Fortschr. Phys. 34 (1986) 12, 817-827

Do CQuark-Stars’ Exist? l)

P. BHATTACHARJEE

Space Physics Group, Tata Institute of Fundamental Research Homi Bhabha Road, Bombay 400005, India

Abstract

Possible existence of quark-matter in dense neutron-stars is discussed using Quantum Chromo- dynamical equation of state for cold degenerate qnark-matter.

1. Introduction

Hadronic matter a t sufficiently high temperature and/or high density may undergo a phase transition to a system consisting of asymptotically free quarks and gluons [l]. In this paper, I shall discuss such a phase transition in a high density system predominantlj- consisting of neutrons with a view to examining the possibility of existence of quark matter (QM) in the core regions of stable (neutron) stars of sufficiently high central density. Here, by quark matter, I mean a possible form of high density hadronic matter in which the hadrons are so closely packed that the quark-structures of neighbouring hadrons have overlapped so that the hadrons are not identifiable as distinct objects and consequently, the degrees of freedoin of the system are those of quarks rather than of the individual hadrons. Moreover, the quarks in quark-matter are ‘deconfined’ unlike the quarks inside a hadron - the long-distance interactions of quarks which bind quarks inside hadrons are screened [2] due to many-body effects in a dense system of quarks. The quarks in quark-matter, therefore, interact only through the short range ‘asymptoti- cally free’ interactions.

In the following, we shall consider degenerate systems of neutron-matter and quark- matter a t zero temperature (T = 0). The typical fermi momentum of neutronsand, a posteriori, of quarks in a degenerate system of neutrons or quarks at density as avail- able in neutron stars, is expected to be on the order of few hundred MeV and above, which is very large in comparison to the typical thermal energy kT - 0.1 MeV for a ternperature T - lo9 K. Therefore, in discussing neutron stars, the temperature, for all practical purposes, can be taken to be zero. The following discussions are, therefore, not directly applicable to hot quark-gluon plasma expected to be formed in ultra high energy heavy ion collisions [3] or in the early universe [4].

I shall first describe the method to obtain the critical density above which neutron matter would be in the quark-matter phase. I shall then discuss construction of ex- plicit models of ‘quark-stars’ - cold degenerate stars containing quark-matter in their

l ) Based on invited talk given by the author a t the “Topical Meeting on Quark-Gluon Plasma”, held at the Variable Energy Cyclotron Centre (VECC), Calcutta, December 2-3, 1985.

818 P. BEATTACHARJEE, Do Quark-Stars Exist

cores. The construction of quark-star models will be done by numerically solving Tol- man-Oppenheimer-Volkoff equation [ 51 of hydrostatic equilibrium in general relativity, the input for which is the equation of state of matter throughout the whole volume of the star. I shall discuss the question of stability of tJhese quark-stars and conclude by briefly discussing the observational consequences, if any, of existence of stable quark-stars, and some related issues.

a. Phase Transition from Neutron-Matter to Quark-Matter

Za. Equation of state (EOS)

To study phase transition, we need the EOS, i.e. the relationship between energy density and pressure for neutron-matter as well as for quark-matter. The EOS of neutron- matter in neutron-stars is not precisely known. A wide variety of EOS exists in the litera- ture [6]. There are, generally, two classes of these EOS’s: (a) EOS based on non-relativistic potential niodels :

In this class of models, one solves the many-body Schrodinger equation with a pheno- menological nucleon-nucleon potential obtained by fitting N-LY scattering data to an assumed potential which is typically taken to be a sum of Yukawa potentials with various ranges and strengths, e.g., the so-called Reid potential [7]. A hard-core repulsive potential, presumably, due to w-meson exchange among nucleons, has also been in- corporated in some cases [6]. (b) The other class of EOS consists of field theoretical models where one considers [8] scalar-, vector-, and, in some cases [9] tensor-meson exchange among nucleons and solves the various field equations in the mean-field-t heory approximation. In the following dis- cussion, I shall only consider two representative neutron-matter EOS belonging to the class (a) above. These are : EOS given by BETHE and JOHNSOX (their model VH) (BJVH) [lo] and the EOS given by Pandharipande and Smith (PS) [ l l ] . For details of these EOS, see ref. [6].

The quark-matter EOS can be obtained from the thermodynamics of a quark gas a t T = 0, the interaction dynamics of quarks being described by Quantum Chromodynamics (QCD). Owing to the property of asymptotic freedom of QCD, the quarks at high density (i.e. a t close separation) interact very weakly allowing the met hods of perturbative QCD to be applied to calculate the thermodynamic quantities of a quark gas. Moreover, the gluons mediating the interaction between quarks acquire an effective mass in a dense system of quarks. In fact, this is what is responsible for the screening, mentioned earlier, of the long-distance confining forces between quarks. This removes the peculiar infra- red problem associated with masslessness of gluons in perturbation theory.

Detail discussions of application of perturbative QCD to calculate the thermodynamic quantities of a quark gas have been given by several authors [ 121. For our purpose here we shall follow the method of CHAPLINE and NAUENBERG [13] who include the effect of quark-quark interaction to second order in the quark-gluon coupling strength gc in calculating the therniodynamic quantities of quark gas at T = 0. Throughout, we shall use natural units h = c = 1.

The energy density ( c ) of a system of qnarks a t T = 0, and upto second order in ge is r 131

Fortschr. Pliys. 34 (1986) 12 819

where pfi is the fermi momentum of the ith flavoured quark; pFi is related to the corre- sponding number density ni through the relat,ion

p p , = (“%li)1/3. (2)

In equation (l), the quantity a, is

gc2 a, = - 16n (3)

The first term, 3/4$ P ; ~ , in eq. (1) is just the energy density of an ideal fermi gas of

quarks at IT = 0. The term proportional to a, is the so-called exchange-energy contribu- tion which comes from two quarks interchanging positions in the Fermi sea by exchanging one virtual gluon.

The quark flavours relevant for our discussions are the u, d and s quarks. An equi- librium number density of s quarks is likely to be present in the quark-matter phase even if the original baryon phase contained no strange baryons. The reason for this is that a t the high densities of interest, the fermi momentum of d quarks in the degenerate quark gas may be much higher than the strange quark mass and in this case it becomes energetically favourbable for the d quarks at the top of the fermi sea to transform them- selves into s-quarks via the weak interaction process u + d + u + s. The charm quark is too massive to be present in any significant number a t the densities of interest to us. We shall, therefore, consider only u, d and s quarks. Moreover, since at high densities the fermi energies of the quarks are large compared to their masses, the quark masses may he neglected as a first approximation.

We shall consider a composition of the quark-matter which is an equal mixture of u, d and s quarks, so that

1

nu = nd = n, = nB, (4)

where nB is the baryon-number density. Note that in the quark-matter phase the baryon- number is carried by quarks - each quark carrying a baryon number 1/3. Eqs. (2) and (4) imply

p p , = pfd = pp, = (n2nB)’lY = pF, say. (5)

Kow, i t is well-known that in QCD, the quark-gluon coupling strength gc is not a ‘con- stant’. Rather, it is, what is called a ‘running’ coupling constant which satisfies a certain renormalization group equation and the value of g, depends on the energy-scale a t which it is measured. In a many-body system of interacting quarks a t T = 0, the rele- vant energy scale is provided by the typical fermi momentum of the quarks. Thus for a quark-gas a t T = 0, the quark-gluon coupling strength g, is a density dependent quan- tity through its dependence on the fermi-momentum. For a three-flavour quark gas satisfying eqs. (4) and (5), g, is given by [13]

where Af is the ‘QCD parameter’ appropriate for a fermi gas of quarks a t T = 0. Presumably, AF is of the same order of magnitude as the familiar QCD parameter AQcD N (300-500) MeV [14]. Because of the uncertainty in the valueof AF, we shall consider various different values of A, in discussing neutron-quark phase transition. Note from eq. (6) the asymptotic freedom (ac -+ 0) propert,y of the quark-gas at high density (i.e.,

820 P. BHATTACHARJEE, Do Quark-Stars Exist

large pF). Eq. (6) can be rigorously derived [151 from t,he renormalization group equation for the thermodynamic potential of an interacting quark-gas. Introducing the dimen- sionless quantity

P F (n2nB)1’3

AdF L4F ’ x = - =

so that 7E

a, = - 181nx’

One can write the energy density in eq. (1) as

(7)

Keeping in mind t,hat z defined in (7) is a fundion of the baryon number density n,, the thermodynamic pressure P can be obtained from eq. (9) as

d 3 3 P = --(V&)=- --AF4X4 dV 4372

Equations (9) and (10) constitute the required EOS for a massless quark gas containing an equal mixture of u, d and s quarks at T = 0.

2 b. Phase transition parameters

The phase transition is studied by using Gibbs’ criterion. At T = 0, the appropriate phase of matter a t a given pressure is the one which has the lower Gibbs’ energy per baryon, pe :

(11) P f E

iue = -- nB

The value of P for which pe of quark-matter and neutron-matter become equal is the transition pressure. It is found that for most reasonable values of the parameter A=, the phase transition is of first-order, i.e. there is a discontinuity in the energy-density of matter a t the transition pressure. A first order hadron-quark phase transition is prob- ably not entirely unexpected - a discontinuity also shows up in the theoretical be- haviour of d.c. ‘colour’ conductivity which changes nonanalytically [16] from zero in a gas of nucleons to a nonzero finite value in the quark phase. For a discussion of the order of the phase-transition and what numerical simulations in the lattice version of QCD have to say about the phase transition, see ref. [ 17, 181. The quantities E~ and eq - the energy density of neutron-matter and quark-matter respectively, a t the transition pressure can be read off from the EOS curves (i.e. P vs E curves) for the two phases. The difference ( E ~ - eN) > 0 is, of course, the latent heat corresponding to a first order phase transition (see Fig. 1).

From eqns.(7), (9), (10) and (l l) , the expression for the Gibbs’ energy per baryon, pe, in the quark matter phase is

Fortschr. Phys. 34 (1986) 12 82 1

/ / : W (L a / 1

I

I 4 / , ' /

a

m W (L a

-1

GlBBS ENERGY PER BARYON(PB) ENERGY DENSITY ( € 1 ( a ) ( b )

Fig. 1. Schematic diagram illustrating the Gibbs' criterion for phase transition. The solid and broken curves in figures (a) and (b) correspond respectively to neutron-matter and quark-matter. The point of intersection of the two curves in fig. (a) gives the tran- sition pressure which is then used to read out the values of eN and eQ (see text) from fig. (b)

From a chosen EOS for neutron-matter, we can also obtain the quantity ,uB in the neu- tron-matter phase. The points of intersection of the P vs ,uB curves for the two phases give the transition pressures (PT). Let pa and @Q denote the matter densities of neutron- matter and quark-matter respectively, a t the transition point. If BJVH [lo] EOS is chosen for neutron matter, one finds [13] PT = 2.4. dyn . cm-2, eN = 5.6 . 1015 g x ~ m - ~ and pQ = 1.12 - 10l6g ~ m - ~ for the case AF = 300 MeV and PT = 3.68 . dyn x cnir2, ex = 7.39 + 1015g. ~ m - ~ and Po = 1.85 . 10l6g. forthe case ( I F = 400MeV. One gets somewhat lower values for all the above quantities if one uses Pandharipande- Smith (PS) [ l l ] EOS instead of BJVH EOS to describe neutron matter. The reason for this is that PS is the stiffer of the two EOS's, i.e. it gives a greater value for pressure a t a given density than the BJVH EOS. Specifically, for PS EOS, the values are: PT = 4.1 X dyn - and @Q = 3.24. loi5 g . cme3 for the case i l F = 300 MeV and PT = 1.09 * dyn . em-', pa = 3.0 . loi5 g . ~ r n - ~ and pQ = 9.92 x 1015 g . for the case of A, = 400 MeV. The quark matter EOS is made stiffer by lowering the value of A,. Correspondingly, for a given neutron-matter EOS, a stiffer quark-matter EOS (lower value of A,) yields lower values for transition pressures and densities. For A, below 200 MeV, there is no phase transition at all [ 131. For modification of the various phase transition paraineters in the case of massive quarks, see ref. [19].

p N = 1.8 . loi5 g a

3. Quark-star Models and Their Stability

We now come to the main purpose of our discussion, namely, to construct quark-star models and study their stability.

3 a. Quark-star models

The mass M and radius R of an equilibrium configuration of a spherically symmetric st.atic star is given by the solution of the Tolman-Oppenheimer-Volkoff (TOV) equation

822 P. BHATTACEARJEE, Do Quark-Stars Exist

[ 51 of hydrostatic equilibrium in general relativity :

(13) W r ) -- - GM(r)e(r ) il I ;(y [1 + 4nr3P(r)] [ 2GM(r)]- l

3 dr r2 M ( r ) r

where M(r) is the gravitational mass contained within a sphere of coordinate radius r and P(r) and e(r ) are, respectively, the pressure and matter density a t the coordinate radius r. The boundary conditions are specified by requiring that the centre be free of singularities and the pressure and the density at the surface are fixed: M(0) = 0, dP/drl,,o = 0, P(R) = Psurface. Usually, one takes P(R) = 0, or P(R) = P(esurface), where esurface = 7.86 g is the density of solid 56Fe well-known to be the constituent of the surface of neutron stars. The equilibrium configurations are parametrized by their cen- tral densities eo. Starting with a given value of Po, one numerically integrates the coupled differential equations (13). The value of r a t which P = 0 gives the radius R of the star. The corresponding mass M(r = R) is the gravitational mass of the star. Any equilibrium configuration whose central density is above eo (see section 2(b) above) contains quark- niatter in the core and will, henceforth, be called a ‘qiiark-star,’.

Starting from the centre ( r = 0) of such a star, as one goes radially outward, there will be a point r = TQ at which the density will suddenly fall discontinuously from @ Q to eN, the pressure a t this point being PT, the transition pressure. Note that the pressure always remains continuous and has a non-zero radial gradient a t this point, i.e. dP/drl,=ro40 < 0. In fact, eqns. (13) imply that dP/drlr+o < 0. The radius r = rq may be called the radius of the ‘quark-core’ of the star. The quark-core must have a sharp boundary; a boundary or transition region of finite width across which the density could vary continuously in the interval eN < e < @ Q in which quark-matter and neutron-matter could coexist in chemical equilibrium a t a constant pressure (i.e. dPldr = 0) is not possible because the constant pressure condition, dP/dr = 0, cannot be satisfied at any point except a t the centre ( r = 0) (see equations (13)). However, since the pressure always has zero gradient at the centre (r = 0), a star whose central density lies in the range eN < eo < eo would have quark-matter and neutron-matter coexisting in chemical equilibrium, but, that, only at the centre ( r = 0) and a t all other points the matter must be in the neutron- matter phase.

The EOS of matter in the various density ranges are taken as follows: 1. For e > Po, the EOS is given by eqm. (9) and (10). 2. For 1.6 - 1014 g - (3111-3 < e < phi, various different EOS can be used. I shall, in this

talk, show the results only for the case when the BJVH [lo] EOS is used. For densities below 1.6 + 1014 g - the following ‘standard) combinations of EOS

are used: 3. For 4.3 . loll g . the EOS is given by BAYM, BETHE

and PETHICK [20]. 4. For 1.15 . 103 g . cm-3 < e < 4.3 . 10l1 g . the EOS is given by BAYM, PE-

THICK and SUTHERLAND [21], and 5. For 7.86 g .em13 < e < 1.15 - lo3 g . the EOS is given by FEYNMAN, METRO-

POLIS and TELLER [22]. The various different EOS are, of course, smoothly matched on to one another at the

edges of their ranges. All the above mentioned EOS are numerically listed in detail by BAYM, PETHICK and SUTHERLAND [21]. See also CANUTO [el.

< e < 1.6 . 1014 g .

Fortschr. Phys. 34 (1986) 12 823

With the above specification of the EOS of matter throughout the whole density range inside a star, one can now obtain, by means of numerical solutions of eqns. (13)) various equilibrium configurations of cold, non-rotating stars. In figures 2 and 3, I show some of the results [23] of these numerical integrations.

1015 1 O l 6 1 017 10’8 1 019 1020

P, ( g c n i 3 )

Fig. 2. Mass ( M ) versus central density (po) curves for equilibrium configurations ob- tained by using quark-matter EOS with AF = 300 MeV (curve I) and AF = 400 MeV (curve 11) together with the Bethe-Johnson VH EOS for neutron-matter. The broken portions represent the quark-star configurations. The double arrowed horizontal lines represent t,he configurations whose central densities lie within the density interval of neutron-quark phase transition

3b. Stability 01 quark-stars:

The solutions of eqns. (13) represent only equilibrium configurations. A given equilib- rium configuration may, however, be in a state of stable or unstable equilibrium under radial perturbations. We are, of course, interested in the stable configurations. The necessary condition for stability of the equilibrium configurations against radial pertur- bations is [24] that

- > 0. dM deo

In Fig. 2, we have a plot of M vs Po (see the figure caption for details). The ‘quark-star’ configurations ( P o > pQ) lie on the broken section of the curves. The conventional neu- tron stars (eo < en) lie on the solid section. The neutron-star configurations lying on the left of the peak (in the solid curve section) satisfy condition (14) and are the well-known stable neutron-star configurations. As is well known, a stable neutron-star can only have a certain maximum inass (Mmax) and maximum central density (pO,max) corresponding to the values of M and po at this peak. Of course, the maximum mass and central density are model dependent quantities depending on what EOS one chooses to describe neutron- matter.

3 Fortschr. Phys. 34 (1986) 12

824 P. BEATTACRARJEE, Do Quark-Stars Exist

239 -\&/\ I 1015 1.76 x 10 l5

\ 1.85 x 70fe ;;

0 1 I 2.61 x ' t

1.888 70 lB - 2.85~10 l8 , . ..

2.23170~~

I 0.91 I I

6 8 10 km 1 R

Fig. 3. Mass ( M ) versus radius (8) curves for the same equilibrium configurations (para- metrized by the central density in units of g . as in fig. 2. The quark-star configu- rations again lie on the broken sections of the curves. The numbers above and below the arrows a t the beginning of the broken sections (marked by x ) indicate the jump in density a t the neutron-quark phase transition points

Coming now to the quark-star configurations (Po > P o ) [broken sections of the curves], we notice that both in curve I and curve 11, there is a family of equilibrium configura- tions lying between the first minimum and the first maximum (of the broken sections) which satisfy condition (14), and are therefore potential condidates for quark-stars. These objects have central densities roughly two to three orders of magnitude higher than the central density of the maximum mass conventional neutron-star. The maximum quark- star mass is less than the maximum neutron-star mass in the given model. On closer scrutiny, however, these quark-star configurations turn out to be unstable against radial perturbations. The point is that, although condition (14) is necessary, it is not the suffi- cient condition for stability of general relativistic equilibrium configurations against radial perturbations. The question of stability of general relativistic stars against radial perturbation is an extensively studied subject and comprehensive discussions have been given among others, by BARDEEN et al. [25] and by THORN [26]. The method is based on the analysis of the mass ( M ) versus radius (R) curves for the equilibrium configurations parametrized by central density. Without going into details, I list here the following necessary and sufficient criteria for stability [25, 261 :

(a) Each extremum of the M va R curve is associated with a change from stable to un- stable, or vice-versa, of a radial mode of oscillation.

(b) A previously stable (unstable) mode becomes unstable (stable) if the M vs R curve passes, with increasing central density Po through an extremuni in a counter-clockwise (clockwise) direction.

Fortschr. Phys. 34 (1986) 12 825

(c) The mode which changes stability is the lowest even (odd) one if dR/deo < O(dR/de, > 0) in the neighbourhood of the extremum.

The M vs R curves for the equilibrium configurations of fig. 2 are displayed in fig. 3. As in fig. 2, the quark-star configurations lie on the broken sections. The stability cri- teria listed above clearly show that all quark-star configurations are unstable against radial perturbations. Moreover, the quark-star Configurations which satisfy condition (14) all have at least two unstable modes of radial oscillations. The conclusion from the above analysis is, therefore, that quark-star configurations are unstable.

In the above, I have shown the results for a particular neutron-matter EOS. However, although use of different neutron-matter EOS gives different estimates for the neutron- quark phase transition parameters, and also different values for the maximum neutron- star mass, most ‘reasonable’ EOS seem to lead to the same general conclusion that quark- stars are unstable [27, 23, 281. For example, the above conclusion remains unchanged [28] when one uses a more ‘realistic’ EOS for neutron-matter such as the one given by CANUTO, DATTA and KHLMAN [9], which takes into account the effect of exchange of spin-2 fO-meson among nucleons a t high densities, in addition to the scalar and vector meson exchanges.

The quark-matter EOS also suffers from uncertainties associated with our imprecise knowledge of the value of the parameter (1, which fixes the quark-gluon coupling strength gc in a degenerate system of quarks. Moreover, although perturbative QCD is likely to be valid a t very high densities, its validity a t low densities close to the phase transition densities where aC becomes close to unity, may be open to question. For a discussion of non-perturbative instanton effects in the QCD EOS of quark-matter, see Shuryak [29].

Thus, although it seems unlikely that stable quark stars exist, a‘definite answer has yet to come. FECHNER and JOSS [30], for instance, have chosen the relevant parameters in such a way that the resulting quark-matter EOS yields a neutron-quark phase tran- sition density which is less than the maximum central density limit of stable neutron stars for a chosen neutron-matter EOS. Hence in their model, quark-stars are stable and the maximum mass neutron star and some of the lower-mass ones are actually quark- stars, i.e., they contain quark-matter in their cores. However, it is open to question as to how realistic is their EOS for quark-matter. For some recent discussions of quark-stars with higher-order QCD correction to the EOS of quark-matter, see BHATTACHARJEE [31].

4. Discussions and Conclusions

Is there any observational way to settle the question whether or not neutron-stars ac- tually contain quark-matter in their interiors? One way would be to study neutron-star cooling by neutrino emission. The dominant neutron-star cooling mechanism is neutri- no emission as long as the surface temperature is > los K. Below this surface temperature, photon emission overtakes the neutrino emission process. It has been shown [32] that neutrino production in quark-matter occurs a t a much faster rate than in neutron-matter. Comparing the observed upper limit on the surface temperature with that predicted from theoretical neutrino emission calculations, one could, in principle, determine whether or not a substantial fraction of the material inside a neutron-star of known age is in quark- matter phase. Unfortunately, it so happens [33] that enhanced cooling by neutrino emis- sion can also be achieved if a substantial fraction of matter inside a neutron-star is in a pion-condensed phase [34]. The latter is a possible phase of dense neutron-matter in which an equilibrium abundance of negatively charged physical pions appearing through the process, n -S p + 5c-, a t densities a t which (,un - ,up) > m,c2 - 139.6 MeV (the p7s are the chemical potentials) form a bose-condensate. Therefore, from surface temperature

3*

826 P. BHATTACHARJEE, Do Quark-Stars Exist

observations alone, one is not likely to be able to distinguish a neutron-star with a quark- matter core from a neutron-star with a ‘pion-condensed’ core even if these objects exist.

Another point to be mentioned here is that masses and radii of stable quark-stars, if they exist at all, as in the model of FECHNER and Joss [30], for example, typically fall in the same ranges as those for conventional neutron star models. Similarly, the allo- wable range of surface redshifts of quark-stars is also comparable [30] to that of con- ventional neutron star models. Observationally, therefore, it seems to be difficult indeed, at the present stage to settle the question of existence of quark-matter core in neutron- stars.

WITTEN [35] has suggested that if at all a stable neutron-star develops a core-region made of quark-matter, then this quark-matter core will gradually absorb the ordinary neutron-matter at the boundary of the core and convert it into quark-matter until al- most the whole star is converted into quark-matter. Such a star may be called a ‘pure’ quark-star (PQS). Since the pressure a t the surface of the star must be zero, for a PQS to exist, the quark-matter at zero pressure must be stable, i.e., it must remain as quark- matter (QM) and not undergo a phase transition to ordinary nuclear- (or neutron-) matter. In other words, the energy-per-baryon ( & / n B ) in the QM phase at zero pressure must be less than the nucleon mass. Non-strange QM (i.e. QM containing only u and d quarks) at zero pressure cannot have E / n B less than nucleon mass, because, if it did, then ordinary nuclei would spontaneously turn themselves into this QM phase, which is not observed. This argument, however, does not apply if the QM is ‘strange’, i.e., if it con- tains s quarks in addition to u and d quarks. Strange QM a t zero pressure could have & / n B less than nucleon mass and yet ordinary nuclei would not spontaneously turn into strange QM-phase simply because of absence of s quarks in nucleons. Thus while & / n B of non-strange QM at zero pressure must always be greater than nucleon mass, irrespective of whether or not PQS exist, the existence of PQS implies that QM at zero pressure must be strange and that strange QM at zero pressure must have &InB less than nucleon mass. Whether or not this can actually be so depends on the value of the parameter AF which governs the QCD EOS of QM. In the model of Chapline and Nauenberg, for example, strange QM with an equal mixture of u, d and s quarks (all assumed massless) can have &Ing less than nucleon-mass if AF < 207 MeV. The value of aC a t the density a t which the pressure vanishes is or, g 0.55 with A , = 207 MeV. Clearly, i t is arguable whether or not perturbative QCD would be valid with such ‘large’ value of or,. Nevertheless, the possibility of existence of strange pure quark stars cannot be ruled out a t this stage.

One may wonder, how strange quark-matter will appear a t all in a system of dense neutrons which contain only u and d quarks. It is certainly true that neutron-matter, even at very high density, cannot directly pass over to strange QM phase even though the latter may have less Gibbs’ energy per baryon ( p B ) than ordinary neutron-matter. However, dense neutron-matter may first undergo a phase transition to non-strange QM at a high density when p B in the non-strange QM falls below that in the neutron-matter phase. As mentioned earlier, the strangeness content will then subsequently develop due to strangeness-violating weak-interaction of quarks, such as the process u + d --+ u + s. Therefore quark-matter, if a t all it exists, is very likely to be ‘strange’. For discussions of some interesting astrophysical consequences of existence of strange pure quark-stars, see the recent paper of BAYM et al. [36].

I thank Dr. Bhaskar Datta, Dr. Shivaji Raha and Dr. Bikash Sinha for interestingdiscussions. I also thank Dr. Bikash Sinha for hospitality a t the VEC Centre, Calcutta.

Fortschr. Phys. 34 (1986) 12 827

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