Volume 229, number 1,2 PHYSICS LETTERS B 5 October 1989

STRANGE QUARK MATTER AND THE M E C H A N I S M OF CONFINEMENT

Somenath CHAKRABARTY a, Sibaji RAHA a and Bikash SINHA b a Saha Institute o f Nuclear Physics, 92, A.P.C. Road, Calcutta 700 009, India b Variable Energy Cyclotron centre, 1/AF, Bidhan, Calcutta 700 064, India

Received 27 April 1989; revised manuscript received 12 July 1989

It has been suggested in the literature that strange quark matter may be the true ground state of QCD. We investigate the model dependence of such speculations and show that the conclusions depend on the nature of the confinement mechanism.

Properties of strange quark matter have become a topic of high current interest, especially since Witten's speculation [ 1 ] some time ago that the phase sepa- ration induced by a first-order cosmic phase transi- tion may provide an explanation for dark matter within known physics. Several authors [2,3] have studied the possibility that nuggets of strange quark matter might be stable and thus become viable can- didates for dark matter. Lacking a trustworthy de- scription of confinement for quark matter at very large baryon densities (chemical potentials), most authors have fallen back on the phenomenological bag model [4 ] which a priori assumes that within the boundary of the bag, the quarks are asymptotically free.

Our motivation in writing this letter is to analyse how far the properties of quark matter are affected by such a drastic assumption about the confinement mechanism. With recent results from detailed calcu- lations on the lattice [ 5 ], it has now indeed become clear that quark matter does not become asymptoti- cally free immediately after the phase transition from hadronic matter even if it is a first-order transition; it approaches the free gas equation of state rather slowly. In this context, the bag model is thus an in- adequate description of confinement. There exist in the literature, however, other phenomenological de- scriptions of confinement which purport to remedy this situation. Indeed, a density-dependent quark mass approach to confinement was proposed several years ago [6-8 ] which, although arbitrary and with- out any real support from an underlying field theory,

was successful in fitting experimentally extracted val- ues of some thermodynamic variables - namely the velocity of sound [ 9 ], over large ranges of tempera- ture [ 10,11 ]. In the following, we therefore study the properties of strange quark matter, describing con- finement through the variable effective quark mass; as we shall see, conclusions about the stable configu- ration of quark matter are altered from those of ear- lier authors [ 1-3 ].

The "dynamical" density dependent quark mass approach has been widely discussed in the literature; for the sake of completeness, though, let us review very briefly the basic idea. Confinement is mimicked through the requirement that the mass of an isolated quark becomes infinitely large so that the vacuum us unable to support it. Thus, for a system of quarks at zero temperature (globally colour neutral, of course), the energy density tends to a constant value while the mass tends to infinity, as the volume increases to very large sizes. Treating the system of quarks as a Fermi gas (degenerate for each flavour), the effective mass mq changes with the density in the following manner [6,10-12]

B (1) mq ~ nq

where B is the constant energy density in the zero density limit, nq the total quark number density, counting all flavours. Extrapolating relation (1) to very large densities implies that the effective quark mass becomes negligible in accordance with our ex- pectations from asymptotic freedom and restoration

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Volume 229, number 1,2 PHYSICS LETTERS B 5 October 1989

of chiral symmetry. B, for our purpose, is a parame- ter, similar to the big constant, except that it does not lend itself to be interpreted as an external pressure.

This prescription, initially used for only the light quarks u and d, may easily be extended to include heavier quarks - strange for our present purpose. At low densities, eq. ( 1 ) applies equally well for strange quarks too. On the other hand, the consensus is that strange quarks have a non-negligible current mass, 150 MeV, while the current masses for (u)p and (d) own quarks may be ignored. We thus write:

B /'Hu, d ~, - - ,

nq

B ms~ - - + m ° , (2)

nq

where rn o denotes the strange current mass. The cur- rent mass is taken to be zero for u, d quarks.

The importance of studying strange quark matter lies in the fact that for charge neutral systems, a uds configuration is more stable than a ud configuration at high densities; thus is because the presence of strange quarks extends the flavour phase space and as a result, the system can lower its Fermi energy by distributing the energy over this larger phase space. This is achieved through the following weak reac- tions [2,13]:

d ~ u + e - +ge ,

u + e - ~ d + v e ,

u+d~-~ u + s ,

s ~ u + e - +9~. (3)

ultimately yielding chemical equilibrium. An imme- diate consequence is the set of constraints

/is =/td =-/t,

#s = ~ /= ~u "~ Ue , ( 4 )

where the notation is standard. We assume that the neutrinos leave the system freely and thus have zero chemical potential (absence of Pauli blocking).

Requirement of charge balance further yields

~nu=~nd+ lns+ne, (5)

where n~ refers to the number density of species i

( i= u, d, s, e). The baryon number density na is, obviously,

na=~(nu +na+ns) , (6)

which naturally remains unchanged in the process. Note that (5) and (6) imply nu = nn + ne.

From eqs. ( 4 ) - (6), it is manifest that there is only one chemical potential which is independent. This can be fixed by minimising the total free energy density e of the system, which is given by

e = E (ff~i + n i ~ i ) , ( 7 ) i

where t2i is the thermodynamic potential for species i, for i=u , d, s, we have

1 { 2 a i= - ~ k#,(,ai -m2)'/2(I.t~ -~m2i )

+ ~-m 4 In #i+~(#~rn~ - m ~ ) ) , (8)

and

#4 Qe = - 12rt 2 . (9)

Where, in (8), the m~ are defined through (2). The number densities n~ are related to the respective ther- modynamic potentials I2~ ( i= u, d, s, e) by the canon- ical relation

ni = - - Ofli " ( 1 O)

By suppressing the relevant quantities with sub- script s, the framework for the ud system follows, naturally. Now, solving for g from the criterion 0e/ 0/t = 0 for any given nB, we can calculate the equilib- rium configuration of the uds or ud systems. Clearly there is no a priori rule for fixing the parameter B (vide eq. ( 1 ) ) in our case. Since, however, it is known that baryonic matter around normal nuclear density no [ ~ ( 110 MeV) 3 ] exists in the hadronic phase, we require B to be such that the ud system is unbound, i.e., e/na> nucleon mass mN ( ~ 9 4 0 MeV). This constraints B to be bigger than (250 MeV) 4. On the other hand, we are interested in the possibility that strange quark matter may be bound, so that the en- ergy per baryon in the uds system should be less than

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mN at some density. This constrains B to be less than ~ (400 MeV) 4.

In fig. 1 we thus present our result, for m ° = 150 MeV and B t / 4 = 2 6 5 MeV. The energy per baryon is plot ted as a function o f (baryon densi ty) t /3 . As can be readi ly seen, the uds system has a m i n i m u m at na ~ ( 2 2 0 MeV)3, which corresponds to about 8 t imes the normal nuclear densi ty no with a b inding energy ~ 110 MeV. This contradicts the expectat ion o f ear- lier authors [ 1-3 ] that the ground state o f strange quark mat ter occurs at about the nuclear density. The comments about the physical in terpre ta t ion of the paramete r B notwithstanding, we have, for the sake of comparison, included in fig. 1 results for B~/4=235 MeV (dot ted l ines) . This par t icular value o f B (bag constant ) refers to the es t imate of Hasenfratz et al. [ 14 ] from charmonium data. As can be readi ly seen, the ud system becomes jus t bound at the equi l ibr ium configuration. Fig. 2 represents the strangeness con- tent ns/n, as well as the electron densi ty ne/n, as a function of n ~/3. Dot ted curves, as in fig. I, refer to B1/4=235 MeV. Higher values o f m ° ( ~ 300 MeV)

make the uds system also unbound and hence o f no interest in the present context.

We therefore, conclude that a deta i led unders tand- ing o f the confinement mechanism is essential before reliable conclusions about the propert ies of bulk quark mat te r can be reached. The results above, in spite of the arbi t rar iness o f the conf inement scenario used here, contain, in our opinion, the essential physics. The lower l imit on B, fixed through the requirement that the ud system is unbound can be made compat- ible with baryon mass spectroscopy by introducing a finite-size correction through the casimir energy term [4]: Zo/R. The ces correct ions can be included [12] and work in this direct ion is in progress, to be re- por ted in a future publ icat ion [ 15 ]. Also worthy of not ice in this connect ion is the argument of Bethe et al. [ 16 ] that the study of strange matter , as a possible ground state of nuclear matter , using per turbat ive Q C D is not reliable. They instead propose a system of closely packed hyperons which is energetically more favourable than ord inary nuclear mat te r at large cou- pling constants. These arguments apply to the case

2000 i i i ~ ; ; ;

t // ° m s = 15o MeV

MR " "... . . . . . . " " . ' ' ' " ~ .

5 0 0 1 5 0 2 0 0 2 5 0 :300 :350 8n o

n~/g ( M e V )

Fig. 1. Energy per baryon in the equilibrium configuration of the quark phase as a function of baryon number density. The upper curve is for two-flavour quark matter and the lower one for three-flavour (strange) matter. MN o n the vertical axis denotes the nucleon mass. Solid curves refer to B ~/4 = 265 MeV and the dotted curves are for B ~/4 = 235 MeV. (For a definition of B ~/4 and m o, see text. )

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Volume 229, number 1,2 PHYSICS LETTERS B 5 October 1989

(D

II

1"0

0 ' 8

0 - 6

0"4

0 "2

0"0

I I I I I ! I

- m 0 = 150 MeV. $

- ~ E~/4 = 2 6 5 MeV. ( s o l i d )

2 5 5 MeV (do t ted ) "".. i = s

" ' " " " " ~ . . . . . . . . . . . . . i = e

I I I I I I

150 200 ~ 2,50 300 8n o

,, ( M e V )

3 5 0

Fig. 2. Strangeness and electron content (relative to the number of u quarks) in the equilibrium configuration of strange quark matter as a function of baryon number density. Parameters and notation are the same as in fig. 1.

where the equi l ibr ium configurat ion of the uds sys-

tem is expected tO occur a round no. Our t rea tment is effectively non per turba t ive [ 10-12 ] and moreover , our results show that the ground state o f the uds sys- tem is shifted away from no.

Taking the present results at face value, their im- pl icat ion can, however, be far reaching. I f strange nuggets [ 2,17 ] are at all fo rmed at the cosmic phase t ransi t ion, their s tabil i ty is strongly inf luenced by the densi ty in the nuggets. Rel iable calculat ions of the evapora t ion rate from the nuggets using full non- equi l ibr ium react ion kinetics is an involved task which must be per formed but unfor tunate ly remains a future prospect . In an equi l ibr ium f ramework [2 ], nonetheless, the evapora t ion rate r turns out to be lower for higher densi ty in the nuggets (rocn~2/3). Following the calculat ions o f Alcock and Farh i [ 2 ] at the zeroth level, we f ind that the m i n i m u m baryon number for the nuggets to be stable turns out to be ~ 10 50. This number is a drast ic overes t imate as the analysis of ref. [ 2 ] ignores f lavour matching at the surface o f the nugget which, when accounted for, should significantly lower this number (by 6 -9 or- ders of magni tude) , as has been argued by Madsen et

al. [ 18 ]. Even though the m i n i m u m baryon number is still large, this is much lower than the total number o f baryons within the hor izon at t empera tures about 100 MeV ( ~ 1049). Thus, i f the strange nuggets have a chance to be formed as a result o f the cosmic phase separat ion [ 1 ], they might indeed be stable and com- prise a viable scenario for the dark mat te r within known physics. On the other hand, since the nuggets will have to have very large densities, it is doubtful whether they will be found, i f one does not want to t ampe r with the current ideas about baryon number generation.

The authors would like to thank Dr. Charles Alcock and Dr. Bhaskar Dat ta for helpful conversations.

References

[ 1 ] E. Witten, Phys. Rev. D 30 (1984) 272. [2] C. Alcock and E. Farhi, Phys. Rev. D 32 ( 1985 ) 1273. [3] E. Farhi and R.L. Jaffe, Phys. Rev. D 30 (1984) 2379. [4] T. De Grand et al., Phys. Rev. D 12 ( 1975 ) 2060. [5]A. Ukawa, in: Proc. Quark Matter '88 (Lenox, MA,

September 1988 ), to be published.

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[7] K.A. Olive, Nucl. Phys. B 190 ( 1981 ) 483. [8] D.H. Boal, J. Schachter and R.M. Woloshyn, Phys. Rev. D

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