Incompressibility of strange matter

Strange stars calculated from a realistic equation of state (EOS), that incorporate chiral symmetry restoration as well as deconfinement at high density show compact objects in the mass radius curve. We compare our calculations of incompressibility for this EOS with that of nuclear matter. One of the nuclear matter EOS has a continuous transition to ud-matter at about five times normal density. Another nuclear matter EOS incorporates density dependent coupling constants. From a look at the consequent velocity of sound, it is found that the transition to ud-matter seems necessary.


Introduction
An exciting issue of modern astrophysics is the possible existence of a family of compact stars made entirely of deconfined u,d,s quark matter or "strange matter" (SM) and thereby denominated strange stars (SS). They differ from neutron stars, where quarks are confined within neutrons, protons and eventually within other hadrons (hadronic matter stars). The possible existence of SS is a direct consequence of the so called strange matter hypothesis [2], according to which the energy per baryon of SM would be less than the lowest energy per baryon found in nuclei, which is about 930 MeV for F e 56 . Also, the ordinary state of matter, in which quarks are confined within hadrons, is a metastable state. Of course, the hypothesis does not conflict with the existence of atomic nuclei as conglomerates of nucleons, or with the stability of ordinary matter [3,4,5].
The best observational evidence for the existence of quark stars come from some compact objects, the X-Ray burst sources SAX J1808.4−3658 (the SAX in short) and 4U 1728−34, the X-ray pulsar Her X-1 and the superburster 4U 1820−30. The first is the most stable pulsating X-ray source known to man as of now. This star is claimed to have a mass M * ∼ 1.4 M ⊙ and a radius of about 7 kms [6]. Coupled to this claim are the various other evidences for the existence of SS, such as the possible explanation of the two kHz quasi-periodic oscillations in 4U 1728 -34 [7] and the quark-nova explanation for γ ray bursts [8].
The expected behaviour of SS is directly opposite to that of a neutron star as Fig.(1) shows. The mass of 4U 1728−34 is claimed to be less than 1.1 M ⊙ in Li et al. [7], which places it much lower in the M-R plot and thus it could be still gaining mass and is not expected to be as stable as the SAX. So for example, there is a clear answer [9] to the question posed by Franco [10]: why are the pulsations of SAX not attenuated, as they are in 4U 1728−34 ?
From a basic point of view the equation of state for SM should be calculated solving QCD at finite density. As we know, such a fundamental approach is presently not feasible even if one takes recourse to the large colour philosophy of 't Hooft [11]. A way out was found by Witten [12] when he suggested that one can borrow a phenomenological potential from the meson sector and use it for baryonic matter. Therefore, one has to rely on phenomenological models. In this work, we use different equations of state (EOS) of SM proposed by Dey et al [1] using the phenomenological Richardson potential. Other variants are now being proposed, for example the chromo dielectric model calculations of Malheiro et al. [13].  In nuclear physics incompressibility is defined as [14] K = 9 ∂ ∂n n 2 ∂ε ∂n ,   where ε = E/A is the energy per particle of the nuclear matter and n is the number density.
The relation of K with bulk modulus B is K has been calculated in many models. In particular, Bhaduri et al [14] used the nonrelativistic constituent quark model, as well as the bag model, to calculate K A as a function of n for the nucleon and the delta. They found that the nucleon has an incompressibility K N of about 1200 MeV, about six times that of nuclear matter. They also suggested that at high density K A matches onto quark gas incompressibility.

The velocity of sound in units of light velocity c is given by
The simple models of quark matter considered in [1] use a Hamiltonian with an interquark potential with two parts, a scalar component (the density dependent mass term) and a vector potential originating from gluon exchanges. In the absence of an exact evaluation from QCD, this vector part is borrowed from meson phenomenology [15]. In common with the phenomenological bag model, it has built in asymptotic freedom and quark confinement (linear). In order to restore the approximate chiral symmetry of QCD at high densities, an ansatz is used for the constituent masses, viz., where n 0 is normal nuclear matter density and ν is a parameter. There may be several EOS's for different choices of parameters employed to obtain the EOS. Some of them are given in the table (1). There α s is perturbative quark gluon coupling and Λ is the scale parameter appeared in the vector potential. Changing other parameters one can obtain more EOS's. The table (1) also shows the masses(M G ),radii (R) and the number density(n s ) at star surface of maximum mass strange stars obtained from the corresponding EOS's. The surface of the stars occur at the minimum ε where pressure is zero. The general behaviour of the curves is relatively insensitive to the parameter ν in M(n) as well as the gluon mass, as evident from the figure(3). In figure (4), K with three values of M Q implying different running masses, M(n), is plotted as a function of the density expressed by its ratio to n 0 . Given for comparison, is the incompressibility K q of a perturbative massless three flavour quark gas consisting of zero mass current quarks [14] using the energy expression given in [16] to order α 2 s .   [14,16]).
It can be seen that as M Q decreases, the nature of the relation approaches the perturbative case of [14]. At high density our incompressibility and that due to Baym [16] matches, showing the onset of chiral symmetry restoration. In EOS1 for uds matter, the minimum of ε occurs at about 4.586 n 0 . nucleation may occur at a density less than this value of n. This corresponds to a radius of about 0.67 f m for a baryon. For EOS1 we find K to be 1.293 GeV per quark at the star surface.
It is encouraging to see that this roughly matches with the compressibility K N so that no 'phase expands explosively'. In the Cosmic Separation of Phase scenario, Witten [12] had indicated at the outset that he had assumed the process of phase transition to occur smoothly without important departure from equilibrium. If the two phases were compressed with significantly different rates, there would be inhomogenieties set up.
But near the star surface at n ∼ 4 to 5 n 0 the matter is more incompressible showing a stiffer surface. This is in keeping with the stability of strange stars observed analytically with the Vaidya-Tikekar metric by Sharma, Mukherjee, Dey and Dey [17].
The velocity of sound, v s peaks somewhere around the middle of the star and then falls off. We show it for three different EOS in fig.(5) with parameters given in Table (1 Next we turn to the model of Zimanyi and Moskowski [18], where the coupling constants are density dependent. It was shown [19] that the quark condensate derived from this model via the Hellmann-Feynman theorem is physically acceptable in this model. Details may be found in [19] but the essence is that in the more conventional Walecka model the condensate increases with density whereas in QCD a decrease is expected. In a recent paper [20] Krein and Vizcarra (KV in short) have put forward an EOS for nuclear matter which exhibits a transition from hadronic to quark matter. KV start from a microscopic quark-meson coupling Hamiltonian with a density dependent quark-quark interaction and construct an effective quark-hadron Hamiltonian which contains interactions that lead to quark deconfinement at sufficiently high densities. At low densities, their model is equivalent to a nuclear matter with confined quarks, i.e., a system of non-overlapping baryons interacting through effective scalar and vector meson degrees of freedom, while at very high densities it is ud quark matter. The K N M at the saturation density is fitted to be 248 MeV . This EOS also gives a smooth phase transition of quark into nuclear matter and thus, conforms to Witten's assumption. Interestingly enough, the transition takes place at about ∼ 5n 0 . The KV model does not incorporate strange quarks so that comparison with our EOS is not directly meaningful. However it is quite possible that the signal results of the KV calculations mean that quark degrees of freedom lower the energy already at the ud level and once the possibility of strange quarks is considered the binding exceeds that of F e 56 . An extension of KV with strange quarks is in progress 1 . Note that in our EOS we also have strange quarks reducing the value of K. Fig (9) shows the v s as a function of n B /n 0 for both kinds of EOS. The EOS with quarks shows a lowering of v s .

Discussions.
In our model the density is large even at the surface (4.586 n 0 for EOS1) so that the quarks in the fermi sea have high momentum on the average. Therefore, they often come close together. It was shown by Bailin and Love that in certain colour channels [21], the short range one gluon exchange spin-spin interaction can produce extra attraction, forming diquarks. In s-state the pair can be formed only in flavour antisymmetric state. The flavour antisymmetric diquark may form in colour symmetric (6) and spin symmetric (triplet) state, or in colour antisymmetric (3) and spin antisymmetric (singlet) state. The potential strength of the latter is 6 times the  Figure 9: The velocity of sound in pure nuclear matter and in quark-nuclear system as a function of density ratio. For pure nuclear matter at 9n 0 the sound velocity is too close to that of light c, whereas for quark-nuclear system it is much less, about 0.5 c.
former. The N-∆ mass splitting is obtainable from the one gluon exchange potential, the delta function part of which is smeared to a Gaussian, H i,j = − 80.51 σ 3 (λ i .λ j )(S i .S j ) e −σ 2 r 2 ij .
In the above, the strength of this spin dependent part is taken from Dey and Dey(1984) [22]. The pairing energy of ud pairs in the spin singlet and colour3 state is −3.84 MeV [23] 2 . Our model does not predict diquarks that are permanent -since the quarks must have high momentum transfer if they are to interact strongly with a force that is short-range. The formation and breaking of pairs give rise to endothermic and exothermic processes leading to fast cooling and superbursts respectively.
We also note that the lowering of energy is not very large, as compared to the approximately 42 MeV energy difference per baryon (930.6 F e 56 to 888 MeV ), seen in EOS1. So we do not expect any drastic change in the incompressibility or the velocity of sound due to diquark pairing, nor do we expect a phase transition to a colour-flavour locked state.
We are grateful to the referee for inviting a comment on quark pairing in the strange matter scenario.