of the integral in Eq. (25) is sym- metric with respect to permutations of momenta p~, P2 and P3 = P~ - k. Thus the ladder graph of Fig. 2a gives the same contribution as the crossed graph of Fig. 2c; also the contributions of ~_0 graphs of Fig. 2b and 2e coincide. One can see that after all the possible isospin states of the nucleons and pions are included we obtain the set of Pauli blocking graphs to give the contribution corresponding to subtraction of the two-pion exchanges between three nucleons belonging to Fermi sea - Fig. 4.

For the Pauli-blocking part of the graph of Fig. 2a we obtain

4k 4 1 ~p =

h6(k) [k 2 - ( k q ) ] / m

• [ 0 q o F - - k t ) + 0 q o F - l p : - k l ] . ( 4 1 )

The 0-functions single-out the intervals of integration in momentum k:

0 < k < PF -- P~ (42)

- - - and

--1 < ti < 1;

with ti - (p~k) [pi[.lk[

3(p0 < < l

with

PF -- Pi < k < pF + pi (43)

2pik

We are ready now to find the total contribution of the order p5. Taking into account all the possible isospin states of nucleons and pions, note that the diagrams of Fig. 3 containing the independent identical particle-hole loops (pp and nn pairs in the graphs of Fig. 2a,c) contain the additional factor 1/2. Thus we obtain

S(p) = 6Sa + 6S~ - (--2)Sb -- ( -8)S~ (44)

with the indices corresponding to those of Fig. 2. Eq. (44) can be presented also as

S(p ) = 6 S f + 12Sp - ( -2)S /~ - (-10)Sp~ (45)

460

~ . 0 i i

~ , 0

2 . 0

0 . 0

- 2 , 0

- 4 . 0

~2

-6.0

I. i I , O. 00 0 . 2 0 O. "40 0.60

t I 0.80

i 1

I, O0

rn~/pF

F i g . 5. T h e func t ions ~ l mad (/9 2 ( see the text for the defi-

n i t ions)

with indexes f , p labelling the free nucleon propagator and the Pauli-blocking part. Index e denotes the contribution of the exchange graphs.

In notation of Eqs. (10)-(12) we present

~2(mTr/PF) = 6Vf + 12Vp + 2Vfe + 10Vpe (46)

with the terms in right-hand side corresponding to those in Eq. (45). They are the functions of the ratio rn~/pF. The function qo2 is presented in Fig. 5. At PF = Pgo

vy = 1.32; Vp = - 0 . 4 0 ; vy~ =0.51; vpe = 0.07; ~2 = 4.86. (47)

Note that the integral in the right-hand side of Eq. (39) converges for the point nucleons. However the account of the finite size of the latter changes the numerical value strongly. Taking into account Eq. (33) we find

Of = 772- ~m J ~dk �9 k s In kk-+ qq e_2k2/A2CNNf]. (48)

Comparing the integrals in Eqs. (39) and (48) note that in the first of them the region k > q ~ PF provides the half of the total contribution at ra~ = 0. The account of the finite value of m~ increases the role of the region k > PF where the influence of the exponential factor becomes important. Say, at PF = PF0 we find

v f = 0 . 5 1 ; v p = - 0 . 2 9 ; ;vy~=0.16; vp~=0.09. (49)

The functions ffz) and ~-(z) defined by Eqs. (12) and (12a) contain not only the terms of the order z 2 but also the higher ones. Indeed, the region k > A provides the contribution of the order m/A to the function Cy leading to the terms of the order z 3 to the function t(z). The amplitudes of Pauli blocking contributions obtain the terms of the order p~/A 2. Thus we can present

2 ( p ~ n S(p, A) = PPF E b~ (50) \ A ]

n=0

with the coefficients b~ being the functions of the ratio m~/pF. The coefficient A,~ of Eq. (12) are related to b~ by equation

< ~ ~ An = b~(m, /pso) -~- (51)

Approximation of the function S(p, A) by the power series of Eq. (50) gives

A3 = - 7 . 9 4 ; A4 =4.12; A5 = - 0 . 3 1 (52)

a3 = -8.60; a4 = 5.12; a5 = -0.07.

Thus at z = 1

N = tN1 v = 0.73; TffN 1.31. (53)

Here the lower index denotes the two-baryon intermediate state while the upper one shows the particle on the mass shell.

The contribution of the pion poles

The pion poles contribute to the graphs of Fig. 2a-d pro- viding the values of the same order. However the contributions of Fig. 2a,c cancel each other to large extent. The sum leads to additional factor k /m << 1 in the function

- see Eq. (27). Thus the main contribution comes from the exchange graphs of Fig. 2b and 2d. For the first of them

1 j" dk k)e_2kZ/A2 r = ~ ~ ( q , (54)

with

@(q, k)

2k - q } • In 2k + q + q3k - 2k3q " (55)

461

Note that at k << q ~ ~ ozk 4 while at k >> q ~ ~ ilk4; the values of the coefficients are c~ = 19/6; /3 = - 5 / 2 . Hence, the integral over k in Eq. (54) diverges at the upper limit for the point nucleon and on the lower limit for the chiral pion. In both cases we face the logarithmic divergence. After we single out the contribution/3 In(A/q)+ c~ ln(q/rn~) the remaining part of the integral of Eq. (55) is a smooth function of the parameter rn~/q with weak dependence on the latter. Including also the contribution of the diagram of Fig. 2d we find

UNN(Z ) z 3 ( - 2 . 4 1 1 n A = - - + 1.46 + 0.53 In PF (56) rn~r rr~r j

with the three terms being determined by the regions m~ << k << A; k ~ PF and m~ << k << PF. Eq.(56) can be presented as

t~NN(Z) Z3 ( -1"881n ~ ) = + 0.53 In z (57) T~Tr

g = 0.36 G e V .

The composition of Eqs.(52),(56) and (57) gives for the total contribution of N N intermediate states

A3 = - 9 . 7 4 + 0 . 5 3 1 n z ; A 4 = 4 . 1 2 ; A 5 = - 0 . 3 1 (58)

a3 = - 1 0 . 4 0 + 0 . 5 3 1 n z ; a4 = 5.12; a5 = -0 .07 .

A t z = l we find

tNN = --1.07; rNN = --0.49. (59)

5. The intermediate states N / A

In this case we also present the result as the sum of the contributions of the nucleon and pion poles.

1. The contribution of the nucleon pole is

S = pz3(12vf + 24vp - 20vy~ - 40vpe) (60)

with vk describing the 7r ~ graphs with the dependence on pz 3 being singled out. The function vy is the result of the integration of the function

( 2 ) 2 4k4 (61) ~)a = h6(z~ + [k 2 - (kq)]/rn) CNzxrl

with h = (k 2 + rn~)V2; the factors r] a and CNz~ are defined by Eqs. (5) and (35).

The integral for the function r defined by Eq. (27) is finite even for the point nucleon being saturated by k (mz3)V2 ~ 530 MeV. However the finite size of the baryons influences the numerical value of the function q~. Thus the latter should be calculated as

qS, = 4 (2702 h6(k) A + k2/rn e CNzar 1 (62)

(we put mza = m in the kinetic energy terms k2/2ra). Fol- lowing Sect. 3 we put A2 ~ 0.67 GeV.

The exchange contribution (Fig. 2b) has a more complicated structure since the function ~9 depends not only on k but also on q.

Orb = ~fbl + ~fb2(q) (63)

with q~fb~ = @b(q = 0). We find for the first term

1 ~fb I = 2 ~ fa (64)

while

~fb2 = 4 (27r)2z~

x l + k ~ - - ~- 1+

The integral is saturated by k ~ q ~ PF. After the integration over Pi it provides the contribution of the order papF/A to n.

Eqs. (62) and (65) give in the chiral limit rn~ = 0

vf = 0.65; vy~ = 0.325 + 0.033z. (66)

For the Pauli-blocking contribution of Fig. 2a

~ a = ( 2 ~ 2 4k 4 -- \ 3 J " hU~ O(pF - tPl - kl)CNa~7. (67)

The same reasons as in the case of N N intermediate state lead to the equal contributions of the 7r ~ graphs of Fig. 2a and 2c and also of those of Fig. 2b,d,e. In the chiral limit

24 v p - ~vp~ = 1.17z. (68)

Finally, we come to the contribution of the nucleon pole

rNNA = Z3(1.35 + 0.51Z)U. (69)

Similar calculations for the observable value of rn~r give

tNzx = Z3(0.43 + 1.13z)u. (69a)

(65)

4q2k 2 In CNzxr].

2. The contribution of the pion pole

As in the case of N N intermediate states, the graphs of Fig. 2a and 2c contain the pion poles of the third order. In the limit A >> A they contain the logarithmic terms In A/A which cancel each other. For the realistic values of A -~ 0.7 GeV In A/B _~ 1 and the sum provides the finite value with

~a 1 6 / d k k6 [ 1 1 h ]

= 9- (270 2 h6(k) ~ - ~ + 2 (h + h ) 2

• e-2k2/A2CNzS~] (70)

while q5 c can be described by the same expression by chang- ing the sign of A.

The exchange graphs contain the terms which are dominated by large k >> q N PF. They can be calculated in the limit q = 0, while the account of finite values of q provides the higher order terms in powers of PF. We can use Eq. (63) with

_ 8._5 / ~ dk . k 6 eb~

9 J (271") 2 h 6

(, • ~ - ~ + 2 (h + A)~ e--2k2/A2CNarl (71)

8 1 qSb2 -- 9 (2702 CNz~I J(q) (72)

462

while

J(q) = l f dkh6k7 [, 2 k - q ( q2 q4 ) q

+ ~ ~ 7 - ~ + 2 h

Since momentum q varies in the finite interval 0 < q < 2pF we present

E q (74) J(q) = c,~ r~=l

As one can see from Eq. (73), the function J(q) depends on the ratio q/A. The latter is of the order of unity at q ~ PF0. Since the integral over Pi is dominated by q ~ PF, we have no reasons to expect a priori the quick convergence of this contribution to ec at PF close to PFO. However it appears to be sufficient to include the first four terms of the series only. In the chiral limit m~ = 0 we find

el = - 2 . 2 0 ; e2 = 1.45; e3 = - 0 . 6 1 ; e4 =0.11. (75)

These values give good accuracy at q ~ PF0. After the contribution of the graphs of Fig. 2c,d is cal-

culated in the same way, we obtain

t~va = ( ' 2 . 3 3 z 3 - 0.01Z 4 -- 0-38Z 5

--0.13Z 6 + 0.05zT)u (76)

with u defined by Eq. (35). In the chiral limit

7-~vz~ = ( - 6 . 7 3 z 3 - 1.33z 4 + 2.32z 5

- 0 . 6 9 z 6 + 0.10zV)u, (77)

The large difference in the values of the coefficients in Eqs. (76) and (77) is caused by the large contribution of the region k ~ A in the integral over k which is sensitive to the value of m,~.

Joining Eqs. (76) and (77) with Eqs. (69) and (69a) we find for the total contribution of N A intermediate states

t N zi~ = z3(--1.90+ 1 .12z-- 0.38z 2

--0.13z 3 + 0.05z4)u (78)

and

TN z~ : z3( -5 .38 - 0.82z + 2.32z 2

- 0 . 6 9 z 3 + O.10z4)u. (79)

If z = 1

tNa = --1.24u; rNzX = 4.47u. (80)

6. The intermediate states A A

The leading contribution to the expansion in series of pv is determined now by large pion momenta k >> PF for both baryon pole and pion pole contributions. Thus we treat them simultaneously. The general expression is

S = pz3(6v~ - 4Vb + 6Vc - 8va) (81)

with v,~ being the contributions of rr ~ graphs corresponding to Fig. 2n with the dependence on pz 3 ~ p2 being singled out.

We adduce the expressions for the contributions of the diagrams of Fig. 2 in the chiral limit. For the ladder graph of Fig. 2a with both delta and pion poles contributing we obtain

8 ; dk k 2 + 9 / 8 k A + 3 / 8 A 2 ~b~ = 2-7 a ( - ~ ) 2k A(k + A)3

e -2ki/A2 C a a r l (82)

with CZXA = CNNU 2. The contribution of the diagram of Fig. 2c is determined by the pion pole only:

1 f dk k 5 k 2 + 4 k A + A 2 q~ = 9 (2~r) 2 (A + k) 4

e -2k2/A2 Czszsr]. (83)

For the exchange graphs Eq. (63) is true. For the graph of Fig. 2b we find

5 f dk k(2k "F A) e_2k2/A2 ~lb = 1 ~ - - (2702 A(k + A) 2 CAZr/ (84)

and

J(q)Cz~zxr] (85) r -- 162(2702 ;

/ d 2k + A J(q) = - _

x 1 + 5q2 5q 1 - + 4k 2 4k ~ 5 - ~ q 2 1 n 2 ~ q J "

(86)

Using similar equations for the graph of Fig. 2d we approx- imate the functions J(q) by Eq. (74).

Finally we come to the expression for the contribution of B A intermediate states

tAA = Z3(0.86 -- 0.06Z + 1.45Z 2 -- 0.77Z 3

+0.12Z4)t/2, (87)

while in the chiral limit

rzxzx = z3(1.46 - 0.07z + 1.14z 2 + 0.99z 3

--0.43zg)u 2. (88)

A t z = l

tZXA = 1.60U2; raz~ = 3.09u 2. (89)

7. The total contribution of the single-pion and two-pion states

Now we join Eqs.(58),(78) and (86) to find the final equation

t(z) = z~l + z3q)z

+0.53z 3 In z + z3T(z) (90)

with

T(z) = - 9 . 7 4 - 1.90v + 0.86v 2

+z(4.12 + 1.12v - 0.06v 2)

+z3(-0 .31 - 0.38u + 1.45u 2)

+z3( -0 .13u - 0.77u 2) + z4(0.05 + 0.12u2). (91)

463

Since in the broad interval of the values of Fermi momenta PF near the saturation value PF = PRO (m~/pF ~ 0.5) the function ~z has the broad plato, we can put here PF ~ PFo in the arguments of the functions. Using the values ~1(1/2) -1 .1 ; ~2(1/2) ~ 4.9 we find at u = 1

t(z) = - 1 . 1 z + 4 . 9 z 2 + z 3 ( - 1 0 . 8 + 0 . 5 1 n z

+5.2z + 0.8z 2 - 0.9z 4 + 0.2z 4) (92)

and

~(1) = -1 .7 . (93)

Under the same assumptions

T(z) = - 1 . 1 z + 4 . 9 z 2 + z 3 ( - 1 4 . 3 + 0 . 5 1 n z

+4.2z + 3.5z 2 + 0.3z 3 - 0 . 2 z 4) (94)

and T(1) : - 2 .8 . (93')

Note that the difference between the values of t and "r in the final result is smaller than in the partial contributions calculated in Sects. 4-6.

Recall that the terms of the order z 4 and the higher ones in the right-hand sides of Eqs.(92) and (94), corresponding to the contributions ~ p7 to the function n(p), present the approximation of the contribution of the two-nucleon initial states to t~, if z is of the order of unity - compare Eq. (13). For z << 1 (i.e. p F / A << 1) right-hand sides of Eqs.(92),(94) can be treated as the expansion in powers of Fermi momentum of the total contribution to t~(p). In this case the terms

z 4 are the highest one which can be accounted for since the contributions ~ z 6 corresponding to those of the order p3 to t~ come also from three-nucleon states.

8. The account of the multinuclear configurations

These effects modify both nucleon and pion functions. In the first case the manifest themselves through the dependence of kinetic energy on the effective mass of the nucleon m*. We find a noticeable dependence on m in the terms

N N tNN(T~gN) only. They are proportional to m. The value of the ratio ra*(p) /m at p = P0 varies between 0.9 and 0.7 in realistic models. Using Eq. (53) we obtain that the account of the effect adds gt ~ ( -0 .1 ) - ( , 0 . 2 ) to the value ~(1). In the expansion in powers of Fermi momentum the effect manifests itself in the higher order terms, of expansion of t~(p), starting with the term of the order p%.

The function S(p) in the Hartree approximation (e.g. ig- noring the exchange terms) can be obtained from Eq. (7) illustrated by Fig. 1 if the dependence of the propagators and of the vertex functions on the density p is included. The result of the calculation contains certain functions of the nucleon density. The values of the latter are available at P = P0 only. Thus we can present only the estimations at the saturation value of density.

We use the well-known formulas for the pion propagator in medium D,~ and for the vertex function FNN~ [15, 26]

D ~ n I -= lr 2 - - k 2 - - m~z _ H(ko, k) (95)

with the pion self-energy

H(ko, k) = Ifp(ko, k) + [Is(ko, k). (96)

If the pion energy ko is small enough, the s-wave term is [27]

H.~ - (TP f~ (97)

while the p-wave contribution can be presented as

Hp = -k2X(ko, k). (98)

The single-loop pion susceptibility X(k0, k) and the vertex function FNN~ are renormalized by the correlations in medium

xo(ko, k) X(ko, k) =

1 + 9 (p))~0(/%, k) F

I')vN~ = 1 + 9'(p)xo(ho, k ) (99)

The function 9!(p) accounts the correlations in the particle- hole interaction. Unfortunately, the value 9NN(PO)! ~ 0.7 for

/ N N interactions and the estimation 0.3 < 9Nzx(Po) < 0.7 for N A interactions is the only data on the function available nowadays [16].

The account of the pion susceptibility in the lowest order of perturbative expansion corresponds to the calculation of the diagrams of Fig. 2a,e. If the pion self-energy H does not bring new singularities to the function D ~ , we can still calculate the contribution of the diagram of Fig. 1 as the residue in the pion pole. In this case N A excitations dominate and k0 ~ k _> A. At these values of variables IX(k0, k)l < 0.1 at p ~ p0 [15]. Hence the corrections to the contributions of the pion poles of the diagrams of Fig. 2 are small.

The function D ~ contains also the cuts connected with the particle-hole interactions in which the baryons come to the mass-shell. For Az5 intermediate states k0 ~ A, k >> A. At these values X << 1 [15] and the medium provides small corrections. For N A intermediate states the leading values of h are of the order (mA)U2 ~ 2pF0. Putting ! = 0.7 9 N N [15] and neglecting the contribution of delta to the pion

N susceptibility we find X ~ 0.5 and ~ N N ~' 1.2. It is curious that the account of multinucleon correlations brings the value

N of tNN closer to it's chiral l imi t "INN - see Eq. (69)J For two free nucleons in intermediate states the values of k ~ PF and ho ~ p~ /2m become important. Using the general equation for the function X(ko, k) we find N ]~NN ~ 2.7�9 It should be

N compared to the contribution of vf Eq. (49) - to tNN which N provides the value t, N N = 3.1.

The total contribution of all the Pauli-blocking contributions is contained in the diagram of Fig. 1 with the nucleon being described by the second term in the right-hand side of Eq. (16) - see Eq. (18). The result depends strongly on the exact values of ~ ! ra*. 9 N N , 9 N A , U, (7 and This becomes clear if we note that most of the nucleons carry the kinetic energy e close to eF = p%/2m. Hence we can put k0 = 0; since k < 2pF, the expansion of the function X(0, h) in powers of k/2pF presented in [15] is true. The value X(0, 0) is known to be close to unity [15]. Putting 9~va = g~-N = 0.7 and u = 1 [15] we find X = 0.94 and t = - 7 . 6 while the sum of the Pauli blocking contributions described by the diagrams of Figs. 1 and 2 is t = -4 .6 .

I The meaning of the indexes is given in Eq. (53)

464

Thus at this values of parameters the account of multinucleon configurations into the Pauli-blocking graphs provides additional contribution ~t = -3.0. However the value of lSt[ increases strongly for the smaller values of 9~va. If 9~VA = 0.3 we find X = 1.04 and t = - 1 3 with 6t = -8.4. Note also that since I1 - )d << 1 the contribution depends strongly on the scalar part of the pion self-energy Hs.

9. The limit of low densities

At very low values of densities with

P F < < m ~ ; p < < 2 . 1 0 - 2 F m -3 (100)

the behaviour of t~ with p changes. The contribution of the single-pion exchange graph is quenched since the pion momentum is k << PF << m,~. Thus it's contribution to t~ is of the order p8/3. The leading nonlinear contribution comes from the two-pion exchange.

The contributions of intermediate states N A and A A are determined by Eqs.(78) and (87). It is sufficient to include the terms of the order z 3 only. The contribution of the nucleon pole in the case of N N intermediate states can be calculated as

N 6 3 ; dk SNN = -~ppFm d (270 2

k4 e--2k2/AZcNNI ] ,~ 4.2pZ 3 (101) (k 2 + m2) 3

N tNN = 4.2Z 3.

The pion pole adds the first two terms of Eq. (56). The last term of this formula comes from the region m~ << k << PF in the case PF -> m~ and does not emerge at PF < m~.

Hence we obtain

t (Z) ,'~ 0 .8 Z 3. (102)

10. Summary

In this paper we tried to calculate the quark condensate qq in nuclear matter. We wanted to find the dependence of the function a(p) = (NMIqqlNM) on the density of the matter as well as the value a(p0) at the saturation point p = P0.

We carried out the calculations for the case of the sym- metric matter. We assumed that the interacting nucleons add to the gas approximation [2, 4, 5] the interaction of the operator ~q with their pion field. The assumption, i.e. the possibility to neglect the interaction of the operator ~q with vector mesons is based on the large averaged value (Trlc]qlTr). This reflects the Goldstone nature of the pions.

Considering the expansion in powers of Fermi momentum (at PF -> m~) we find the model to be exact until we deal with the terms of the order p6 ~ pa. Indeed in the gas approximation [2] we obtain a(p) ~ p ~ p~. The Pauli blocking terms of Fig. 1 provide the contribution of the order/94 [2]. The leading contribution of the two-pion states is of the order p5 being determined by the large distances of the order pF 1 between the nucleons. The interactions with the heavier mesons take place at the smaller distances and manifest themselves in the terms of the order p6 ~ p2.

Hence the contributions of the o r d e r i 94/3 and p5/3 can be obtained in the model-independent way if the density p is small enough. They are determined by the interaction of the operator qq with the pion field. Coming from the large internucleon distances, they can be obtained with the finite size of the nucleons being neglected. Note that if the finite value of the pion mass m~ is taken into account, these terms become the functions of the ratio m~/pF - Fig. 5.

In the present paper we included N N , N A and AA intermediate states. The model was used in a number of works were the two-pion exchanges were considered [7, 13, 14]. The N A and AA states contribute to the terms of the order p6 and to the higher ones. The calculation of these terms yields the introduction of certain model-dependent parameters. The integrals corresponding to the mass-shell pions are divergent in the limit of point nucleons. They can be regular- ized by the account of the finite size of the latter - Eq. (33). Thus the values depend on the values of A1,2. There is also an uncertainty in the numerical value of the coupling con- stant f~lva. Note, finally, that in certain quark models the value of d = (AlqqlA) - (NI6~qlN) is finite. This may also influence the contributions of the order p6.

Until the density is small enough re(p) can be expanded into power series in Fermi momentum. Treated in this way the right-hand sides of Eqs. (92) and (94) present the total contribution to t~(p) up to the terms z 4 corresponding to the terms ~ 237 ~ p2pF in the expansion of t~. The multinucleon effects manifest themselves starting from the terms ~ p%. In any case the three-nucleon initial states contribute as p3 p 9 .

The picture becomes more complicated for the values of density p being of the order of the saturation value P0. The values of the integrals saturated by k ~ PF are altered strongly by the account of finite size of nucleons. Thus the contributions of the order p5 f ~ p5/3 are cancelled to large extent by the higher order terms. Also the exchange diagrams with N A and A A intermediate baryon states depend on the ratio q/A ~ p F / A which is of the order of unity at PF PF0. Hence the higher order contributions are numerically of the same order as the p5/3 terms.

The final result is composed of the contributions of sev- ern regions of values of pion momenta. For the N N intermediate states the values k ~ pF are the most important. In the case of N B and AA intermediate states we can single out the regions k ~ PF, k ~ A and k ~ (raA) 1/2.

The function t~(p) is shown in Fig. 6. The decreasing of the value of with density corresponds to the tendency of restoration of chiral symmetry. We see the nonlinear contributions to a to make the tendency weaker. At the saturation point p = P0 the nonlinear terms diminish effectively the value of (N[qq[N) ~ 8 by the value of 1.7 if the gas approximation formula (4) is used. Note that while the contributions of the order p4 and p5 are multiplied the certain functions of the ratio m~/pF, in the contributions of the order p2 the value of m~ is compared to the other parameters which do not depend on the density. Thus it is instructive to present the values of the higher order terms also in the chiral

2 0. In this limit the value (N[glq[N) diminishes limit ra~r = effectively by the value 1~-] = 2.8.

These results are not very sensitive to the exact values of the parameters u, A2/A1 and d - Sect. 3. Note that we

-.5,0 i I 1 I I i I I 1 7

x . l O s �9 G e V - S

[

1

-I0,0

-15.0

-20.0

-25, 0

-30.0

-35, 0 0,0

465

I i J * [ i ! i I r [ r i

0.2 O,-I 0.6 0=8 1.0 1,2 l,"l

p/po

Fig. 6. The dependence of the averaged value of the condensate (lq on the density of nuclear matter. The line 1 presents the gas approximation at (N[CtqiN) = 8. The curves 2 and 3 illustrate the account of nonlinear effect by the functions ~(z) and -c(z)

put Aa/AI = 1; u = 1 and d = 0. If we take u = 0.9 the value of t(1) diminishes by the value 0.15. Since the size of A-isobar is larger than that of nucleon, it is reasonable to analyze the values A2/A1 < 1. Putting A2/AI = 0.9 we diminish t(1) by the value 0.1. The variation of the value of it[ due to the possible finite value of d [24] does not exceed dA2/~IA ~ 1/6.

The part of the multinucleon configuration contribution can be accounted for by replacement of the nucleon mass m by the effective mass m*. This leads to a small additional contribution ( -0 .1) - ( -0 .2) to the value t(1).

A tempting project of the account of multinucleon contributions in the pion susceptibility technique cannot be con- sistently carried out. The reason is that the numerical results are sensitive to the exact values of the parameters which describe the correlations of baryons. We can state only that the results available in this technique (Sect. 8) did not meet contradictions with the perturbative calculation of Sects. 2-7.

Since the final result is the superposition of contributions of the various regions of pion momenta k, it cannot be tied directly to the "effective number of pions" [7]. The latter value is determined by the large values of pion momenta.

In the limit case of very small densities p << 0.02 Fm -3, corresponding to PF << m~ the leading correction to the gas approximation equation is of the order pa. In this case all the interval r a t < k < A of the pion momenta contributes. In this case it is easier to follow the relation of the result to the calculation of the paper [7]. However the qualitative comparison cannot be carried out since these values of densities are not analyzed in ref.[7]. In any case, the sign of the effect corresponds to that coming from ref.[7].

Hence, at very low densities the nonlinear terms are proportional to p~. They increase the tendency of restoration of the chiral symmetry, leading to the larger effective value of {NrqqlN ) if the gas approximation formula (4) is used. The magnitude of the effect is very small, it does not exceed 1- 2%. At larger values of density the leading nonlinear term

is of the o r d e r p4/3. The sign of the effect changes. If the value of the density is close to the saturation value P0 the higher order terms contribution is numerically of the same order as that of the leading term ~ p4/3. The function t~(p) can be approximated by the power series of Fermi momenta containing the terms from p3 till p~. At p = P0 the nonlinear terms, which account the interaction in the system, effectively diminish the value of (Nl(tql N) by 10-20%. This account of the interaction in the nuclear matter lowers down the restoration of chiral symmetry in the latter while we increase the density.

Our model does not include the interaction of the operator ffq with vector mesons. This interaction takes place at the distances of the order of the nucleon size. Thus it yields the account of there quark structure. Note, however, that since the vector meson exchange leads to the repulsion between nucleons, the interaction of the operator qq with vector mesons provides the negative contribution to re(p), increasing the contribution of the pion field.

The slowing down of restoration of chiral symmetry is in agreement with our earlier results [2, 4, 5] and also with conclusions of refs.[3, 8]. The opposite sign of the effect was predicted in the papers [6, 9]. The relatively small values of the nonlinear terms were obtained also in all the previous publications on the subject [2]-[6], [81-[11].

This work was preceded by very useful discussions with Prof.G.E.Brown and also with Prof.M.K.Banerjee, and with Prof.C.M.Shakin. The work was supported by the Russian Fund for Fundamental research - grant #95- 02-0375-a. One of us (E.D.) received also the partial support by grant ISFM9H000 of International Science Foundation.

R e f e r e n c e s

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(1989) 680.

466

3. M.Ericson, Phys.Lett.B 301 (1993) 11. 4. E.G.Dmkarev, E.M.Levin, Nucl.Phys. A511 (1990) 679;

Progr.Part.Nucl.Phys. 27 (1991) 77. 5. E.G.Dmkarev, M.G.Ryskin, Nucl.Phys. A578 (1994) 333. 6. G.Chanfray, M.Ericson, Nucl.Phys. A556 (1993) 427. 7. B.L.Friman, V.A.Pandharipande, R.B.Wiringa, Phys.Rev.Lett. 51

(1989) 763. 8. G.Chanfray, M.Ericson, M.Kirchbach, Preprint LYCE No.9321; Mod.

Phys. Lett. A9 (1994) 279. 9. T.D.Cohen, R.J.Furnstahl, D.K.Griegel, Phys.Rev. C45 (1992) 1881.

10. M.Lutz, S.Klimt and W.Weise, Nucl.Phys. A542 (1992) 521. 11. L.S.Celenza, A.Pantziris, C.M.Shakin, Wei Dong Sun, Phys.Rev. C45

(1992) 2015; L.S.Celenza, C.M.Shakin, Wei Dong Sun, and Ziquan Zhu, Phys.Rev. C48 (1993) 159.

12. K.Saito, A.W.Thomas, Phys.Lett. 327B (1994) 9. 13. R.B.Wiringa, R.A.Smith, T.L.Ainsworth, Phys.Rev. C29 (1984) 1207. 14. Th.A.Rijken, V.G.J.Stoks, Phys.Rev. C46 (1992) 73. 15. T.Ericson and W.Weise, "Pions and Nuclei", Oxford, 1988.

16. A.L.Fetter, J.D.Walecka "Quantum theory of many-particle systems" N.Y.1971.

17. T.P.Cheng, Phys.Rev.D 13 (1976) 2161. 18. T.P.Cheng, R.Dashen, Phys.Rev.Lett. 26 (1971) 594. 19. U.Wiedner et al., Phys.Rev.Lett. 58 (1987) 648. 20. J.Gasser, H.Leutwyler, M.E.Saino, Phys.Rev.Lett. B253 (1991) 252. 21. J.Casser, M.E.Saino, A.Svare, Nucl.Phys. B307 (1988) 779. 22. H.Leutwyler, Phys.Lett. B42 (1974) 431. 23. V.Bernard, R.L.Jaffe and U.C.Meissner, Phys.Lett. B198 (1987) 92. 24. T.Hatsuda, T.Kunihiro, Phys.Rep. 247 (1994) 224. 25. M.Ericson and M.Rosa-Clot, Phys.Lett. B188 (1987) 11. 26. A.B.Migdal, "The theory of finite Fermi-systems", Moscow, 1983. 27. G.E.Brown, Nucl.Phys. A522 (1991) 397.

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quark condensate in the nuclear matter

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