STABILITY OF BULK AND QUASI-BULK STRANGE QUARKMATTER AND STRANGE STARS

Somenath CHAKRABARTYSaha Institute of Nuclear Physics, 1/AF Bidhan Nagar, Calcutta-700 064, India

Using a relativistic version of the Landau theory üi ernii liquids and a dynamical density dependentquark mass approach to confinement, the stability of bulk and quasi-bulk strange quark matter (SQM)has been studied . Using the same model an equation of state for SQM at T = 0 is derived . Thestability and some global properties of strange stars have also been investigated using this equation ofstate .

1 . INTRODUCTIONThe SQM is an almost flavour Symmetric mixture

of up (u), down (d) and strange (s) quarks and a smallnumber of electrons to ensure overall electrical chargeneutrality. It was argued by Witten' some years ago,that a flavour symmetric SQM may be the absoluteground state of bulk hadronic matter near normal nu-clear density at zero temperature and pressure. Thisspeculation was investigated by a number of authors23using a phenomenological bag model of confinement .Stable SQM (quark nuggets) may be produced at thetime of the quark-hadron phase separation induced bya first order cosmic phase transition . However, it wascritizcd strongly' by estimating the evaporation rate inthe hot Universe. It has been shown that as the Uni-verse cools down to the temperature T - ?OMeV, SQMevaporates almost completely.

The other possibility of SQM formation is at thecore of a neutron star or if the density of the star issufficiently high that the whole star imay be convertedto a quark star .

The paper is organized in the following manner :We have studied the stability of bulk and quasi-bulkSQM in sections 2 and 3 respectively. The stabilityand some global properties of strange stars (SS) havebeen investigated in section 4, while section 5 containsthe conclusions and discussions .

2 . BULK STRANGE QUARK MATTERLacking a trustworthy description of confinement

for quark matter at very large b?ryon densities (chem-

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Nuclear Physics B (Proc . Suppl .) 24B (1991) 148-151North-Holland

ical potentials), most authors have fallen back on thephenomenological bag model, which assumes that with-in the bag quarks are asymptotically free . But the re-cent results from lattice QCD calculations 5 show thatquark matter does not become asymptotically free im-mediately after the phase transition even if it is a firstorder transition; some hadronic degrees of freedom stillexists immediately after the phase transition to QuarkGluon plasma. It approaches the ideal gas equation ofstate rather slowly. In this context the bag model isthus an inadequate description of confinement . Thereexist in the literature, however, other phenomenologicaldescriptions of confinement . One ofthem is a dynamicaldensity dependent quark mass nîodel, which was pro-posed long agos and is free from the above mentioneddemerits of the bag model . Although it is arbitrary andwithout any real support from underlying field theory,it was successful in explaining many experimentally ex-tracted thermodynannic variables' . According to thispicture quarks have small masses inside a hadron andvery large masses outside . Therefore vaccum is unableto support a free quark and as a result for an isolatedhadron, this model leads to absolute quark confinement .

Treating the system of quarks as a fermi gas in-teracting thiough the self consistently generated colourfield, we parametrize the variation of masses of non-strange and strange quarks in the following manner?

Mu ,d - B ,

m, - ms -f- B

(1a, 6)nQ

nQwhere B is the constant energy density, or quark num-

ber density, nq -* 0, me is the current mass of the squark (150-300 MeV), whereas the current masses of uand d quarks are assumed to be zero (which is justi-fied by the restoration of chiral symmetry at very highdensity).

In SQM, the strange quarks are produced throug:.the weak processes

d-+u+e- +ve, u+e_-~d+ve,u+dHu+s, s-*u+e- + ve

(2)

An immediate consequence of the dynamical chemi-cal equilibrium is the set of constraints Fe e ---- ILd :--p (say) and Ftu = p - Fe e , where p;(i = u, d, s, or e) isthe chemical potentials of the species i . Here neutrinosare assumed to leave the system freely.

The chemical potentials of the flavour i (i = u, d,and s) is given by

(3)

where the interaction term Okf; has been derived inRef. 8 using the Landau theory of Fermi liquids . Elec-trons are assumed to be massless and noninteractingand therefore pe = kf. .

Now using the conditions imposed by chemicalequilibrium, global electrical charge neutrality and theconservation of baryon number of the system, we cansolve numerically for the chemical potentials pi, (i ---Z., d,,3, and e) .

Then one can obtain Eo(EsQ;rtlnB), the energy perbaryon and pressure P for SQM from the theriaody-namic relationse . The variation of co with nB for both

1 .8

1 .5

0 1 .2_

0-9

S. Chakrabarty/Stability of bulk and quasi-bulk strange quark matter anti strange stars

(kf; + m2i + Akh)i/a

0.5k,

3 6 9 12 15ni, l no

FIGURE 1Variation of energy per baryon (co) with the baryonnumber density nB expressed in terms of normal nu-clear density (no) . Lower curve is for uds matter. andupper curve ib for ud matter, B114 = 197MeV andmo, = 150MeV .

SQM (uds) and rionstrange quark matter (ud) areshown in Fig . 1 . The choice of the parameter B is suchthat the ud system becomes just unstable (E ud/nB isslightly greater than MN, the nucleon mass) . The val-ues of B to satisfy the above condition are (265MeV) 4

and (197MeV ) 4 for noninteracting and interacting casesrespectively.

As can be readily seen, the uds system has a min-imum at - 8no in the noninteracting case and at -5no for the inteacting case . Both these results contradict the expectation of the earlier authorsl -3 , that theground state of SQM occurs at about the nuclear den-sity. For stable SQM-, the upper limit of :noo is found tobe 300MeV.

3. QUASI-BULK STRANGE QUARK MATTER

Recently Blackma.n and Jaf e 9 have investigatedthe stability of strangelets using the phenomenologicalbag model . In this section we shall re-investigate thesame possibility using the model proposed in section2.

Assuming the -trangelets to be spherical, the ra-dius can be written as R = rOA3 , where A is the totalnumber of baryons present and ro is an unknown pa-rameter. Then from eqn. (1), we have

10

149

mu ,d = C1ro i me = m; + C1ro

(4a, b)

where Cl = s7rB .

Doing the same exercise as in section 2, we haveinvestigated the stability of strangelets . The range ofB for which only the strangelets are stable is 160McV <-B 1 j 4 < 200MeV .

Knowing FL's and ro (which is very close to 1fm,0.969fm < ro < 1 .00068fm), the electrical charge tomass ra%.io of the strangelets can be obtained from theexpression

Z

2nu - nd + neA nu +nd+na

This ratio has been evaluated for vari&us .-� ( < J ::rv )and B. Unlike the stable nueleus for which  -- i ,here it is extremely small (< 10'Z) . This is one ofthe peculiarities of the stable strangeletc, distiiil güishingthem from stable heavy nuclei .

150 S . Chakrabarty/Stability of bulk and quasi-bull. strange quark matter and

Introducing the finite size effect (surface effect), wehave the surface energy term9,10

Eaurf = 47rrôaA3

where a is the surface tension of the strangelet andis < 12MeV/fn` and is a function of B. Hence theminimum value ofthe baryonnumber present in a stablestrangelet is given by

1 47rr2oAcr =

0

(6)(MN -Eo)

which is also a function of B. Therefore for a stablestrangelet, the baryon number A should remain withinthe range Ac , < A < Arnax , where A.., is the maxi-mum number of baryons which can be present in stableSQM before the system becomes gravitationally unsta-ble, and this number is independent of B .

4 . STRANGE STARSStrange stars may be formed if the density of the

remnants of supernova explosions is sufficiently high, sothat neutrons may overlap and form a cluster of quarkmatter- Subsequently strange gnark4 ?r^ produced vythe weak deacy of nonstrange quads (see eqn. (2)) .

To study the stability and some global propertiesof SS we consider it to be a spherically symmetric non-rotating object of SQM, and its stability is u0verar_-c?by the general relativistic equation of hydrostatic equi-libfium, the :;ell known Tolman Oppenheimer Volkov(TOV) equation 2,3,10-1`. Using the equation of stateof the stable SQM (which also satisfies the restrictionsimposed by special relativity) for constant B and m0,as discussed in section 2, one can solve the TON' equa-tion. iôr 81 4 = 197MeV, and m° = 150MeV, we haveseers that the maximum mass and the correspondingradius of SS are M = 1 .367Me and R = 8.3km whereMe is the solar mass . These results are marginallyconsistent with the existence of PSR 1913+ 16 (M =1.41 f 0.06M(D) 13 . The period of rotation can easilybe obtained by equating the centrifugal force with thegravitational attraction and is found to be r = 0.535ms.Therefore most cf the strange stars are extremely fastrotating pulsars (in the sub-millisecond regime) . Un-

strange stars

like a neutron star in the case of SS there is no lower1-imit for the radius R . For M << Ma the major rolein the stability of SS is played by the QCD interac-tion, unless the surface effect becomes more importantas in the case of strangeletslo . In this case R - Mi .As M increases and becomes > 1Me, gravity starts todominate over QCD; and ultimately it becomes gravi-tationally unstable. The central and surface densitiesin this case are p, -- 11p o and ps -- 2.6po respectively,which are within the range of density for which SQM isa stable configuration .

5 . CONCLUSIONSWe therefore conclude, that the confinement model

followed in this paper is able to explain the stability ofbulk and quasi-bulk SQM and also strange stars . Thismodel can also predict some theoretical estimate of theSS parameters which can be measured experimentally.

ACKNOWLEDGEMENTSThis work was done in collaboration with Bikash

Sinha, Sibaji Raha and Bhaskar Datta . I am al ,,j grate-ful to Gautam Bhattacharya for some useful help in using TEX word processor . The work was supported inpart by CSIR ; Govt . of India, grant no . 7/1(401)/91-EMR-II.

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