24 June 1999

Ž .Physics Letters B 457 1999 261–267

Strange quark matter with dynamically generated quark masses

M. Buballa 1, M. Oertel 2

Institut fur Kernphysik, TU Darmstadt, Schlossgartenstr. 9, 64289 Darmstadt, Germany¨

Received 2 November 1998; received in revised form 21 January 1999Editor: J.-P. Blaizot

Abstract

Ž . Ž .Bulk properties of strange quark matter SQM are investigated within the SU 3 Nambu–Jona-Lasinio model. In thechiral limit the model behaves very similarly to the MIT bag model which is often used to describe SQM. However, whenwe introduce realistic current quark masses, the strange quark becomes strongly disfavored, because of its large dynamicalmass. We conclude that SQM is not absolutely stable. q 1999 Published by Elsevier Science B.V. All rights reserved.

PACS: 12.39.Ba; 12.39.FeKeywords: Strange quark matter; Dynamical quark masses

Ž .The properties of strange quark matter SQM andin particular the conjecture that SQM could be theabsolute ground state of strongly interacting matterw x1,2 have attracted much attention in nuclear physicsand astrophysics. Whereas the empirical stability ofordinary nuclei excludes the existence of absolutely

Ž .stable non-strange quark matter NSQM this argu-ment does not hold for SQM because the decay ofnuclei into SQM would involve higher-order weakinteraction processes, associated with a very long

w xlifetime. In 1984 Farhi and Jaffe 3 investigated thequestion of absolutely stable SQM within the MIT

w xbag model 4 . Treating the bag constant and thestrange quark mass as free parameters they found areasonable window for which SQM is stable and

1 E-mail: michael.buballa@physik.tu-darmstadt.de2 E-mail: micaela.oertel@physik.tu-darmstadt.de

NSQM is unstable compared with an 56Fe-nucleus.Almost 15 years later bag model calculations still

Ž w x.play a key role in this field for reviews see 5–8 .w xIn Ref. 9 we have investigated non-strange quark

Ž .matter within the Nambu–Jona-Lasinio NJL modelw x10 in mean field approximation. For vanishingŽ .current quark masses and with the appropriatechoice of parameters we found that quark matter inthe NJL mean field behaves very similarly to quarkmatter in an MIT bag. For the description of SQMwe have to introduce finite quark masses, at least inthe strange quark sector. In this case the correspon-dence between the two models becomes less accu-rate, with the NJL mean field behaving in a muchmore complex way. This is related to the fact that thebag constant in the NJL model is not a phenomeno-logical input parameter, like in a bag model, but adynamical property, which has its origin in the spon-taneous breaking of chiral symmetry. This motivated

w xus to extend the model of Ref. 9 to investigate bulk

0370-2693r99r$ - see front matter q 1999 Published by Elsevier Science B.V. All rights reserved.Ž .PII: S0370-2693 99 00533-X

( )M. Buballa, M. OertelrPhysics Letters B 457 1999 261–267262

properties of strange quark matter within the NJLmodel.

The three-flavor NJL model has been discussedw xby many authors, e.g. 11–21 . In this article we

w xadopt the Lagrangian of Ref. 21 :

82 2

LLsq iEuym qqG ql q q qig l qŽ . Ž .ˆŽ . Ý0 k 5 kks0

yK det q 1qg q qdet q 1yg q .Ž . Ž .Ž . Ž .f 5 f 5

1Ž .

Here q denotes a quark field with three flavors, u, dŽ u d s .and s, and three colors. m sdiag m ,m ,m is aˆ 0 0 0 0

3=3 matrix in flavor space. We restrict ourselves tothe isospin-symmetric case, mu s md. The La-0 0

Ž . Ž .grangian contains a U 3 =U 3 -symmetric four-L R

point interaction term and a six-point interaction,which is a determinant in flavor space and which

Ž .breaks the U 1 -symmetry. G and K are constantsA

with dimension energyy2 and energyy5, respec-tively.

The model is not renormalizeable and we have tospecify a regularization scheme for divergent inte-grals. For simplicity we use a sharp cut-off L in3-momentum space. Besides L we have to fix fourparameters, namely the coupling constants G and Kand the current quark masses mu and ms . In most of0 0

our calculations we will adopt the parameters ofw x 2 5Ref. 21 : L s 602.3 MeV, GL s 1.835, K L s

12.36, mu s 5.5 MeV and ms s 140.7 MeV.0 0

These parameters have been determined by fittingf , m , m and m X to their empirical values, whilep p K h

the mass of the h-meson is underestimated by about6%.

Due to the interactions the quarks acquire dynam-ical masses m which are in general different fromi

their current masses mi and depend on temperature0

and density. In the following we consider quarkmatter at Ts0 and zero or non-zero baryon number

1 1 Ž .density r s n s n q n q n , with n sB B u d s i3 3

² † : Ž .q q . Then, in mean field Hartree approximation,i i

the dynamical quark masses are solutions of the gapequation

i ² : ² :² :m sm y4G q q q2 K q q q q , 2Ž .i 0 i i j j k k

Ž . Ž .with q ,q ,q being any permutation of u,d,s .i j k² :The quark condensates q q are given byi i

3 mL i2² :q q sy p dp . 3Ž .Hi i 2 2 2p p (m qpFi i

Ž 2 .1r3They depend on the Fermi momenta p s p nF i i

and again on the dynamical masses m . Thus, foriŽ . Ž .given quark number densities n , Eqs. 2 and 3i

have to be solved self-consistently. After that we cancalculate the energy density ´ and the total pressurep of the system:

p3 Fi 2 2 2(´s p dp m qp y ByB ,Ž .Ý H i 02p 0isu ,d , s

2 2(psy´q n m qp . 4Ž .Ý i i F iisu ,d , s

Here B denotes the bag pressure,

3 L 2 2 2 i2 2( (Bs p dp m qp y m qpÝ H ž /i 02p 0isu ,d , s

2² : ² :² :² :y2G q q q4K uu dd ss , 5Ž .i i

<and B sB . The latter was introduced inn sn sn s00 u d s

Ž .Eq. 4 in order to ensure ´sps0 in vacuum.Ž .The existence of a bag pressure term in Eq. 4

suggests that there is a certain connection betweenthe present model and standard bag model descrip-tions of quark matter. Let us have a closer look atthis point. It is useful to begin with the chiral limit,i.e. mi s 0 for all flavors. With L, G and K as0

specified above, chiral symmetry is spontaneouslybroken in vacuum with equal dynamical masses mu

sm sm s 310.6 MeV and a bag pressure B sd s 03 Ž57.3 MeVrfm . There are other solutions of the gap

.equation but they are energetically less favored.Ž .Now we consider SU 3 -symmetric quark matter,

n sn sn sr . At low densities the dynamicalu d s B

quark masses decrease with density but they remainfinite. However, when r exceeds a critical value ofB

;0.9r chiral symmetry gets restored and the quark0

masses and condensates vanish. Consequently thebag pressure is also zero in this regime and we get

Ž .from Eq. 4

9 34 4´s p qB , ps p yB . 6Ž .F 0 F 02 24p 4p

( )M. Buballa, M. OertelrPhysics Letters B 457 1999 261–267 263

This means the system behaves exactly like a gas ofmassless non-interacting quarks with three flavorsand three colors inside a large MIT bag with a bagconstant B . It should be kept in mind, however, that0

Ž .Eq. 6 only holds in the regime of vanishing dynam-ical quark masses. For the present parameters it turnsout that the minimal energy per baryon numberErAs´rr is indeed found inside this region.B

Ž .Thus Eq. 6 is appropriate to describe the system inthe vicinity of the point of stability.

For comparison we consider non-strange isospin-3symmetric quark matter, n s0, n sn s r , stills u d B2

in the chiral limit. In this case strange and non-strangequarks behave differently with density, and chiralsymmetry is only partially restored at high densities:When r exceeds ;1.0r the non-strange quarksB 0

become again massless but the strange quark remainsmassive with a mass m) which is 154.2 MeV in ours

example. Hence the bag pressure remains also finite,BsB) s3.7 MeVrfm3, and the system now be-haves like a bag model for a gas of massless quarkswith two flavors and three colors but with an effec-tive bag constant B sB yB) , which is smallereff 0

Ž .than the bag constant B in the SU 3 -symmetric0

case.Thus, in contrast to a naive bag model, the effec-

tive bag constant which describes the behavior ofNJL quark matter depends on the flavor compositionof the matter. This is also shown in Fig. 1 wherevarious properties of isospin-symmetric quark matterare plotted as functions of the fraction of strangequarks, r sn rn . With the present parameters sta-s s B

ble quark matter exists in the region 0-r -0.48.sw xAs discussed in Ref. 9 these solutions correspond to

the high-density phase of a first-order chiral phasetransition in equilibrium with the non-trivial vacuum.Throughout this paper we will denote the dynamicalquark masses of these stable matter solutions by m)

i

and the corresponding bag pressure by B). In Fig.1a the solid line indicates the energy per baryonnumber, ErA of stable quark matter. m) and thei

effective bag constant B sB yB) are shown ineff 0

Fig. 1b and Fig. 1c, respectively. For 0.12-r -0.48s

the system behaves like a bag model with masslessquarks and a bag constant B sB . For r -0.12eff 0 s

the effective bag constant is somewhat smaller andŽ .the strange quarks become massive. In panel a we

also show the bag model result for ErA with a bag

Ž .Fig. 1. Properties of isospin-symmetric n s n stable quarku dŽmatter as a function of r s n rn in the chiral limit parameterss s B

w x i . Ž .of Ref. 21 , but with m s 0 . a Energy per baryon number.0Ž .The result of the present model solid line is compared with a bag

model calculation with bag constant B s 57.3 MeVrfm3BM

Ž . Ž . Ž .dotted . b Dynamical masses of the strange solid and non-Ž . Ž .strange quarks dotted . c Effective bag constant.

Žconstant B sB and massless quarks dashedBM 0.line . For 0.12-r -0.48 it agrees of course exactlys

with the NJL result, but also for r -0.12 the devia-s

tions are relatively small. In Fig. 1 all NJL curvesstop at r ,0.48. For r )0.48 the NJL quark mat-s s

ter is unstable against evaporation of free massives-quarks. In fact, this kind of phase separation isvery typical for asymmetric matter and has been

w xdiscussed for nuclear matter 22,23 as well as forw xmodels of the deconfinement phase transition 24,25 .

However, the evaporation of free quarks is of courseunphysical and reflects the missing confinement ofthe NJL model.

With finite current quark masses mi things change0

considerably. Fig. 2 shows the same quantities asFig. 1, but now with mu smd s 5.5 MeV and ms

0 0 0w xs 140.7 MeV 21 . We now find stable quark

matter for 0-r -0.79. In vacuum the dynamicals

masses of the up- and down-quarks are increased by; 60 MeV to 367.6 MeV and that of the strangequark by ; 240 MeV to 549.5 MeV. Perhaps moreimportant for us is the well-known fact that the

( )M. Buballa, M. OertelrPhysics Letters B 457 1999 261–267264

ŽFig. 2. The same as Fig. 1 but with finite current masses parame-w x. Ž .ters of Ref. 21 . In panel a the result of the present model

Ž .solid line is compared with bag model calculations with bagconstant B s105.2 MeVrfm3 and m s m s 5.5 MeV, mBM u d s

Ž .s 140.7 MeV dotted or m s m s 52.6 MeV, m s 464.4u d sŽ .MeV dashed-dotted .

quark condensates do not completely vanish even athigher densities. As a consequence, the dynamicalquark masses in stable quark matter, m) , are stilli

i Ž .well above the current masses m see Fig. 2b . For0Ž .non-strange isospin-symmetric quark matter r s0s

we find m) s 52.6 MeV and m) s 464.4 MeV.u s

Since on the other hand the Fermi energy in thissystem is only 361.1 MeV the conversion of anon-strange quark into a strange quark is obviouslyenergetically not favored. Here the notion of dynami-cal quark masses is crucial: In a naive bag model onewould assign the strange quark its current masswhich in our example is more than 200 MeV lowerthan the Fermi energy.

The difference becomes obvious in Fig. 2a wherethe energy per baryon number of stable quark matteris plotted as a function of r . The solid line indicatess

the NJL result while the dotted line corresponds to abag model calculation with B s 105.2 MeVrfm3

BMŽ .the effective bag constant at r s0 and quarks withs

m sm s 5.5 MeV and m s 140.7 MeV, ouru d s

current quark masses. Whereas the latter has a mini-mum at a finite fraction of strange quarks, r s0.27,s

the former is a strictly rising function of r : addi-s

tional strangeness is always disfavored in the NJLcalculation. In Fig. 2a we also show the result of abag model calculation with m sm s 52.6 MeVu d

and m s 464.4 MeV, our dynamical quark massessŽ .at r s0 dashed-dotted line . This curve rises evens

steeper with r than the NJL result. The reason iss

that in the NJL calculation the dynamical strange) Ž .quark mass m drops with r see Fig. 2b . There-s s

fore the penalty for adding strangeness becomes) Žsmaller than it would be with a constant mass m rs s

.s0 , even though this effect is partially compen-sated by the fact that the effective bag constantB sB yB) rises with r for not too large r .eff 0 s s

B is shown in Fig. 2c. It varies much strongereff

with r than in the chiral limit and is much larger ins

magnitude. Already at its minimum, at r s0, it iss

about twice as large. To major extent this is thereason why the energy per baryon number at thispoint is almost 200 MeV larger than in the chirallimit. Both, the magnitude of the effective bag con-stant and its strong variation with r , have theirs

origin in the fact that the bag pressure is verysensitive to the dynamical quark masses. As a conse-quence the vacuum bag pressure B is now 291.70

MeVrfm3, five times the chiral limit value, and,since the strange quark mass in stable matter varies,the corresponding bag pressure B) varies stronglywith r .s

It should be also noted that, even for fixed r ,s

quark masses and the bag pressure are density de-pendent and the values for m) and B shown ini eff

Fig. 2 only correspond to the densities of stablequark matter. As soon as one moves away from theminimum of ErA, m and B yB change. Thus,i 0

whereas in the chiral limit NJL quark matter couldbe well described in terms of an MIT-like bag modelwith vanishing quark masses and a single bag con-

Žstant at least within a larger range of densities and.strangeness fractions , it behaves in a much more

complex way when we include finite current quarkmasses.

So far, focusing on the bag model aspect, wemade some simplifying assumptions which we haveto abandon for a more realistic study of SQM. Inparticular we have to take into account weak decays.This implies that we have to include electrons andŽ .in principle neutrinos. To large extent we will

( )M. Buballa, M. OertelrPhysics Letters B 457 1999 261–267 265

w xadopt the model of Farhi and Jaffe 3 , with the MITbag replaced by the NJL mean field. With the usualassumption that the neutrinos can freely leave the

Žsystem, the matter is characterized by four three.quark- and one electron- chemical potentials. They

are related to the corresponding densities in thestandard way:

1 3r22 2 2 2n s m ym u m ymŽ . Ž .i i i i i2p

m3e

for isu ,d ,s and n s . 7Ž .e 23p

Here we neglected the electron mass. For the quarksone has to keep in mind that the masses entering ther.h.s. have to be calculated from the gap equationand are therefore density dependent themselves.Treating the electrons as a free gas, energy densityand pressure of SQM are given by

m4 m4e e

´ s´q , p spq , 8Ž .SQM SQM2 24p 12p

Ž .with ´ and p as defined in Eq. 4 .In chemical equilibrium maintained by weak in-

teractions only two of the four chemical potentialsare independent:

m sm sm qm . 9Ž .d s u e

Furthermore we demand charge neutrality,2 1n y n qn yn s0 . 10Ž . Ž .u d s e3 3

Thus the system can be characterized by one inde-pendent quantity, e.g. the baryon number density r .B

Ž .Our results are shown in Fig. 3. Panel a showsthe energy per baryon number ErAs´ rr as aSQM B

function of r . The solid line corresponds to theB

model described above, the dotted line to non-strangequark matter, where m was set equal to zero bys

hand. Obviously the strangeness degree of freedomis only important at densities above ; 4 r . This0

Ž .can also be seen in panel c where the fractionsr sn rn of the various particles are plotted. Sincei i b

Ž . Želectrons dotted line play practically no role and.are hardly visible in the plot the fraction of u-quarks

Ž . Ždashed-dotted is fixed by charge neutrality Eq.Ž ..10 to r , 1r3. For r-3.85 r the fraction ofu 0

Ž .strange quarks, r , solid is zero and thus r musts dŽ .be the remaining 2r3 dashed . For r)3.85 r r0 s

becomes non-zero and r drops accordingly.d

Fig. 3. Properties of charge-neutral quark matter with electrons asŽ w x.a function of baryon number density r parameters of Ref. 21 .B

Ž . Ža Energy per baryon number in chemical equilibrium m sms d. Žsm qm , solid line and for non-strange matter m s0,m su e s d

. Ž . Ž .m qm , dotted . b Dynamical quark masses m dashed-dotted ,u e uŽ . Ž .m dashed and m solid . The dotted line denotes the chemicald s

Ž .potential m sm . c r s n rn for the different particles d i i BŽ . Ž .species: up quarks dashed-dotted , down quarks dashed , strange

Ž . Ž .quarks solid , and electrons dotted .

The fact that there is no strangeness at lowerdensities is again due to the relatively large mass ofthe strange quark. The dynamical quark masses areplotted in Fig. 3b as functions of r . For comparisonB

Žwe also show the chemical potential m sm dotteds d.line . As long as m Fm the density of stranges s

quarks is zero. For r Q2 r , m drops with density.B 0 s

This can be mainly attributed to the decrease of² :² :uu dd due to the rising density of non-strange

² :² :quarks. At higher densities uu dd is practicallyzero and m stays almost constant until n becomess s

<² : <non-zero and causes ss to drop. This behavior israther different from standard parameterizationswhich have been used in the literature to study SQMand which depend on the total baryon number den-

w xsity r only 26,27 .B

At those densities where strange quarks exist,they lead to a reduction of the energy, as can be seenby comparing the solid curve in Fig. 3a with thedotted one. However, the minimum of ErA does not

( )M. Buballa, M. OertelrPhysics Letters B 457 1999 261–267266

lie in this regime, but at a much lower density,r s2.25 r . Here we find ErA s 1102 MeV.B 0

Compared with the energy per baryon in an ironnucleus, ErA, 930 MeV, this is still very large. Inthis sense our results are consistent with the empiri-cal fact that stable NSQM does not exist. However,since the energy of strange quark matter is evenhigher, our calculation predicts that also SQM is notthe absolute ground state of strongly interacting mat-ter.

This result is very robust with respect to changesof the model parameters. Obviously stable quarkmatter with finite strangeness is only possible if thein-medium strange quark mass m) is much lowers

than the value we obtained above. The easiest way toachieve this is to choose a lower value of the currentmass ms . If we leave all other parameters un-0

changed, ms should be at most 85 MeV if we require0

n /0 at the minimum of ErA. With this value wesŽobtain much too low masses for K and h m ,K

.390 MeV, m , 420 MeV , while ErA is stillh

Ž .relatively large 1075 MeV . If we want to comedown to ErA, 930 MeV we have to choose ms s0

10 MeV or, alternatively, ms s 25 MeV and mu s0 0

md s 0. This is of course completely out of range.0

We could also try to lower m) by choosing asŽ ² :² :smaller coupling constant G. Since uu dd is

already very small at the densities of interest, m) iss.almost insensitive to K . However, with a lower G

the vacuum masses of all quarks drop and corre-spondingly the vacuum bag pressure B . This causes0

the minimum of ErA to move to lower densities. Itturns out that the corresponding chemical potentialm) drops even faster than m) and it is thus nots s

possible to obtain stable SQM in that way. To avoidthe strong density decrease we could increase thecoupling constant K while decreasing G, e.g. insuch a way that the vacuum masses of u- andd-quarks are kept constant. In order to get n /0 ats

the minimum of ErA we have to lower GL2 to 1.5and to increase K L5 to 21.27, almost twice thevalue of our original parameter set. For these param-eters the energy per baryon number is still 1077MeV. On the other hand we cannot further decreasethe ratio GrK because that would flip the sign ofthe effective qq coupling in the pseudoscalar-flavorsinglet channel, which is dominated by the combina-

2 Ž² : ² : ² :.tion 2Gq K uu q dd q ss . In that case3

there would be no solution for the hX-meson in

vacuum. Hence there seems to be no realistic way tofind absolutely stable SQM within our model.

We could ask whether this result can change if weintroduce additional interaction terms to the La-

Ž .grangian Eq. 1 In particular vector interactions areknown to be important in dense matter. However,since vector mean fields are repulsive, the energy perbaryon number will be even larger than before andSQM remains strongly disfavored compared withordinary nuclei. The same is true for the medium

w xeffects described in Ref. 28 which also tend toincrease the energy. In fact, these effects are comple-mentary to ours because they are most importantŽ .and most reliable at high densities, whereas ourdynamical quark masses are most important at low

w xdensities. In Ref. 19 the chiral phase transition wasstudied within a ‘‘scaled’’ NJL model. There is nosix-point interaction in this model, but the differentflavors are coupled through a dilaton field. Thecoupling was found to be weak except for rather low

w xchoices of the gluon condensate 20 . This makes itvery unlikely to find absolutely stable SQM withinthat model, although a quantitative study of thisquestion has not yet been done.

In summary, we investigated strange and non-strange quark matter within the NJL model. In thechiral limit the model behaves very similarly to theMIT bag model which has been mostly used in theliterature to describe SQM. For realistic model pa-rameters including finite current quark masses theNJL model shows a more complex behavior. This isdue to the fact that bag pressure and effective quarkmasses are dynamical properties of the model andare therefore in general density dependent. We findthat the dynamical mass of the strange quark staysvery large at the relevant densities. As a consequenceSQM is not favored compared with NSQM in themodel and there seems to be no chance to find SQMwith energies per baryon number lower than in ordi-nary nuclei. This almost rules out the original idea ofabsolutely stable SQM as a simple consequence ofthe Pauli exclusion principle. Of course we cannotcompletely exclude that more sophisticated mecha-

w xnisms, like color superconductivity 29,30 , in partic-ular the recently discovered mechanism of color-

w xflavor locking 31 , change our results. This is, how-ever, beyond the scope of this letter.

( )M. Buballa, M. OertelrPhysics Letters B 457 1999 261–267 267

Acknowledgements

We would like to thank J. Schaffner-Bielich forhaving drawn our attention to the topic of strangequark matter.

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