Z. Phys. C - Particles and Fields 44, 129-138 (1989) Zeitschrift P a r t i d e s ~r Physik C

and �9 Springer-u 1989

Quark core stars, quark stars and strange stars F. Grassi* NASA/Fernmilab Astrophysics Center, Fermi National Accelerator Laboratory, Box 500, Batavia, IL 60510-0500, USA

Received 22 February 1989

Abstract. A recent one flavor (zero temperature) quark matter equation of state is generalized to sever- al flavors. It is shown that quarks undergo a first order phase transition. In addition, this equation of state depends on few parameters, one in the two flavor case, two in the three flavor case, and these parame- ters can be constrained by phenomenology. This equation of state is then applied to 1) the hadron- quark transition in neutron stars and the determina- tion of quark star stability, 2) the investigation of strange matter stability and possible strange star ex- istence.

1 Introduction

The possibility that quark matter might exist in stars has never stopped to generate a lot of interest. It was first suggested by Ivanenko and Kurdgelaidze in 1969 [1] that a quark core might exist in some types of stars. The following year, I toh 1-2] computed the max- imum mass of quark stars, neglecting totally interac- tions between quarks. This mass, as in the perfect neutron gas of Oppenheimer and Volkoff of 1939 [3], was too small, ~ 10-3 Mo"

A few years later in 1976, after the MIT bag model was invented, Brecher and Caporaso 1-4] showed that, by making use of it, higher maximum masses could be obtained for quark core stars. However, when first order corrections in the strong coupling constant were included in the quark MIT equation of state, as done by Baym and Chin [5] and Chapline and Nauenberg [6], the density at which the quark phase should appear was much higher than the maximum

* On leave from D.A.R.C., Observations de Paris, F-92195 Meudon, France

central density reachable by stable neutron stars. (However [7], both these papers made use of a high value of the coupling constant, obtained when fitting hadronic spectra and, as the coupling constant was held fixed, the equation of state contained large loga- rithms involving the density.)

In parallel to this phenomenological approach, in 1975, Collins and Perry [8] demonstrated that asymp- totic freedom which holds for large momentum transfers, also holds at high density, e.g. perhaps, in- side neutron stars. Then Keister and Kisslinger [9], using a perfect gas equation of state, and Chapline and Nauenberg [10], starting from an expression of the energy density computed to first oder in the-den- sity dependant-coupling constant, concluded again that the quark-hadron phase transition would occur at too large a density, so that the possibility of having a pure quark phase inside neutron stars could be ruled out. However, these calculations were a little crude for the first one and not completely self-consis- tent for the second one (see [11]). In 1978, it was shown independently by Kislinger and Morley [7] and Freedman and McLerran [11] - and checked in a different approach by Baluni [12] - that when an expansion of the quark matter energy in the den- sity dependant coupling constant was done, the quark-hadron phase transition was predicted to occur at densities lower than the maximum central density for a stable neutron star and quark core stars could exist.

In addition, it was also noticed by Freedman and McLerran, that this result still hold if one added a vacuum constant to the equation of state - thus defin- ing an improved MIT bag, without divergent loga- rithms when a proper choice of the subtraction point was made. This was checked by Fechner and Joss [13], with and without vacuum constant, in the case of a first or a second order phase transition between hadrons and quarks, and for several nuclear-matter equations of state. They also showed that besides

130

quark core stars, stable pure quark stars could exist. These stars would form a third family of stable dense objects beside white dwarfs and neutron stars with or without quark core with higher central density [14].

In addition to these studies, a number of other approaches have been tried. The equation of state of bulk quark matter was computed either starting directly with the pure QCD Lagrangian and some approximation [15-17], or after modifying this La- grangian to include confinement, for instance by ad- ding a coupling to a scalar field as in the SLAC bag model for hadrons (see [18, 19], and to a lesser extent [14]).

Recently, a radically different picture of dense ob- jects has been suggested: neutron stars might actually not be neutron stars. In 1984, Witten [20] pointed out that the true ground state of hadrons might be strange matter and not 56Fe (strange matter is bulk quark matter containing roughly equal numbers of u, d and s quarks plus a small admixture of electrons to insure charge neutrality). It may be conjectured that this matter has an energy per baryon lower that that of ordinary nuclei, and would thus be absolutely stable. If this is true, then what we usually think of as being neutron stars might actually be strange stars, i.e., stars made of strange matter. Contrarily to quark core stars and quark stars whose mass-radius or mo- ment of inertia-mas relationships are quite similar to that of neutron stars, Haensel et al. [-21] and Alcock et al. [-22], showed that strange stars have mass-radius and moment of inertia-mass curves quite distinct from that of neutron stars - though in the range of masses typically observed, namely 1.4 M o, they are fairly similar. Moreover, strange stars may not have a crust and this could give rise to interesting new phenomena [22]. However one should keep in mind that regard- less of whether or not strange stars have a crust, it is not clear how they could produce glitches; if no model of strange stars producing glitches could be found, this would seriously weaken the stable strange matter hypothesis.

In a previous paper, we gave a method to derive the equation of state of a one flavor gas of quarks, by starting from a Dirac Lagrangian plus a potential term, to take quark interactions into account. All the thermodynamical functions could be expressed as powers of the quark mass, which was kept as an un- known parameter. Qualitatively, it was shown that the quark plasma would undergo a first order transi- tion as the density rises, from a state of massive parti- cles, to a state of particles of decreasing mass. It was also shown that immediately after the transition, the plasma would be in a collective bound state (the ener- gy per particle would be smaller than their mass).

This might perhaps be interpreted as the fact that the quarks just start to leave the inside of the hadrons. The aim of this paper is two-folded. First we will get quantitative results. To do that, we will generalize our method for deriving the equation of state, to sev- eral flavors (Sect. 2), and we will show how the param- eters, one in the two flavor case (Sect. 3), two in the three flavor case (Sect. 4), can be constrained using phenomenology. Second we will apply this equation of state to some physical cases: quark core stars and quark stars in Sect. 5, strange matter stability and strange stars in Sect. 6. Our results are summarized and discussed in Sect. 7.

2 Equation of state

The equation of state that we will use to describe quark matter is a generalization to several flavors, of a one flavor quark matter equation of state derived elsewhere [23, 24J. For pedagogical purposes, we first recall the results for the one flavor case, since those for the several flavor case can be derived in a similar fashion but are a little less straightforward to under- stand.

Because it is not known how confinement arises in QCD, we suppose that interactions between quarks can be reproduced by a phenomenological instanta- neous potential*. For reasons already mentioned in [23], this interquark potential may be assumed to have the following structure

V(r) = Vv(r ) ~(1)~)t~(2)__ Vs(r ) 1(~) I(D 2) (2.1)

where

Vs(r)=Vc(r ) 1~ 1) 1~ 2)

with Vc(r), a confining potential, linear for instance, dominating at long interquark distances, and where

Vv(r)=VG(r)~l)~ a(2),

with VG(r), the one gluon-exchange potential, dom- inating at short interquark distances. (1D designates the unity matrix in Dirac space and lc, the unity ma- trix in color space. The superscript (1) refers to one

* Instantaneous potentials might be used at zero temperature and nuclear matter density, because three flavors of massless, non-inter- acting quarks, have a chemical potential of 162 MeV, this is to be compared with constituent quark masses of several hundreds of MeV

of the quarks interacting via V(r) and (2) to the other.) The quark (Dirac) equation of motion reads

(ir 1~- m 1D lc) G(X, y) = 64(x-- y) 1 o 1~

+ i~d3zG(x, z; y, z+)V(lx- z l)[=o :~o (2.2)

where the notation z + means that the time z ~ is infinitesimally greater than z ~ In order to be able to solve (2.2), one can make the Hartree approxima- tion

G(x, y; z, t)= G(x, z) G(y, t). (2.3)

Thus, after a Fourier transform, (2.2) becomes an equation involving the one-particle Green function only

[p l~--m 1D lc--Un] G(p)= lo 1~ (2.4)

where the Hartree field UH is given by the matrix

U . = ~ lo 1~ (2.5)

with

U n'~ -~d3zVc([x-zl)'Wr[-iG(z, z +) 1o 1~]. (2.6)

In expressions (2.5 6), the one-gluon exchange poten- tial has disappeared as is the case for any color-depen- dent potential in the Hartree approximation (see [23] and references therein).

To see more clearly what the effect of our approxi- mation (2.3) is, (2.4) may be rewritten as

[70 pO_7p_mn] G(p)= 1 (2.7)

where by definition, ran-m+ Us n. From this equa- tion, one sees that the interactions of a given quark with all the others give simply rise to a change in its mass (in this approximation).

The one-particle Green function G can then be computed by standard methods [25 27] and used to calculate the Hartree field (2.5). One obtains the fol- lowing self-consistent equation for the change of mass U H, as a function of #

U~ = --/~(#) �9 ~(Ug, #) (2.8)

where

2kt (2.9a) F_kt.=-- d3r .r ( ) [ [V,~( )]=r~(U~=_2#,#)

pF d3p mH ~(Ufl,/~)=6 ! (2~) 3

-- 2 rc 2 mn OF ~/P~ + m~ -- m~ log (PF + k m H / J

(2.9b) and

Pr (Us H , #) = if # __> mH (2.9 c) otherwise

131

(in (2.9a), the equality is obtained by imposing the constraint [U~[ <2# , otherwise pair creation occurs, see [23, 24]. This renders Us ~ independent of the shape of the interquark potential).

In the case of seveal flavor quark matter, one should replace (2.2) by

(ir 1~-mq 1 n i~) Gq(x, y)= 64(x-y) lu 1~ +iIdaz • Gq, q ' (x ,z;y ,z+)V(Ix-zl) l=o=xo . (2.10)

q'=u,d,s . . .

Then proceeding as for the one flavor case, it is easy to show that

Us n = -- r(,L)" Y, ~q(Us n, Uq) (2.11) q=u,d,s, . . .

where /~(~0 - Sd 3r [ V.(r)]

2#L - ~ ~q(U~ = - - 2 ~ L, #q) (2.12 a)

q=u,d,s, . . .

"c F d3p m~ ~ q ( U ~ l , #q ) =-- 6

o J (2n) 3 ~

- - q q q2 _~_ 2rc2 mH pr[/p m~ 2

--mq21~ (2.12b)

and

Pqf(u~t, t~q)={~O #2--m~l 2 if otherwise'l~q>mq (2.12c)

/

In the above expressions, #L designates the smallest among the Fermi energies of the various quark fla- vors; pairs of this flavor are created preferably to screen the interquark potential.

If can be seen that the Hartree field Un is the same whatever the flavor, and the effective masses m q =mq + Us ~, which play a somewhat similar part to running masses, will all have the same decrease.

Once (2.11) is solved, similarly to the one flavor case, the quark contribution to the equation of state will be obtained from

P~ d3p ,- pZ+m~mq]

3 ( 3 " 2 2 - - \ _ _ J , . , q ] / n q + m q /2 - ~ 2 _ 2 3 t 'F V PF H I

q=u,d,s .... "ll, [ .

/ m~2 ~ p~2 +k--~+m~rnq)[PvV/]2-+rnq2/2

m~/2 l~ ~ / ] ~ q 2 m q 2 (2.14a)

2 \ mu / 3 3

132

PQ= q = u , d , s , . . .

3 ( q 3 = L pl/ +mf/6

q = u , d , s . . . . 7~

+{m~2-mq m ~[-~ ~/P~2 +m~2

2 " q] t" 2

m~2 lo - [P} + ~ ~ l ) 2 ~k m~n }JJ"

pv d3-v v z q2 --p2/3--m~tmqql

(2.14b)

P~ dap n o = ~ 6 o~ (2~) 3

q = u , d , s , . . .

= ~, nq= Z p~3 rc 2 . (2.14c) q = u , d , s . . . . q = u , d , s . . . .

If electrons mus t be included to insure charge neutral- ity, their contr ibut ion to the equat ion of state will be taken to be that of a perfect Fermi gas

E e -- 47C2 (2.15a)

p~ =-12~2 (2.15b)

H e - - 3rC2. (2.15 c)

Let us now see how to apply this model in concrete cases.

else

#d = #- (3.2 b')

One sees that there is just one independent chemical potential #, and that #d is greater or equal to #, so ut7 pairs will be created rather than d d pairs, i.e. in (2.11) and (2.12a), L=u.

Equat ion (2.11) for the change of mass U~ ma y be solved numerical ly as a funct ion of the only inde- pendent chemical potential #. In Fig. 1, the effective mass mu (i.e. the initial mass plus its change), is shown as a funct ion of the Gibbs energy per ba ryon G =- ~ #a nq/nQ/3 = pu + 2 pd. Its general behaviour

q - u , d , s , . . .

is quite similar to that of the one flavor case, so one expects that the quark plasma will undergo a first order phase transition. This transit ion (represented by the solid line) corresponds to a passage from a state of massive particles to a state of particles of decreasing mass - thus mimicking the onset of asymp- totic freedom. In order to check that such a t ransi t ion actually takes place and compute the Gibbs energy per particle at which it occurs, the pressure as a func- t ion of G is plot ted in Fig. 2 a. The the rmodynamica l ly preferred state (i.e. G min imum for a given p) is p = 0 f rom* G = 0 to G = 1 . 5 7 5 mu, and then p starts to in- crease abruptly. So there is indeed a first order phase transi t ion and it happens when G = 1.575 mu. In order to know at which density this corresponds, the ba ryon density has been depicted as a funct ion of G in Fig. 2b. One sees that when G = 1.575 mu, the density

3 Conversion of neutron matter to two flavor quark matter

In the case of neu t ron mat te r undergoing a phase transit ion to qua rk matter, weak interactions do not have time to settle, so there are only two flavors of quarks. Assuming charge neutral i ty (we have in mind the s tudy of the interior of neu t ron stars), one gets

e(~nu--lnd)=O. (3.1)

Hence, denot ing

# , = # (3.2a)

and using expressions (2.14c) for densities and (2.12c) for Fermi momenta , we get the following relation be- tween #d and #

if # _>_ m, = md #d = [22/3# 2 q- (1 -- 22/3) m 2"] 1/2 (3.2b)

I , , , , I , , , , I , , , , I | . . . . . . . . . . . . . . . 2�9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Muslve/" qu~ka

[ I I I I I l l 0 .5 1

G/m~

�9 "1"" 1

/

qu~ks of """-. d~re~ivg m ~ a ~

I [ I I I I I 1.5

Fig. 1. Plot of the effective mass as a function of the Gibbs energy per particle in the case of two flavors of quarks with similar mass (dotted line). A phase transition from the massive quark state to the phase of quarks of decreasing mass takes place (solid line)

* The fact that the pressure is zero in the interval [-0, 1.575] is due to the fact that we assume T=0

is n~=0.0085 m*. We do not show the energy density .003 per baryon versus the baryon density because it is very similar to the one flavor case: immediately after the transition, the quarks are in a collective bound .oo2 state, (eQ/nn< 3m,), which we may perhaps interpret as a manifestation of the fact that they just start to o01 "go out" of the baryons.

In what precedes, all the quantities have been computed in unit of the quark mass, m,, which is the only parameter of the two flavor model. It is in fact possible to find a lower bound for this parameter: n, should be greater or equal to the nuclear matter -oo~ density, so we must have

3 /'It X D I u 2> F/ . . . . . . t t . - 1 . 2 8 . 1 0 6 M e V 3 (3.3)

hence

m , > 532 M e V . (3.4)

The fact that we obtain 532 MeV as a lower bound for the u-d constituent mass - usually thought to be of the order of 340 MeV is an indication that our model is reasonable but crude. (A better lower bound might perhaps be obtained through the inclusion, for instance, of the Fock terms.)

It is also possible to obtain a higher bound for m,, as follows. In Fig. 3, p(e) is plotted. The behaviour of this curve is very different from the one which one would obtain by doing a perturbative calculation. One expects that the transition density, nt = 0.0085 m, should be smaller than the density, ?/pert, at which quark matter start to be describable with a perturba- tire equation of state. Ideally one would like to known np~rt from experiment. In practice, this quan- tity is not k n w o n but we can get an estimate of it by solving

67~ ~,(/~2) = (33 - 2Ns) In (I~/A) --- 0 . 9 9 9 . . . (3.5)

(This is the expression of the running coupling con- stant in the case of quarks of mass much smaller than ~.)

If we take A to be 200 MeV for instance and N s = 3, the solution of (4.5) is # = 402 MeV. This corresponds to a baryon density (for non-interacting particles) t/pert ~ # 3 / 3 g2 = 2 .19" 106 M e V 3. So , t h e following c o n -

s t r a i n t should be satisfied

nt x m 3 < nv~t = 17.8 106 MeV 3 (3.6)

* As a consequence, the integral (2.14b) for the pressure, is zero for a whole range of chemical potentials (/~q =< m~), therefore of Gibbs energy. In reality, T r 0 and the pressure is approximate ly zero

--1002

a

�9

O1

.008

ff "~ .006

004

002

b

133

1 ' ' ' ' l ' ' ' ' l ' ' ' ' l ' / ' '

M ~ i v e / / / qu~km

i .:.:..i // /

0 .5 I 1.5 G/m.

i L , , , I ' ' ' ' I ' ' ' ' I ' '

Qu~ks of / /

/..././/

/ /

.... /

...:

"L --~

M~slve qu~ks \ ',

~ L , t , I , , ~ , I , , , , 'l , , ,- O .5 1 1.5

C/m~

Fig. 2.a Plot of the pressure as a function of the Gibbs energy per particle. The transit ion is seen to be first order and occurs w h e n G/m, = 1.575. It is associated to the change o f mass (see solid curve in Fig. 1). b Plot of the baryonic density as a function of the Gibbs energy per particle (dotted line). W h e n G/m,= 1.575, the density jumps to n/m~ =0.0085 (solid line).

"~uE 3x10 ~a

~.xl0 I~

SxlO 13

/ ,/

4xlO la /

/ /'

10 '~ .//

/ MIT

l O *~ , , , , , , , , , l , , , , A , , , , l , , J , I j , ,

2~ 14 14 J4 2.5XI014 3X10 ]4 3.5XI0 ~xl0 ~4.Sx10 5X1014 S.SxI014 6XI014 e/e (g orn- )

Fig. 3. P l o t o f t h e p r e s s u r e a s a f u n c t i o n o f t h e e n e r g y d e n s i t y f o r

two quark flavors (solid line). For comparison , the MIT equat ion of state has been plotted as well (dotted line)

134

hence,

m, < 636 MeV. (3.7)

This upper bound is compatible with the lower bound (3.4); m, must belong to the interval [532 MeV, 636 MeV] and accordingly, the transition density n t lies in the interval [n . . . . . . tt., 1.71 n . . . . . . tt.]. Note that while the nuclear matter density is (rather) well known, the calculation of npert is very unaccurate. Had we taken A = 500 MeV instead of 200 MeV, we would have obtained an upper limit for the u mass equal to 1592 MeV and an upper limit for n, equal to 26.78 n . . . . . . t t . " It is in fact possible to find an upper bound for m,, in a much simpler way, by using prop- erties of neutron stars, as we will see in Sect. 6.

4 Three flavor quark matter in chemical equilibrium

We have seen in the previous section that neutron matter undergoes a phase transition to two flavor quark matter. This two flavor quark matter will then, on a weak interaction timescale, transforms to three flavor quark matter and reach chemical equilibrium, via the following reactions

d+-+u+e+Ve (4.1 a)

s*-~u+e+~e (4.1 b)

s +u~--,d+u. (4.1 c)

This implies that

#a = #s = # (4.2 a)

# , + # e = # (4.2b)

and overall charge neutrality requires that

e (2 n , - �89 n d - �89 n s - ne)= 0. (4.3)

Thus again, there is only one independent chemical potential, which we choose as being #. In addition, one sees that #, = # - #e is smaller or equal to #, there- fore u~ pairs will also be created preferably. Note that in the three flavor case, there are two parameters: m = m, and r =-mJm,. Since m, and ms are constituent masses, we expect that r ~ 500/340 = 1.47.

Equation (2.11) for Us H and (4.3) can be solved numerically simultaneously, as a function of #. Once this is done, the thermodynamical quantities (pres- sure, energy density, baryon density, electronic den- sity) can be computed by using (2.14 a-c) and (2.15 a~z). The behaviour of these various functions is rather similar to that of the two flavor quark matter, so we do not show them. (This does not mean that there will be another phase transition, from ordinary quark

matter to strange quark matter : once u and d quarks start to appear with a given density, they will be grad- ually depleted, the pressure needs not vary abruptly.)

5 Quark core stars and quark stars

Now that we have an equation of state, expressed as a function of m, and r, and bounds or orders of magnitude for these parameters, we may try to apply it to stellar objects. First let us consider neutron stars with high central density, thus possibly containing a quark core.

Usually, to study such objects, one starts with two equations of state, one for the neutron phase and one for the quark phase. Then to see if a first order transi- tion is possible, a Maxwell construction may be done to determine the transition pressure; under this pres- sure, the neutron equation of state must be used, above, the quark one. If a second order transition is assumed, the two equations of state are matched at some transition pressure. In our case, a first order transition is already embodied in the equation of state that we computed, so in principle we do not need a second equation of state. However, because we ne- glected temperature, the whole neutron (massive quark) phase, is at zero pressure. We may circumvent this by replacing the p = 0 branch of our equation of state by the Baym-Bethe-Pethick equation of state [28]. In Fig. 4, this has been matched at the transition density computed in Sect. 3, to our quark equation of state. For comparison, we also represented the usual matching of the Bethe-Johnson equation of state [29] to the Baym-Bethe-Pethick equation of state. (We chose m , = 532 MeV, i.e. a quark-hadron phase transition occurring at a low density

n . . . . . . t t . " In the case where m, would have a higher value, the matching would simply occur at a higher density.)

In addition, in Fig. 4, at high density, our quark equation of state is matched smoothly with the equa- tion of state of massless u-d quarks and massive s quarks, computed (for simplicity) to order zero in the strong coupling constant. We have taken the s mass to be 250 MeV because if mE ~ 0 then U~ ~ - m , and m~-(r+U~) m , = 2 5 0 MeV (for our choice of m, = 532 MeV and r = 1.47). Our equation of state pro- vides a bridge between the equation of state for quarks confined inside hadrons at low densities, and the equation of state of asymptotically free quarks at high densities.

In order to calculate the effects of the appearance of a quark phase in the deep interior of neutron stars, the matched equation of state has to be used in the solution of the so-called Oppenheimer-Volkoff sys-

1 3 5

-#

1.4x1014 ~ i i ~ I i 1 i ~ ] , , , , , , , ,

1.2x1014

1014 a,=O.

m.=~50. : ! 8 x l O j3 ./

x l O la : __

4xI0'3 O

2x lOll

0 I014 2xi014 3xlO 14 4x1014 5xlO" e/m~ 4

Fig. 4. Plot of the pressure versus energy density for u, d, s quarks and electrons in chemical equilibrium, with m . = 5 3 2 MeV and r = 1.47 (solid curve labeled Q3). It is matched (dotted line) to a perturbative equation of state at high density and to the Feynman- Metropolis-Teller plus Baym-Pethick-Sutherland plus Baym-Bethe- Pethick equation of state at low density. For comparison, the usual matching with the Bethe-Johnson equation of state is also shown (dashed-dotted curve)

3 . 5

a

3 . 5

2 .5

2

tern. These are the differential equations of static equi- = ~.~ librium in general relativity and read (see for example [303)

JE (r) + 4 ~r3p(r) (5.1a) p' (r) = - - G (~ (r) + p (r)) r (r-- 2 GJf(r))

(5.1b) o b

(5.1c) 10

J f ' (r)==_ 4 rcr2e(r)

2p' V t - -

e + p

These equations have to be solved with the boundary conditions

p(r=O)=p(~3 JZ(r=0)=0 v (R) = 1 -- 2 GJ/I (R)/R.

The integration of this system is stopped when a radi- us R is reached where the pressure becomes lower or equal to zero. A unique solution exists for each value of the central density ~c. In Fig. 5 a, the mass- central density relation is presented. One sees that it has two extrema: as usual the first corresponds to neutron stars and the second to quark stars. As can be seen in Fig. 5 b, the curve for the mass-radius relation curls counterclockwise after the neutron star peak, so these quark stars have unstable oscillation modes [31]. However, the so-called neutron stars of the stable peak, consist of a very massive quark core, surrounded by a thin shell of more conventional neu- tron matter.

6

~ 4 ! 0

are~oren r

/ S t r a ~ k s ~ Q u a r k ~ a r

~ a k r - - - -

Strange s~r : p e a k

-- �9 Ouark core s ~ r " . . - . . . - " . . . . . .

Quark s ~ r peak

1 - -

. . . . . . . . m.=532., r=L47

< . . . . . . m~=708., r=L47

. . . . . . . . m~=595., r~l.30

z~ ., , , , I , , , , I , , , , I , , , ~

14 15 16 1 pc'g[ c m 3~7 18 l o g

19

___, ' ' ' .... I ' ' ' ' I.........,~i..i~7.46. ' ' ' I ' ' ' ' I ' ' ' ' I ' ' ' ' I ' ' ' 1

- 10'*

~ _ " < . . . . . . mm=532., r=1.47

- - < . . . . . . . . . . . . . . . m , ~ ? o s . , r=l.4?

5 10 15 20 25 30 35 40 R

, , , , , , ,i, , , , i , , ~, i, , , , i, ,, ,i , ,~

m.~532., r= 1.47

. . . . " /

. . . . . . . . . . .

, , , , I , , t , l , , , , [ , , , , I , , ~ , l . . . . I , , , , -

.5 1 1 .5 2 2 .6 3 c M/Mo

Fig. 5.a Plot of the mass-central density relation for the matched

equation of state for two values of m. and m = 1.47 (solid line). Plot of the mass-central density relation for a stable strange matter equation of state for m. = 595 MeV and r = 1.30 (dotted line), b Plot of the mass-radius relation for the matched equation of state and for a stable strange matter equation of state (see a for curve designa- tion), e Plot of the moment of inertia-mass relation for the matched equation of state and for a stable strange matter equation of state (see a for curve designation)

(5.2)

If the neutron star is slowly rotating (which is usually the case for pulsars, see e.g. [32]), they per- turbe the metric only slightly and one has just one more Einstein equation to solve [33, 34]

1 d (raj~r)+4~rCB=O r a dr

1.6

where

j(r) = e-~/2 l/1 _ 2 GJ/ (r)/r.

This second order equation must be integrated with the boundary following conditions

e5 (0) = constant

R do3l ~5(R) + - ~ ~ r R =~2

(where ~2 is the angular velocity of the fluid in the star with respect to remote stars). Once this equation is solved, the moment of inertiy inertia is obtained from

R 3

, (53,

In Fig. 5 c, the moment of inertia-mass relation is plot- ted.

Because our equation of state is very stiff (see Figs. 3 and 4), the maximum masses and maximum moments of inertia reachable may be fairly high: Mm,x~3 .35M o and Imax~9.7"1045gcm 2 if m, = 532 MeV. For comparison, a neutron star with a Walecka equation of state at high density, has Mma x

~2.6 M G and Im~x~3" 1045 g cm 2 - see e.g. [32, 35]. The precise location of the maxima in Fig. 5 a~c, de- pends sensitively on the matching with the neutron equation of state: the higher the transition density, i.e. the higher the value of m,, the lower these quanti- ties. In fact, this provides us with a simple way to find an upper bound to m,: requiring that the maxi- mal mass of neutron stars be at least 1.4 M o, we get that m, < 760 MeV.

6 Strange matter stability and strange stars

As suggested by Witten [20], the true ground state of matter may be matter formed of u, d and s quarks because at T = 0

G(u, d)p=o _-> Gnucl .... p=o (6.1 a)

1.4

1.2

1

550

136

< - - 2 10 -a, 0.37

< - 2 10 -4, 0.72

< - - 5 I0 -s, 0 .84

~ . ~ - - 10 -8 0 9 5

< - - - 0., 1.

600 650 700 750 m .

Fig. 6. Contour in the (mu-r) plane inside which strange matter would be more stable than ordinary matter. The numbers on the right indicate respectively the value of the hadronic electric charge per baryon and the strangeness per baryon

(ordinary matter is formed of nucleons and not of u-d quarks) but it is possible that, due to the Pauli principle,

G(u, d, s),=0 < Gnucl .... p=O- (6.1 b)

This hypothesis has been studied more closely in the framework of the MIT bag equation of state by Farhi and Jaffe [-36]. They concluded that there exist rea- sonable windows in the ranges of their parameters (the strange quark mass, the strong coupling constant and the vacuum constant), for which inequalities (6.1a-b) are satisfied. More precisely, as they ex- plained, the following criteria for strange matter sta- bility are fulfilled

G (u, d)p = o > GFe + 4 MeV = 934 MeV (6.2 a)

and

G(u, d, s)p=o < GF~ = 930 MeV. (6.2b)

It is possible in the context of our equation of state, to see if these last inequalities can hold. We saw in Sect. 3, that G(u,d)p=o=l.575xm,, so that (6.2 a) holds if and only if

rn, > 593 MeV. (6.3 a)

As for inequality (6.2a), if r = l for instance, G(u, d, s),=o = 1.3825 m,, so it is satisfied if only if

m, < 673 MeV. (6.3 b)

Other values for r lead to other upper bounds for In,. In Fig. 6, we have depicted the surface in the (m,, r) plane, for which strange matter would be more stable than ordinary matter. As can be seen, if r ~ 1.47

(which is expected as mentioned in Sect. 4), the chance that strange mat ter be the true ground state of had- rons is small. A similar conclusion was reached in a different approach by Bethe et al. [37].

If one assumes though for instance r ~ l . 3 and m , = 595 MeV, then strange matter would be more stable than ordinary matter at zero pressure. Strange stars might then exist, they could be formed either because after the neutron-quark transition, s quark would appear in the core and this strange matter could eat the neutronie mat ter in the star up to its surface [20], or via some other more exotic processes as suggested in [22]. Their properties have been stud- ied in the context of the M I T bag equation by Haensel et al. [21] and by Aleock et al. [22]. In Fig. 5a-c, we plotted for our equation of state, the results of the integration of the Oppenheimer-Volkoff system (5.1 a-c) and of the equation for the moment of inertia (5.2), in a case where strange matter would be more stable than nuclear matter. Our results are in qualita- tive agreement with those of the authors mentioned above. M(ec) has a vertical asymptote, at ec= 5-1014 g cm -3 (corresponding to stars with zero pressure, i.e. for which gravity is not important) and it reaches a max imum beyond which no stable configuration can be found: the second strange star peak in Fig. 5 a is unstable, like the quark star peak of the previous section. The mass-radius relation is different from that of neutron stars for low mass (because then gravity does not matter and M o c R 3, while for low mass neu- tron stars, the gravitational pull is small and the star can expand far), but is fairly similar in the range of masses (1.4 Mo) typically observed.

7 Conclusion

In this paper, we generalized to several flavors, a one flavor zero temperature quark mat ter equation of state derived recently [23, 24]. It was shown that the main features of the one flavor case still hold. Precise- ly, as density increases, the quarks undergo a first order phase transition from a state of massive parti- cles to a state of particles of decreasing mass - thus mimicking the onset of asymptotic freedom. Also, im- mediately after the transition, the quarks are in a col- lective bound state, thus suggesting that they just started to go out of the hadrons.

Because of screening through pair creation, our results depend on just one parameter, the u quark constituent mass, in the two flavor case, and two pa- rameters on the three flavor case, the u quark constit- uent mass and the ratio of the s mass and u mass. While parameters in usual quark mat ter equations of state (MIT, SLAC . . . . ), are usually fitted to repro-

137

duce static properties of hadrons (mass spectrum, magnetic momenta , mean radii...), we fitted ours by using properties of the hadron-quark phase transi- tion: n . . . . . . tt. ~ nt ~ F/pert" This lead us to mu~ [532 MeV, 636 MeV].

This quark mat ter equation of state was then ap- plied to the study of the interior of neutron stars. We found that neutron stars would have a big quark core and that their inertia parameters could be similar or larger than in usual models of high density matter. How large they really are, depends on the exact tran- sition density, i.e, on the value of mu. By requiring that the max imum mass of neutron stars be compati- ble with observations (Mmax > 1.4 Mo), we found that m, < 760 MeV. In this model, quark stars are not sta- ble.

This equation of state was also applied to the in- vestigation of strange matter stability. It was shown that within the range of values expected for our two parameters, strange matter should not be more stable than ordinary matter at zero pressure (contrarily to what may be predicted within the f ramework of the M I T bag model [36]). One may however make a choice of parameters such as to ensure strange matter stability. In that case, two families of strange stars (i.e. stars entirely made of stable strange matter), can be found. The first one would be qualitatively similar to that of [21] and [22], which were studied in the context of the M I T bag equation of state. The second one corresponds to unstable objects.

I t seems therefore that a potential based approach for the quark mat ter equation of state leads to sensi- ble qualitative and quantitative results. However, as a final remark, we would like to recall that this equa- tion of state can be, and probably should be, im- proved (for details see [23]).

Acknowledgements. The author wishes to thank L. McLerran and A. Olinto for helpful discussions when writing this paper. This work was supported by the DOE and the NASA (at Fermilab) and a Lavoisier fellowship (France).

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