Nuclear Physics A462 (1987) 791-802 North-Holland, Amsterdam

S T A R S O F S T R A N G E M A T F E R ?

H.A. BETHE l, (3.E. BROWN 2 and J. COOPERSTEIN 2

Institute for Theoretical Physics, University of California, Santa Barbara, CA 93106*, USA

Received 25 June 1986

Abstract: We investigate suggestions that quark matter with strangeness per baryon of order unity may be stable. We model this matter at nuclear matter densities as a gas of close packed A-particles. From the known mass of the A-particle we obtain an estimate of the energy and chemical potential of strange matter at nuclear densities. These are sufficiently high to preclude any phase transition from neutron matter to strange matter in the region near nucleon matter density. Including effects from gluon exchange phenomenologically, we investigate higher densities, consistently making approximations which underestimate the density of transition. In this way we find a transition density Ptr ~ 7po, where P0 is nuclear matter density. This is not far from the maximum density in the center of the most massive neutron stars that can be constructed. Since we have underestimated Ptr and still find it to be ~7p0, we do not believe that the transition from neutron to quark matter is likely in neutron stars. Moreover, measured masses of observed neutron stars are ----1.4 M®, where Mo is the solar mass. For such masses, the central (maximum) density is Pc < 5p0. Transition to quark matter is certainly excluded for these densities.

I. I n t r o d u c t i o n

W i t t e n 1) r e c e n t l y r e v i v e d in te res t in s t r ange m a t t e r 2), sugges t i ng tha t q u a r k m a t t e r

w i th s t r a n g e n e s s p e r b a r y o n o f o r d e r un i ty , " s t r a n g e m a t t e r " , m a y be s table . F a r h i

a n d Jaf fe 3) e x p l o r e d in s o m e de ta i l t he s tab i l i ty o f such ma t t e r , b o t h fo r b u l k m a t t e r

a n d fo r i n t e r m e d i a t e b a r y o n n u m b e r 102~ < A<~ 107. Th is w o r k sugges t s tha t t he re

a re s u b s t a n t i a l " w i n d o w s " in the r anges o f t he r e l e v a n t p a r a m e t e r s ( c o l o r c o u p l i n g

c o n s t a n t as , s t r ange q u a r k mass ms, b a g c o n s t a n t B, etc.) fo r w h i c h s t r ange m a t t e r

is s table . I n a r e c e n t f u r t h e r i n v e s t i g a t i o n o f s t r ange q u a r k stars 4), it is c l a i m e d tha t

t he t r a n s i t i o n f r o m n e u t r o n m a t t e r to q u a r k m a t t e r m a y o c c u r in n e u t r o n stars at

dens i t i e s n o t far a b o v e n u c l e a r m a t t e r dens i ty . T h e a b o v e c l a ims a re at v a r i a n c e

wi th ea r l i e r resul t s o f B a y m a n d C h i n s) a n d o f C h a p l i n e a n d N a u e n b e r g 6) w h i c h

g a v e the t r a n s i t i o n at m u c h h i g h e r dens i t i e s , i f at all. We, t h e r e f o r e , r e inves t i ga t e

th is ma t t e r .

In this n o t e we s h o w tha t t he e n e r g y o f s t r ange q u a r k m a t t e r fo r dens i t i e s n e a r

n u c l e a r m a t t e r dens i t y c a n n o t be r e l i ab ly e s t i m a t e d u s ing p e r t u r b a t i v e Q C D , A t this

1 Permanent address: Newman Lab, Cornell University, Ithaca, NY. 2 Permanent address: Physics Dept., State University of New York, Stony Brook, NY 11794; Supported

in part by USDOE Contract DE-AC02-76ER 13001. * This research was supported in part by the National Science Foundation under grant No. PHY

82-17853, supplemented by funds from the National Aeronautics and Space Administration.

0375-9474/87/$03.50 O Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

792 H.A. Bethe et al. / Stars o f strange matter

density relevant momenta in the gluon-exchange interactions are - A , the QCD scale-breaking parameter. Therefore, the leading log expression

6~r as - (33 - 2 N f ) In ( q / A ) ' (1.1)

where Nr is the number of flavors, makes no sense, the logarithm being near zero. The correct behavior of as for momenta corresponding to distances

r - A - 1 ~ 1 fm (1.2)

is not known; it is more appropriate to consider strong coupling in this regime where confinement occurs. Indeed, the phenomenological strong coupling constant

o~s is 2.2 for the MIT bag model 7), where the confinement range is - 1 fm, For a time it was hoped that factors like (2~r)-1 conspired so that one could use perturbation theory for such a large as, but then the Lamb shift, of order as, was evaluated 8) and it turned out to be 0.9 a J R per quark, as large as the kinetic energy which is of zero order in as. We will, however, later find that the correction in the chemical potential goes as a/2~r.

For infinite quark matter corrections have been calculated 9) to second-order in as- The energy per quark is

asNf B eQ=~kf 1+ + Nf ln + - - , (1.3)

nQ

where B is the bag constant and nQ is the quark density. In order to obtain the pressure we must assume a density dependence for as. We choose

as = a ~O)( k~O>/ kf) , (1.4)

(0) and k~f °> where as are the values of ~s and kf, respectively, at nuclear matter density. This dependence on k? I follows from the phenomenological behavior of as, roughly ocR, in bag model calculations which clearly refer to the nonperturbative regime (1.2). This behavior of ~s comes about because R -I gives the only energy

scale in bag model fits. Therefore, energies from gluon exchange must go io) as as/R. The bag model radius, like the R-matr ix radius in nuclear physics is, in some sense, just a construct, and one ought to be able to describe the physics with a range of R. One can thus fit the ground-s ta te- -decouple t splitting of the strange baryons. (We will not use the N - A splitting, much of which comes from the pion cloud) with different R, provided one chooses the same ratio as/R. Ground-state energies by themselves give a less sensitive determination of ors, because other parameters; e.g., zero-point energies, can be varied. However, Carlson et al. 11) find as ~ I with baryon radii R - 0.64 fm, to be compared with the as = 2.2 of De Grand et al. 7) using radii R = 1-1.1 fm. This is a somewhat faster drop in as than linear in R, but other effects are coupled in. It is clear 11) that a drop in as as k r I with increasing

H.A. Bethe et al. / Stars of strange matter 793

density includes effects of higher order, since the leading log would go as [In ( k f / A ) ] - ~ . As we go into the perturbative regime at higher density, our rate 0£ decrease will continue to be faster than the logarithmic one, and we will therefore underestimate as in this regime. This leads to an underestimate of the transition

density. Given the dependence of as on kf, we can calculate the pressure per quark as

Z - - 0eQ --~kf 1 - Nf ln zr +1.02Nf+6.75 - - - (1.5) nQ 0In p nQ"

We shall later be concerned mostly with the chemical potential /ZQ = eQ + P / n Q

r : " ' l] ~I~Q = k r 1 + a s + Nf ln -0 .48Nf+6 .75 . (1.6) L .~¢" o ' r r C "rr

Note that the bag constant B does not occur in ~Q.

The value a~ °)= 2.2 has been obtained from phenomenology with the MIT bag model 7). The a 2 corrections have not yet been calculated for the bag, so this must

be viewed as the as which fits phenomenology in the region kf~-k~ °) in a linear approximation. I f we use this value, we should drop the quadratic terms.

Although the coefficients of as and a 2 in eq. (1.6) are less than unity, the series

shows no sign of converging for the large oes = 2.2 appropriate to nuclear matter density and it is clearly unwarranted to use it for this value. We shall therefore turn to empirical quantities to obtain quark energies in the long wavelength (low density) regime.

In this note we begin from the argument that color must be neutralized over a range r - 1 fm. In other words, the quarks within a sphere of radius r - 1 fm should,

as a whole, be colorless. This follows from the fact that the known string tension is T - - 1 GeV/fm. Thus, it would cost ~ 1 GeV, which is a very large energy compared with the energy differences we shall be discussing, to pull a quark - 1 fm out of a

colorless ensemble. (The quark transforms as a 3 in in the color SU(3). I f the total ensemble of quarks is colorless, then the remaining quarks, after one is pulled out, must t ransform as a 3. Thus, the energy for pulling a quark out of a colorless ensemble is the same as that for pulling a quark and antiquark apart.)

In sect. 2 we argue that quark matter at densities roughly that of nuclear matter density should look like a collection of close packed A-particles. The wave function of the A is taken from the M I T bag model, the bag serving as a Wigner-Seitz sphere in the many-body system. The energy of the A is, however, taken empirically, not calculated. As we go over to denser systems by adding quarks, we move to the perturbative formulae eqs. (1.5) and (1.6). We hope that in the region p ~ 8po, where kf has doubled, that these are valid. We keep, however, only the corrections linear in oLs, as discussed earlier.

In sect. 3 we give our conclusions.

794 H.A. Bethe et al. / Stars o f strange matter

2. Strange quark matter

Arguments have been made 12) that the confinement region in the case of the nucleon is substantially smaller than that given by the MIT bag model, because of compression of the bag by pressure due to the pion cloud. The coupling of this cloud is smaller in the case of strange baryons, since the coupling of strange quarks to K-mesons is down by a factor of ( m ~ / m K ) 2 from that of nonstrange quarks to pions. Thus, the argument used for nucleons cannot be made in the case of strange baryons. Indeed, phenomenology, especially the magnetic moments of the strange baryons, favors a bag radius R ~ 1.1 fm 13), consistent with the earlier MIT descrip- tion. Now r0, the radius of a sphere contains one baryon, equal to 1.15 fm for nuclear matter at normal densities. Therefore, a system of close-packed A-particles, each with an up, down and strange quark, has nearly the same baryon density, as nuclear matter. We take the system of close-packed spheres, each containing a A particle as our model for strange matter at nuclear matter density. In the correspond- ing wave function, the strong gluon-mediated interactions are accounted for by the imposition of confinement (color neutrality) within each sphere. Each sphere can be viewed as a Wigner-Seitz sphere for the strange matter. The pressure on each bag is zero, by construction, so the bag boundary conditions are suitable for this.

The advantage of this method is that the strong gluonic interactions are taken into account, as nearly as we can mimic, by observed quantities. For large distances r - 1 fm, nature confines color, and we impose this confinement on our wave function. From experiment we know the A-mass to be

mA = 1116 MeV. (2.1)

We take this to be the energy per baryon of strange matter at nuclear matter density,

vis.,

e = mA = 1116 MeV. (2.2)

Here a radius of R A = 1.15 fm equal to the ro for nuclear matter has been assumed; since the A-energy is minimum at its equilibrium radius, small shifts about it give only negligible energy changes.

Possible interactions between A-particles via, e.g., boson exchange are neglected in this limit. They are expected to be small, substantially less than interactions between nucleons 14), and inclusion of these would not substantially change the

argument. The energy of A-matter would, however, be expected to be substantially higher

than (2.2) because of the zero-point motion of the A's. The center of mass of a free A will be in a plane wave state, actually with momentum P = 0 for lowest energy, The center-of-mass motion of a A in a crystal or a liquid will be constrained; a simple estimate gives -100-200 MeV for the additional energy. (We are grateful to Leonardo Castillejo for raising this point.)

H.A. Bethe et al. / Stars o f strange matter 795

Since the pressure on the A-particle is zero, the A being the equilibrium configura- tion of u, d, s quarks, the chemical potential of these three quarks is equal to e, i.e.,

31~Q = mA = 1116 MeV(p -~ P0)- (2.2a)

This is far above the chemical potential for neutron matter, which is 15)

/~N --~ m N ---- 940 MeV (p = Po) ; (2.2b)

i.e., the neutron matter chemical potential is, to within a few MeV, just the neutron mass. Thus, there is no possibility in the region p - Po of joining neutron matter to strange quark matter, which would require equality of chemical potential and pressure in the two phases.

As noted earlier, arguments based on perturbation theory cannot be trusted.

Already for as = 0.9, second-order corrections in eq. (1.3) are as large as the 1st-order ones, and the unknown higher order ones would be expected to vitiate any con- clusions from perturbation theory. Imposit ion of color confinement to handle gluon effects seems more reasonable to us at densities p - po.

Let us now calculate the chemical potential ~ of these three-quarks of three colors put into the system without regard for color confinement, ignoring gluonic correc- tions for the moment, keeping only their kinetic energies. For the moment we treat the strange quark as massless; later we will correct for the mass. We have a degeneracy of 18:2 for spin, 3 for color and 3 for flavor. One easily finds

1.33 /-£Q ~- k f = , (2.3)

ro

where ro is again the radius of a sphere containing one baryon (3 quarks). For r0 = 1.15 fro,

3/ZQ = 684 MeV. (2.4)

This is substantially less than (2.2) and may explain why calculations 4) which do not impose color confinement find transitions from neutron to quark matter not far above nuclear matter density. This energy would be raised by a factor 3~/2= 1.44

by decreasing the degeneracy by 3, making the three-quark object colorless. Correc- tion for a strange quark mass of ms = 150 MeV, increases the energy another 82 MeV.

As will be clear from our results, especially those shown on figs. 2 and 3, adding

perturbative gluonic corrections to the kinetic energy (2.4) will, provided a s is chosen to be sufficiently small, allow for an apparent transition from neutron to quark matter. In fact, as in refs. ~,3), arguments based on either this procedure or total neglect of gluonic corrections can even show strange matter to be bound. As stated above, we believe these results based on weakness of the interactions, to be incorrect.

Having described strange quark matter at nuclear matter density as a collection of close packed A-particles, we go on to denser matter, now adding quarks of all

796 H.A. Bethe et al. / Stars o f s trange matter

three colors in p lane-wave states. We therefore use

3kf 1 (2.5) /~ = 3/~Q = . 2~rJ

with as given by (1.4). As noted earlier, we use the linear formula with phenomeno-

logical density dependence , rather than the perturbative (1.6) with quadrat ic terms. The pressure is

P = 171/z~3p0- B. (2.6)

Here /zQ = n Q / 3 p o , three quarks being needed to make up a baryon.

For nuclear matter we adopt a very stiff equat ion o f state

eN-- mn = 60 (U -- 1.7) M e V , (2.7)

suggested 15) by the mean-field EOS for neu t ron matter. Here e is the energy per

nucleon; u = P / p o . Eq. (2.7) is to be used for p ~ 3 p o . Inclusion o f short-range correlat ions would soften this equat ion substantially. The mean-field EOS is slightly

stiffer than the (very stiff) Be the - Johnson Model I [ref. ~6)], which we also show in

fig. 1. We shall return to a discussion of the equat ion o f state in the next section.

By taking what we believe to be a too stiff equat ion o f state, we underest imate the

transit ion density. F rom (2.7) we find the nucleon pressure

0e PN = PO-i-~n p = 60u2p°" (2.8)

300

250

200

E/A(MeV)

150

I00

50

, , . . . . / . , , / ,

/ I II

ABBC "p//llllB~ 5

J~" I II

"f// ×.ill

1 I I I I i I 0 0 2 4 6 8

u=pIPo(with Po=O.16/fm 3)

I0

Fig. 1. Comparison of the ABBCP [ref. 15)] mean-field equation of state for neutron matter with those of Bethe-Johnson 1 and 5. A good fit to the ABBCP equation for u>4 is E / A = 6 0 ( u - 1.7) MeV.

H.A. Bethe et al. / Stars of strange matter 797

The behav ior of P on densi ty here gives an adiabat ic index F = 2 for the high densi ty

regime, which would follow from two-body interact ions, especially those from

to-exchange, which are expected 15) to domina te at high density. F rom (2.7) and

(2.8) we find

/-6 N = eN + P N = mn-- 102 M e V + 120 u MeV. (2.9) P

In fig. 2 we plot/XN versus PN for the mean field equa t ion of state, and 3/ZQ versus

(o)_ 0, 0,96 and 2.2, with as d ropp ing with densi ty as in eq. Po for the choices as -

(14). We base our analysis on the a~ °)= 2.2 curve.

Let us i l lustrate how the t rans i t ion could take place at Ptr = 7/90. On the neut ron-

mat ter curve we look for u = 7. The quark-mat te r curve would have to be moved

an a m o u n t B --- 55 M e V / f m 3 to the left in order to make the two curves intersect at

UN = 7. They would then cross at UQ-=--9.3 giving a factor ~ 1.3 j u m p in the densi ty

at the t ransi t ion. For this value of B, B U4 = 143 MeV, just the value used in the MIT

bag. In table 1 we give values of B for t rans i t ions in the region 3 < UN < 8p0. One

can see that B is u n r e a s o n a b l y small for the smaller UN. There is substant ia l

uncer ta in ty in the value of B and to this extent fltr is unde te rmined .

In fact, at the lowest densi t ies p Q < fin at the t ransi t ion, ind ica t ing that the

t rans i t ion would proceed from strange quark matter to neu t ron matter. This is

cer tainly unphys ica l and results because at these densit ies we should not be using

1800 . . . . . . . . . ' ~ s mean field

1700 neutron motter ~ , ~ . ~ o ~ _ , ~ \ / , , , - " - "='s-=2.2

i j "

,soo F(MeV) s_,~./'s .- 7 . ...o- t s ... f-- " ' ~

'400 ~ - / J e T " ~ = ~"~"~(.-0)=0 -

1500 /,~/'4 s.,, " ~."r

a / X 4 / . ~ f ,2oo Z /" .,5"" ,oo /#//z /7 / , "

I / / I000 //' / ' /' ' ' ' ' ' ' ' " i

0 I00 200 300 400 500 600 PN or PQ+B(MeV/fm 5)

Fig, 2. The chemical potential il~N is plotted against PN for nuclear matter; 3,U,q versus PQ+ B for quark matter. For quark matter, curves are given for a~ = 0 and a~ = 0.96k~°)/kf and a s = 2.2k~°)/kf. The densities can be determined from the values of u = P/Po given along each curve. Crossing of the two curves of/~ versus P can then be achieved by proper choice of bag constant. Thus, for u = 7 on the neutron matter curve, if the quark matter curve for c~ = 2.2k~°)/kf is shifted by B = 55 MeV/fm 3 to the left the two curves

will cross. This gives B t/4= 145 MeV.

798 H.A. Bethe et al. / Stars of strange matter

TABLE 1

Parameters at the transition a s = 2.2 (k~°~/kf)

PrJPo B (MeV. fm -3) PQ/PN P (MeV- fm -3) u (MeV)

3 17.1 0.92 86.4 1198 4 12.3 0.98 154 1318 5 13.0 1.08 240 1438 6 25.1 1.19 345 1588 7 54.8 1.33 470 1678 8 109.2 1.48 614 1978

the perturbative expansion, which begins at /z < m A at u o = 1, but should impose color confinement as we did for the gas of close packed A-particles. This prevents any transition from happening at the lower densities and avoids this unphysical behavior.

Repeating our analysis using BJ1 for the neutron matter equation of state, we find UN = 7, essentially the same results as for the mean field. In ref. 4), using BJ1, a transition at UN--2 was found. However, these authors did not impose color neutrality (nor did they include the decrease of a~ with kf). If we include these effects as sketched above, we find that at such a density B is negative, a reflection that the transition cannot occur until much higher densities. We note that later calculations by Friedman and Pandharipande 16a) gave an equation of state for neutron matter close to B J1. These authors achieve saturation in nuclear matter at P,m by introduction of a density dependence in the two-body intermediate-range attraction. This is essentially what the relativistic correction in ref. 15) does. Work in progress by Wiringa 16b) is giving a neutron matter equation of state close to, but slightly softer than that of Friedman and Pandharipande. There is, therefore, considerable convergence on an EOS close to, but possibly a bit softer than, BJ1.

Had we adopted a different density dependence for as, our curves would have shifted somewhat, but the qualitative result, the pushing upwards of the transition density, would remain. Furthermore, we note that our choice underestimates this

density. We noted that our mean field equation of state, taken from ref. 15), is too stiff.

This is because antisymmetry and short-range correlations between nucleons, both neglected in that mean field description, cut down the to-exchange repulsion that predominates at high densities. These effects multiply 17) the to-repulsion by a factor ----0.5. Some of these effects are roughly taken into account by using a reduced to-coupling in ref. is).

We therefore show in fig. 3 results for B J5, which is somewhat softer than our mean field equation (see fig. 1) and softer than B J1. It can be seen that in the region UN> 4, in which there is any hope for joining, the transition from neutron to quark matter would require a negative bag constant and is, therefore, obviously impossible.

H.A. Bethe et al. / Stars o f strange matter 799

1800

1700

1600

1500 p (MeV)

1400

1300

1200

I100

I000

I I I I I I I I I I I I

a(O)- 2 2

/ ; ' 7 j , , . O - 8 - ~ ' ~ - --$ " -

4.,#- 4 / = ,,-" /~- ,o- 5 / X'.i . / .-" 3/~ ~,, , , ."

/ ' , ,* ,," , ," III 111 I l l I I I I 1 I I 1 I

0 I00 200 300 400 500 600 PN or PQ+B (MeV/fm ~)

Fig. 3. Plot s imi l a r to tha t o f fig. 3 excep t tha t the B J5 e q u a t i o n of s ta te for neu t ron mat te r is d rawn as a sol id l ine.

We believe that this result will apply to reasonable neutron matter equations of

state, but these have not yet been constructed including the many-body effects

discussed in ref. 15). In our considerations we have not included electrons. The number of these is

negligible 3) and would not affect our conclusions. We probably should not continue our considerations to densities higher than

p - 8po because our assumption of rigid nucleons, implicit in the above, would be expected to go wrong. Although the quark core in the nucleon is small, R ~ 0.5 fm [ref. 12)], and may not be too much affected by increasing density, the longer-range

pion cloud would be substantially squeezed by p - 8po. Calculation of what happens at such and higher densities may require a much more detailed treatment of the

nucleonic structure.

3. Discussion and conclusions

Our work has been along the lines of the earlier work by Chapline and Nauenberg 6). We have supplemented this work by using the known A mass to pin down the energy of strange quark matter in the region of nuclear matter densities. Rather than using the leading-log expression for as(q2), we argue that we are in the nonperturbative regime, and employ a phenomenological as which decreases

as k~ -a. Chapline and Nauenberg 6) were unable to get a transition for A < 200 MeV. With

the rapid drop off with kf of our as, our situation corresponds to a small A, in this range. Thus, we have to understand how our procedure allows us to join the curves

800 H.A. Bethe et al. / Stars o f strange matter

for neutron matter and quark matter, fig. 2, to cross without any shift in the latter by B. They interpret B as the difference between ½pe and P. (See their eq. (2.11)), which in our case would be just ~ k f p since there is no as term in P, given our assumed density dependence of as. It has, however, been shown 5) that the bag

constant should occur (with negative sign) in the pressure for the quark-gluon plasma, as Chapline and Nauenberg knew (see their eq. (1.2b)). The chief difference between our procedure and theirs is in our explicit inclusion of the - B in the pressure; this makes the joining of the two curves possible. Using their procedure, we would not get a transition for any density.

Our results are in general agreement with those of Baym and Chin 5), although

our transition densities are somewhat smaller than found by these authors. They used the same as = 2.2 at nuclear matter density, but took it to be independent of density. With our assumed density dependence, our as has dropped a factor of

nearly 2 by Ptr. ThUS we find lower Ptr. Amusingly, these authors found a stable zero pressure phase of quark matter corresponding to a minimum in energy of e - rnnc 2 = -71 MeV at p ---]po- They say "such a strongly bound system of quarks at a density well below that of nuclear matter is, to say the least, not observed". Just such a state will be removed through the local imposition of color neutrality,

as our comparison with the A-particle gas shows. Farhi and Jaffe 3) manage to obtain strange quark matter chiefly because they

employ a small color coupling constant, as = 0 to 0.9. (They call this a¢; in fact the ac they use is as, twice the original 7) ac). In their table I they quote, however, as=2 .0-2 .8 as obtained from bag-model fits to light hadron spectra, with the exception of the a s ~ < 1 from ref. 11). As noted by Farhi and Jaffe, fitting to hadron

spectra invariably couples the output values of ms, as and B ~/4 to those of the more phenomenological additional parameters. In fact, the as<~ 1 of Carlson et al. 11) is

coupled with a B = 353 MeV/fm 3. (For given string tension T, B - a~-l; the ratio of their B to the MIT B = 62 MeV/ fm 3 is substantially larger than the ratio of the as's, however.) It can be seen from fig. 1 that Ptr with the as and B ~/4 of ref. ~1) occurs

for a density -8po . Do strange quark matter cores exist in neutron stars? All measured neutron star

masses are consisent with M - 1.4 M®, with M® the solar mass. Presumably, 1.4 M® is the mass resulting f rom supernova collapse, and this is compatible with theoretical calculations of supernovae. However, a neutron star in a binary system might accrete matter overflowing the Roche lobe of its companion star. This can proceed until the max imum neutron star mass, Mmax is reached, whenceforth implosion is

triggered. The central density of a neutron star depends of course on the equation of state

employed. Obviously if that central density is less than the neutron-quark transition density for the same equation of state there can be no quark core. In table 2 we list the value of the central baryon density in units of po (not the central energy density which is usually quoted), for the M = 1.4 M o and the M -- Mmax configurations.

H.A. Be the et al. / Stars o f strange mat ter

TABLE 2

Neutron star central densities

801

EOS Mm~x ( Pc~ Po) M = M=.x (Pc/Po) ~ = 1.4Mo

REID 1.64 ~8 - 5

B J5 1.65 9.1 4.6

B J1 1.85 8.1 3.6 TI 1.93 ~ 4 --1.8 Mean field 2.22 4.3 -1 .5

These have been calculated by solving the general relativistic hydrostatic equations

by standard means. Even for the softest equations of state used (Bethe-Johnson Model 5 (B J5) [ref. 16)] and Reid 18)) the central density is at most 519o for the

M = 1.4 M® stars. Our arguments about the energy of A-particle matter are certainly sufficient, without detailed calculation, to preclude the transition to quark matter here, and even more for 1.4 M® stars with a stiff equation of state, including B J1.

Turning to neutron stars of maximum mass, the central density is again only about 4po for the very stiff equations of state, the tensor interaction (TI) and the mean field model of ref. 19). This density is well below the value of 7po we calculated in sect. 3 and thus these stars have no quark cores. The other three equations of state have central densities somewhat higher than this; as the equation of state softens the central density for M m a x increases. However we have shown that already with B J5 it is impossible to make the transition to quark matter and this will also be so with other soft equations of state.

With respect to the stiff equations of state which enable a transition from neutron to quark matter, the situation with neutron stars with measured masses, which all lie close to M = 1.4 M®, is particularly clear. In table 2 we see that the largest central densities for such stars would be 3.6/90 for the B J1 equation of state, far below the possible transition density. Of course, for softer equations of state, pc is larger; but for these the transition is impossible (see fig. 3).

Observationally, the identification of the 35 day periodicity in Her (X-1 as being free precession places constraints 20.21) on the equation of state of neutron matter.

Equations of state as soft as Reid are probably excluded, requiring an unusually large rotational rate at the time of formation and an excessive oblateness - 5 % , close to the break up limit 21). The free precession combined with the post-glitch behavior of glitching pulsars, favors a stiff equation of state, such as TI or B J1 [refs. 2o.21)].

Our chief conclusion is that we do not find any tendency for neutron matter to merge into strange matter up to densities achievable in neutron stars. The reason for this, as seen from our arguments about A-matter gas for p - p o , is that gluon interactions are strong and try to maintain color neutrality. It is the large color coupling as which presents the occurrence of strange-quark matter.

802 H.A. Bethe et al. / Stars of strange matter

We are grateful to the Institute for Theoretical Physics at Santa Barbara for warm hospitality while this work was carried out. We wish to thank Ken Johnson for helpful critical remarks and Tom Ainsworth for help.

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