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NEUTRON STARS AND STRANGE BNL—39402 DE37 006746 J. Cooperstein* Brookhaven National Laboratory, Upton NY 11973 and SUNY at Stony Brook NY 11794

We investigate the likelihood that matter with strangeness of order unity resides in stars. In the strong coupling regime near Po this is found to be unlikely. Considering higher densities where perturbative expansions are used, we find a lower bound to be at 7p0 for the transition density. This is higher than the inferred density of observed neutron stars, and thus the transition to quark matter is precluded.

I. Introduction We will deal with the question: does bulk strange matter exist in nature? For this we will draw upon our recent investigation of this subject done in collaboration with Hans Bethe and Gerry Brown1. Strange matter is usually taken to mean deconfined i .lark matter with strangeness of order unity. If such material exists in the universe today, the core of neutron stars is the oi?ly reasonable place to search for it in the bulk. Observationally the problem is difficult. We know that all well measured neutron stars have masses in the range 1.4 ~ M/MQ ~ 1.6, so we have at least a lower limit to the maximum mass. As far as an upper limit is concerned one might be tempted to believe it is not much higher than the above range for two reasons. First there is the fact that Type II supernovae calculations always tend to leave behind a mass at the lower end of the above range, since this is about the iron core mass of the presupernova star, and the explosion should sweep out almost all of the overlying burning shells. Afterwards, however, matter may continue to accrete onto the , especially if it is in a binary system (and the measured masses have only been accurately determined for such systems) and the companion overflows its Roche lobes. The second reason is the

lack of any strong indication of a neutron star with a large mass, say ~ 2MQ. But at any rate, the upper limit is not very precise, there having been only a handful of measured neutron star masses. Radii of neutron stars are not as well known as their masses and other static properties such as surface temperatures may not tell us anything directly about the interior. The most hope for discriminating amongst equations of state may come from evaluation of dynamic properties of neutron stars. For instance cooling rates could give us clues as to the existence of pion condensates or quark matter. Examination of core quakes and glitches can also yield

* Supported in part by the U. S. Department of Energy under Contracts No. DE-AC02- 76CH00016 and No. DE-AC02-76ER13001. m B , '£•;* / DISTRIBUTION OF THIS [iOCUMFjf some (parameterization dependent) constraints. From such an analysis of the free precession of the Hercules X-l , Pandharipande, Pines and Smith2 posited, for example, that soft equations of state such as that constructed from the Reid soft core potential, could be excluded as requiring uncomfortably high values of oblateness and initial angular velocity. However such dynamical studies are not fully developed, and so unfortunately, at the present time, our only "good" constraint is on the maximum neutron star mass; many equations of state can satisfy this criteria. On the theoretical plane evaluating the probability of the existence of matter in neutron stars involves two steps. First one must calculate (estimate) the density at which such a transition would take place, based on the properties of the confined and deconfined phases. Second one must ask what density is achieved at the center of neutron stars. If this density lies below that of the , strange matter will not find a home in neutron stars.

II. Strange Matter The possibility of quark stars is not a new subject. In the late 1970's Baym and Chin3, Chapline and Nauenberg4, Freedman and McLerran0 and others considered this question. A general consensus emerged tljaS the density of the transition would be too high to occur in neutron stars, about 10 po °r greater. Recently Witten6 has revived interest in this question and it has been further explored by Farhi and Jaffe7 and others8. This work floats

upon the observation that one can find a set of the relevant parameters (a3, m3, B), such that strange matter could be stable (either in • «i\\ or for small nuggets). Why nature should opt to dial in such values is far from obvious, *'.' ••• ough it does keep some theorists amused. One recent investigation8 even claims to find a phase transition from neutron matter

to strange quark matter t: just less than 2p0. We have reinvestigated these possibilities in a recent article1 and find agreement with the negative results of the 1970's. Perturbative QCD calculations cannot be used reliably to determine strange matter

properties near p0- Here the leading log expression for the running strong coupling constant,

a. = (33 - 2n/) In (fc/A)

is useless, the momenta corresponding to distances r ~ A"1 ~ 1 fm. The precise behavior

of a.a in this regime is not known; strong coupling must be used in the confinement regime. Consider a system of A particles confined on a lattice. Each A has one up, down and strange quark, and thus has no charge, like the neutron. Chiral lag model calculations indicate the A particle is significantly larger than the due to the absence of a pion cloud providing a ram pressure9. These find R^ ~ 1.15 fm10, just equal to the radius of

a sphere containing one nucleon at p0. Since this is the equilibrium radius of the A, the pressure vanishes at the edge of the bags which can be considered as Wigner-Seitz spheres, and since the A energy is minimized at this radius, small shifts about it give only negligible energy changes. Inclusion of interactions between the A particles are neglected but should also be relatively small and not change our arguments. The energy per A would be equal to its mass, m^ = 1116 MeV, substantially above the energy per nucleon of neutron matter, ~ rn^ ~ 940 MeV. (In fact, constraining the center of mass motion of the A in a or leads to another 100 ~ 200 MeV additional energy, as pointed out to us by Leonardo Castillejo.) Taking the A-matter as the model of equilibrium strange matter with vanishing pressure near pQ, we see there is no possibility near this density of joining neutron matter to strange matter. This would require equality of chemical potential and pressure for the two phases. A general thermodynamic prescription for locating the phase transition is to require continuity of the intensive variables across the phase boundary. Specifically we require

Pt = PN = Pq (2)

where Pt is the pressure at the transition point and "N" and "q" refer to and respectively; and

r M: = /"A = 3Ai? (3)

for the chemical potentials, three quarks being needed to achieve number of unity. The temperatures of the two phases should also be equal of course, but here we assume T = 0. The phase transition (assumed to be first order) can be constructed graphically as

shown schematically in figure 1. Plotting Pr* vs. fj.^ and Pq vs. 3/i? on the same graph, we find the curves cross a: point "t" where eqs. (2) and (3) are satisfied. Since we have the Maxwell relation

the baryon number density discontinuity at tHe transition is given by the change in the slopes at point "t". We must have the quark curve cross from below of course, in order to have our normal world made up of baryonic matter and not quark matter. The discontinuity in the energy density Ae is given by

Ae = Eq - e.v = MtAp = m (pg/3 - pN) (5)

For the neutron matter we employ a simple power law for illustration12:

KNp0 V?N —g — (6a)

PN _ -,

\ I i i i I i ( i I i i i I ill

or 3 FIGURE 1

The dotted line displays Pg + B vs 3^? for the quarks; the line P^ vs fx^ for the neutron matter. The quark curve must be displaced downward by the value B required to get the first order phase transition. where ujq — p;v/po, KN is the compression modulus of neutron matter at the symmetric matter saturation density po = .16 fm3, 7 is the high density adiabatic exponent, and fh^ is

fhN = mN + ^ + £8ym - —T- -r (7)

?he chemical potential is then given by

7 = (eN + PN) lpN = mN+ -zn (3)

We can eliminate p^ in (8) by using (6a) to obtain

7 (9)

giving us the nuclear curve in figure 1. Equation (9) indicates that stiffening the neutron matter equation of state tends to move the transition downward. If this stiffening is done by increasing iCjv, then at fixed the pressure is reduced, so the nucleon jlurve in figure 1 is displaced downward, crossing the quark curve sooner; it is also easy to s/ee that the neutron density, pjy, is also lowered from equation (4). If stiffening is accomplished by raising 7 exemption of eqs. (6-8) shows the effect to be the same as long as UJV > (7 - 1) /7, which is no problem. Thus using a stiff neutron matter equation of state always promotes the transition to occur at lower densities. However, as we shall see later, stiffer equations of state lower the neutron star central density at a fixed star mass, and so there is a pincers effect which gives some stability to conclusions about the occurrence of the transition. We will thus adopt a stiff equation of state, Bethe-Johnson I11 ( BJl) in the following. We need not consider maintainence of,/? -equilibrium, which could only soften the eos. The BJl neutron matter eos is fit sufficiently well by eqs. (6-9) if we use as parameters

KN = 500 MeV, 7 = 2.52, rhN = 961 MeV {BJl) (10)

This gives 52 PN = 3.48 u^ (lla)

5 HN - 961 = 38.3 utf

with the pressure in MeV fm~3 and the chemical potential in MeV. We now turn to the quark equation of state. The most naive form we could use would be the asymptotically free quark with m, = 0 and a bag constant B. This would give an equation of state 3/z, = 689 u\lz (12a)

(126)

where uq = (pq/3) /PQ. Given a value for B, equations (2,3,11,12) can be solved to obtain

m, Pt, and uq, UN at point "t". The result is plotted in Fig. 2, the dashed line showing

Pq + B vs 3fiq for the noninteracting quark eos, and the soli 1 line P/y vs /zjy for the BJl

neutron matter eos. At a given chemical potential m, at the transition, we displace the Pq-rB curve by a distance B downward. In Fig. 3 the B thus obtained is plotted as a function of the neutron density. At vanishing neutron density we would obtain £if = 961 MeV, giving u, = 2.7 and B = 103 MeV. A value B ~ 60 MeV as would seem necessary for MIT bag model phenomenology would not be allowed. Accepting a higher value for B, one would still be hard put to specify the transition. In particular, one could quite easily obtain quark 1 00 J . . .-[ iii. T"l '/I' i 1

/ • // / / * / / • / / 6OO // f/ FREE // INT. / / y / / — // - y Cu 4OO _____ // - y/ BJ 1 . // / / / // / // •"•"• s * 200 s

o , . . 1 1 1 1 I r I , 1000 1200 1400 1600 1800 2000

FIGURE 2 Same as Fig. 1, using the BJl eos. The dashed line is for free quarks; the dotted line for interacting quarks.

300 1 i I i i 1 / I i i i

/ — FREE 200 ~~~— s

s — 1 ! 100 — INT. •'

o 1 I 1 1 1 1 1 l I I l I l 6 8 10

FIGURE 3 The bag constant required for the phase transition as a function of the neutron density, UN = PNPo- matter unphysically at subnuclear densities. At any rate, here precisely is where strongly interacting quart ; must be used so the asymptotically free case has only pedagogical value.

Interactions can be included perturbatively in aa and to first order in the energy per quark

where 2 1/3 l k, = (rr p?/3) = 1.16 u j* (14)

The pressure is obtained by differentiation, and for this the density dependence of as requires specification. We choose a.3 ~ l/kq, or equivalently

(0) -1/3 ,,,v a3 = a\ 'ug ' (lo) where

logical as ~ R behavior in bag model calculations in the nonperturbative regime. The only i energy scale in bag model fits is R~ , and so gluon exchange energies must go as cx3jR. We use the ground state-decouplet splitting of the strange to avoid the pion cloud which gives much of the N — A splitting. For a baryon radius R ~ 0.64 fm, Carlson et a! 14 lo find as ~ 1, while DeGrand et al obtain a3 ~ 2.2 using radii R = 1 —1.1 fm. This drop is somewhat faster than linear in R, but other effects of higher order are involved. Moving to higher densities the rate of decrease will continue to be greater than the logarithmic depen- dence of eq. (l) and thus we will underestimate the transition density. We will choose the

value a, = 2.2^°) from MIT bag model phenomenology. Since this is the as which fits the n a regime near p0 ^ linear approximation the best consistency is to stop before the quadratic terms in eq. (13). Obtaining the pressure and chemical potential, we find

(16a)

As a function of quark density, Pq + B is unchanged, while the chemical potential is increased. Thus the slope in Fig. 2 is lessened and the quark matter curve is displac . to the right, as indicated by the dotted line in Fig. 2. The B obtained is plotted in Fig. 3 with a dotted line and is seen to have a minimum at upj = 5. In Fig. 2 this is at the

point of tangency of the P^ and Pq -+• B curves. Below this point the quark pressure has a greater slope, and thus by eq. (4) quark matter is at a lower density than neutron matter. If taken seriously, this would mean the normal world would be made up of quarks. This is of course absurd, but is the result of employing perturbation theory where it is entirely non-applicable. Figure 3 indicates that there is no way the phase transition could occur for u^ < 6; for a value B = 60 MeV /m~3, such as goes along with the a[ ' = 2.2 of the MIT bag model fits we would obtain u^v ~ 7 at the transition. Thus we find that there is no possibility for the existence of strange quark matter for densities near normal nuclear densities, in the strong coupling regime, and underestimate the density of such a transition to be at least 7po- Whether or not such matter can exist in the centers of neutron stars, therefore, depends on whether such densities are reached

in neutron stars. For the BJl eos, one finds that for a neutron star of M = 1.4 AfQ , the central density is 3.6po and thus cannot have a quark core. For the maximum mass BJl

neutron star, the central density is 8.1p0 and thus could possibly have a small quark core, but is is possible to show its effect would be unnoticeable. However, we have estimated a lower bound to the transition density, and so the occurrence of quark matter would remain unlikely even in this case. If we had chosen a stiffer eos than BJl, the transition density would decrease somewhat, but then the central density would decrease in the neutron star. Likewise a softer eos would have opposite effects on both quantities. Thus, our results are not very model dependent on the neutron matter side. Our conclusion is that strange quark matter will not be found in neutron, or "strange", stars. References 1. H. A. Bethe, G. E. Brown, and J. Cooperstein, Stars of Strange Matter, Nucl. Phys. A (1987)(in press). 2. V. R. Pandharipande, D. Pines and R. Smith, Ap. J. 208 (1975), 507. 3. G. Baym and S. A. Chin, Phys. Lett. 62B (1976), 241. 4. G. Chapline and M. Nauenberg, Phys. Rev. D16 (1977), 450. 5. B. A. Freedman and L. D. McLerran, Phys. Rev. D16, 1169 (1976); B. A. Freedman and L. D. McLeiran, Phys. Rev. D17, 1109 (1978). 6. E. Witten, Phys. Rev. D30 (1984),272. 7. E. Farhi and R. L. Jaffe, Phys. Rev. D30 (1984), 2379. 8. P. Haensel, J. L. Zdunih and R. Schaeffer, (19S5)N. Copernicus Astronomical Center Preprint. 9. G. E. Brown and M. Rho, Comments in Nuclear and (to be published). 10. G. E. Brown, M. Rho and W. Weise (to be published). 11. H. A. Bethe and M. B. Johnson, Nucl. Phys. A230 (1974), 1. 12. E. Baron, J. Cooperstein, and S. Kahana, Nucl. Phys. A440 (1985), 744; Phys. Rev. Lett. 55 (1985), 126. 13. G. E. Brown and M. Rho, Phys. Rev. Lett. 82B, (1979)177. 14. C. E. Carlson, T. H. Hansson, and C. Peterson, Phys. Rev. D27 (1983), 1556. 15. T. A. DeGrand, R, L. Jaffe, K. Johnson, and J. Kiskis, Phys. Rev. D_12 (1975), 2060. BROOKHAVEN NATIONAL LABORATORY MEMORANDUM

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