Quark Matter in the Chiral Color Dielectric Model

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HEP-PH-9503462 err a bini t is imp osed e unrealistic yp eset using REVT t. Sensible re- am T ole and the quarks a, Portugal c Mo del h con nemen uclear pro cesses tri t phase transition is smo oth and Ubaldo T b a, and INFN, Sezione di F a, Italy 44100 ar ys a marginal r^ ar a da Universidade uclei and n err err a, P-3000 Coimbr isic 1 h 30, 1995) oric adiF e uel Fiolhais Abstract (Marc aT ersion of the mo del turns out to giv Quark Matter adiso 12, F , Man a ersion of the CDM for whic artamento de F a, Universit o de Fisic Via Par Dep o{phases v b w in the Chiral Color Dielec e till high densities. The decon nemen and Centr ely to b e rst order. Instabilities of the quark matter and the gap . The t Alessandro Drago othly artimento di Fisic e study the equation of state (EOS) of quark matter at zero temp erature, W Dip using the Color Dielectric Mo delsults (CDM) are to obtained describ e in con nemen the v smo equation are discussed. 24.10.Jv Relativistic mo dels, 24.85.+p Quarks, gluons and QCD in n and unlik results for the EOS.are Chiral massiv symmetry pla a provided byCERNDocumentServer brought toyouby CORE

I. INTRODUCTION

The study of the Equation Of State (EOS) of Quark Matter (QM) has b ecome a fash-

ionable topic in view of the next exp eriments using heavy ions in program at RHIC(BNL)

and at LHC(CERN) [1]. Furthermore the inner structure of neutron stars is now under in-

vestigation: the connection b etween the comp osition of the star and the co oling time, which

can b e measured, allows to discriminate among the various mo dels, indicating the p ossible

existence of a quark matter phase (see, e.g. [2]). The study of the equation of state of matter

at high densities can also give usefull informations to traditional nuclear physics, since one

can search heavy nuclei for precursor phenomena, b oth of the decon nement and/or of the

chiral restoration phase transition (for a review see [3]).

In many mo del calculations of the decon nement phase transition the frame of the MIT

bag mo del has b een used [4,5]. In suchaway a rst order decon nement phase transition is

obtained (apart from sp eci c, ad hoc choices of the mo del parameters) and the decon nement

phase transition coincide with the chiral restauration one. At densities and temp eratures

slightly bigger than the critical ones the right degrees of freedom are already quarks, having

current masses, and p erturbative gluons. There are anyway indications, from lattice calcu-

lations, that at temp eratures bigger than the critical one non p erturbative e ects are still

present in the quark{gluon plasma [6].

In this pap er we study the EOS of QM using the Color Dielectric Mo del (CDM) to

describ e con nement [7]. We shortly review CDM in sect. 2. This mo del has b een widely

used to study b oth the static and the dynamical prop erties of the nucleon. Morover it can

b e used to describ e many{nucleon systems: for a twonucleon system it allows to compute

anucleon{nucleon p otential qualitatively similar to the ones used in nuclear physics [8]; in

the case of a homogeneous, in nite system of nucleons the CDM can b e used to construct a

nonlinear version of the Walecka mo del [9].

The aim of our work is to extend previous calculations of the decon nement phase

transition, where the same mo del has b een used [11]. An imp ortant p oint of our calculation

will b e to x the mo del parameters in order to repro duce the basic static prop erties of

the single nucleon, as was already done in the study of the nucleon structure functions

[12]. We will later use the same parameters to study the EOS of QM, whichwe de ne

as a system of totally decon ned quarks. In suchaway the study of the QM's EOS will

turn out to b e a severe test for the di erentversions of the CDM, and we will b e able to

make some predictions of the prop erties of matter at high densities. Within CDM (with a

double minimum p otential for the scalar eld), we will investigate the p ossibility of getting

a scenario similar to the one describ ed by the MIT bag mo del, with two phases undergoing

a sharp rst order phase transition (sect. 3). Our results show that such a description is

incompatible with the CDM. We than study another version of the CDM (with a single

minimum p otential for the scalar eld), where con nement is imp osed more smo othly, and

we get a sensible EOS for QM, without a sharp decon nement transition.

An imp ortant feature of this decon ned quark matter is that quark's masses are big till

high densities. Chiral restauration and decon nement do not o ccur at the same density.

The reasons whychiral simmetry is restaured so slowly are discussed in sect. 4.

In the last sections we analyze the prop erties of QM, as describ ed in the CDM. In sect.

5 the stability of quark matter is studied, using the technology of the resp onse function. In 2

sect. 6 we consider a gap equation, trying to understand the formation of quarks' clusters.

Finally sect. 7 is devoted to the concluding remarks.

I I. THE COLOR DIELECTRIC MODEL

A. The mo del lagrangian

In this section we shortly summarize the main features of the CDM. For a comprehensive

review see the ones by Pirner [7] and by Birse [10].

We will use a chiral invariantversion of the CDM, as the one used in Ref. [11] and also

in Ref. [12]. The Lagrangian reads

L = i @ + ( + i ~ ~ )

1 1 1

2 2 2

(@ ) U ( )+ (@ ) + (@ ~) U (;~) ; (1) +

2 2 2

where U (;~) is the usual mexican-hat p otential, as in ref. [15]. L describ es a system of

interacting quarks, pions, sigmas and a scalar-isoscalar chiral singlet eld , whose p otential

U ( ) has an absolute minimum for = 0. In suchaway, in the case of the single nucleon

problem, the quarks' e ective mass g= diverges outside the nucleon. The simplest

p otential for is a quadratic one:

2 2

U ( )= M : (2)

Using this p otential, uctuates around zero, reaching this value asymptotically, at large

distances. In the following we will refer to this version of the mo del as the Single Minimum

(SM) one. Another p ossibility is to use a p otential for having a Double Minimum (DM):

2 3

! ! !

4 4

8 6 1

2 2

4 5

1+ M 2 + 1 ; (3) U ( )=

2 2

2 M M

where the absolute minimum is still in the origin and the relative one is in = M . Cho osing

appropriately the parameters it is p ossible to obtain solutions for the single nucleon where

the eld interp olates b etween the relative minimum at the center of the nucleon and the

absolute minimum at big distances. In the following, we will consider for DM only solutions

where has the ab ove discussed b ehaviour. A complete analysis of the various p ossible

solutions of the mo del (in a non{chiral{invariantversion) can b e found in Ref. [13].

In Fig.1 we showtwotypical solutions of the mo del for the single nucleon problem.

As it app ears, in the DM case the transition b etween the interior of the nucleon and the

external region is sharp er than in the SM, where all the elds havea very smo oth b ehaviour.

Corresp ondingly, the kinetic energy contribution (which comes mainly from the quarks) will

b e bigger in DM than in SM. This p oint will b e relevant when studying the EOS of QM.

The Lagrangian is chiral invariant, and it can b e considered a con ning version of the

traditional {mo del. An imp ortant p oint concern the value of the chiral elds, the pion

and the sigma, in this mo del, in the single nucleon case. These elds are always near their 3

vacuum value. This p oint has b een checked out numerically several times and some euristic

explanations have b een prop osed [10,14]. As a consequence, the chiral elds cannot `wind'

around the mexican hat p otential. The pion is thus just a p erturbation and cannot develop

a non{trivial top ology.

B. Fixing the parameters

The parameters of the mo del are: the chiral meson masses m =0:14 GeV, m =1:2

GeV, the pion decay constant f =0:093 GeV, the coupling constant g , and the parameters

app earing in the quadratic (2) or in quartic p otential (3).

The free parameters are xed to repro duce the basic prop erties of the nucleon. At the

mean eld level we use the hedgehog ansatz which is an eigenstate of the so called Grand

~ ~ ~

Spin G = S + I , and is a sup erp osition of various bare nucleon and delta states.

In order to describ e the single nucleon state we p erformed a double pro jection on linear

and angular momentum eigenstates from the hedgehog, whose details can b e found in Ref.

[15].

For the DM version of the mo del we use the sets of parameters of Ref. [16], for whicha

go o d description of the static prop erties of the nucleon was obtained.

For the SM version such a set of parameters was not available in the literature. We

xed the parameters g and M to repro duce the exp erimental value of the average mass of

the nucleon and of the delta, and the isoscalar radius of the nucleon. Cho osing g =0:02

GeV and M =1:7 GeV we got: (E + E )=2=1:112 GeV (exp.val.= 1:085GeV) and

1=2

=0:82 fm (exp.val.= 0:79 fm). These values dep end essentially only on the

N isoscalar

quantity G = gM . A detalied presentation of the single{nucleon prop erties in SM version

of the CDM will b e presented elsewhere [17].

To p erform an exhaustive analysis of the various versions of the CDM, we considered also

the p ossibilityofhaving an e ective mass term for the quarks in which the eld app ears

p p

with a p ower di erent from one: m = g = . Studying this p ossibility in the SM version,

and for p = 2, a go o d description of the single nucleon prop erties can b e achieved using

1=2

g =0:02 GeV and M =1:10 GeV ((E + E )=2=1:102GeV and =0:78fm).

isoscalar

Several imp ortant di erences exist b etween the CDM and the MIT bag mo del. In the

latter mo del, inside the bag the quarks have current masses of few MeV. The bag is stabilized

through the intro duction of a big vacuum pressure, of the order of 150MeV =fm .Perturbative

gluons are considered to b e the right degrees of freedom inside the bag. In the CDM mo del,

in all versions, the e ective quark mass is everywhere bigger than a numb er of the order

of 100MeV, hence chiral symmetry is broken and Goldstone b osons are the right degrees of

freedom. We will come back later to this p oint, comparing the EOS of QM as computed in

the chiral CDM with the one computed in a non{chiral version [18]. In the CDM a vacuum

pressure is also present, coming from the eld: this pressure is roughly constant inside the

4 4

nucleon in the DM case, where it equals U ( = M )=M , whilst in the SM case the

2 2

pressure (r ) dep ends on r . It is imp ortant to stress that in the DM version of the M

CDM mo del the pressure is very small, of the order of few MeV, and this p oint will also b e

imp ortant when discussing the EOS of QM. 4

I I I. MEAN FIELD APPROXIMATION TO THE EOS OF QM

In this section we will study the EOS of QM, using the Lagrangian of the chiral CDM

(1{3).

For QM we mean a system of totally decon ned quarks, describ ed using plane waves,

with the , pion and sigma elds having a constantvalue, given by the Euler{Lagrange

equations.

The total energy of QM in the mean eld approximation is the following:

dk g

2 2

E =12V k +( ) (k k)+VU( )+VU(; ~ =0); (4)

QM F

(2 )

where k is the Fermi momentum of quarks, and are the solutions of the coupled

equations

dU ( ) ( ; )

; (5) = g

dU (;~ =0) ( ; )

= g ; (6)

and the scalar density ( ; )isgiven by

dk g =

( ; )=< (k k): (7) >=12

S F

2 2

(2 )

k +(g= )

In the mean eld approximation, for an homogeneous in nite system, <~>=0,sothe

pionic eld is not contributing (of course it indirectly enters the EOS, b ecause our mo del

parameters are xed in the single nucleon problem, where <~>6= 0). This is drawback

of the mean eld approximation, when applied to homogeneous in nite systems. A wayto

circumvent this problem is discussed by Ghosh and Phatak [19].

It is imp ortant to analyze the b ehaviour of the eld, b oth in the SM and in the

DM version of the mo del, when the CDM is used to describ e a collection of nucleons at

increasing densities. First of all, in the DM mo del a critical density exist, for which the

eld undergo a discontinous jump. In the app endix a pro of of this statement is given,

based on the study of a two{nucleon system for various internucleon distances. For distances

smaller than a critical one, the eld, in the region b etween the twonucleon, will cease to

interp olate b etween the two minima, as discussed in Sec.2B, and will stay near the relative

minimum. This transition cannot b e made continuous. In the DM version of the CDM, the

decon nement phase transition is therefore a discontinous rst order transition. In the SM

version of the mo del nothing similar can happ en. Of course this is not enaugh to conclude

that, in this case, the decon nement phase transition is not rst order. We will study more

in detail what happ ens in Sec.5 and 6.

We will now compare the EOS of QM, as computed in our mo del, with the EOS of

nuclear matter as obtained in the Walecka mo del [20]. In Fig.2 our results for the EOS are

shown.

In the DM case we used the parameter sets [15,16]: 5

i): g =0:059 GeV, M =1:4 GeV, =0:04, =0:06

ii): g =0:029 GeV, M =1:2 GeV, =0:06, =0:06

iii): g =0:0235 GeV, M =1:6 GeV, =0:03, =0:06

As it app ears, the energy p er baryon numberinDMisvery small, its value b eing b elow the

one given by the Walecka mo del for almost all densities. The DM version of the CDM is

therefore unrealistic when used to describ e the EOS of QM. A similar result was obtained in

a calculation p erformed in a non{chiral version of CDM [18]. The reasons for sucha deb^acle

are to b e nd in the very small value of the pressure in DM. If one would add `by hand' a

pressure's contribution of the order of 150MeV= to the energy p er baryon shown in Fig.2

for DM, one will get a result similar to the one obtained using the MIT bag mo del. The

problem is that such a big pressure cannot b e obtained in the CDM, b ecause corresp onds

to a solution for the single nucleon problem where the quark elds are very steep: as it has

b een shown by Leech and Birse [21], the center of mass motion cannot b e pro jected out

consistently in this case, and the mean eld approximation is no more a go o d starting p oint.

In the SM with p = 1 case (see Sect.2B), on the other hand, the mean eld approximation

to the EOS of QM gives a sensible result: the energy p er baryon numb er is bigger in the QM

phase than in the hadronic phase for all densities smaller or of the order of =0:17N=f m ,

the equilibrium densityofnuclear matter. After this density the equation of state of QM

and the one of nuclear matter seems almost equivalent, till densities of the order of 2 ,

after which the energy of QM is smaller than the energy of nuclear matter. Wewould like

to remind that wehave not mo di ed the parameters of the mo del, but we are sticking to

the ones xed to the static prop erties of the nucleon, as discussed in Sec.2.

The last p ossibilitywehave considered is SM with p = 2. Also in this case, the EOS of

quark matter that one obtains after xing the parameters is to o low, as it can b e seen from

Fig.2.

We can conclude from analysis of all the versions of the CDM, that the most realistic

EOS of the quark matter is obtained using the mo del in which con nement is imp osed in

the smo othest way, i.e. SM with p = 1. In the following sections, only this version will b e

considered.

The most relevant feature of the result for SM p = 1, is the wide range of densities

for which the EOS of traditional nuclear matter and the EOS of quark matter are almost

equivalent. The di erence in energy b etween the two phases is of some tens' MeV, only.It

is remarkable that this `almost equivalence' starts at a density of the order of . A natural

interpretation of this result is that in the case of heavy nuclei, some precursor phenomena

of decon nement could b e seen (as swelling, for instance), but no dramatic change is going

to happ en in the system till much higher densities. In Sec.6 we will analyze the p ossibility

that the small energy gap b etween the two phases can b e explained taking into account

correlations in QM, using Bethe{Goldstone equation.

IV. QUARKS' MASS AND CHIRAL SYMMETRY RESTAURATION

We discuss now more in details the dep endence of quarks' e ective mass m = g=

on the density. In this mo del two di erent mechanisms are at work to reduce m : one comes

q 6

from the chiral eld sigma, that moves from its vacuum value f as the density increases

(in the following we will not takeinto accountachiral{symmetry breaking term giving mass

to the pion, b ecause it is irrelevant to what wewant to discuss). The other mechanism that

mo di es m is con nement, through the eld, whichmoves away from zero. As we will

see, this second mechanism is the relevant one till high densities.

We restrict our discussion to the SM p =1version of the mo del, the one which gives

sensible results for the EOS of QM. In this case, eq.(5) can b e formal ly solved, giving

1=3

=( < >) : (8)

This is not really the solution of eq.(5), b ecause the scalar density =< > still dep ends

on through the fermions' mass. It can nevertheless b e used as an approximation to the

real solution, if one neglects the di erence b etween =< >and (this approximation

is not to o bad, b ecause, as we shall see, quarks' masses are not decreasing very fast).

The `Maxican hat' p otential, entering the Lagrangian (1), and parametrized as in Ref.

[15], is

2 2 2 2

U (;)= ( + f ) ; (9)

where m is the mass of the sigma eld. Using the formal solution for given in eq.(8), one

can compute the shift of the sigma eld from its vacuum value

= f + (10)

" #

1=3

1 (gM )

2=3 2=3

( ) C ( ) : (11) =

m f

Substituting in eq.(11) typical numb ers, and in particular the ones we used in Sec.3, the

dimensional co ecient C in eq.(11) turns out to b e small, of the order of 0.1 fm, and is

therefore also small, if compared to f , till very high densities.

In Fig.3 we compare the e ective quark mass, as computed taking into account b oth

the reductions coming from the eld and from the eld (solid line), with the e ective

mass when the eld is kept xed at f (dashed line). At a density of order of 3 , the

e ective mass of the quarks is reduced from its value at by a factor ' 0:8. Comparing

the two lines of Fig.3, one can see that most of the e ect is due to the con ning eld.

V. QUARK MATTER INSTABILITIES

The more direct way to study the instabilities of a many{b o dy system is to lo ok where its

compressibility K b ecomes in nite. The compressibility is related to the pressure through

the following relations

@ (E=N )

P = (12)

@ 7

K = : (13)

Here (E=N ) is the energy p er particle and is the density.

In Fig.4 the compression mo dulus K is shown (here and in the following we will refer

to the SM version of the CDM, only). As it can b e seen, quark matter b ecomes unstable,

in our calculation, at a density smaller than ' =2. This instability is not due to the

inst eq

fact that in this region the energy p er particle of quark matter is higher than that of nuclear

matter, but it is related to the p ossibility of creating undamp ed density uctuations in the

system, without sp ending energy.

To clarify even more this p oint, one can try to repro duce the same result, studying

the collective excitations of the system, which can b e found searching for the p oles of the

propagators of the scalar elds, dressed by the particle{hole p olarization propagator. In the

following we will only discuss the collective states coming from the propagation of the and

the sigma eld, leaving the discussion of the pion propagator, and the related phenomena,

to a future study.

To de ne the propagators of the and the sigma, we expand these elds around their

mean eld value:

=+~ (14)

= +~ : (15)

The mass term reads thus (<~>=0)

( +~) ~

g ~ + ::: (16) = g =g + g g

( +~ )

where wehave expanded the eld till rst order in the uctuation. Since we are lo oking for

the instabilities of the system, i.e. for the situations in which an (arbitrary small) uctuation

around the mean eld develops sp ontaneously and propagates undamp ed, it is enough to

consider a rst order expansion in ~.

From the linearized lagrangian (16), we read the couplings b etween the scalar elds ~

and ~ and the quark elds

g = g (17)

g = : (18)

The masses corresp onding to the ~ and to the ~ uctuations are given by the following

relations

" #

2 2 2

U ( ) g >= M 2g =3M (19) M <

2 3

@ ~ 8

@ U (;~)

m < >: (20)

In the previous equations, the brakets <> indicate mean value in the mean eld approxima-

tion, where the uctuations are set equal to zero. In eq.(19), wehave used the eld equation

for (5), to eliminate the dep endence on the elds. The mass of the ~ is only slightly reduced

from its value at zero{density, b ecause of the slowness of the chiral symmetry restauration

(see the discussion in the previous section).

The mean eld propagators of the scalar elds are thus

2 2

D (q )=1=(q m +i ) (21)

0 ~

2 2

D (q )=1=(q M +i ): (22)

We de ne the scalar p olarization propagator in the following way

d k

(q )=i Tr[G(k)G(k + q)]; (23)

(2 )

where G(k ) is the single quark relativistic propagator in the medium which can b e decom-

p osed in the Feynman propagator and a density dep endent correction

" #

i 1

G(k )=( k +m ) + (k E (k )) (k jkj) G (k)+G (k):

q 0 F F D

2 2

k m +i E (k )

(24)

The e ective quark mass is computed from mean eld equations: m = g = , and E (k )=

2 2

m +k .For a review of the formalism, see Re . [20,22].

In accordance with the mean eld approximation, we neglect the vacuum uctuations

e ects, and we take only the density{dep endent part of the p olarization propagator [22]. In

sucha wayweavoid p ossible complications coming from the ill-de ned zero densityvacuum

state. An explicit expression for can b e found in Ref. [22].

Wenow consider the propagators of the scalar elds, mo di ed by the p olarization propa-

gator insertion, i.e. the propagators in the RPA approximation. The ~ and the ~ propagators

mix up in the RPA approximation. They are the solution of the following coupled equations

g (g D + g D ) + D D = D

~ s ~ ~

0 0

D = D + D g (g D + g D )

~ s ~ ~

0 0

D = D g ( g D + g D )

~ s ~ ~

g ( g D + g D ): (25) D = D

~ s ~ ~

In the previous equation, the dressed propagators corresp ond to various situations in which

mix among themselves via the p olarization the two mean eld propagators D and D

propagator. These propagators can b e decoupled, giving the following RPA propagators 9

D = D (1 g g D )=

~ s ~

D = D (1 g g D )=

~ s ~

D = D g g D =

~ s ~

D = D g g D =

~ s ~ 0

=1(g g D +g g D ): (26)

~ s ~ ~ s ~

The collective excitations of the system corresp ond to the zeros of the denominator (q ).

Instabilities are the collective states at zero energy transferred [22], i.e. the solutions of the

equation

(q =0;~q )=0 (27)

The solutions of this equation are shown in Fig.4, as a function of the transferred momentum

and of the density of the system. In the region enclosed by the line, corresp onding to

the solutions of eq. (27), the system is instable; in particular, when these instabilities

exist for zero transferred momentum the system develops sp ontaneously undamp ed density

uctuations. The range of densities for which the system is found to b e instable (at zero

momentum transferred) coincides with the range found studying the compressibility.

Concerning the interplaybetween chiral symmetry and con nement in the development

of the instabilities, we found again that chiral symmetry plays a marginal r^ole in the mo del:

if one takes into account the ~ propagator only, the instability region shown in Fig.4 is

almost unchanged.

VI. TWO{BODY CORRELATIONS: GAP EQUATION

We will now study two{b o dy correlations, using the formalism of the gap equation [23].

The gap is the di erence b etween the energy of the incorrelated pair and the energy of the

correlated pair. It is thus p ositive if the p otential is attractive. We will use as the residual

interaction the one arising from the exchange of a ~ or a ~ . Of course we are able to take

into account only two{b o dy correlations, and three b o dy{correlations can b e even more

imp ortant.

The gap equation reads [23]

0 0

= (28) < k; kjV jk ; k >

2 2

1=2

) + 2 (

0 0

k k

Here we are following the convenction [23] that an attractive p otential is p ositive. The

p otential V is evaluated b etween incorrelated states, describ ed by plane waves. is the

single{particle energy, mesured relative to the chemical p otential = k + m .We will

F q

not use the relativistic reduction of , b ecause the relativistic corrections are not totally

2 2 2

negligible. Therefore = k + m k + m , where m is the quark mass as computed

k q

F q q

in the mean eld approximation.

As wehave already said, the p otential V arises from the exchange of the scalar elds'

uctuations ~ and ~ . The masses of this uctuations and the couplings b etween the uc-

tuations and the quarks' elds, have b een obtained in the previous section, expanding the 10

lagrangian around the mean eld approximation. Of course this expansion, whichwas su-

cient to study the instabilities of the system, is in general quite a crude approximation. The

results obtained for the gap are thus only the rst step in the study of the dicult problem

of the clusterization.

2 0

The p otential V is a Yukawa p otential, but an extra factor m =E (k )E (k ) has to b e

2 2

included (E (k )= k +m ), b ecause of the cho osen normalization of the quarks' spinors.

Following Ref. [23], since the resulting gap is much smaller than the Fermi energy,

the integrand app earing in eq.(28) is sharply p eaked near = 0, and then ' ,

k k

i.e. the gap is almost indep endent on the momentum.

We show in Fig.4 the resulting gap. It shows a strong dep endence on the density, and

for densities of the order of is already totally negligible.

The gap equation can b e solved analytically for small value of the gap. The result is

2 2

k k =2m

F F

8 exp( ); (29)

q F k

where the matrix element of the p otential is

k sink xV (x)sink xdx: (30)

F k F F F

Since our residual interaction V is always attractive, it do esn't exist a critical densityat

which the gap is exactly zero, and this value is reached only asymptotically.

VI I. CONCLUSIONS

In this pap er wehave studied quark matter, using the non{p erturbative to ol of the CDM

mo del. Let us summarize our main results:

{ of all the considered versions of the mo del, only one gives sensible results, i.e. the one

in which con nement is imp osed in the smo othest way. The other versions of the mo del give

an exceedingly low energy p er baryon numb er for the quark matter.

{ the SM (p=1) version gives an EOS for the quark matter which is almost identical to

the EOS of nuclear matter as computed using the Walecka mo del, for the range of densities

2 . In this range, the di erence in the energy p er baryon number between

eq eq

the nuclear matter and the quark matter is very small. Taking into account the theoretical

incertitude in the xing of the parameters (Sec.2B), this energy di erence is of the order

of 20 MeV. At densities smaller than the energy di erence rapidly increases, and for

densities higher than 2 the quark matter is the energetically most favourable state. An

imp ortant p oint is that the minimum for the EOS of quark matter is at a density of the

order of , and this result do es not dep end on the ne tuning of the parameters.

{ the mass of the quarks remains big (of the order of 100 MeV) till high densities, much

higher than the density at which quark matter b ecomes the ground state. The decon nement

phase transition and the chiral symmetry restauration arise at totally di erent densities.

{ the quark matter (in SM and p=1) b ecomes unstable at low densities, of the order of

=2. The instability can b e obtained b oth from the study of the compressibility (where the

compressibility b ecomes negative the system is unstable) and from the study of the collective

states at zero energy transfer. The two metho d give the same critical density. 11

{ the pro cess of clusterization can b e studied considering the correlations b etween the

particles, b eyond the mean eld approximation. The gap we obtained seems rather small.

Wehave to b ear in mind that wehaveoversimpli ed the problem, by linearizing the resid-

ual interaction (and thus getting an approximate propagator for ~ , go o d only for small

uctuations), and by considering only two b o dy correlations, where the three b o dy ones are

probably the most relevant.

To conclude, wewould like to consider three p ossible applications of the mo del.

{ Co oling of neutron stars (see C.J.Pethick in [3]).

A mechanism called URCA has b een invoked to explain the rapid co oling of neutron stars.

This mechanism pro ceeds via the exchange of electrons b etween neutrons and protons, which

co ol down emitting neutrinos and antineutrinos:

n ! p + e + (31)

p + e ! n + (32)

A minimal fraction of protons is required, in order to ful l momentum and energy conser-

vation. This critical fraction is of the order of 1/9. Using traditional nuclear physics mo dels

to compute the protons' fraction, one gets numb ers slightly smaller than the critical one,

and the URCA mechanism cannot start.

Another p ossibilityistoinvoke the presence of quark matter in the core of the star,

and to consider reactions similar to the one previously describ ed, but with the electron now

exchanged b etween up and down quarks. In this case the problem is that, considering quark

matter as describ ed by the MIT bag mo del, quarks are massless and the phase space is

thus zero. Therefore one would need a massive quark matter phase, and the p ossibilityof

reaching this phase at the density of the core of neutron stars, tipically of the order of 5 .

This situation is actually the one describ ed by the SM (p=1) version of the CDM. URCA

mechanism should therefore b e p ossible, and with an high luminosity,too.

{ Energy released in sup ernova explosion.

Using a traditional nuclear physics approach, the energy released in sup ernova explosion

is generally to o small. A softer EOS could solve the problem, but if one uses e.g. the

MIT bag to study matter at high density, the decon nement phase transition is reached

at densities larger than the one presumably reached in the collapse of the star. The EOS

for matter at high density as computed in the CDM, is softer than the EOS of nuclear

matter, and presumably a similar result will b e obtained when computing neutron matter.

Furthermore the softening starts at densities of the order of 2 .

{ EMC e ect and swelling of the nucleon.

To conclude let us consider the problem of the p ossible swelling of the nucleons emb edded

in a nucleus. If one considers,e.g. electron{scattering on heavy nuclei, one realizes that the

swelling is a sensible mechanism, but it must b e of the order of some 5% in order to b e

realistic. The real problem is thus not the one to obtain a swelling, but to obtain a not to o

big e ect. In other words, the nucleons have not to dissolve when emb edded in a nucleus.

Since the minimum of the EOS of the quark matter is for a density near , only in the

center of heavy nuclei some swelling mechanism can app ear. The exact amountofswelling

dep ends on the precise di erence in energy b etween the quark matter and the nuclear matter

at densities , and is b eyond the p ossibility of the present calculation.

We are now carrying out researches in all the directions previuosly outlined. 12

APPENDIX

In this app endix we discuss the b ehaviour of the eld in the DM version of the CDM,

foratwonucleon system, as a function of their distance d, which is related to the density

of the system / d .We will show that a critical internucleon distance exists, at which

the eld has a discontinuous b ehaviour.

For simplicitywe will omit the chiral elds, the pion and the sigma, from our discussion,

considering the following Lagrangian:

g 1

L = i @ f + (@ ) U ( ) ; (A1)

where the p otential for is the one given in eq.(3) and shown in Fig.5a. Wehave lab elled by

4 4

the relative minimum and by the other value of for which U ( )=U( )=M .

m 1 1 m

We consider a system made of two clusters of three quarks each, with an intercluster

distance d. In the mean eld approximation the total energy of the two{nucleon system is

given by

E =6 +T +E (A2)

2 q

where is the energy of the single quark, T is the energy of the eld asso ciated with its

spatial uctuations and E is the energy coming from the p otential U ( ).

When the intercluster distance d is large, the eld interp olates b etweenanumber

slightly bigger than and a small numb er, smaller than , in the internucleon space (see

m 1

Fig.5b, the solid and dotted line).

When the distance d is reduced, the value of the scalar eld in the internucleon region

increases. We compare nowtwo p ossible solutions: one in which the eld has a minimum

value in the internucleon region, equal to , and a second in which the eld remains

almost constant in the internucleon region (solid and dashed line in Fig.5b). The second

solution has a smaller energy than the rst one. In fact: is smaller, b ecause the quarks

move in a single big well, instead of moving in a double well; T is smaller b ecause is

uctuating less; E is smaller b ecause is not moving (in the internucleon region) through

the relative maximum of U ( ). Since the second solution has a smaller energy, the value

of in the internucleon region will not smo othly increse, as the distance d is decreased,

but will jump, at a certain critical distance, from the b ehaviour describ ed by the solid and

dotted line in Fig.5b to the one describ ed by the solid and dashed line. The internucleon

value of can b e assumed as an order parameter that undergo a discontinuous change as

the density is increased, thus the transition is rst order.

This energy is not the kinetic energy of the eld, which is zero, b ecause is a scalar eld and

it is assumed to b e time indep endent. 13

REFERENCES

[1] Quark Matter '93, Nucl.Phys.A566(1994).

[2] Neutron Stars: Theory and Observation,J.Ventura and D.Pines eds. (Kluwer Acad.

Pub., 1991).

[3] Realistic Nuclear Structure, G.Brown, P.Ellis, E.Osnes and D.Strottman eds.

Phys.Rep.242(1994).

[4] J.W.Clark, J.Cleymans and J.Rafelski, Phys.Rev.C33(1986)703.

[5] R.Tamagaki and T.Tatsumi, Progr.Theor.Phys.Supp.112(1993)277.

[6] J.-P.Blaizot, Quark Matter '93, Nucl.Phys.A566(1994)333c.

[7] H.J.Pirner, Prog.Part.Nucl.Phys.29(1992)33.

[8] K.Brauer, A.Drago and A.Faessler, Nucl.Phys.A511(1990)558.

[9] E.Naar and M.C.Birse, J.Phys.G:Nucl.Part.19(1993)555.

[10] M.C.Birse, Progr.Part.Nucl.Phys.25(1990)1

[11] W.Broniowski, M.Cib ej, M.Kutschera and M.Rosina, Phys.Rev.D41(1990)285.

[12] V.Barone, A.Drago and M.Fiolhais, Phys.Lett.B338(1994)433.

[13] W.Broniowski, M.K.Banerjee and T.D.Cohen, Univ. of Maryland preprints ORO 5126-

298 (1986).

[14] A.Drago, K.Brauer and A.Faessler, J.Phys.G15(1989)L7.

[15] T.Neub er, M.Fiolhais, K.Go eke and J.N.Urbano, Nucl.Phys.A560(1993)909.

[16] M.Fiolhais, T.Neub er, K.Go eke, P.Alb erto and J.N.Urbano, Phys.Lett.B268(1991)1.

[17] V.Barone, A.Drago, M.Fiolhais and U.Tambini, work in progress.

[18] V.Barone and A.Drago, submitted for pubblication.

[19] S.K.Ghosh and S.C.Phatak, J.Phys.G18(1992)755.

[20] B.D.Serot and J.D.Walecka, Adv.Nucl.Phys.16(1986)1.

[21] R.C.Leech and M.C.Birse, Nucl.Phys.A494(1989)489.

[22] K.Lim and C.J.Horowitz, Nucl.Phys.A501(1989)729.

[23] A.L.Fetter and J.D.Walecka, Quantum Theory of Many{Particle Systems McGraw{Hill

1971. 14

FIGURES

FIG. 1. Typical solutions of the mo del for single nucleon problem in the SM (a,b) and DM

(c,d) versions. 15

FIG. 2. EOS of QM and EOS of nuclear matter in the Walecka mo del (dotted line).We show

the EOS of QM in CDM for the DM version of the mo del (dashed lines i) ii) iii) ) and for the SM

version with p=1 (solid line) and p=2 (dot-dashed line) . 16

FIG. 3. E ective quark mass versus density: .... . 17

FIG. 4. Compressibility (dashed line), instability region (solid line), and gap (dotted line) for

the QM...... 18

FIG. 5. Behaviour of eld in the DM version of the CDM. Potential U ( ) (a) and shap e of

eld for internucleonic distance d . 19