NUCLEAR PHYSICS A

ELSEhER Nuclear Physics A639 (1998) 45 1~454~

Hyperon formation constrains the existence of strange quark stars

Shmuel Balberga

aRacah Institute of Physics, The Hebrew University, Jerusalem 91904, Israel

The strange quark matter hypothesis [1,2] suggests that deconfined quark matter com- posed of about equal fractions of up, down and strange quarks could be the true ground state of matter. This will be true if the ground state of strange quark matter (SQM) has a lower energy per baryon than normal nuclear matter, making it absolutely stable.

The SQM hypothesis is of fundamental importance with regard to neutron star struc- ture. If a “seed” of SQM appears in the core of a neutron star it will proceed to consume the surrounding neutrons, which would lead to a burning (exothermal) conversion of the entire star, deconfining the baryon matter into SQM. The end product of this conversion will be a compact object composed almost entirely of SQM, which has been referred to as a “Strange Quark Star” [3,4]. The timescale for such a burning has been estimated to be of the order of minutes [5]. Thus, the SQM hypothesis predicts that some (and maybe all) neutron stars could actually be strange quark stars.

Various processes have been suggested for creating SQM seeds in neutron stars [3], and can be divided into two distinct sources:

?? The absorption of an external source SQM “nugget” formed elsewhere (for exam- ple, from debris of a strange star which has coalesced with a binary counterpart). However, a quantitative estimate of the flux of SQM in the universe is dependent on many assumptions, and thus it is impossible to constrain the flux (various arguments can be made for evolutionary selection effects).

?? A SQM seed could form as an internal source in the core of a neutron star should deconfinement become favorable. The availability of internal sources clearly de- pends only on the structure of the neutron star. Furthermore, assuming the SQM hypothesis is correct, if some internal source is available in all neutron stars, then all existing neutron stars must actually be strange quark stars.

The purpose of this work is to illuminate the role of hyperon formation as an available mechanism for creating SQM in all neutron stars at birth. Conversion of nuclear matter into SQM is suppressed by the weak interaction, even if the latter is more stable, since such a decay would require multiple and simultaneous conversion of many non-strange quarks into strange ones. A significant hyperon fraction can, on the other hand, stimulate the formation of a SQM seed by providing the necessary strange quark content so that deconfinement proceeds through the strong interaction. Note that the same argument holds for kaon condensation, which is not discussed here.

The consensus among recent works is that hyperon formation in neutron stars should begin at rather low densities. It is widely accepted [6-91 that hyperons begin to accumulate

0375-9474/98/$19.00 0 1998 Elsevier Science B.V. All rights reserved. PI1 SO375-9474(98)003 11-X

452~ S. Balherg/Nuclear Physics A63Y (1998) 451c-4.54~

at about twice the nuclear saturation density, ps (po zz 0.16 fme3). The relevant hyperons at low densities (pn 5 3~0) are the C- and A. Their combined abundance yields a strangeness per baryon fraction, ISl/A, that exceeds 0.1 at pn = 2.5~0, and at pi z 3~0 is already N 0.2. This wide agreement is a direct consequence of employing realistic hyperon-nuclear-matter interactions based on hypernuclear data, and is only weakly model dependent [9].

It is noteworthy that hyperon induced deconfinement does not require the baryon strangeness fraction to be equal to that of ground-state SQM, ISl/A M 0.748 [a]. Rather, the condition is only that the phase transition into SQM with a composition identical to that of the baryon matter, e.g. strong deconfinement, be energetically favorable; the SQM then reaches its ground state by series of weak decays.

In the following analysis, strong-interaction deconfinement is examined through bulk deconfinement, which can be shown to set an upper limit condition for formation of quark matter [lo]. A bulk of baryon matter of baryon number density pn and underlying quark composition {qi} will deconfine spontaneously through the strong interaction into quark matter of equal density and composition if the latter has a lower energy per baryon, i.e.,

where E is the energy density, and the subscripts B and Q denote the baryon and quark phases, respectively. Whether or not deconfinement will occur in a neutron star depends on the resulting density of deconfinement, which is the lowest density where Eq. (1) is satisfied. Deconfinement should proceed if this density, which is dependent on the equations of state of the two phases, is reached in the star’s interior.

The quark-matter equation of state is evaluated here with the MIT bag model detailed in [2]. Although this model cannot be taken as a comprehensive description of quark matter, it does provide a quantitative indication for the properties of such matter. The specifics of the bag model are determined by the values of the bag constant, B, and the quark interaction coupling constant, Q,. For each value of crcr B is limited from below so that two-flavor quark matter is less bound than symmetric nuclear matter, since ordinary nuclei do not deconfine strongly. The SQM hypothesis places an upper limit in order for the ground state composition of SQM to be more bound than symmetric nuclear matter. Combining these two conditions constrains the allowed range of values for the bag constant for any given value of cy, [2]. For example, for crc = 0, the value of the bag constant is limited to the range 56 MeV fme3 2 B 5 82 MeV fme3.

Fig. 1 compares the energy per baryon of baryonic matter with hyperons calculated with a model similar to b = y = 4 of [9], to the energy per baryon of quark matter with identical quark composition, using different combinations of B and a,. The up and down quarks were assumed to be massless, and the mass of the strange quark is set to 150 MeV. The deconfinement density corresponds to the point where the energy per baryon in the baryonic phase crosses that of the quark phase.

It is found that all models which allow SQM to be the true ground state also predict that bulk deconfinement occurs at densities lower than 3~s. A deconfinement density of pn 5 2.5~0 is found even for combinations which are borderline for keeping SQM more bound than ordinary nuclear matter, i.e. (B=82, o,=O), or (B=63, c~~=O.3). Even in

S. BalberglNuclear Physics A639 (1998) 451c-454~ 453c

the combination (B = 100, Q, = 0), for which the SQM hypothesis is no longer valid, the deconfinement density is still found to be lower than 3~0. Quark matter models which do not allow spontaneous deconfinement are possible, of course, like (B = 125, a, = 0) shown in Fig. 1. In such models the corresponding binding energy of SQM is significantly larger than ordinary nuclear matter. In this case, some deconfinement might occur, and the resulting state could be coexisting baryon and quark phases [ll].

Figure 1. Energy per baryon of bary- onic matter in equilibrium (solid line), and of quark matter of identical compo- sition and density (dashed lines), as a function of the baryon number density. Quark matter curves correspond to dif- ferent bag models, denoted in brackets as (B, a,) - see text. The arrow marks the density where the first hyperons ap- pear (~0.3 fmw3).

0.2 0.3 0.4 0.5 0.6

PB 6’)

The prevailing effect is that the the nonzero strangeness fraction lowers the energy per baryon in the deconfined phase much more than the in the baryonic phase. This can be simply understood in terms of the Fermi energies, which in the quark phase are much larger than the strange quark mass, while in the baryonic phase the mass differences are of greater significance. If hyperon formation is ignored, the baryonic matter includes only two flavors of quarks, and deconfinement does not necessarily occur at low densities, even for models which are compatible with the SQM hypothesis.

Similar results are found for other models of the baryon equation of state, since the equation of state with hyperons is limited to a rather narrow range of values [9]. In any case a valid equation of state must be stiff enough to support a maximum mass of at least 1.4 A4D (the determined mass of pulsar 1913+16). While the quark bag model is clearly a simplified description of quark matter physics, it seems that the margin it allows for low-density deconfinement into SQM is large enough, so that the qualitative results are unlikely to be dependent on the quark matter model as well.

The immediate consequence is that the baryons in the cores of neutron stars with a central density greater or equal to 2.5-3~0 should deconfine into SQM, if such matter is indeed absolutely stable. The SQM will then convert to its equilibrium composition, and proceed to convert the entire star into a strange quark star. Equations of state for high-density matter generally require a central density of pC > 3~0 to support a mass of 1.3-1.5 A4,, which is the range of observed neutron star masses. This is true even for equations which disregard hyperon formation, and is pronounced when hyperons are taken into account, since the inclusion of more baryon species softens the equation of state. Hence, the most likely conclusion is that if the SQM hypothesis is correct, then

454c S. Balberg/Nuclear Physics A639 (1998) 451c-454~

all neutron stars should be strange quark stars, and that conversion into strange quark matter takes place immediately at birth. Alternatively, if the SQM hypothesis is false there are no strange quark stars, but, in any case, a mixture of both classes is ruled out.

Could all neutron stars be strange quark stars? Current observations of neutron star masses and limits on their radii are inadequate to distinguish a neutron star with a baryonic interior from a strange quark star - at 1.4 M, both objects would be similar in size. It has been suggested [la] that glitch phenomena in pulsars may serve as indication against strange quark stars. Glitches are sudden changes in the rotation frequencies of pulsars, and current models suggest that they originate from the inner crust of the pulsar. These models require a glitching pulsar to have a relatively large crust, which a “normal” neutron star can hold but a strange quark star cannot. Glendenning and Weber have argued [13] that glitches could originate even in the very low-mass crust of strange quark stars, but up to date no model for strange quark star glitch has been suggested. While this argument cannot serve as decisive proof, it does lean towards the interpretation that at least glitching neutron stars are not strange quark stars.

In conclusion, it seems that on the basis of hypernuclear experiments hyperons should be included in supernuclear density matter equations of state, and that all neutron stars should have central densities which allow for formation of a significant hyperon fraction. In turn, this result provides a robust mechanism for the creation of strange quark matter in all neutron stars, if such matter is indeed the true ground state of baryonic matter. If so, all neutron stars should convert at birth to strange quark stars - a possibility which is difficult to reconcile with observed pulsar phenomena. Hence, widely accepted evaluations of hyperon formation in neutron stars [6-9] may serve to set a severe constraint on the existence of strange quark stars and the validity of the strange quark matter hypothesis.

The author is grateful to Avraham Gal for extensive guidance and advice. This re- search was partially supported by the U.S.-Israel Binational Science Foundation grant 94-68.

REFERENCES

1. A.R. Bodmer, Phys. Rev. D 4 (1971) 1601; E. Witten, Phys. Rev. D 30 (1984) 272. 2. E. Farhi and R.L. Jaffe, Phys. Rev. D 30 (1984) 2379. 3. C. Alcock, E. Farhi and A. Olinto, Astrophys. J. 310 (1986) 261. 4. For a review see N.K. Glendenning, Compact Stars, (Springer, New York, 1996). 5. A.V. Olinto, Phys. Lett. B 192 (1987) 71; Nucl. Phys. B (Proc. Suppl.) 24 (1991) 103;

H. Heiselberg, G. Baym and C.J. Pethick, Nucl. Phys. B (Proc. Suppl.) 24 (1991) 144. 6. N.K. Glendenning, Astrophys. J. 293 (1985) 470. 7. R. Knorren, M. Prakash and P.J. Ellis, Phys. Rev. C 52 (1995) 3470. 8. J. Schaffner and I.N. Mishustin, Phys. Rev. C 53 (1996) 1416. 9. S. Balberg and A. Gal, Nucl. Phys. A 625 (1997) 435. 10. S. Balberg, astro_ph/9708134. 11. N.K. Glendenning, Phys. Rev. D 46 (1992) 1274. 12. M.A. Alpar, Phys. Rev. Lett. 58 (1987) 2152. 13. N.K. Glendenning and F. Weber, Astrophys. J. 400 (1992) 647.