A study of meson dynamics in relativistic models of nuclear matter

Nuclear Physics North-Holland

AS26 (1991) 623-673

A STUDY

OF MESON

Joaquin

DIAZ DARC,

DYNAMICS NUCLEAR ALONSO’

IN RELATIVISTIC MA’ITER

and Armando

Observaioire

PEREZ

MODELS

OF

CANYELLAS’

de Paris, 92190 Meudon, France

Received 2 October (Revised 22 October

1989 1990)

Abstract: We analyze the dynamics of mesons in nuclear matter, as resulting from two relativistic models for the meson-nucleon interaction (D and Walecka model). The covariant Wigner function techniques are used in order to describe a plasma of nucleons in thermodynamical equilibrium, and to analyze the quasi-meson spectra in the medium. The models are solved in the mean-field and Hartree approximations, as first-order approximations to the solution of a hierarchy of kinetic equations describing the collective behaviour of the plasma. The small perturbations around both ground states give, in every case, the dispersion relations for the mesons. A numerical study of the different branches and the evolution of the quasi-meson masses with density is performed. This analysis shows that the mean-field ground state is unstable in some density ranges and should evolve towards a new spatially-structured state. We also show that vacuum polarization effects eliminate such instabilities, although they introduce a new instability related to“tachyonic modes”, arising at large momenta. For such momenta, the structural properties of the nucleon should be taken into account. We show that the “tachyonic instability” disappears with the introduction of a simple monopolar form factor for the nucleon. Finally, we study the linear response of the plasma and analyze the screening effects on the interaction.

1. Introduction In the last years, relativistic lagrangian models have been extensively used in the analysis of nuclear matter and many aspects of the nuclear interaction ‘). Historically, the sigma model 2-4) played a relevant role within this context. The approximate chiral symmetry of the nuclear interaction and the PCAC hypothesis manifest in this model in a simple and elegant way. This model has been used in the analysis of many

topics

in nuclear

matter,

such as the study

of the possible

existence

and

behaviour of abnormal states 5V6),pion condensation 7-9), etc. More recently, other lagrangian models, which describe the nucleon-nucleon interaction through the exchange of several kinds of mesons have been introduced in the literature. In this framework, the pioneering model of Walecka lo), using the exchange of a-scalar and w-vector mesons in the treatment of the nuclear interaction, gives a satisfactory description of nuclear matter saturation. In order to approach to other aspects in this field, models where the interaction is described through the exchange of a more ’ Also: Departamento de Fisica Teorica C-XI, Facultad de Ciencias, Universidad Autdnoma de Madrid, 28049 Madrid, Spain. 2 On leave from: Departamento de Fisica Teorica, Universidad de Valencia, 46100 Burjassot (Valencia), Spain. 03759474/91/$03.50

@ 1991 - Elsevier

Science

Publishers

B.V. (North-Holland)

J. Diaz Alonso, A. Perez Canyellas

624

rich meson

sector have been introduced.

not at present in terms

a renormalizable

of meson

matter saturation,

exchanges (approximate)

Nevertheless,

lagrangian which

/ Meson dynamics

as far as we know, there is

model describing

could

account,

chiral symmetry,

the nuclear

simultaneously,

interaction for nuclear

PCAC, and rrN scattering

ampli-

tudes. In the Walecka

model,

when pions are added

in an isospin-invariant

way, one is

led to an overestimation of the TN scattering amplitudes in two orders of magnitude ‘). This difficulty can be overcome by adding a non-linear coupling term between the (+ and rr fields of the form orr2, but the chiral symmetry is not implemented in this way. Moreover, in this case the pion acquires an affective mass in the mean-field approximation which becomes imaginary at low densities, and introduces a “tachyonic” pole in the quasi-pion propagator Ii). The chirally invariant a-model gives a satisfactory realization of the approximate chiral symmetry of the nuclear interaction and PCAC. It also gives a good qualitative description of VN scattering. However, the nuclear saturation cannot be fitted within this model in the mean-field or Hat-tree approximations*. Even for these relatively simple models, there is not in the literature a detailed analysis of the propagation properties of the meson sectors in the particular situation of a high-density plasma, as would be the case in the description of nuclear matter. The analysis of the u- and w-meson modes in the framework of the Walecka model in the mean-field approximation has been performed recently by Lim and Horowitz 4’). Also, the instabilities of the mean field 21) and Hartree 21S36)ground states against static meson modes have been analyzed. The main purpose of this work is an extensive study of meson propagation for the two models mentioned above, in both the mean field and Hat-tree approximations. The limitations of the lagrangian models in the description

of all the known

properties of the actual nuclear matter could be regarded, in principle, as a serious caveat when interpreting our results as a satisfactory description of the actual meson dynamics. However, such a study provides an important insight about the dynamic structure contained in the basic lagrangians, the nature of the approximations introduced, the influence of the vacuum effects, etc. Moreover, as we will show later, some of the results which are obtained from our analysis are qualitatively the same in both models. In particular, this is the case for most of the pion propagation features. This suggests that such features could be rather independent of the underlying lagrangian model. In the following sections, we first study the exchange of scalar mesons and pions within the framework of the o-model lagrangian. In contrast with the exactly chirally-symmetric (++ w model analyzed in ref. ‘), the chiral symmetry is explicitly broken here by a small linear term in the lagrangian. Because of this feature, the l However, taking into account many-body effects, which provide a net repulsion contribution which varies rapidly with density, close to the saturation point, the empirical properties of nuclear matter at nuclear density can be precisely fitted in the u-model context “).

J. Diaz Alonso, A. Perez Canyellas / Meson dynamics

pion

no longer

substantially short-range meson

behaves modified

repulsion

propagation

is also included

as a massless in the medium.

between

nucleons

in the context into the Lagrangian

Goldstone In order

boson,

and the pion

to analyze

and saturation

of the Walecka of the latter

625

the incidence

properties,

model. model

dynamics

of the

we also consider

The r-meson in order

is

exchange

to compare

the

pion propagation in both cases. In sect. 2 we introduce the lagrangian models, and we obtain the corresponding field equations. Next, we define the statistical device (that is, the Wigner operator, the Wigner function and the many-body correlation functions), and introduce the hierarchy of kinetic equations satisfied by all these objects. The formalism of the covariant Wigner function 24-26) provides an alternative to the usual Green function techniques

in the statistical

analysis

of dense matter,

and keeps a close analogy

with

the conventional methods of kinetic theory. The problems of renormalization of divergences due to vacuum fluctuations and vacuum polarization contributions have effects in this formalism is been solved 15,22), and the treatment of the temperature a simple and straightforward generalization of the ground-state methods. In sect. 3, we define the thermodynamical equilibrium in general and obtain the set of kinetic equations for this case. We briefly explain the method of solution for this hierarchy of equations, developed in ref. r4), as an expansion into many-body correlation functions. Next, we introduce the mean-field and Hartree approximations as the lowest orders in this expansion. The infinitives associated to the contributions of the Dirac sea (vacuum fluctuations, neglected in the mean-field approximation) are renormalized through a counterterm procedure developed elsewhere, which is equivalent to the usual methods in quantum field theory 16-18). Recently, a two-loop calculation for the Walecka model has been performed i9). Such a calculation introduces strong corrections to the Hartree results and indicates that the loop expansion is, perhaps, not convergent. Nevertheless, we must emphasize that, in our scheme, the mean-field and Hartree approximations correspond to the first order of expansions into many-body correlations, which differ from the loop expansions for higher orders. The corresponding second-order analysis in the correlation expansion is still lacking. Moreover, the results of ref. “) suggest that the Hartree approximation might be the dominant contribution to this expansion in the high-density limit. Consequently, the conclusions of ref. r9) do not necessarily invalidate the interest of the results obtained here. In sect. 4 we show that the solution to the kinetic equations which are obtained by taking into account two-body correlations and neglecting the higher order ones, is possible when two-body correlations are assumed to be small, The solution obtained in this way is shown to be equivalent to the linear response analysis of the plasma around the mean-field ground state. In this way, we obtain the dispersion relations for the propagation of quasi-a and quasi-r mesons (or equivalently, the corresponding one-loop meson propagators in the medium) in the a-model case. The vacuum polarization effects are neglected in this section, and we perform a

626

J. Diaz Alonso, A. Perez Canyellas / Meson dynamics

detailed

analysis

of the semiclassical

dispersion

relations

obtained

in the mean-field

approximation (small perturbations around the mean-field ground state). Sect. 5 is devoted to the renormalization of vacuum polarization effects. We obtain the renormalized ation

dispersion

and counterterm

relations

procedures,

and in refs. ‘34) ( c h’ira l-’mvariant

for the u-model developed

procedures).

following

different

regulariz-

in refs. ‘*,14) (non-chiral procedure), The expressions for the quasi-meson

renormalized propagators obtained within these schemes are compared. We find that, in the chirally-renormalized cases, they are the same that result from the self-consistent one-loop calculations *,13). This result shows the nature of the oneloop calculations, which are equivalent to the linear response theory around the Hartree ground state. The numerical analysis of the renormalized dispersion relations for the a-model in the Hartree approximation will be compared to the semiclassical results. As we will show, the introduction of the vacuum corrections eliminates many of the pathologies on the meson propagation which are present in the semiclassical case. In sect. 6, we analyze the dispersion relations obtained from the Walecka model, solved in the Hartree approximation. First, we show that, when the pions are introduced in the model through the usual isospin-invariant Yukawa coupling, the pion dispersion relations are only slightly modified with respect to the a-model case (when renormalized with the corresponding non-chiral procedure) by the presence of the o-field and the associated changes of the model parameters. Next, the fluctuations of the w-field are taken into account. As we shall see, the mixing between the w- and a-fields introduces additional modifications on the propagation of the latter modes. We perform a study of the dispersion relations for the C- and w-mesons, which decouple into a pure-w transverse part and a mixed longitudinal part, and we compare the propagation properties with the corresponding situation when mixing is absent. In sect. 7, we outline the study of the screening effects in the plasma, in the framework of the linear response formalism. We introduce the effective coupling constants for the fields created by nuclons in a given quantum state in the medium, and we analyze the particular case of the screening of a nucleon field in a puremomentum

and spin state lk, s). We conclude

that these screening

effects, as calcu-

lated in a linear response analysis, are self-consistently contained in the mean-field and Hartree approximations. For nucleons outside the Fermi sphere (or insideholes), we show the behaviour of the effective coupling constants, which decrease with increasing density, and vanish asymptotically. Finally, we conclude in sect. 8 with a summary and some remarks. 2. Dynamical

and statistical equations of the model

The lagrangian describing our quantum system is the usual of Gell-Mann and Levi 233)which can be written in the form: L = LD + L,,. + L,.i. + L,.

a-model

lagrangian

(2.1)

J. Diaz Alonso, A. Perez Canyellas / Meson dynamics

Where

L,, is

the Dirac massless

lagrangian

L, = $[$ya* L,,,

is the dynamic

for the baryons -aJy*]

part of the free-mesons

627

(protons

and neutrons):

.

(2.2)

(o. and r) lagrangian:

Lf,__=$ [aa*am*+am. am]. L,,i, is meson

the lagrangian

mass terms, which

for the meson-meson

(2.3)

self-interaction,

containing

also the

reads:

Ls.i.=-&[a(U*2 + m2)+ b( a*2 + n2)2]+fa*

)

(2.4)

where the last term, proportional to u*, is the usual explicit symmetry term. Finally, LI is the fermion-meson interaction lagrangian given by:

breaking

L, = g&(u* + y57. mr)+.

(2.5)

The constants a, b andfare related to the U- and n-masses; g is the nucleon-meson coupling constant and it is the same for both meson degrees of freedom. As is well known, associated to this lagrangian are the baryon current density, the electric current density, the isospin vector current density and the energymomentum tensor. As a consequence of the explicit symmetry breaking, the isospin axial current, given by: Ap = $[&y”$n$]

+&

[aa*u*- c+*a@p]

is only partially conserved. The divergence of this current symmetry breaking term coefficient through:

is related

(2.6) to the explicit

PA,, = fm, and in this way the PCAC hypothesis is satisfied in this model 23). As usual, the true vacuum is the minimal energy state, located at U* = U, approximation, V, is obtained by minimizing the classical meson potential, the stable root of the equation: (u:+

a/2b)v,

= 2rf/b.

(2.7)

.

In our and is

(2.8)

By translation of the origin of the o-field to this value, the fermion field acquires a mass which is a function of the constants Q, b andf: By identifying this mass with the free-nucleon mass, one obtains a condition relating the values of the parameters of the model. Moreover, we can obtain the expressions of the (+- and n-masses in terms of a and b, which read as: m’, = a +6bm2/g2,

(2.9)

m’, = a +2bm2/g2.

(2.10)

628

J. Diaz Alonso, A. Perez Canyellas / Meson dynamics

We can now identify

these masses with the “physical”

In sect. 3 we shall define

the values

of the model

values

of the meson

parameters

to be used

masses. in the

explicit calculations of this work. The explicit symmetry breaking termf; in eq. (2.4), will be small in the nucleon mass scale. In this way the axial current density (2.6) remains almost conserved and the rrN scattering reproduced ‘) by the lagrangian (2.1). The w-exchange

can be now introduced

under

amplitudes the usual

will be qualitatively form:

m~w”O,]+g,~yw’w,*,

Lv= -:[;F~“F,,-

(2.11)

where (2.12) The field equations physical

obtained

from the lagrangian

(2.1) and (2.11) in terms of the

fields* are: i~(a-ig,&)~-[m-g(~++Y5~~~)]~=0,

(2.13)

i(a+ig&)&+$[m-g(G+$r.-ir)]=O,

(2.14)

[O+mZ,]~+P,(~,6)=4~g~~,

(2.15)

[O+mZ,+P,(~,~)]~=4.rrg~~57(i,

(2.16)

[(O + m2,)8: +a”a,]h,” For the pure c+-model, the polynomials

= 4rgU&$.

in the u and n-mesons

(2.17) fields are:

P1(&, 6) = -6b(m/g)&*+2bG3-2b(m/g)&*+2b&*-f,

(2.18)

P2(&, &) = -4b(m/g)6+2bG2+2b&*,

(2.19)

and g, = 0. For the Walecka

model

we have P1 = P2 = 0.

At this level, we introduce the one-fermion relativistic in terms of the Fermi field operators as *‘,14): E(x,p)

= (273)-4

Wigner

d4R e-‘pR$(x+iR)@&x-fR),

operator,

defined

(2.20)

where the tensor product is to be understood in both the spin and isospin spaces. From eqs. (2.13)-(2.20) one can obtain the following dynamical equations relating the Wigner operator and the meson fields: [y(a--2@)+2im]&(x,p) =2i

I

d4ke-ik”[g&(-k)+gy5~.&(-k)+g,yG(-k)]fi(x,p-~k), (2.21)

l

In what follows,

the symbol

over characters

means

a quantum

operator.

629

J. Diaz Alonso, A. Perez Canyellas / Meson dynamics

= -2i and the equations

J

d4k eik”~(x,p-~k)[g~(k)+gy57.

for the meson

fields in terms of the Wigner

operator

h, Jd4p J J

[O+m~+P,(G,&)]G=4~gTr

Tr

one-particle

quantities

Wigner

(i) Baryon

mentioned

operator.

J

Isospin

.

J

d4p yP( 1 + r’)@(x, p) + (e/4r)[

of the

(2.27)

&‘#‘G2 - 7;2d’*7;1]

.

(2.28)

vector current:

J

V’=iTr

d4py~~E(x,p)+(l/4rr)[~xa~~].

(iv) Energy-momentum

J

d4p y’“&x, p)

in terms

are:

current:

.?r = $e Tr

fcL” = Tr

expressions

J

fields are the

(2.26)

can now be written

current:

(ii) Electric

(iii)

above

(2.25)

.

d4x eeik"A(x)

The corresponding

_?g = Tr

(2.24)

d4py@fi(x,p).

On the right-hand side of eqs. (2.21) and (2.22) the k-dependent Fourier transforms in four dimensions, defined as:

The conserved

(2.23)

d4py5r$x,P),

[(O+m2,)S~+a’“a,]~“=4~g~

(2.22)

are:

P>,

[Cl+m~]G+P,(&,&)=47rgTr

A(k) = (27rp4

,

ii(k)+g,y&(k)]

(2.29)

tensor:

d4ppPy”fi(x,

p) + (1/4~)[(#‘&*)(a”G*)+

(Y&)(8”&)

- (a”&Aa”GA)]

+(g““/8~)[~~(&*~+8~)+b(&*~+i$~)*-m~~~ - (a,G*)(a”cT*) - (aa+) . (a”&) + (aJG*)(a”LY)] (v) The isospin

- gP”“fG* .

axial current:

Ap = ii Tr

J

d4p -y“~y~rg(x, p) + (1/47r)[&Y‘&*

- $*#‘$I

(2.30)

,

(2.31)

J. Dim Alonso, A. Perez Canyellas / Meson dynamics

630

where the traces momentum

(Tr) are taken

associated

So far we dealt quantum

and so forth. density

with quantum

field theory.

corresponding matrix

over both the spin and isospin

indices.

The energy-

to the w-field has also been included.

macroscopical To achieve

in the form in which

quantities

in describing

such as the energy

this, we need to perform

describing

statistical knowledge the Wigner function

operators

Now we are interested

the

ensemble

density,

the average

of possible

states

they arise in a

the behaviour

of the

electric

with respect

compatible

charge to the

with

our

of the system. Let p^be this density as:

matrix. Then we can define

~11,

(2.32)

F(x, P) = tr [i+(x,

where the symbol “tr” means now the trace over the space of states of the system. In the same way we can introduce the mean values of the meson fields through: a(x) = (G(x)) = tr [&9(x)],

(2.33)

n(x) = (h(x))

(2.34)

= tr [+(x)1,

w*(x) = (h@(x)) = tr [j%P(x)].

(2.35)

Other useful tools for describing the kinetic and statistical behaviour are the many-body correlation functions. The two-meson correlation defined

of the model functions are

as: C&(x,

x’) = (~‘(x)a’(x’)>-(~‘(x))(~‘(x’))

)

(2.36)

where xi denotes any of the fields C? (for i = 0), 8 (for i = 1, 2, 3) and & (for i = 4). Some of the properties of the correlation functions are given in 28,40). In the same way, we introduce the meson-nucleon correlation functions: Cl&, C&(x’,

x’, P) = (Ux)@(x~,

PN-Gwx~w,

x,p) =(fi(x’,JJ)P(x))-(~i(x))(~(x’,p)).

PN,

(2.37) (2.38)

As in the cases of the scalar plasma and the isospin-invariant model “,14), the statistical averages taken over the fundamental operatorial equations (2.21)-(2.25) lead to a system of kinetic equations relating the Wigner functions, the mean values of the meson fields and the many-body correlation functions. This system is the starting point of the generalized relativistic quantum BBGKY hierarchy for the model. Higher-order equations of this hierarchy can be obtained by averaging the eqs. (2.13)-(2.17) multiplied by the appropriate operators 25*22,14). The infinite chain of kinetic equations can be broken through an appropriate ansatz which is related to the particular physical situation. For states near the thermodynamical equilibrium, one can introduce small parameters (which give the order of the correlations) and solve the hierarchy to a given order in this parameters. By taking into account the two-body correlation functions (and neglecting higherorder ones) one obtains most of the useful information about the macroscopic state of the plasma 14). In this work we shall restrict ourselves to this approximation.

J. Diaz Alonso, A. Perez Canyellas / Meson dynamics

3. Thermodynamical In this section Hat-tree

we analyze

approximation.

w-field

is present

We impose condition

The generalization

to the global

that

equilibrium

the thermodynamics

is straightforward

of the a-model

of this analysis

and has been developed

thermodynamical

every macroscopical

631

equilibrium

quantity

must

in the relativistic

to the case where elsewhere

state of the system

be invariant

the

10,1,27).

under

the

space-time

That means that single-point functions must be constants, whereas translations*. two-point functions can only depend on the difference of the space-time coordinates. So we have:

m, PI = F(P),

(3.1)

a(x)=a,

(3.2)

p(x)=?l,

(3.3)

Cf*(x, x’) = Cf*(x-x’), &(%

x’, P) = Cl&

CF*(X, x’,p) = The thermodynamical matrix:

where

k

equilibrium

-X’,P)

CL&-x’,p)

is also defined

,

(3.5)

.

(3.6)

by the grand-canonical

p^= exp [-p(A

-&.~~fi~)]/Z,

operator,

$i are the different

is the hamiltonian

(3.4)

density

(3.7) conserved

charges

and

pi their associated chemical potentials; 2 is the partition function. By averaging eqs. (2.21)-(2.25) with this density operator, and using the global equilibrium conditions (3.1)-(3.6) and Fourier transformation defined by eq. (2.26), one obtains

the first equations

[~~--++g(a+y57.~)]F(p)=-g

of the BBGKY

I

hierarchy

in equilibrium,

given by:

d4k[C,,(k,p+;k)+y5FC”,,(k,p+;k)], (3.8)

F(p)[yp-m+g(a+y5T.~)]=-g

d4k[C,,(k,p+5k)+C”,,(k,p+~k)y5T”l, I (3.9)

l Obviously, this condition excludes spatially structured states (pion condensates and others) as possible lower-energy configurations of the system. Nevertheless, as we will see in the next sections, the analysis of small perturbations around this thermodynamical equilibrium state, allows for the study of its stability and the identification of the eventual more stable spatially structured states. The translational invariance hypothesis must be relaxed when considering such states. This analysis, however, is beyond the scope of the present work.

J. Diaz Alonso, A. Perez Canyellas

632

/ Meson dynamics

Oo*+ao*+2b~*(a*~+~~)+6ba*C,,(O)+2ba*C”,”,(O)+4bC~,(O)~“-471-f (3.10)

+4brbC”,b,(0)

= 4n-g Tr

d4p y5+‘F(p)

,

(3.11)

J

where three-body correlations have been neglected, in agreement with our previous considerations. In these equations (+* is a c-number, related to the mean value of the scalar field through: a*=w-m/g. In obtaining

eqs. (3.8) and (3.9) we have assumed Re C:,(x,

where

(3.12)

Re stands

the condition:

p) = Re C6(-X,

p) ,

for real part. As we shall see this relation Re Cfc,(x)=Re

It has been shown that these conditions

(3.13) implies:

CcA(-x).

(3.14)

are consistent,

in equilibrium,

at least to

the order in which the two-body correlation functions are taken into account and the higher order ones are neglected 14). The kinetic equations for the meson-fermion correlation functions are obtained from eqs. (2.21) and (2.22) by multiplying by the appropriate meson operators and averaging with the p^ density matrix (3.7). The resulting equations are: [Y(p+tk)+g(a*+y5~.~)]Cl,(k,p)=-g[Cl,(k)+yS7”C~~(k)]F(p-~k), (3.15) C:dk,~)h(~-;k)+g(a*+y~~.

= -gF(p++k)[C:,(k)+ In the same way we can obtain, for the meson-meson

correlation

[-k2+a+2b(

=)I +T:,,(k)]

.

(3.16)

from eqs. (2.23) and (2.24), the kinetic functions,

which in compact

notation

equations read:

a*‘+?r2)]C~A(k)+4bhj[a*C~,(k)+&T’,”,(k)]

= 4ng Tr

d4p T’Ci,(k,

p) ,

(3.17)

I where r’ stands for Is if j = 0 and y5ri if j = 1, 2, 3. The system of eqs. (3.8)-(3.11), (3.15)-(3.16), and (3.17) is now closed and can be solved, in principle, but an exact solution is not possible in general. We can try to solve it approximately as an expansion in the two-body correlation functions, which are assumed to be small for high-density matter near the equilibrium. In this

J. Diaz Alonso, A. Perez Canyellas / Meson dynamics

way the lowest order in the expansion

is the Hartree

in neglecting

in these

two-body

approximation the action

correlations

of the mean-field

be uniformly

distributed.

the perturbations

created

fermion

the Hartree

undergoes

which

fields are treated distribution,

which consists

Physically,

in the plasma

by the other nucleons,

The mean meson

to the uniform

approximation,

equations.

is based on the idea that the nucleon

633

as classical

introduced

mainly

are assumed

to

fields, and

by the local field

of other nucleons in its neighbourhood (many-body correlations) are neglected. Under these conditions, the dynamics of a quasi-nucleon in the plasma is given by the Dirac-like

equations. [iya-m+g(a+y57.~)]cC,=0,

(3.18)

where CJ and n are here constant classical mean-fields. The dynamical eqs. (3.15)-(3.17) are now trivially tions),

and eqs. (3.8), (3.9) reduce

On the other self-consistent

hand,

satisfied

(vanishing

correla-

to:

r7/P-“+g(a+y5~.~)1F(p)=0,

(3.19)

F(p)[yp-m+g(cr+$~.n)]=O.

(3.20)

the mean

values

of the meson

fields satisfy

the following

equations: au* + 2ba*( u*2 + d)

-471-f = 4?rg Tr

d4P F(p),

(3.21)

d4p y5r”F(p).

(3.22)

I ~rP+2bd(a*~+a~)

=4rg

Tr I

From eqs. (3.19) and (3.20), and the definitions (2.20), (2.32) and (3.7) we can calculate the Hat-tree equilibrium form of the Wigner function 14). &(P)

= [PY -&?(a* - $7

* ?r)l{~[fi(~)+fi(~)lZz+~[fi(~) -fi(p)lr

where u, = n/ r3, and I2 is the 2 x 2 unit matrix. given by:

The functions

* a,],

(3.23)

fi( p) and f2(p)

are

f,(p)=(2~)-36(p2-M2)[(~~+~~)cos2~e+(~,”+~~)sin2~e-H(-po)], (3.24) f2(p)=(2~)-38(p2-M2)[(f2npf+flnp)

sin2@+(L!~+0,J

cos2@-H(-po)], (3.25)

where the 0’s are the Fermi

factors

defined

as:

fJg= H(*pdU

+ exp @[E(P) 7 PJ)I-’ ,

(3.26)

0,’ = H(*tpd{l

+exp @[E(P)

(3.27)

r

PJ))-’ ,

634

J. Diaz Alonso, A. Perez Canyellas

and H(x)

is the Heaviside

p-L, are the quasi-proton quasi-particles

step function.

/ Meson dynamics

p is the inverse

and quasi-neutron

chemical

temperature

potentials,

and pi, and

respectively.

The

are on the mass-shell: E(p)=-.

M2 = g’( (+*2+ n’) )

The last terms in eqs. (3.24) and (3.25) correspond

(3.28) to the quasi-particle

distribution

in the Dirac sea (vacuum distribution). These vacuum contributions lead to divergences in the calculation of the macroscopic quantities which must be renormalized. The renormalization procedure in the Hartree approximation has been developed elsewhere 15,i4). The method leads to the same results as in the self-consistent one-loop Green function calculations *729). Divergences appear in the self-consistent equations for the mean values of the classical fields eqs. (3.21) and (3.22), and in the fermionic part of the energymomentum tensor eq. (2.30). Because of the presence of y5 in the lagrangian, the use of a dimensional regularization technique is delicate and we have regularized them through a cut-off method 14). As a consequence energy-momentum

of the whole process, tensor becomes:

+ where V(a”,

V(a*,

the final expression

of the renormalized

gP”“/@n)ua*,=r> ,

m) is the meson-plus-vacuum

(3.29)

part, given by:

1r)=a(c+**+~‘)+b(u*~+n~)~-87~~~*+(4g~/rr)c+~(0~+~~)-(4g~/3~)a~ +(3g4/2~)(a2+~2)2-(6mg3/~)a(cr2+a2) -(8mg3/3r)03+(m2g2/rr)((+2+n2) +(6m2g2/~)~2-(2m3g/~)o-(M4/~)

with a general

counterterm

lagrangian,

In (M/m)‘,

(3.30)

or

V(o*, ~)=~(a*~+~~)+b(a*~+~~)~-8r$r*+(3M~/2~) -(M4/n-)

In (M/m)2-2M2m2/n+m4/2r

(3.31)

if a chirally invariant counterterm lagrangian is used ‘*13).In absence of more stable spatially-structured states, such as pion condensates and others, the mean-value of the T-field in the translational invariant stable ground-state vanishes: (n)=O.

(3.32)

We shall fix the model parameters in the two cases considered. For both, the nucleon and pion masses will be fixed to m = 939 MeV and m, = 138 MeV respectively. Next, in the first case (the a-model), the v-mass and the coupling constant

J. Diaz Alonso, A. Perez Canyellas / Meson dynamics

635

will be fixed to representative values which are obtained from the fitting of nucleonnucleon scattering experimental data in vacuum, as given in ref. 31).These values are: g*=14 3 In this case, the plasma ground model

state, which

shows

disappears

we chose the following

a Lee-Wick

phase

(3.33)

transition

in the Hat-tree approximation

in the mean-field “). For the Walecka

values:

m, = 550 MeV ,

g~(m/m,)‘=

m, = 500 MeV .

183.3,

m, = 783 MeV , gt(m/mw)2=

114.7.

(3.34)

With these values, saturation is attained in the Hartree approximation at pfO= 1.42 fm-‘, with a binding energy Eb = -15.46 MeV. At this point, we end the analysis of the Hartree approximation. This is the first-order approximation to the solution of the set of eqs. (3.8)-(3.11) and (3.15)(3.17). In the next sections we shall analyze the second-order approximation to this solution in the two-body correlation expansion. We shall obtain, in particular, the propagation modes for the quasi-mesons in the plasma. Evidently, such modes will be dependent on the structure of the coupling and the values of the parameters in the lagrangian. Consequently, in looking for the actual behaviour of mesons in nuclear matter, we are faced with a problem of choice, because, at present, there is not a unique model giving a satisfactory description of all the known properties of the nuclear interaction. As we shall see later, the choices leading to a good description of the scattering data and those which lead to the good empirical properties of nuclear matter, give qualitatively the same results for the quasi-pion propagation at meaningful densities. Only small quantitative differences arise between both cases, at least in this approximation. Consequently, the pion dynamics is not very sensitive to this choice. The propagation of the scalar modes is also slightly modified by the presence of the fluctuations of the w-sector, which must be introduced in order to account for saturation at nuclear density. The only important modifications, introduced by the mixing between in the propagation of the zero-sound modes.

the w- and c+-mesons, will manifest

4. Beyond the Hartree approximation We now return to eqs. (3.15)-(3.16) and replace there the one-fermion Wigner function by its Hartree (first-order) expression (3.23). In this way we obtain a system of equations for the two-body correlation functions which can be explicitly solved. As mentioned at the end of the previous section, the corresponding solution will give us the second-order approximation to the complete solution of the system (3.8)-(3.11) and (3.15)-(3.17) in the two-body correlations expansion. The details of the method have been developed elsewhere 14), and we give here only the principal

636

J. Diaz Alonso, A. Perez Canyellas

steps. The extension of this introduced will be performed In order

to obtain

approximation

analysis when in sect. 6.

this solution,

boundary

the equations

fields and the fermion

with eqs. (2.21)-(2.25), fields and the Wigner

the fluctuations

we must interpret

and define the appropriate

As a first step, we shall obtain of the meson

/ Meson dynamics

the physical conditions

satisfied

distribution

of the

meaning

G(x) = an+

its Hartree

values.

&(k, p)[y(p

+w&‘dSd

(4.1)

&l(X),

(4.2) (4.3) in the small perturbations

= -g[G,(k)+ $7. +,(k)l&(p -+I,

gad= -g&b

-&k) - m +

Starting

for the meson

+ &(x, P) ,

and we conserve only the terms which are linear and i,. The resulting equations are: - m

of this

by the small perturbations

around

G(x) = G,(x),

[y(p+$k)

are

to be imposed.

which are the fundamental evolution equations operator, we make there the replacement: @(x, P) = F,(P)

w-sector

+%)[~~(k) + 7’~. h,(k)],

[-k*+u+6b(m/g-a,)‘]G,=4rgTr

(4.4) (4.5)

d4p&(k,p),

(4.6)

d4py5rs1(k,p).

(4.7)

I [-k2+a+2b(m/g-m,)2]&1=4rrgTr

@,, G1

I

This system must be interpreted as the field equations for the excitations in the plasma around the Hat-tree equilibrium-ground state. In particular, the G1 and Gir, operators must be interpreted as the quasi-meson field operators in the medium. On the other hand, from the definitions of the correlation functions (2.36)-(2.38), and the eqs. (4.1)-(4.3) defining the quasi-meson fields, we find the following relations: Psw'))

CL(x

-x’,

GF(X

-x’, PI = K(xu%x’,

C&(x-x’)

P> = &(x,

= &(x)Xj(x’))

,

(4.8)

P)> ,

(4.9)

)

(4.10)

provided the identification (6) = uH is made. This shows that, for a system in thermodynamical equilibrium, the Fourier transforms of the two-meson correlation as the one-particle Wigner functions functions C,, and C,, should be interpreted for the quasi-u and quasi-m mesons, respectively 14). Moreover, the kinetic equations satisfied by these correlation functions can be obtained from the excitation-field equations (4.4)-(4.7) and the relations (4.8)-(4.10). It is straightforward to verify that these kinetic equations coincide with eqs. (3.15)-(3.17) as it should be.

J. Diaz Alonso, A. Perez Canyellas / Meson dynamics

The perturbation

to the Hartree-Wigner

operator

631

can be explicitly

obtained

from

eqs. (4.4) and (4.5), and reads: &(k,p)

=

-g[S,(p+tk)~(k)F,(p-~k)+F,(p+bk)~(k)S,(p-~k)l,

(4.11)

where c&k) = c?,(k)+ 7’1. iiil,

(4.12)

&(P)=(yP-m+gaEi-’ (4.13)

=(‘YP+m-g%)l(P2-M2). This expression must be inserted in eqs. (4.6) and (4.7) in order to obtain of field equations for the quasi-meson propagation:

the system

(4.14)

D,(k)&=[-k2+a+6b(a,-m/g)2-g217,(k)]~I(k)=0, m/g)2-gZII,(k)]&,(k)-ig217~(k)u,xii,(k)=0.

D,(k)&+-k2+a+2b(q,-

(4.15) is a 3 x 3 matrix. is:

Notice that D,(k) scalar polarization n,(k)

= 87r I

d4p [A(+)

+A(+)

Here, u, is the constant

isovector

-fiW(p’-fk’+

-h(-)

(0, 0, 1). The

M2)l(kp) ,

(4.16)

with

A(*) =J;(p*fk), and J(p) are given polarization isotensor 17,(k)

= 8~

by eqs. is: d4p

and the anti-symmetric n;(k)

= 87~

(3.24)

and

i=l,2,

(3.25).

The

(4.17) diagonal

part

[fi(+)+f2(+)-fi(-)--f2(-)l(p2-tk2-M2)l(kp),

of the pion

(4.18)

part is: d4p

[fi(+)-f2(+)+fi(-)-f2(-)l(p2-tk2-~2)l(kp).

(4.19)

In the same way, from the definitions (4.8)-(4.10), and eqs. (4.14) and (4.15), or directly starting with eqs. (3.15)-(3.17), we can obtain the kinetic equations satisfied by the two-boson correlation functions. These equations are, in fact, the same equations (4.14), (4.15), satisfied by the quasi-meson fields: R(k)C:,(k) [D,(k)]&(k)

= 0,

(4.20)

= 0.

(4.21)

638

J. Diaz Alonso, A. Perez Canyellas / Meson dynamics

The compatibility relations

conditions

for the propagation

(with (&>n = 0) both meson

for eqs. (4.20) of the quasi-meson

modes

decouple D,(k)

and

(4.21) give us the dispersion

modes.

In the present

and the dispersion

relations

=o

situation reduce to: (4.22)

P,(k)l = 0.

(4.23)

The only information we can obtain on the two-meson correlation functions from the system (4.20) and (4.21) is that these functions are on the “mass-shells” given by eqs. (4.22) and (4.23). For their explicit determination we must impose appropriate boundary conditions, associated to the thermodynamical equilibrium situation in which such quasi-particles propagate. In defining these boundary conditions, the first step is to quantize the quasi-meson field equations (4.14) and (4.15) and define, in the associated Fock space, the grand-canonical density matrix corresponding to the thermodynamical equilibrium. This density matrix allows for the calculation of the mean values in eqs. (4.8)-(4.10) and leads to the final expressions of equilibrium two-body correlation functions. The quantization of a non-local system of field equations of the generic form (4.14) and (4.15) and the explicit calculation of the correlation functions has been developed in refs. 8*‘4), and the generalization to the present case is straightforward. We shall not insist here on this point. We shall return now to the analysis of the dispersion relations (4.22) and (4.23). As in the cases of the pure scalar plasma “) and the pion-nucleon system r4), the polarization “tensors” (4.16) and (4.18) contain a vacuum polarization part which diverges, coming from the vacuum part of the distribution function F,(p). We shall neglect these vacuum polarization contributions in this section and we shall analyze the “semiclassical” dispersion relation thus obtained. In the next section we shall deal with the vacuum effects and their renormalization, and we will analyze the corrections to the semiclassical dispersion relation and the meson propagators obtained here. The analysis of this section concerns the u-model case only. The next figures are a graphical representation of the numerical study of eqs. (4.22) and (4.23), for a chosen

set of values

of the Fermi

momentum,

PF/m = 0.1,

0.4, 0.8 and 1.2. All these plots correspond, for the sake of simplicity, to the zero-temperature case, although calculations at non-zero temperature can be done without additional difficulties. In fig. la, b, we show the quasi-meson branches for the propagation of the quasi-u and quasi-r modes, respectively, in symmetric nuclear matter, calculated in the mean-field approximation. There we observe two different kinds of curves. The ones appearing inside the light cone are just the analogues of the free-mesons mass-shell branches in vacuum, given by: -k*+m:=O,

(4.24)

-k?+mt=O,

(4.25)

J. Diaz Aionso, A. Perez Cartyellas / Meson dynamics

639

1

0i 0 Fig. 1. Branches for the propagation of the quasi-meson modes in the w, /kj ptane [part (a) quasi-u modes, part (b) quasi-v modes], obtained from the a-model in the mean-field approximation. Here w (time-like component of k”) and lkl are in units of the respective free-meson masses. See commments in the text for these branches.

but they are modi~ed by the polarization of the medium. We shall cat1 them “normal branches”. These curves have an hyperbolic behaviour for large values of k (~/lkl--, 1). Aside the normal branches, there exists another kind of modes (lower part of the figures). As we shall see later, these modes correspond to a pathology of the mean-field approximation and disappear when vacuum polarization contributions are taken into account. They are similar to the ones encountered in the pure-scalar

.I. Diaz Alonso, A. Perez Canyellas

640

plasma

22), and in the Walecka

of an instability

in the mean-field

by minimizing

the energy-density

where “cp” represents

/ Meson dynamics

model 2’,41), and are indicative ground

state. Indeed,

functional

the components

through

of any meson

of the appearance

the ground

state is obtained

the equation:

field. The analysis

stability of the ground-state solutions of eq. (4.26) ((50)~) requires sign of the eigenvalues of the matrix *l):

of the local

the study of the

(4.27) which is the second variation of the energy functional. In our case, because of the translational invariance, this matrix is diagonal in momentum space. It is given by: (4.28) In the regions of the w, (kl plane where one or more eigenvalues become negative, the mean-field ground state becomes unstable under the corresponding perturbations. In particular, for negative eigenvalues corresponding to modes with w = 0, Ikl# 0 (this happens on the o = 0 axis, in the regions inside the lower branches), a new more stable spatially structured ground state appears, which in the present case is a “static isospin wave”. In order to see the behaviour of this instability of the ground state under the w = 0 modes, we have plotted on a Pr, \kl diagram the regions where the eigenvalues of the matrix (4.27) and (4.28) are negative (fig. 2).

Pflm 0.0

0.1

0.2

0.3

Fig. 2. Regions of instability of the mean-field ground state in the (kl, PF plane (in units ofthe free-nucleon mass). Inside the solid line is the region of instability against a-modes. The dashed line gives the frontier of instability against rr-modes.

J. Diaz Alonso, A. Perez CanyeNas / Meson dynamics

641

Fig. 3. Effective quasi-meson masses (plasma frequencies) in units of the free-meson masses for (a) the quasi-r and (b) quasi-n modes, as functions of the Fermi momentum (in units of the free-nucleon mass), calculated in the o-model, for both the semiclassical approximation (Hart.), and the perturbative approximation (Pert.). The discontinuities are related to the Lee-Wick phase transition and to the instability of the quasi-meson modes against the decay into nucleon-antinucleon pairs, when the quasi-meson effective masses become twice the free nucleon mass (in the perturbative case), or twice the quasi-nucleon effective mass (in the mean-field case).

642

J. Diaz Alonso, A. Perez Canyellas / Meson dynamics

For densities below the normal-abnormal phase transition density, the instability region concerns only the o-modes (except for a small zone of instability for r-modes near this transition

density).

This instability

of the Van der Waals

phase

density,

concerns

nearly

the instability the same region

transition.

coincides

Beyond

the u- and

with the unphysical

the normal-abnormal

n-modes

and,

in both

region transition

cases,

occupy

of the PF, Ikl plane.

We can analyze now the behaviour of the quasi-meson effective masses in the plasma as density increases. These effective masses are given by the “plasma frequency”, wPl, defined as the value of w at the point Ikl = 0, for the normal branches. This has been done in fig. 3a, for the quasi-o mesons and in fig. 3b for the quasi-n mesons, the curves in the perturbative case (M = m) are also plotted for comparison. We notice first that both approximations give the same results at small densities but differ strongly as density increases. Next, we observe the presence of discontinuities. In the perturbative calculation, the discontinuities occur when the effective quasi-meson masses becomes twice the free nucleon mass. It must be interpreted as an instability of the medium against the production of nucleonantinucleon pairs from the quasi-meson decay. In the mean-field approximation, the situation is slightly more complicated. The discontinuity occurs when the quasi-meson effective masses becomes twice the quasi-nucleon efective mass, which decreases with density. The interpretation is now the same: instability of the medium against the decay of the quasi-mesons in quasi-nucleon-antiquasi-nucleon pairs, but now the instability density coincides with the transition density from the normal state to the abnormal one. Clearly, this coincidence is due to the sharp jump of the quasi-nucleon effective mass at this density, which falls at this point under the threshold of the pair production instability (2M < wPl). We end here the al.alysis of the semiclassical dispersion relations and we shall consider, in the next section, the corrections introduced by the vacuum contributions. As we shall see there, the vacuum polarization effects modify strongly the analysis of this section, in general, in a stabilizing sense. 5. Vacuum polarization and renormalization In this section

we discuss

the renormalization

of the divergences

which

appear

in the polarization expressions, eqs. (4.16)-(4.19). The method is the generalization of the to the present case of the one developed in refs. 22,‘4). After the replacement quasi-nucleon distribution functions, given by eqs. (3.24)-(3.27), in eqs. (4.16)(4.19), it is easily seen that 172(k) is finite, while &(k) and 17,(k) can be separated into matter and vacuum contributions:

where the subscripts

K(k) =17,,(k) +17,,(k),

(5.1)

n,(k) = 17,m(k)+Krv(k),

(5.2)

m and v stand

for matter

and vacuum,

respectively,

and the

J. Diaz Alonso, A. Perez Canyellas / Meson dynamics

vacuum

contributions

diverge.

One has explicitly:

I;T,,( k) = 16~

d4p LK,(p+&kt.b

17,,(k)

d4p [f,(~+k~)-f,(~-~~)l(p*-~k*-~~)/kp,

= 16a

643

-$W(p2-~~2+~Z)l~~,

(5.3)

(5.4)

where: L?(P) = -(27d-‘IcPmP*-

(5.5)

M2)

is the vacuum part of the quasi-nucleon distribution function (3.24)-(3.27). By integrating the angular variables in eqs. (5.3) and (5.4), we obtain the following expressions:

where the integrals

17,,(k)

= -(4/~2)[M2-4k2]12(k)+(4/~2)11,

(5.6)

K,(k)

= (k2/~*)~212(k)+(4/~*)1,

(5.7)

,

II and I*(k) are: co dx x*/m,

I,=2rr

(5.8)

I0 cc I,(k)

[I and the function

O(k*, M2)

dx (x2+@-“*-

=27r 0

1 ,

(5.9)

O(k*, M2) is given by the integral: ‘x f3(k*, AI*) = 8(y) = y

dx/[(x2+y)=],

(5.10)

I0 with: y=l-k2/4M2. This function can easily be computed, both has the following relation between the function de(y)ldy

analytically and numerically. 0(y) and its derivative:

= [Y - @(Y)~/[~Y(Y - 111.

One

(5.11)

As it is readily seen, I,(k) and I, give divergent contributions to the vacuum polarization which must be renormalized. The method of renormalization was discussed in ref. 14) for a simple pion-nucleon system. Here the question is more involved. By analyzing the structure of the divergences in the vacuum polarization (5.6) it is easily seen that they can be cancelled by the addition of counter-terms to the original lagrangian, in the form

$[(ab)(ad)+(air). (a~)]+sa(a**+~*)+~b(~**+~*)*,

(5.12)

J. Diaz Alonso, A. Perez Canyellas

644

which preserve

the chiral

three conditions

in order to determine

the conditions

symmetry.

Within

/ Meson dynamics

this renormalization

counter-terms

scheme,

4*1).We can impose,

that the poles of the u and rr renormalized

propagators

we need

for example, arise on the

measured values of the meson masses, and that the residue at the pion propagator pole be unity. These conditions imply that the renormalized dispersion relations must reduce

in vacuum

and the effective the vacuum

to the usual

coupling

polarization

expressions

-k’+m$=O,

(5.13)

-k’+mi=O,

(5.14)

constant

for the pions

in the plasma,

+g2&(k)l(k2-

d)l

which is related

to

through g%, = g’/[l

must reduce

free-meson

in vacuum

(5.15)

to g2, that is g&(k)k2=m:,pF=0

(5.16)

= g2.

By this procedure, the effective coupling constant for the sigma mesons in the plasma is fixed, and does not reduce in vacuum to g2. In what follows, we shall call this the chiral-1 renormalization procedure. The expressions of the renormalized vacuum polarizations, in this case, are: 17,,(k)=8(M2-~k2)~(k2,M2)+(6M2-k2)ln(M/m)2+(Z,/g2)k2 - aFIg

(5.17)

- (6&/g4)M2,

~,Jk)=-2k28(k2,M2)+(2M2-k2)ln(M/m)2 + (ZFlg2)k2 where the finite constants

(5.18)

- aFlg2 - (2&/g4)M2, are determined

by the conditions

rrJk’=mi;

M2=m2)=0,

IT,,(k2=mt;

M2=m2)=0,

d17,,,/ak2(k2= m’,; M2= m’) =O. We also

tried

a second

renormalization

scheme,

based

(5.19) on adding

the same

counterterms as in (5.12) (we shall call it the chiral-2 procedure). Now, we impose the same two constraints on the pion renormalized propagator as in the chiral-1 case, and replace the condition on the sigma propagator by the constraint that the at one-loop level 13). threshold TN scattering amplitude Ap’ remains unchanged The expressions for the vacuum polarization tensors are given by eqs. (5.17) and (5.18), where the finite part of the renormalization constants are now determined by the conditions

Il,,(k2=m~;

M2=m2)=0,

a17,Jak2( k2 = m’, ; M2 = m’) = 0, a17,,,/aM2(k2 = m’,; M2 = m’) = 0.

(5.20)

J. Diaz Alonso, A. Perez Canyellas / Meson dynamics

All these conditions

are imposed

on the pion vacuum

polarization

the structure of the dispersion relation for the sigma modes order to minimize the vacuum effects, we have envisaged renormalization. non-chiral

Indeed,

invariant

and all possible

our model

lagrangian

sigma-pion

can be considered

tensor

terms compatible

only, and

is fixed in this way. In a third procedure of

as a particular

model with a pseudoscalar coupling

645

pion-nucleon

case of a coupling,

with renormalizability.

In

this case, we can add a counterterm part of a general form to the original lagrangian and determine all the counterterms uniquely, in such a way that the contributions of the vacuum polarization tensors to the “physical” constants of the model in the renormalized one-loop dispersion relations must vanish, thus minimizing the vacuum one. effects ‘*13). We shall call this procedure the “nonchiral” These calculations are rather tedious, although straightforward. We will simply give the final expressions for the renormalized dispersion relations, which now can be written

as:

-k*+a+6b(a,--k*+a+2b(a, Where

n,,(k)

and

17,,(k)

m/g)‘- m/g)‘are

the

g217,dk) - (g’/ ~)17,,(k) = 0,

(5.21)

g2K,(k)

(5.22)

finite

- (g’h)L(k)

vacuum

polarizations,

= 0. given

by the

expressions:

&(k)=(2k2-8m2)e,+8(M2-k2/4)e(k2,

M*)

+8(m2-m~/4)(mf,-k2)0,k+(6M2-k2)ln(M/m)2

-2g2a2[6+m~/m2+16m2e,,+8(m2-m~/4)m*~,,,] -(M*-m*)[6-m~/m2+8e,+8(m2-mt/4)e,,], 17,Jk)=2k2[8,-B(k2,

(5.23)

M2)+m2,e,k]-2m4,e,k

+(2M2-k2)ln(M/m)2-(2-mfJm2-2m~e,,)(M2-m2) - 2g2a2(2 + mfJm2 - 2m~m2eTmm) . In writing

down eq. (5.23), we have introduced

e, = e(m’,, m')

(5.24)

the notations: ,

e vrn= aO(k2, M2)/dM2, eTk = ae( k*, M2)/dk2, e wmm= a*e( k*, M~)/~(M”)*.

(5.25)

All these derivatives are to be calculated at the point k* = m$, M = m. For that purpose, repeated use of eq. (5.11) can be done. Similar definitions have been made in eq. (5.24), with the replacement m, + m, everywhere. We shall now continue with the analysis of these dispersion relations. In fig. 4a, b we show the branches associated to the propagation of the quasi-w and quasi-r

J. Diaz Alonso, A. Perez Canyellas / Meson dynamics

646

I

I

I

0.L I 2

I 6

I L

I

1.2 0.8

0.1 OS-1 0

1

I

I

I 8

8

10

IZl/m, I

12

30

OL--J0

I 5

1

,

I

I

II t

10

I

I

1

15

25

30

Fig. 4. Same as fig. 1, in the reno~aiized approximation; (a) and (b) co~espond to sigma and pion modes, respectively obtained with the non-chiral renormalization procedure. (c) and (d) the same modes obtained with the chiral-1 procedure. (e) and (f) are obtained from the chiral-2 procedure. The non-chiral results show two kinds of time-like branches. The lower branches correspond to the normal meson modes, the upper ones to the heavy-meson modes. For high densities, both kinds of branches mix together. The zero-sound branches in (a) and (c) are present in the non-physical region of the Van der Waals phase transition. The “tachyonic” branches (also the heavy-meson branches) are due to vacuum polarization effects. The parameter is the Fermi momentum in units of the nucleon mass.

L Diaz Ahso,

A. Perez Canyelias / Meson dynamics

647

Fig. 4-continued

modes, respectively, corresponding to the solutions of eqs. (5.21) and (5.22) in the (rl, lk/ plane. These figures are tu be compared with figs. la, b, in order to see the changes introduced by the vacuum polarization effects. We sum-up the following features: (i) First, we observe the presence of the normal branches which are very similar tu the corresponding branches obtained in the semiclassical approximation. There are not important qualitative changes introduced by the vacuum effects fur these branches. Quantitative changes are a&o small outside the interval near the phase

648

J. Diaz Alonso, A. Perez Canyellas / Meson dynamics

01

II

I

0

2

,

30

,

L

I

6

, , I ,

I

I

8

I

I

10

I

I 12

,

w/m, Pf/m=O.l d

KJK,, ,,,Iy] 10

15

20

25

30

Fig. 4-continued

transition density, which is strongly modified by the vacuum fluctuations (disappearance of the Lee-Wick phase transition). (ii) As in the semiclassical case, the zero-sound branches for the quasi-u are yet present in the unphysical range of density, inside the Van der Waals phase transition region, where pressure is decreasing with increasing density. In this region the ground state is yet unstable, but, clearly, this instability does not represent any difficulty for the model. It is more important to notice the disappearance of the

J. Diaz Alonso, A. Perez Canyellas

lower branches

present

As a consequence, in physical ranges taken (iii) vacuum

in the semiclassical

/ Meson dynamics

calculation,

649

for both the o- and rr-modes.

the instability of the ground state associated to these branches of density disappears when the vacuum polarization effects are

into account. We note

the appearance

polarization

terms,

of two kinds

in this approximation,

of branches

which

are related

and which are present

to

for both

(T- and rr-modes. First of all, upper branches associated to massive meson modes. It is not clear, at the present, whether these branches do correspond to some physical reality or come from a pathology of the one-loop-order calculations of the vacuum polarization. In fact, such branches are encountered in similar calculations in a simple scalar coupled plasma **), in the pion-nucleon system with isospin invariant coupling 14), and in the one-loop calculations in quantum electrodynamics. In the latter case, the corresponding massive-photon mode propagates in vacuum with an unphysical value of the mass, of the order of exp (l/a) in units of the electron mass, where (Y-A is the fine-structure constant. Moreover, the analysis of the asymptotic behaviour of the quasi-boson propagator, shows that the one-loop calculation of the polarization tensor cannot be extrapolated 32) for too large values of k2 (depending on the strength of the coupling), and these branches seem to be an artifact of the approximation ‘l). Nevertheless, as shown in ref. 33), in presence of very strong magnetic fields, massive quasi-photon modes with physically reasonable masses can be excited, due to vacuum polarization effects. Our guess is that the physical or unphysical character of such vacuum-related modes, is a question which must be decided by the experiment, at least in cases where the underlying field theory is associated with an actual physical system. The other branches related to the vacuum polarization appear in the space-like region of the w, (kl plane, and we shall call them “tachyonic branches”. The intersection of these branches with the w = 0 axis appears always at large values of k and introduce a new instability of the Hartree ground state at all densities. In fig. 5 we have plotted the regions of instability of the Hartree ground state in the lkl, Pr plane for both the u- and rr-modes. As in the Walecka model *I), we see that the tachyon pole occurs for: (kJ>m>M=m-g(a),

(5.26)

while the one-loop calculation cannot be extrapolated much farther than k*- M* in this model*. Moreover, for such values of the momentum, the point-particle * This results from the analysis of the asymptotic behaviour of the one-loop propagator for large k [ref. 32)]. The validity of the one-loop approximation is limited by the condition g log lk2/MZI < 1, where g is the coupling strength. In the case of QED the approximation remains satisfactory for large values of k[ - m exp (137)], owing to the weakness of the coupling. Nevertheless, for our strong-coupling models this conditions shows that the one-loop results cannot be extrapolated for values of k beyond a few times the fermion mass,

650

J. Diaz Alonso, A. Perez Canyellas / Meson dynamics

I/---A _ / I 0 -’ ’ 0.0 0.2

\ \ 1 \

I

I

0.L

I 0.6

I

I 0.8

Pf/m

1.0

Fig. 5. Regions of instability of the Hartree ground state, in the Ikl, PF plane (in units of the free-nucleon mass) due to the presence of the tachyonic and zero-sound branches obtained with the non-chiral renormalization procedure. The solid line gives the frontier of the instability region against the rr-modes. The dashed line is the frontier of instability region against the a-modes. When the form factors are introduced, the tachyonic instability disappears but the zero-sound instability remains in the unphysical region of the Van der Waals phase transition.

approach becomes inadequate, and the structural properties of the baryons should introduce vertex effects 36) which play a role. In this way, even the introduction of a simple phenomenological monopolar form factor given by: f(k)=(A’-&/(A’-k2)

(5.27)

removes the tachyon branches for values of A s 1700 MeV for both the u- and r-modes, and the instabilities of the Hartree ground state disappear. Nevertheless, the zero-sound branches for the a-modes and the associated instability in the unphysical regions of the Van der Waals phase transition still remain. Now, the question is whether the instabilities in the semiclassical approximation can also be eliminated by such a form factor. A detailed analysis shows that, in this case, the instabilities disappear at high densities but remain unchanged at low densities. The results shown in fig. 2 are then dramatically modified by the nucleon structure when density increases. Let us now return, for comparison, to the dispersion relations obtained with other renormalization procedures*. Fig. 4c, d gives the sigma and pion branches, respecl

The corresponding

equation

of state is calculated

consistently

with these renormalization

schemes.

J. Diaz Alonso, A. Perez Canyellas / Meson dynamics

tively, when the dispersion In this case, the normal

relations branches

They approach

the k2 = 0 diagonal

these principal

branches,

modes

show a zero-sound

sical region For arise two and

are renormalized are present as density

there are space-like behaviour

of the equation

within the chiral-1

in the vacuum increases, branches

651

and then disappear.

Aside

which, in the case of sigma

for PF/ m < 0.45, corresponding

of state (where the pressure

procedure.

and at low densities.

is decreasing

to the unphywith density).

higher densities, these branches become tachyonic branches as in fig. 4a, but at lower momentum. For the pion modes in the space-like region, there are tachyonic branches at low densities which lose the contact with the w = 0 axis mix together when density increases. At higher densities, we found a single

tachyonic branch. The heavy-meson branches are present for both the sigma and pion modes, and have the same qualitative behaviour than the ones obtained with the non-chiral renormalization procedure. For sigma modes, a second heavy-meson branch with lower mass is present. For pion modes, a new meson-like branch appears at high densities, which has the behaviour of a new principal branch. Fig. 4e, f corresponds to the (T- and r-modes obtained from the dispersion relations with the chiral-2 renormalization procedure. Owing to the fact that the renormalization conditions in vacuum are imposed on the pion polarization only, the sigma modes do not show principal branches at low densities, but a time-like branch appear at higher densities, which can be considered as the remnant of the principal branches. Indeed, at high densities the matter polarization contribution dominates over the vacuum polarization ones, and the behaviour of the semiclassical and renormalized approximations to the meson propagation become similar. The qualitative structure of the space-like branches is also strongly modified when compared to the results of fig. 4a. Nevertheless, the qualitative structure of the heavy-meson branches remains unchanged. Concerning the pion modes (fig. 4f), we found a normal branch whose mass slowly increases with density from PF/m = 0 to 0.7 and then grows faster, owed to the first two conditions (5.20) imposed on the pion vacuum polarization tensor, and to the dominating behaviour of the matter polarization at high densities. The structure of the space-like branches is also very different and more involved than in the non-chiral

case. In particular,

we found new instabilities

state at low momenta, which now cannot qualitative structure of the heavy-meson case.

of the renormalized

ground

be eliminated through a form factor. The branches remains unchanged also in this

The main conclusion of the comparison between the propagation behaviour obtained from the different renormalization procedures studied here, is that the more satisfactory results are obtained from the non-chiral procedure. Indeed, in this case, normal meson modes can propagate in the plasma at any density and, what is more important, the instabilities of the ground state related to the space-like branches arise at large momenta, out of the range of applicability of the model and of the approximation to the description of the actual nuclear medium. As discussed

652

J. Diaz Alonso, A. Perez Canyellas

-

/ Meson dynamics

\ Pf/m_

0

0.0

I!1 0.2

““‘I

0.4

0.6

0.8

“111 1.0

1.2

I

l.L

8-

6-

*\ - ,+-Chirall U"1"'1"'("'1'," 0.0 0.2

0

I q/m 0.4

0.6

0.8

1.0

Fig. 6. Same as fig. 3, now for the renormalized self-consistent approximation (Hart.), and the renormalized perturbative approximation (Pert.) with the non-chiral renormalization procedure. The discontinuities related to the Lee-Wick phase transition and to the instability of the quasi-meson modes against the decay into nucleon-antinucleon pairs have been eliminated by the vacuum effects. In (b), the effective quasi-pion mass curve obtained in the Wale&a model coupled to pions has been added for comparison. The results obtained from the chiral-1 and chiral-2 renormalization procedures are also shown.

J. Diaz Alonso, A. Perez Canyellas / Meson dynamics

above, such instabilities nucleon

structure.

instabilities

This is not possible

arise at small momenta.

are well adapted symmetry

can be easily eliminated

to the treatment

(see ref. 13)), where

in the chirally

the pion

but they lead to an odd behaviour

by taking into account renormalized

In fact, the chiral-renormalization of the vacuum remains

653

the extended

cases, where the procedures

effects in the case of exact chiral a massless

when the symmetry

boson

is explicitly

at any density, broken

in the

model and the pion modes become massive. In both the non-chirally and the chirally renormalized cases, the equation of state remains unphysical up to very high densities (from PJrn = 0.02-0.45, corresponding to four times the nuclear density in the chiral case, and from PF/m = 0.08-0.4 in the non-chiral case). The meson branches obtained in this range of densities (which is the one of interest in the study of nuclear matter) are therefore unphysical. When density increases, in the physical range, the results obtained with the different methods approach each other, and become the same in the high-density limit. Fig. 6a, b are plots of the effective quasi-meson masses (plasma frequencies for the principal branches) obtained in the renormalized-Hartree and perturbative cases, with the non-chiral renormalization procedure, and also with the chiral-1 and -2 renormalization procedures. In the non-chiral case, as in the semiclassical calculations (fig. 3), they are increasing functions of the density, except for the low-density region where they can slightly decrease. The main characteristic now is the disappearance of the discontinuities. This is related to the elimination of the Lee-Wick phase transition due to the vacuum fluctuations, and to the stabilizing effect of the vacuum polarization on the quasi-meson decay at wPl = 2M. When density increases, the effective

quasi-u

and quasi-r

masses continuously

grow and, as can be easily seen

from eqs. (5.21)-(5.22) and (5.23)-(5.24), their ratio goes to one. This is a consequence of the manifestation of the chiral symmetry “a la Wigner” in the highdensity limit, as will be discussed in sect. 7. The corresponding perturbative results, also with non-chiral renormalization, are plotted for comparison in these figures. In the case of chiral-1 renormalization procedure, the effective quasi-meson masses equal the free-meson masses at zero density and decrease to zero at low density, when the principal branches disappear. Some of these principal branches are present in the still physical region of the equation of state in the low-density region. Next, the range of density going to PF/m = 0.45 is not of interest, since it is inside the non-physical region of the equation of state. At high densities (where the matter polarization dominates), the pion branch reappears and the associated effective mass increases with density and behaves asymptotically like in the non-chiral case. For the chiral-2 case, the principal branch for the sigma modes is present at high densities, and the associated effective mass increases and has the same asymptotic behaviour as in the non-chiral case. The principal pion branches are present at any density, and their associated effective mass remains nearly constant at low densities. At high densities, this effective-mass curve increases and behaves similarly to the one obtained in the non-chiral case.

J. Diaz Alonso, A. Perez Canyelias

654

6. Scalar-vector So far, the numerical with a choice scattering

calculations

of the model

data in vacuum.

/ Meson dynamics

model with pions

have been performed

parameters However,

leading this model

so we can ask how the meson

in the pure a-model

to a satisfactory cannot

description

account

propagation

case, of the

for saturation

is influenced

of

nuclear

matter,

when we

consider analyzed the pions with the gives a

a model in which this property is achieved. For this purpose, we have the dispersion relations in the framework of the Walecka model, where have been introduced through the usual isospin invariant Yukawa coupling, same value of g, as in the sigma model. As mentioned above, this model satisfactory fitting of the saturation of nuclear matter but leads to an

overestimation of the rrN scattering amplitudes in two orders of magnitude. In obtaining the renormalized dispersion relations for the quasi-mesons in this case, there is not mixing between the r-polarization tensor and the polarizations for the other mesons. Consequently, the pion dispersion relations have the same analytic expressions (5.22), with the replacement a +.2b( u,, - m/g)‘+ In fig. 7 we have plotted the principal branches describing the propagation of the quasi-pion modes, as obtained from this modified Walecka model in the renormalized Hartree approximation. The branches are computed for three values of the Fermi momentum: PF= 0.3; 0.4 and 0.5, corresponding to three values of the density: The quasi-pion branches saturation density (n,), 2.37n0 and 4.63n,, respectively.

rni.

obtained

in the a-model,

L

0.3

and for the same densities,

H

T” --

--

2

O.L_

in the same approximation

’

’

Oo---+--

_-_I IRl/m,

I

4

I

I

6

I

I 8

10

Fig. 7. Comparison between the principal branches for the quasi-pion propagation obtained from the modified Walecka model (including pions) in the relativistic Hartree approximation (solid lines), and from the u-model in the same approximation, renormalized with the non-chiral procedure (dashed lines).

J. Diaz Alonso, A. Perez Canyelias / Meson dynamics

with the non-chiral parison. models,

renormalization

procedure,

have

655

also been

plotted

for com-

We note a good quantitative agreement for the branches obtained in both at least in this meaningful range of densities. The same agreement is ob-

served in fig. 6b, where we have superposed frequency)

as a function

the quasi-pion

of the Fermi momentum,

model in the renormalized obtained from the m-model.

effective

also obtained

mass (plasma

within the Walecka

Hartree approximation, to the corresponding curve The quasi-pion effective mass obtained in the perturba-

tive approximation (dashed line in fig. 6b) is the same for both models. In fact, the pion dynamics is changed in the Hartree approximation due to the presence of the effective mass M in the expression of the polarization tensor, which takes different values for different models at the same density. In a perturbative calculation, the effective mass M is replaced by the free-nucleon mass M, and so the pion dynamics at a given density remains the same for both models. As one of the major conclusions of this calculation, the pion dynamics in the plasma does not seem to be very sensitive to the choice of the model parameters, at least in the physically meaningful range of densities*. This result is also confirmed by the analysis of the pion propagation as obtained in the framework of a model similar to the present one, to which a non-linear UT coupling has been added 34). This model is the simplest one allowing the simultaneous fitting of the saturation and the vacuum r-nucleon scattering phenomenology ‘). In this case, the non-linear c~rr coupling modifies the form of the polarization tensors given in eqs. (5.23) and (5.24), but in the meaningful range of densities (from --n, to -4n,), it introduces only small quantitative corrections to the present results. We shall now analyze the modifications introduced by the fluctuations in the o-field on the u-meson propagation and the effects of the UOJmixing on the w-meson propagation itself. To that end, we shall study the dispersion relations of the cand w-mesons in the framework of the Walecka model. Such dispersion relations have been obtained by Chin 29) in the one-loop perturbative approximation. The dispersion relations in the mean-field approximation at T =0 can be obtained by replacing

the free nucleon

mass m in the Chin expressions,

mass M = m-g(v). The analysis approximation has been performed

by the effective nucleon

of the u- and o-modes in this semiclassical partially in ref. *I) and more recently by Lim

and Horowitz 41). Some effects of the vacuum polarization on the stability of the ground state against static transverse and longitudinal modes have been analyzed in refs. 21X36). Nevertheless, a detailed analysis of the w- and a-modes in the Hartree approximation is still lacking [see ref. ‘) and references quoted therein on this question]. Here we shall obtain the wu dispersion relations in the renormalized Hat-tree approximation using the Wigner function formalism, we shall then analyze the meson modes in some detail and compare the results with those obtained in absence of mixing. The method follows the same steps as in sects. 4 and 5. The kinetic equations for l

Evidently,

the renormalization

procedures

must be chosen

consistently

in all models to be compared.

656

J. Diaz Alonso, A. Perez Canyellas

/ Meson dynamics

the small perturbations of the distribution function (the analogues of eqs. (4.4) and (4.5)) are*: MP-Sk)

-

MlF,(k PI = k,v,(k)

around

the Hat-tree ground

+ tw,(k)l~,(~+~k),

F,(k P)[yU’+%) - Ml = -6-dJ’-ikkww,W where now P” =pp +g,(op); eqs. (3.23)-(3.27) effective nucleon The equations (4.6) and (4.7)),

F,(P)

is

+ iw,(k)l

the equilibrium

Wigner

,

function,

d4PFI(k,P), I

[k’k”-(k*-m2,)g’““]o,.(k)=-4rg,,,Tr The solution

(6.1)

with the conditions (3.12) and (3.32), and M = m-g,(a) mass. for the perturbations of the meson fields (the analogues are obtained from eqs. (2.23) and (2.25): [-k2+m~]a,(k)=4rrgUTr

state

(6.2) given by is the of eqs.

(6.3)

(6.4)

of (6.1) and (6.2) has the same form as in eq. (4.1 l), if we replace there: g+(k)

+ g,ar(k)+

g,ywr(k)

.

(6.5)

The form (4.11) of F,(k, P), when replaced in eqs. (6.3) and (6.4) leads, after a straightforward calculation, to the wave equations for the propagation of the quasi-o and quasi-u modes, which read: L-k*+

d-gYLW)l~~(W

= g,gJTL(kh(k),

[kc”k”-(k2-m~)gC”“+g2,n~“(k)]w,,(k)=-g,g,lT~”(k)a,(k)? where the polarization

(6.6) (6.7)

terms are given by

I&,(k) = 8n

d4P [(P-;k)(P+$k)+M2]X(k,

d4PPw2(k,

17:_,(k) = 167rM

P),

P),

(6.9)

I d4P[PwP”-~(P2-~k~kv-~k2-M2)g~v]E(k,P)

17Cv( k) = 167r

(6.8)

(6.10)

I and (6.11)

~(k,P)=[f,(P+~k)+_f,(P+~k)-f,(P-~k)-f,(P-$k)]/kP,

where f,(P) and f,(P) are given by eqs. (3.24) and (3.25) with 0 = 0. The above wave equations remain valid at any temperature 37). In the T = 0 case, after replacing M+ m, we recover the one-loop perturbative equations studied by Chin 29). l In this section field operators.

and in sect. 7, we shall suppress,

for convenience,

the superscript

for the perturbed

651

J. Diaz Alonso, A. Perez Canyellas / Meson dynamics

Without ation

loss of generality,

terms

defined

in

we can perform

eqs.

the explicit

(6.8)-(6.10),

in

the

(k’, 0, 0, k3). In this frame the only non-vanishing

calculation

particular

components

of the polariz-

frame

where

k=

of the polarizations

are :

n:a,,(k), n:“,(k), II:(k)

170,0(k),n:3(W, n:(k), =17:(k),

and eqs. (6.6), (6.7), can be written D”(k)

0

(6.12)

IL(k)

as a 5 x 5 matrix

equation:

0

Do3(k)

Do4(k)

w:(k)

0 0

w:(k) w;(k)

0

0

0

D22(k)

0

D3’(k)

0

0

D33(k)

D34(k)

w;(k)

D4’(k)

0

0

D43(k)

D44(k)

c+,(k)

0

D”(k)

0

where the elements

of the 5 x 5 matrix

D(k)

(6.13)

=0

are:

D44(k) = -k2+mt-gf&(k) , Do4( k) = -D4’( k) = -gcgJ7”,,( D34(k) = -D43(k)

k) ,

=gcgJ&(k),

Doo( k) = k2 + rnt + g$Z7”,“(k) , Do3( k) = D3’( k) = k”k3 + g;II”,‘( k) , D33( k) = ko2 - rni + gtI722( k) , (6.14)

D”(k)=D22(k)=k2-m~+g~17~1(k).

The dispersion relations for the propagation of the different quasi-meson modes are now obtained by equating to zero the determinant of the square matrix D(k). This determinant

factorizes

into three terms:

=

ID(k)/ = D”(k)DZ2(k)

Doo( k)

Do3( k)

Do4( k)

D3’(k)

D33(k)

D3”(k)

[ D4’( k)

D43( k)

D44( k)

=0

(D’1(k))2A(k)

The first two factors lead to dispersion which read:

.

relations

1 (6.15)

for pure vector-transverse

modes,

(6.16)

[k2-m~+g~ZT~1(k)]2=0, whereas

the determinant

part: A(k)=O,

gives the dispersion

relations

for mixed

scalar and longitudinal-vector

(6.17) modes.

658

J. Diaz Alonso, A. Perez Canyellas / Meson dynamics

Before we enter in the analysis of the vacuum polarization and vacuum vacuum

of these expressions, its renormalization.

we shall study the question As in the last sections, the

part of the distribution function in eqs. (3.24)-(3.27) introduces a divergent polarization contribution which must be renormalized. For the scalar

polarization

17,(k),

this vacuum

contribution

has the same structure

and so the results of sect. 5 can be immediately

used here. In particular,

as in eq. (5.3), the expression

of the scalar finite vacuum polarization in the present case is given by eq. (5.23), obtained with the non-chiral renormalization procedure. As it can be easily seen from eq. (6.9), there mixing polarization part nEW( k).

is no vacuum

contribution

to the

In obtaining the vacuum contributions to the vector polarization tensor 17EY(k), we proceed as in sect. 5. First of all we must regularize the vacuum divergent contribution in eq. (6.10). Using dimensional regularization, the vacuum polarization tensor can be decomposed into two parts: a divergent (for E + 0) part, which has the form: n::,(k) where the projector

ACLUis defined

= (4/3r&)k2AP”(k),

(6.18)

by:

ACLU(k)= g’““- k’k”/k*, and a finite vacuum

contribution

(6.19)

which reads:

~~~,(k)=(4/3~)[k2ln(M/m)-2M2+(2M2+k2)8(k2,M2)+~]AC”“(k). (6.20) Here f3(k2, M2) is the function defined in eq. (5.10) and r is an arbitrary finite constant. This constant must be determined through the renormalization conditions. Following the same criterion as in sect. 5, the w-wave eq. (6.7), must reduce in vacuum to the free-wave equation: (6.21)

[kpk”-(k*-m2,)gp”]w,(k)=0. From this on-shell

criterion,

the constant

r becomes:

r=2m2-(2m2+mZ,)e(m2, The addition

to the lagrangian

of a counterterm L cc =ZF’““F PY

m2,).

(6.22)

with the form: (6.23)

allows us to cancel the divergent contribution to the vacuum polarization, which now reduces to the finite part (6.20). By replacing the effective mass (M) by the free-nucleon mass (m) in this formula, we recover the form of the vacuum polarization tensor in the perturbative approximation, obtained by Chin 29). The analysis of the dispersion relations in the semiclassical and renormalized perturbative approximations has been performed in ref. *“). More recently, the mean

J. Diaz Alonso, A. Perez Canyellas / Meson dynamics

field dispersion

relations

study of the meson

have been studied

in the Hartree

will be published

elsewhere

free fermion

in ref. 41). We shall concentrate

modes which results from the renormalized

we have obtained

mass as energy

In fig. 8a, we have plotted

and momentum

dispersion

at T = 0.Finite-temperature

approximation

37). Owing to the presence the branches

659

units obtained

of the mixing,

on the relations analysis

we chose the

for the next drawings. from the transverse

part of the

dispersion relation (6.16). The parameter is the Fermi momentum, and it goes from saturation density up to the value corresponding to 4.6 times this density. We observe the presence of the normal quasi-meson modes, aside the “tachyonic branches” and the “heavy-meson branches”. As in the model analyzed in sect. 5, the last two modes are related to the vacuum polarization terms and are not present in the semiclassical (mean-field) approximation 41). We mention that in the mean-field approximation, they also appear space-like branches for the transverse modes which cut the w = 0 axis, which are analogous to the ones obtained in sect. 4 (see fig. la, b). These branches also introduce instabilities in the mean-field ground state and (as shown in fig, 8a) disappear when vacuum polarization terms are considered. This question has been studied in detail in refs. 2’P36). Fig. 8b shows with more detail the principal branches of fig. 8a. The effective quasi-meson mass for these transverse vector modes grows with density. All branches behave as quasi-hyperbolas which go asymptotically to o = (k(. Fig. 9a is a plot of the meson modes obtained from eq. (6.17). These modes are obtained at saturation density. The dispersion relation mix up, in this case, the w-longitudinal modes and the o-modes. Consequently, we found two branches of each class, which can be interpreted as the remnants of the w- and g-modes after mixing. There are also two small zero-sound branches which form a loop. Fig. 9b gives the detail of the principal branches. the same value of w as does the principal

The upper branch cuts the (kJ = 0 axis at branch of the transverse-vector modes at

saturation density in fig. 8. This allows the identification of this upper branch as the remnant of the longitudinal vector modes after mixing. The same identification can be made

between

the lower

“heavy-meson”

branch

and the left “tachyonic”

branch. The rest of the branches can be identified as the modified a-modes after the mixing. The comparison between these modes and those of fig. 4a shows that no qualitative modifications are introduced by mixing on the propagation of the a-mesons. As we shall see subsequently, this assertion concerns also the zero-sound modes associated to the o-field. In fig. 10 we have plotted the zero-sound branches corresponding to the mixing between the w-longitudinal and u-modes. We have also included in this drawing the zero-sound branches associated to the w and u fields in absence of mixing. These latter branches have been obtained as solutions of eq. (6.17) when the mixing polarization terms, ZTgti(k), is assumed to vanish. The form of these curves suggest that the loop formed by the zero-sound mixed branches is the remnant of zero-sound modes of the longitudinal o-field, which are strongly inhibited by the mixing. It

.I. Diaz Alonso,

06 00

02

Fig. 8. Branches for the vector transverse Hartree approximation. The parameter is the damping regions for Pr/ m = 0.3 and detail

A. Perez Canyellas

OL

f Meson dynamics

0.6

0.8

1.0

modes, obtained from the Walecka model in the renormalized the Fermi momentum in units of the baryon mass. In part (a), PF/m = 0.5have been plotted (dotted lines). Part (b) gives a of the principal branches.

also suggests that mixing is responsible for the disappearance of the zero-sound modes associated to the a-field. Nevertheless, this last conclusion must be shaded. Indeed, as shown in ref. 22), the zero-sound modes for the scalar field, as calculated in an approximation which is consistent with the approximation used in obtaining the e.o.s. (the renormalized dispersion relations, if the e.o.s. is obtained in the renormalized Hartree approximation, or the semiclassical dispersion relations, if

J. Diaz Alonso, A. Perez Canyeilas / Meson dynamics

661

0.8

0.t

0.2

0.0 0.0

0.2

0.6

0.6

0.8

1.0

Fig. 9. Branches for the mixed w-longitudinal and o-modes obtained from the Walecka model in the renormalized Hartree approximation at saturation density. Part (b) gives a detail of the principal branches and the zero-sound loop. The dotted lines give the limits of the damping regions.

the e.o.s. is obtained in the mean-field approximation), are present only in the non-physical region of the e.o.s., and completely disappear outside this region. This is confirmed by the results of sect. 5 (see fig. 4a) and it is the reason for the disappearance of the scalar zero-sound modes in fig. 10, which corresponds to the physical zone in the e.os. In order to emphasize this point, we have performed the calcufation of the same mixed zero-sound modes for Fr/rn = 0.1, which is a value

J. Diaz Alonso, A. Perez Canyellas / Meson dynamics

662

&’ :

0.2 -

/

,,/

/

/

.’

/ ..’

/:’ / : :.

/:

/

4.’

/.

/

/

P, /m=0.30

/

. /

/

/

’

/

/

/ __----------.

u_ MODE ‘\

0.1

0.2

‘1’

1 ‘I” 0.3

’ 1 I I a ” 0.L

IGl/m 0.5

”

\ \

a .’

_

0.6

Fig. 10. Zero-sound branches obtained from the Walecka model in the renormalized Hartree approximation at saturation density for the longitudinal mixed modes. The zero-sound branches for the c-modes and w-longitudinal modes in absence of mixing have also been plotted, as well as the particle-hole damping region.

of the renormalized Hat-tree e.o.s. inside the unphysical region of the Van der Waals phase transition. In this case, the loop zero-sound branches disappear, and there is only a zero-sound branch which corresponds to scalar modes. The appearance of the scalar zero-sound modes in fig. 10, when the mixing effects are neglected, must be associated also to the inconsistency between the solved dispersion relations and the approximation used in the calculation of the e.o.s. We conclude that, in a consistent

calculation,

there

are no zero-sound

modes which can be associated of the e.o.s., and this fact is independent

to of

the scalar field in the physical region the mixing with the w-sector. In ref. 29), Chin shows that the mixing has inhibitory effects on the zero-sound modes, specially at low densities. Although the Chin argument is obtained from a perturbative approximation, the same effects are observed in fig. 10, in the Hartree approximation, on the vector-longitudinal zero-sound modes. In fig. 11, we have plotted the behaviour of the effective quasi-meson masses (plasma frequencies) with density, for the principal branches of the transverse (T) and longitudinal-mixed (L) modes. The transverse modes and the remnant of the longitudinal vector modes in the mixed dispersion relation, have the same effective masses in the meaningful range of densities (going here up to 4.6 times the nuclear density). The behaviour of the effective mass for the scalar modes must be compared to the results of the Hartree approximation from the a-model shown in fig. 6a. The only qualitative difference is the small decreasing behaviour, at low densities, of

663

J. Diaz Alonso, A. Perez Canyellas / Meson dynamics 1.0,

I I I I, _ R/m

) I

I,

(

I,,

,

,

,I,

I,

I,

I,

Fig. 11. Effective quasi-meson masses for the longitudinal (L) and transverse (T) modes, obtained from the Walecka model in the renormalized Hartree approximation, as a function of the Fermi momentum.

the effective

mass

obtained

from

the a-model,

coming

from

the non-linear

uu

couplings, which reduces the meson mass at low densities (see eq. (4.6)). This comparison confirms the small incidence of the mixing with the o-sector on the normal branches for the a-modes. We note also that the pathological behaviour encountered for the effective quasi-meson masses in the mean-field approximation in ref. 41), w h’ic h were interpreted as due to Pauli blocking on NN excitations, disappear in the renormalized Hartree approximation, where all vacuum contributions are consistently included. To end this section, let us make some comments on the stability of the system we studied. We follow here the analysis of ref. 35). The behaviour of the dispersion relations for both the principal and the heavy-meson branches indicates that they must be considered as evanescent modes. On the other hand, the presence of the “tachyonic branches” should be associated to the existence of a nonconvective instability [see also refs. 21336)and the discussion of sect. 5 about the nature of this instability]. Nevertheless, as mentioned in sect. 5, such branches arise at high momentum transfer, where the structural properties of the nucleon invalidate the point-like particle approach, and consequently, the associated instability is unrealistic. In sect. 5, we have verified that the introduction of a simple monopolar form factor eliminates these “tachyonic branches”. The same result can be obtained in the present case, as shown in ref. 21). Concerning the damping of the different modes, their analysis requires the obtention of the imaginary parts of the polarization terms (6.8)-(6.10). These imaginary parts arise from the insertion of the complete Feynman propagator in eq. (4.11), in

J. LXaz Afonso, A. Perez Canyelias ,/ Meson dynamics

664

the place of S,(P). The result of such calculation coincides with the corresponding expressions obtained in ref. 29),in the perturbative approximation, if the replacement m + M is made there. The analysis of this imaginary part has been also performed in ref. 41) for the mean-field approximation. In our case, the imagina~ part of the vacuum polarization tensors should be included. All these imagina~ parts divide the w, ]k] plane in several regions, where the decay of a mode is allowed or forbidden by the Pauli exclusion principle. The imagina~ part of the vacuum poia~zation defines the curve w = (k2+ 4M2)3j2 . Above this curve, a mode sea. Nevertheless, the energy a negative-energy baryon to in the more economic case),

(6.24)

could decay into a NR pair in absence of the Fermi and momentum necessary far a mode in order to raise a positive-energy state of momentum P (parallel to k, are related by the equation

w=[(P-k)Z+M”]“/2t(P2+1W2)1’2,

(6.25)

but the pair creation is blocked if ]PJ < F,,. The value of o in eq. (6.25) as a function of ]P} for a fixed k, reaches a minimum o = (kZ+4M2)‘/* at IP] =$/ki_ Consequently, if PF> $lk/, the only possibility for a mode to decay into a pair arises in the region above the curve o=[(P,-~k~)2~~2]“‘2~(P~+M2)1’2, (6.26) whereas if lJr:< $1kl, the decay into a pair arises in the region above the curve (6.24). In the w, Jk] plane, both curves are tangent at /kl= 2& and the curve (6.26) lies always above curve (6.24). Consequently, in this plane, the modes which lie in the time-like region under curve (6.24) with fkl> 2P,, or under the curve (6.26) with lkl<=‘,, are undamped because of the Pauli blocking. Other time-like modes are damped by pair creation. In the space-like region of the m, lk] plane, the imaginary part of the matter polarization defines the region I[(PF-jkl)2+

MZ]1’2- (P:-t-M2)“*~
(6.27) where a mode can decay into a particle-hole pair. Outside this region, the Pauli blocking in the Fermi sea prevents the decay, and the modes are undamped. From these elementary considerations, we can discuss the damping of the different modes obtained in this section. On fig. $a we have introduced the limits of the damping regions for PF = 0.3 and PF = 0.5 (dotted lines). We see that the transverse heavy-meson branches are damped by the NR pair creation whereas the transverse normal meson branches are undamped in this range of densities. We have verified that for higher densities the normal branches enter the damping region for some value of Ikl. The tachyonic branches are undamped for small values of ]k] and enter

J. Diaz Alonso, A. Perez Canyellas / Meson dynamics

the particle-hole

damping

hold for the normal

region

branches

for large values

of Jkl. The same considerations

and the heavy-meson

branches

tudinal modes (see fig. 9a). Concerning the zero-sound the region of particle-hole decay, whereas the upper values

of (kl and enters

qualitative

behaviour

this region

of the zero-sound

when

(see fig. 9b). In fact, the

in this renormalized

to the one encountered in the mean-field approximation have found is that vacuum effects lower the threshold existence

of the mixed-longi-

modes, one of them is inside mode is undamped for small

(kJ increases

modes

665

case is similar

41). The only difference we value of the density for the

of these modes.

7. Linear response and screening effects We shall now analyze the screening effects of the nuclear plasma on the field created by a particle in a given state. First, we shall introduce the method, in detail, for the scalar-coupling model 22). The generalization to other meson interactions is straightforward. Consider a nucleon in a state [cp) introduced in the nuclear medium at equilibrium. The scalar field created by this particle is assumed to be small and to obey to the linearized

equation: PJ+

The presence of this test particle tion F,,(p) through: Rx, Also, this modification becomes:

changes

(7.1)

dlqc =%++Fw~ in the state 1~) modifies

P) = F,,(P) the total

the equilibrium

distribu-

+ SF(x, P) . scalar

(7.2)

field in the plasma,

which

now

(7.3)

u,,,=(a)+u,+6a,

where (u) is the mean value of the scalar field obtained in the Hartree approximation, up is the field created by the nucleon in the state IV), before screening, and &r is the correction

due to the modification

of the equilibrium

distribution

(screening

field). If we assume this correction to be also small, we can linearize eqs. (2.21)-(2.23) (in the particular case of the simple scalar coupling) around the Hartree equilibrium, and obtain the following equations relating 6F and 8~:

CTQJ-+~)-WW~ aF(k

~)=-gF,,(p+ik)[a,(k)+6~(k)l,

P)[Y(P+tfk)-Ml= [-k2+mfG]Sa(k)=4rg

-gF,,(p-~k)[a,(k)+6a(k)l, tr

I

d4p6F(k,

p).

(7.4) (7.5) (7.6)

666

J. Diaz Alonso, A. Perez Canyellas / Meson dynamics

The explicit form of W(k, p) obtained from eqs. (7.4) and (7.5) is the same as in eqs. (4.11) and (4.12), where the combination of the meson fields is now: 4(k)=c~~(k)+tb(k).

By inserting relating

this expression

in eq. (7.6) we obtain

field So(k)

the screening

(7.7) the following

linear

field a,(k):

to the perturbating

[-k2+m:+g217,(k)]6a(k)=g211,(k)u,(k),

where the polarization The total (screened)

is given by eq. (4.16). field created by the nucleon

eqs. (7.1) and (7.8) we obtain scalar source: [-k2+

the following

(7.8)

in the state 1~) is uV + 60: From

equations

relating

the total field to the

g217,(k)l[~,(k) + Wk)l = 4~drpl&blcp)k,

m$+

equations

(7.9)

where the index k in the source term stands for the Fourier transform. The comparison of eqs. (7.1) and (7.9) leads to the definition of the effective coupling constant for the quasi-a field, accounting for the screening effects in the plasma: g,,,(k)

=

g{l -

g2K#)lW2+

g’~,Wl~.

m2,+

(7.10)

If the same analysis is performed for both the v and rr couplings in the framework of the complete a-model lagrangian, we obtain the following expressions for the effective coupling constants: g,rrm(k) = g,&k) where the expressions

@,(k)E’(k) ,

(7.11)

,

(7.12)

= gZ,(kP,‘(k)

for the E’s are: E,(k)=

-k2+a+6b(m/g-(a))‘,

(7.13)

Z,,(k)=

-k2+a+2b(m/g-(a))2,

(7.14)

and D;‘(k), D;‘(k) are the quasi-a and quasi-T propagators, obtained from the respective dispersion functions defined in eqs. (4.14) and (4.15). A particular case of this analysis is the study of the screening effects on the field created by a nucleon in a well-defined momentum and spin state 1cp)= )p, s)( 1pi > PF). In this case, the form of the source terms in eq. (7.1) can be explicitly calculated. Using the results on the Fock space developed in ref. 14), one obtains straightforwardly: (p, sl~,@\p, s)=(2r)-‘M/~&%+(2rr-~2M

I

d4qS(q2-M*)O(-q’), (7.15)

J. Diaz Alonso, A. Perez Canyellas / Meson dynamics

where the second

term in the right-hand

tion to the scalar density. transform created

is proportional by a particle

side corresponds

This expression to the Dirac

is space-time distribution

in the (p, s) state before a, =-

1

to the Dirac-sea independent

6(k).

screening

667

contribu-

and its Fourier

Consequently,

the field

is:

@@(k)

(7.16)

27r2 r&&Z The screened

field is obtained

from eqs. (7.8) and (7.9):

a,+s~={l-g2~~(0)/[m2,+g2~,(0)]}~~. The effective

coupling ge,

constant =

&A1

obtained

from eq. (7.10) reduces

-g’n,(0)l[m2,+g2~,(O)I}.

(7.17) to: (7.18)

In the calculations on the complete m-model we obtain, in the same way, the following expressions for the effective coupling constants among the quasi-nucleons, out of the Fermi sphere, and the quasi-a and quasi-r fields:

g,ff,=gm2,/[a+6b(mlg-((T))*-g2~~(0)1, &ff, =sm~/[a+2b(mlg-(a))*-g2n,(0)1. In these formulae

the polarization

“tensors”

(7.19) (7.20)

are taken in the limit k + 0. Because

this limit is not unique, it must be specified in accordance to the particular physical situation. In fig. 12 we have plotted the evolution with density of the effective

Fig. 12. Effective coupling constants (in units of the bare coupling constant g of the v-model) describing the screening effects of the medium on particles in a defined momentum state out of the Fermi sphere ([It, s), Jkl> PF), as functions of the Fermi momentum (in units of the free-nucleon mass). The curves are obtained in the semiclassical linear response approximation. The discontinuities are due to the Lee-Wick phase transition. When density increases, both effective coupling constants vanish asymptotically.

J. Diaz Alonso, A. Perez Canyellas / Meson dynamics

668

coupling

constants

for the quasi-u

and

quasi-r

mesons

in units

of the vacuum

coupling constant. They are calculated in the mean-field approximation (neglecting vacuum contributions), and the k+ 0 limit is taken in the o-axis direction (jkj +O first, o + 0 next) in order to obtain

the appropriate

[see ref. ‘“)I. The discontinuities

are due to the normal-abnormal

The

behaviour

of both

effective

coupling

coupling

constants

to the conserved

is very

phase different.

currents transition. This

is a

manifestation of the amplification of the explicit chiral symmetry breaking in the medium, when fluctuations around the ground state are included, which distorts the original equality between the couplings of nucleons to the u- and T-mesons in vacuum. As density increases, both coupling constants approach to zero and the equality of the nucleon-meson couplings is asymptotically recovered. This is a consequence of the restoration of the symmetry in the high-density limit* (this asymptotic behaviour is independent of the direction from which the limit k + 0 is taken, as can be easily checked analytically). Fig. 13 is a plot of the same effective coupling constants as calculated in the renormalized case (using the non-chiral renormalization procedure). The evolution with density is now smooth, because of the disappearance of the Lee-Wick phase transition, due to the vacuum contributions. As in the semiclassical case, the effective couplings vanish asymptotically when density increases, In this case, the effective coupling constants are in units of their values at zero density,

which are different

from the vacuum

coupling

constants.

0.8 -g,f,(P,)/g,,,(O)

0.6 -

0.1 -

0.2 -

0.00

0.05

0.10

0.15

0.20

0.25

0.30

Fig. 13. The effective coupling constants, in units of the PF = 0 coupling constant (see text), calculated in the renormalized linear response approximation. The discontinuities of fig. 12, related to the Lee-Wick phase transition, have been eliminated by the vacuum effects. l In this limit, the symmetry is realized “a la Wigner”. In fact, in this limit, the effective quasi-nucleon mass vanishes, and the ratio between the quasi-o and quasi-r effective masses goes to one (see sect. 5).

J. Diaz Alonso, A. Perez Canyellas / Meson dynamics

The reason vacuum

whereas

for that, is that the renormalization

is given by the on-shell

the effective coupling

in vacuum Because inside the (7.20) also

point

669

for the coupling

constants

in

conditions:

g(k’=mi,

PF=O)=g,

(7.21)

g(k’=mt,

PF=O)=g,

(7.22)

constants

for particles

in the Jp, s) states must reduce

to the renormalized coupling constant in the point k2 = 0. of the Pauli blocking, we can not add external particles to the plasma Fermi sphere (in states Ip, s) with (pj < PF. However, eqs. (7.19) and describe the behaviour of the effective coupling constants for holes). We

can ask, nevertheless, whether the effect of the screening on the particles composing the plasma, as calculated from the linear response, must modify the thermodynamical behaviour obtained in the mean-field or Hat-tree approximations. In order to clarify this point, we return to the pure scalar-coupling model. Assume the N-particle plasma to be in the mean-field or Hat-tree ground state with a given value of the Fermi momentum. The associated effective mass is M = m -g(a). The kinetic equations for the equilibrium Wigner distribution in this case reduce to: r -Yp-

Ml~eq(P) = Fe,(P)[YP- Ml = 0.

(7.23)

If we add, now, a particle on the Fermi surface (that is, in a state 14,s), with (41- PF), the field created by this particle, before screening is a,,,, as given by eq. (7.16). The initial particle distribution is modified by the presence of the test-particle field, and becomes Feq + SE The new distribution obeys to the following equations, obtained from the general kinetic equations:

MI[Fe&)+ WP)I = -dF&)+

[V-J [Fe&)

+

W~)l[v

Wp)li-cr,,, + 6~1,

(7.24)

-Ml = -g[Feq(p) + Wp)l[~q,s + 6~1.

(7.25)

These equations are exact in the mean-field or Hartree approximations. The kdependence disappears because the S(k) distribution factor in the source term. The eqs. (7.24) and (7.25) for the modified distributions are identical we replace the effective mass M by the new effective mass: M’= M -g[a,,,

+tb]=m-~[((+)+cT~,,+c%],

to eq. (7.23) when

(7.26)

and, consequently, the perturbed distribution F(p) + SF(p), is on the mass-shell p2 = M12. The same argument can be developed for the dynamical equations of the nucleons in the mean-field or Hartree approximations. The dynamics of nucleons before the introduction of the perturbing particle is governed by the Dirac equation: (iya-M)+=O, and in presence

(7.27)

of the new particle: (N-MM-g(a,,,

+Lb)t,b=(iyd-M’)+=O.

(7.28)

610

Consequently, the introduction

J. Diaz Alonso, A. Perez Canyellas / Meson dynamics

the Fock space in presence of this particle,

both cases is in the mass of the corresponding sphere with N quasi-particles ground

particle

is, as before

space. The only difference quasi-particles.

between

If we fill the Fermi

of mass M, in states 1q, s), or with N + 1 quasi-particles

of mass M’, in the same momentum the self-consistent

of the perturbing

the Dirac-Fock

states

and spin states, we shall obtain

of the mean-field

or Hartree

in both cases

approximations

for

systems of N or N+ 1 particles, respectively. The screeening correction to the N-particle equilibrium distribution gives the N + 1 -particle equilibrium distribution and, thus, the screening effects associated to the nucleons inside the Fermi sphere, as calculated in the linear response approximation, are already contained in the mean-field or Hartree approximations (they reduce to the self-consistent modification of the effective mass of the quasi-nucleons). A more detailed analysis of the screening effects in nuclear matter is in progress 39). In particular, the screening effects on the pion-exchange part of the static twonucleon potential in relativistic nuclear matter have been analyzed in ref. 34).

8. Concluding

summary

We have performed a fully relativistic quantum analysis of meson propagation for two different QHD-models in a many-body framework. We first considered the approximate chirally invariant a-model. The study has been also extended to the case of the Walecka model. The analysis has been performed using the techniques of the covariant Wigner function. In order to solve the system of kinetic equations near the thermodynamic equilibrium, a method of expansion into correlation functions has been used. The first order of this expansion gives the well-known thermodynamic behaviour of the plasma in the mean-field and Hartree approximations. The second-order approximation is obtained by considering the two-body correlations and neglecting higher order ones. The kinetic equations have been solved to this order, under the assumption of small two-body correlations, and we have deduced the dispersion relations for the propagation of the quasi-meson modes. Two different approaches have been considered in this context, leading to the different approximations to the quasi-meson propagation behaviour. These solutions correspond, respectively, to the analysis of the linear perturbations around the mean-field ground state (semiclassical approximation) and the Hartree ground state (renormalized approximation). The more satisfactory behaviour is obtained in the last one. These second-order solutions lead to the same results as the usual one-loop calculations in each case. From the analysis of the semiclassical dispersion relations, we have found a pathological behaviour of the mean-field ground state, which becomes unstable, and must relax to a more stable, spatially-structured ground state. Next, the vacuum polarization has been considered and renormalized within three different procedures. The analysis of the renormalized dispersion relations with a general renormalization

671

J. Diaz Alonso, A. Perez Canyellas / Meson dynamics

procedure

which

restores

the approximate

one-loop level, by an appropriate disappearance of the semiclassical

chiral

symmetry

of the model

at the

choice of the renormalized parameters, shows the instabilities and the appearance of a new instabil-

ity of the Hartree ground state for static modes at large values of Jk(, related to vacuum polarization effects. This new instability arises beyond the range of validity of the approximation,

and of the point-particle

approach

as well. Even a simple

monopolar form factor, accounting phenomenologically for the nucleon structure at short distances, allows for the elimination of such instabilities and the associated tachyonic branches. We also found time-like branches which are the normal meson branches, representing the usual on-shell propagation of the quasi-mesons in the plasma, and the heavy-meson branches, related to the vacuum polarization contributions. We have performed a detailed analysis of the evolution of all branches with the thermodynamic conditions in both approximations, and conclude that vacuum fluctuations and vacuum polarization have important effects on the meson propagation in this model. The renormalized dispersion relations have also been obtained with two different chiral renormalization procedures. In these cases, the inconsistency between the explicit symmetry breaking and the chiral symmetry preservation of the procedure introduces a pathological behaviour in the meson propagation at low densities, as compared with the results obtained in the first case. Nevertheless, this pathological behaviour arises mostly in the non-physical region of the equation of the state and, for high densities, the results obtained with all renormalization procedures become similar. The quasi-meson dynamics has been also analyzed in the framework of the Walecka model, in the relativistic Hartree approximation. For the sake of comparison with the a-model case, a n--meson sector has been added to the lagrangian, through an isospin-invariant Yukawa coupling. The results for the u- and n-modes are qualitatively the same in both models, for the meaningful range of densities (between n, and -4~). The effects of mixing between the (+- and longitudinal-w-modes have been also analyzed. While such effects are qualitatively not significant for most of the modes, we have found an important inhibition of the longitudinal-o zero-sound modes in the mixed case, in agreement with previous results. Finally, we have analyzed the screening effects within the context of the linear response

formalism.

In this way, we have obtained

the expressions

of the effective

coupling constants, which describe the effects of screening on the interactions between quasi-nucleons (in a given state) and the mesons. For the particular case of nucleons in a fixed momentum and spin state (1s s), with (kl >pF), we have obtained the behaviour of the a-nucleon and rr-nucleon effective coupling constants with density. We have shown that, at high densities, both couplings become of the same order and vanish asymptotically. This result is a manifestation of the restoration of the chiral symmetry in the high-density limit, where it becomes exact in the Wigner sense. Thus, the screening of the nuclear interaction in the medium, as calculated in the linear response approximation in this model, becomes more and

J. Diaz Alonso, A. Perez Canyellas / Meson dynamics

672

more effective as density increases. Finally, we have shown that the screening effects, obtained in the above way, were already self-consistently contained in the mean-field and Hartree dynamics calculation

approximations,

by the screening performed

and corrections effects

cannot

to the mean-field be obtained

from

or Hartree the linear

thermoresponse

here.

The authors are indebted to B.L. Friman, R. Hakim, P.A. Henning, H. Sivak and V. Vento for useful discussions and suggestions. A.P.C. acknowledges the support of CAICYT (SPAIN), research project AE-88-0021-04. J.D.A. acknowledges the kind hospitality of the G.S.I. Theoretical Group, where a part of this work has been developed.

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