QQuark cores in extensions of the MIT Bag model
Salil Joshi, Sovan Sau, and Soma Sanyal
School of Physics, University of Hyderabad,Gachibowli, Hyderabad, India 500046
Abstract
Recent observations of massive pulsars having masses of the order of two solar mass pose a newchallenge for compact objects such as hybrid stars and neutron stars. Extensions of the bag modeland the Nambu-Jona-Lasino model have been used to model these stars to get higher mass stars.Quark matter has been predicted in the cores of these massive stars. In this work we show thatan extension of the bag model, with a chemical potential dependent bag parameter can lead to anisentropic phase transition in the core of the neutron star. Our model shows that an EoS havingall three quarks u , d and s would lead to massive stars with stable quark matter. We find thatthe mass of the stars not only depends on the bag constant but also on the mass of the strangequark. The mass - radius ratio which determines the redshift values on the surface, indicates thatit is possible to obtain stable self bound strange quark matter stars with reasonable values of thebag pressure which correspond to the recent observations of large mass stars. Keywords: strange matter, stability, Equation of state. a r X i v : . [ nu c l - t h ] J u l . INTRODUCTION Among the various hypothetical compact objects discussed in the literature, the quarkstar and the hybrid star hold a very important place, as they are objects capable of generatinggravitational waves. The detection of gravitational waves by recent experiments raise thepossibility of experimental determination of these special stars. Stable quark stars had beenfirst predicted in ref.[1]; this was followed by various other studies on quark stars [2]. Thepossibility of quark matter inside the cores of neutron stars has also been discussed [3–5].Recent observations of relatively massive pulsars like PSR J1614-2230 and PSRJ0348+0432,have further contributed to the challenges presented by these stars.The Equation Of State (EoS) of these stars present a challenge in the current scientificscenario. The most commonly used EoS is the bag model [6–9]. Apart from the standard bagmodel there have been discussions in the literature of several extensions of the bag model [10].Recent studies of hybrid stars have shown that it is possible to have stars with masses of theorder of two solar masses and higher. These higher mass limits are usually set by observationsof pulsar glitches. The masses of these stars (PSR J1614-2230 and PSRJ0348+0432) [11, 12]can be successfully modelled in extensions of the bag model[13]. In these studies, the authorhad used a density dependent bag constant. Such density dependent bag parameter has alsobeen used by other groups to study strange stars [14]. Various modified bag constants havebeen used to understand the quark hadron phase transition in baryon dense environments.It has been established that bulk strange matter can be stable [15]. They can be formed bythe trapping of strange matter due to phase transitions in the early universe or by the slowburning of the core of the neutron star which gradually changes it to a quark star [16]. Thusstrange bulk matter can exist as cores in neutron stars, as quark stars or as hybrid stars.Apart from these there have been discussions of a color flavor locked phase in the core of theneutron stars and strange neutron stars [17]. The Nambu Jona Lasino model has also beenused to show that the core of massive neutron stars may contain a region of quark hybridmass [18]. All these models have been motivated by various observational data on the massand radius of these stars.There is also the question of the phase boundary between the confined and the deconfinedphases. The nature of the quark hadron phase transition usually gives the kind of boundarybetween the two phases. In recent times, there have been quite a few models where a fixed2ntropy per baryon has been suggested at the phase boundary [19]. The different stages ofcooling of proto-neutron stars are modelled as isentropic phase transitions. This has beendone using the Maxwell construction. Other than this, the isentropic phase transition isalso considered for neutron stars at birth using detailed simulations [20]. The isentropicquark-hadron phase transition had been considered for the extended bag model previouslyin the case of relativistic heavy ion collision as well as the early universe [21, 22]. It has beenconjectured that an extended bag model with a temperature and baryon chemical potentialdependent bag pressure can give an isentropic phase boundary between the two phases. Suchmodified bag models have been discussed in some detail in the literature . In this work weuse the method introduced in ref. [21] by Leonidov et. al. to see whether such extension ofthe bag models can lead to stable strange bulk matter with mass and radius consistent withexperimental observations. Our aim is to demonstrate that an extension of the bag modelcan explain the current observables related to large mass stars with a deconfined quark gluonplasma at its core.Here we have characterized the properties of these massive objects using the bag constantand the mass of the strange quark. Though we have studied other parameters too, these twoproperties are the ones that we focus on. Since the strange quark mass is already constrainedfrom other experiments, this helps in obtaining very specific and limited ranges for ourcompact objects. The quark matter that we get in this model is energetically favourable overthe Fe crystal. The other requirement of any such model is the gravitational stability. Thestandard way of checking the stability is through the Tolman - Oppenheimer - Volkoff (TOV)equations. The partition function usually gives the energy density, pressure and numberdensity of the quark gluon plasma. These are then substituted in the TOV equations. TheTOV equations are routinely used to study the stability of strong gravitational objects likethe neutron stars, the hybrid stars and the quark stars. The TOV equations correspondingto the EoS of the modified bag model are then solved numerically to obtain the mass radiusratio corresponding to various different chemical potentials at different temperatures. Wefind that the EoS leads to structures that are quite stable for a large range of chemicalpotentials at low temperatures.We also study an experimental observable that is directly related to the compactness ofthe star. The mass and radius of the star allow us to calculate the surface red shift of thestar. This can be measured through direct observation and is well documented for most3tars. There are several massive stars which are considered as candidates for strange stars.We calculate the red shift from our model and find that it is within the range of severalcandidates for strange stars. We will discuss these candidates in detail in section IV of thepaper.In section II we present the EoS of the extended bag model used for modelling thequark cores of the strange matter in more detail. We have divided this section into threesubsections. In the first subsection, we discuss the bag model EoS with two massless flavorsof quarks, the u and the d . In the second subsections, we discuss the EoS with three masslessflavors of quarks, the u , d and the s . In the third subsection we discuss the EoS where thestrange quark is taken to be massive compared to the other two quarks. In section III,we discuss the TOV equations and their numerical solutions, In section IV, we present theresults and discuss the possible candidates which fit our model. Finally we present ourconclusions in section V. II. THE EXTENDED BAG MODEL
We first briefly discuss the extended bag model that we use in our calculations. Theoriginal MIT bag model was the first phenomenological model which successfully modelledthe phase transition of the quark gluon plasma (QGP) state to the hadronic phase. The phasetransition was a first order phase transition which led to inhomogeneous baryon densitiesin the plasma [1]. The equation of state consisted of a bag constant B which was actuallythe pressure difference between the two phases. Leonidov et. al [21] modified the bagmodel by changing the bag constant to a variable dependent on chemical potential µ andtemperature T . Their model was consistent with the Gibbs equilibrium criteria for a phasetransition. The model also conserved baryon number and entropy at the phase boundary.Following Leonidov’s approach others also modified the bag constant by making it dependenton the chemical potential and temperature [22]. The discovery of the massive neutron starsindicates that the EoS leading to the quark cores in the neutron stars has to be reasonablystiff. We find that the isentropic phase transition in the core with a chemical potential andtemperature dependent bag constant does provide this stiffness to the EoS. The entropy perbaryon number is also maintained at the boundary. We find that it is possible to have selfbound strange quark matter in the core within reasonable values of bag pressure. Though4n isentropic phase transition is not a strong requirement for phase transitions in theseconditions, the grand canonical partition function used by Leonidov [21] and Patra [22] dogenerate the stiffness that is required to have stars with a mass greater than 1 . M and radiiof the order of 12 kms.We use their grand canonical partition function and calculate the number density andthe pressure of these stars. For massless quarks, both two flavors ( u and d ) as well as threeflavors ( u , d and s ), the energy density and pressure is given by, (cid:15) = N c N f π T + µ T + µ π ) + π N g T + B ( µ, T ) − ∂B ( µ, T ) ∂µ T − ∂B ( µ, T ) ∂T µ (1)Here N c and N f are the number of color and flavor degrees of freedom. P = N c N f π T + µ T + 12 π µ )+ π N g V T − B ( µ, T ) (2)In our model, we consider the EoS in the limit of high baryon density. As was shown byLeonidov [21], in this limit, the thermodynamical quantities can be written as the sum of twoparts. One part gives the zero temperature contribution and the second part gives the finitetemperature contribution. When we consider the s quark to be massive, these contributionschange significantly. Hence in the next subsections, we will first consider the quarks to bemassless and obtain the mass and radius relations for the three flavor case. Then we willconsider the strange quark to be massive and obtain the mass and radius relations for thatcase separately. For the two flavor case, the plasma is found to be meta stable. A. Case 1: Two massless flavors ( u and d ) We first look at the two flavor u and d case. Here we do not consider the leptons butmaintain charge neutrality by assuming that the number of d quarks is twice that of the u quarks. As has been shown in ref.[23], the plasma stability parameters are only weaklydependent on lepton contributions in the case of the two flavor plasma hence the leptoncontribution to the charge neutrality can be ignored. The general way to construct the EoSis to describe the EoS in each of the two phases and then perform the Maxwell constructionto join the two phases along their common boundary. We follow Leonidov’s approach in5odifying the bag constant. This means that at the phase boundary the following equationmust hold, S q − ∂B ( µ,T ) ∂T n q − ∂B ( µ,T ) ∂µ = S H n H (3)Here the suffix ”q” denotes the thermodynamical variables in the quark phase, while thesuffix ”H” denotes the same variables in the hadronic phase. The modified bag’s constantin our case is the following, B ( µ, T ) (cid:39) B + µ T − µ T θ H (4)where θ H = ( µ − m H ) / and m H is hadron mass.While the core consists of the quarks, the other side of the boundary is the hadronicphase. In this case, the hadronic phase consists of a non-interacting neutron - pion gas. Thehadronic partition function, is based on a hard core neutron neutron repulsion. Due to this,it is necessary to divide any thermodynamical quantity for point like particles by a volumefactor [21]. The volume factor is given by, (1 + πr n ), where r n is the radius of the neutrons.For the case of the two flavor massless quarks, a change in the volume factor does not giveany significant change in the results. However,in the case of three flavor massless quarks, achange in the volume factor affects the values of the chemical potential of the quarks andthe value of the corresponding bag constant for which the star is stable. The importantquantity that can bring in a change in the volume factor is the radius of the neutron. Wewill discuss this later when we discuss the three flavor massless case. As mentioned in theintroduction, we need to check the gravitational stability of the massive objects. For this weneed to solve the TOV equations using the pressure obtained from our EoS. So the pressure,in our case turns out to be, P = 13 ( (cid:15) − B ) − (cid:18) µ T θ H − µ T θ H (cid:19) . (5)Apart from the gravitational stability, to form a stable plasma state the energy per baryonnumber of the quark gluon plasma has to be calculated. For the two flavor quark matterat low pressures, the energy per baryon number should be larger than that of the nuclearplasma (which is around 940 MeV) and also a F e crystal which has the minimum bindingenergy per nucleon [23, 24]. Substituting all these constraints, we find the chemical potentialis constrained between 313 . . /f m . However, the plasma in this case is meta stable and one cannotgive a definite mass and radius to this plasma, hence we introduce the strange quark andlook at the plasma composed of three massless quark flavors. B. Case 2: Three massless flavors ( u , d and s ) We now consider the plasma to have three massless flavors of quarks. We treat the plasmaas a degenerate Fermi gas with equal numbers of quarks of different flavors. This and thefact that the plasma is beta equilibrated gives the necessary charge neutrality to the plasma.The only change in the expressions for the bag constant and the pressure come from thefact that the number of flavors have been increased to three. As of now the assumption of µ u = µ d = µ s holds as all the flavors are considered to be massless. The bag parameterbecomes, B ( µ, T ) (cid:39) B + 32 (cid:18) µ T − µ T θ H (cid:19) (6)and the corresponding pressure equation is given by, P = 13 ( (cid:15) − B ) − (cid:18) µ T θ H + 18 µ T θ H (cid:19) (7)Here B is the bag parameter at T = 0. For this case too, we have to check the stabilityof the plasma. Since the number of flavors has increased, the energy per baryon will alsochange accordingly. The preferred state of a stable plasma comprising of all the quark flavorsshould have energy per baryon less than that the nuclear energy of 940 MeV so that theplasma does not hadronize. Similarly, the energy per baryon should be less than the bindingenergy of the F e crystal. The lower energy per baryon will make the quark gluon plasmathe preferred state for the bulk matter [23, 24]. As mentioned before, the stability of theplasma puts a constraint on the values of the chemical potential as well as the bag constant.We see that the stability of the plasma increases with smaller bag constant values.Again similar to the previous case, since the hadronic phase is also present, there are someparameters in the hadronic phase which also contribute to the stability of the plasma. Wefind that in the three flavor case, the radius of the neutron which contributes to the volume7 P c * ( P a ) B (MeV) r=0.5 fmr=0.6 fmr=0.7 fm FIG. 1: Core pressure for stable masses at different bag constants for different values of the neutronradius. correction in the neutron - pion gas does change the stability values of the core pressure.This is because we have equated the pressure of the hadronic phase and the quark phase toobtain the bag constant. We have plotted the pressure in the core for different values of bagconstant for different values of the neutron radius for which the quark plasma is stable infig 1. As can be seen from the figure, the increasing bag constant decreases the pressure inthe core.We thus find that the change in the volume corrections to the non-interacting neutron-pion gas affect the limiting values of the pressure and the density at the core. In all the cases,we have considered we have kept the hadronic phase to be the same as we are interestedin the quark phase. We find that as long as the hadronic part consists of a neutron - piongas, the only parameter that affects the stability of the plasma is the radius of the neutronthrough the volume corrections. Though this effect also occurs when the strange quark massis taken into account, the dominance of the strange quark mass leads to very small variationsin the core pressure due to the change in the volume corrections.
C. Case 3: Massless u , d and massive s quarks If we consider the s quark to be massive then we cannot have µ u = µ d = µ s . So it isimportant to consider the first order correction terms to the temperature and the entropy.As in the previous case, strange matter is modelled as a Fermi gas of up, down, and strangequarks. This time the charge neutrality of the system is maintained by the electrons. Weak8nteractions ( β equilibrium) will then lead to µ s = µ d = µ and µ u + µ e = µ [25]. The numberdensities are related by, n u − n u − n s − n e = 0. This means that effectively we are dealingwith only two chemical potentials µ u and µ s .To deal with high baryon densities and low temperatures, finite temperature correctionsare added to the zero temperature terms. This means that, P s (cid:39) P s + P s T + P s T (8) n s (cid:39) n s + n s T + n s T (9) S s (cid:39) S s T + S s T (10)The zero temperature terms are given by, P s = 16 π ( µ s θ s ( θ s − m s m s ln ( µ s + θ s m s )) (11) n s = 2 θ π (12)Here m s is the mass of the s quark and θ s = (cid:112) ( µ s − m s ). At high baryon densities and lowtemperatures ∂B∂T dominates over ∂B∂µ , based on that at the phase boundary one can obtainthe bag constant as, B ( µ, T ) (cid:39) B + ( µ u + µ d − µ s θ s ) T − µ s T θ H ( µ u + µ d + 23 θ s ) (13)Once the bag constant is obtained the pressure can be expressed as, P = 13 ( (cid:15) − B µ e π + m s π ln ( µ s + θ s m s ) − ( µ u + µ d − θ s )( µ s T θ H ) + µ s θ s π ( θ s − m s − µ s θ s T θ H − µ s π ( µ s θ s + m s θ s ( m s − µ s ) ) + µ s T µ s θ H ( µ u + µ s + 2 θ s θ H ))) (14)As we can see apart from the chemical potentials of the u and s quark, the main pa-rameters are the s quark mass, the temperature and the bag constant. The s quark massis already constrained by other experiments. So though we do get large stable mass starswith a smaller value of the s quark mass, we do not vary the s quark mass below 90 MeV. Generally this is the lower limit of the s quark mass from various other sources [26]. Thetemperature is also below 1 MeV. The constraint on the temperature comes as the entropyper baryon number has to be continuous even in the bulk. Increasing the temperatureviolates this continuity and hence we are constrained to temperatures below 1 MeV. The9 V a r i a t i on µ u (Mev)Strangeness fractionElectron fractionBaryon number density FIG. 2: The rate of change of the electron fraction, the strangeness fraction and the baryon numberdensity with respect to the chemical potential of the u quark. electron fraction in the core increases in this temperature range reaching a maximum at1 MeV, consequently the strangeness fraction goes down. So we find that lower the tem-perature, higher is the strangeness fraction in the core. However, the strangeness fractionsaturates to a value of 0 .
33 and becomes independent of temperature at lower values.In the previous section, we had seen that some parameters in the hadronic phase alsoaffect the stability of the quark cores. We had found that changes in the volume correctionsto the neutron - pion gas cause variation to the core pressure. In this case, the chemicalpotential of the quarks u and s along with the mass of the strange quark dominate the corepressure. In fact as the chemical potential of the u quark is increased, the electron fractionand the strangeness fraction required for a stable core goes down while the baryon numberdensity increases to maintain the stability. The rate of increase(decrease) of the electronfraction, the strangeness fraction and the baryon number density has been plotted against thechemical potential of the u quark in fig 2. Generally a larger value of the chemical potentialof the u quark ( µ u ) results in more stable quark cores provided the strange quark mass ison the lower side. This is seen by calculating the energy density of the quark core, whichhas to be more stable than the F e nucleus. For all the values in fig 2, we get stable quarkmatter. The stability is checked by calculating the energy per baryon in the plasma. Thuswe have found that it is possible to have energetically stable quark matter using three flavorsof the quark gluon plasma. As these are massive objects, we know that gravitation playsan important role in establishing the stability of these stars. The macroscopic quantities of10ass and radius are thus calculated using the Tolman - Oppenheimer - Volkoff equations[27]. In the next sections, we briefly describe these equations and their solutions.
III. THE TOLMAN - OPPENHEIMER - VOLKOFF EQUATIONS
We consider the TOV equations for a spherical and isotropic metric. We assume that thecore of the quark star has a uniform density in all the directions. The TOV equations arethen given by, dPdr = Gr ( (cid:15) ( r ) + P ( r ))( M ( r ) c + 4 πr P ( r )( rc − GM ( r )) (15)and dMdr = 4 πr (cid:15) ( r ) c (16)where the mass density ρ ( r ) = (cid:15) ( r ) c and c (cm/sec) is the speed of light and (cid:15) ( r ) is the energydensity of the plasma in MeV. The TOV equations are solved numerically with appropriateboundary conditions to obtain the mass distribution of the stars. The solution also gives usthe gravitationally allowed values of the mass and radius of the stars. We solve the TOVequations numerically for the high baryon density regime. We find that stable stars arepossible for various values of the bag constant. Detailed results and graphs are shown in thenext section.Apart from the mass radius ratio, we would also like to look at a direct observable thatis affected by the compactness of the star. One such observable is the surface redshift valuesof the stars. It is a parameter that corresponds to the redshift experienced by a radiallypropagating photon travelling from the star’s surface to infinity. It can be directly calculatedfrom the mass - radius ratio of a star. z s = (cid:18) − GMRc (cid:19) − / − M / M O • R (Km)B=130 MeVB=135 MeVB=140 MeVB=145 MeVB=158 MeV
FIG. 3: M (in solar masses) Vs R at different bag constants for massless quarks. IV. RESULTSA. Mass-radius ratios with three massless flavors ( u , d , s ) Since the two flavor plasma only gives us a metastable plasma we do not solve the TOVequations for the two flavor plasma. In the case of the three massless flavors of quarks wevary the various parameters and do a systematic search to put constraints on the variousparameters of the model.For massless quarks, we do get large mass values close to 2 . M with a radius of 14 km.This occurs for lower bag constants. The strange quark is considered massless here andtherefore we get the mass values greater than two solar masses. We then obtain the surfacered shift values for these large mass stars and the resultant plot is shown in fig 4. As seenfrom fig 4., we obtain redshift values of 0 . M or higher. It has beenpredicted that PSR J0348+0432 has a radius of about 12 kms and mass of 2 . M [11] witha red shift value of 0 .
4. So our model with massless quarks is able to reproduce the mass -ratio required to have stable quark stars as seen from the surface red shift data. To obtainthe large mass stars, our bag constant should be around 130 −
140 MeV.
B. Mass-radius ratios for the massive strange quark
We now present the results of the TOV equations with the pressure obtained from theEoS corresponding to the massive strange quarks. The mass of the quark puts a constraint12 .050.10.150.20.250.30.350.40.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 Z s M/M O• B=130 MeVB=135 MeVB=140 MeVB=145 MeVB=158 MeV
FIG. 4: Red shift values for cores with massless quarks. M / M O • R (Km)B=130 MeVB=135 MeV
FIG. 5: M (in solar masses) vs R at different bag constants for m s = 100 MeV, µ u = 100 MeVfor a u-d-s massive plasma on the quark chemical potentials. We plot the mass radius ratio for stable stars for twodifferent masses of the strange quark. Since we know that the larger mass stars would occurin the range of bag constant values between 130 −
140 MeV, we give the results for twovalues within this range.As we see from fig 5 and fig 6, we get larger mass with smaller values of strange quarkmass. So if the star has massive strange quarks, it’s size will be constrained by the strangequark mass. We have plotted the redshift values for different masses of the stars.Here since we have constrained the strange quark value to a lower limit of 90 MeV, wedo not get massive stars with masses greater than 2 M . We can get such values if we lowerthe strange quark values to around 75 MeV. We find that stars with masses of the order of13 M / M O • R (Km)B=130 MeVB=135 MeV
FIG. 6: M (in solar masses) vs R at different bag constants for m s = 150 MeV, µ u = 100 MeV fora u-d-s massive plasma Z s M/M O• B=130 MeVB=135 MeV
FIG. 7: Redshift values with the plasma having a massive strange quark of mass m s = 100 MeV Z s M/M O• B=130 MeVB=135 MeV
FIG. 8: Redshift values with the plasma having a massive strange quark of mass m s = 150 MeV . M would typically have surface red shift values around 0.22. This has also been seenin other models [11, 12]. Thus the observational constraints given by red - shift values areall satisfied with different values of strange quark mass in our model. We now discuss someobservational data in support of our model.There are many observational instances of stars with these kind of parameters. The XTEJ1739-285 with a mass of 1 .
51 solar mass and 10.9 km radius [28], is often referred to asa quark star. It is rapidly spinning and has a red shift ranging from 1 . . −
135 MeV gives red shifts in this range. The EXO0748-676 with a mass of 2 . . .
35. The EXO 0748-676 can be a quark star with a bag constant of 135 MeV as perthis model. For stars like PSR J1614-2230, the gravitational redshift values are in the range0 . − .
5, such high redshift values occur for the quark plasma with massless quarks. Aswe have mentioned before the mass - radius ratio and the red shift values depend cruciallyon the masses of the strange quark.
V. CONCLUSIONS
In conclusion, we have looked at bulk strange matter using a bag constant which isdependent on chemical potential and temperature. The bag constant in this model hadoriginally been derived by Leonidov et. al. for two massless flavors of quarks. In this workwe have extended their model to include the strange quark. Though an isentropic phasetransition is not essential in the core of the star, we show that such a phase transitioncan lead to stable quark matter inside the stars. We have done a systematic study of theparameters of the model. We have shown that the presence of the massless strange quarkincreases the stability of the plasma and the mass limits of these stars. The stability of theplasma limits our bag constant to the lowest value of 130 MeV. As is already known, thestability of the quark matter increases with decrease of the bag constant. However, in thiswork, we have also shown that apart from the bag constant, it is the mass of the strangequark that determines the radius and mass of the massive object. Since the mass-radiusratio determines the surface redshift of the star, we have also obtained the surface red shiftvalue for both massless and massive strange quark stars. We have shown that our model15an reproduce the mass- radius ratio and the surface red shift of stars like PSR J1614-2230and PSR J0348-0432 for bag constant values in the range 130 −
140 MeV. The mass of thestrange quark is important and we get larger mass stars for lower values of strange quarkmass. Though we have maintained the lower value of strange quark mass at 90 MeV dueto constraints from other sources, there are previous studies where the strange quark massis considered from 80 MeV [30] onwards. Lowering the strange quark mass will result inhigher mass stars. Thus we have established that it is quite possible to have a stable quarkgluon plasma bulk phase in the core of a massive star with no other exotic phases.Apart from PSR J1614-2230 and PSR J0348-0432 already mentioned before, both mass-radius ratios and the red shifts of XTE J1739-285 and EXO 0748-676 can be explained byour model. Thus we do find a large number of candidates with larger masses and surface redshifts which fit our model. We have not considered the rotational effects of this model in thiswork, we have studied the stability of the quark mass only in the bulk. We have establishedthat it is possible to model masses of the order of two solar masses using extensions of thethree flavour bag model with a s quark whose mass essentially determines the mass-radiusratio of the star. Our future plan is to study the rotational effects of these stars.AcknowledgmentsS.J would like to acknowledge partial financial support (through long term project scheme)from the UGC Networking Resource Center, School of Physics, University of Hyderabad. [1] E. Witten, Phys. Rev. D 30, 272 (1984);[2] S. Banerjee, S. K. Ghosh and S. Raha, J. Phys. G26, L1 (2000);[3] J. C. Collins and M. J. Perry, Phys. Rev. Lett. 34, 1353 (1975);[4] G. Baym , Physica A, 96, 131-135 (1979);[5] E. Alvarez and J. M. Ib´a˜nez, Astron. Astrophys., 98, 390-396 (1981);[6] A. Rosenhauer, E. F. Staubo, L. P. Csernai, T. Øverg˚ard and E. Østgaard, Nucl. Phys. A,540 (1992) 630-645;[7] M. Kutschera and A. Kotlorz; Astrophysical Journal, v 419, 752 (1993);[8] G. H. Bordbar, M. Bigdeli and T. Yazdizadeh, IJMPA vol 21, No 28, 5991 (2006);
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