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ADP-04-19/T600 JLAB-THY-04-25 Liquid-gas phase transition in nuclear matter including strangeness 5 0 P. Wanga, D. B. Leinwebera, A. W. Thomasa,b and A. G. Williamsa 0 2 aSpecial Research Center for the Subatomic Structure of Matter (CSSM) and Department n of Physics, University of Adelaide 5005, Australia a J bJefferson Laboratory, 12000 Jefferson Ave., Newport News, VA 23606 USA 7 2 We apply the chiral SU(3) quark mean field model to study the proper- 2 ties of strange hadronic matter at finite temperature. The liquid-gas phase v 6 transition is studied as a function of the strangeness fraction. The pressure 5 of the system cannot remain constant duringthe phase transition, since there 0 7 are two independent conserved charges (baryon and strangeness number). In 0 a range of temperatures around 15 MeV (precise values depending on the 4 model used) the equation of state exhibits multiple bifurcates. The difference 0 / in the strangeness fraction f between the liquid and gas phases is small when h s t they coexist. The critical temperature of strange matter turns out to be a - l non-trivial function of the strangeness fraction. c u n PACS number(s): 21.65.+f; 12.39.-x; 11.30.Rd : v i X Keywords: Liquid-gas Phase Transition, Strange Hadronic Matter, r Chiral Symmetry, Quark Mean Field a I. INTRODUCTION The determination of the properties of hadronic matter at finite temperature and density is a fundamental problem in nuclear physics. In particular, the study of the liquid-gas phase transition in medium energy heavy-ion collisions is of considerable interest. Many intermediate-energy collision experiments have been performed [1] to investigate the un- known features of the highly excited (hot) nuclei formed in such collisions [2,3]. Theoret- ically, much effort has been devoted to studying the equation of state for nuclear matter and to discussing the critical temperature, T . Recently, Natowitz et al. obtained the lim- c iting temperature by using a number of different experimental measurements [4]. From these observations the authors extracted the critical temperature of infinite nuclear matter, T = 16.6 0.86 MeV [5]. We can expect that further experiments may eventually yield the c ± limiting temperatureofhypernuclei andthecriticaltemperatureforinfinitestrangehadronic 1 matter. Itisthereforeinteresting tostudy theliquid-gasphase transitionofstrangehadronic matter theoretically. Exploring systems with strangeness, especially with large strangeness fraction, has at- tracted a lot ofinterest in recent years. Such a system has many astrophysical and cosmolog- ical implications and is indeed interesting by itself. There are many theoretical discussions for both strange hadronic matter [6,7] and strange quark matter [8]- [11]. However, most discussions are at zero temperature. The properties of strange hadronic matter at finite temperature have not been studied very much yet. Unlike symmetric nuclear matter, for strange hadronic matter, there are two conserved charges, baryon number and strangeness. Glendenning [12] first discussed the phase transition with more than one conserved charge in general and applied it to the possible transition to quark matter in the core of neutron stars. Mu¨ller and Serot [13] discussed asymmetric nuclear matter, which has two conserved charges (baryon number and isospin), using the stability conditions on the free energy, the conservation laws and the Gibbs criterion for the liquid-gas phase transition. The liquid-gas phase transition of asymmetric nuclear matter was also discussed in effective chiral models in Refs. [14,15]. It was found that the critical temperature decreases with increasing asymmetry parameter α. Forstrangehadronicmatter,themethodissimilartothatforasymmetricnuclearmatter. In both cases there are two conserved charges. Recently, Yang et al. [16] used the extended Furnstahl-Serot-Tang (FST) model [17] to discuss the liquid-gas phase transition of strange hadronic matter. The original FST model satisfies the SU(2) chiral symmetry and was first applied to study nuclear matter [18]. In the extended FST model [16], the author found a so called critical pressure above which the liquid-gas phase transition cannot exist. As a result no critical strangeness fraction was obtained for a given temperature. This critical pressureexistsinfinitenucleitogetherwithalimitingtemperature, T . However, ininfinite lim hadronic matter, physically, there should be no such critical pressure, since the pressure of the system can be much higher than the critical pressure. For strange matter with f = 2, s i.e. pure Ξ matter, from Fig. 1 of Ref. [16] one sees that the pressure does not increase monotonically with density. This means that at this temperature, pure Ξ matter can be in liquid-gas phase coexistence. Therefore, the binodal p µ diagram at temperature T = 10 − MeV should terminate at f = 2. We will reconsider this problem and show how the critical s strangeness fraction can be obtained. To study the properties of hadronic matter, we need phenomenological models since QCD cannot yet be used directly. The symmetries of QCD can be used to constrain the hadronic interactions and models based on SU(2) SU(2) symmetry and scale invariance L R × have been proposed. These effective models have been widely used to investigate nuclear matter and finite nuclei, both at zero and at finite temperature [18]- [20]. Papazoglou et al. extended thechiraleffective models toSU(3) SU(3) , including thebaryonoctets[21,22]. L R × As well as models based on hadronic degrees of freedom, there are some other models based on quark degrees of freedom, such as the quark meson coupling model [23,24], the cloudy bag model [25], the NJL model [26] and the quark mean field model [27], etc.. Recently, we proposed a chiral SU(3) quark mean field model and investigated the properties of hadronic matter as well as quark matter [28]- [32]. This model is quite successful in describing the properties of nuclear matter [28], strange matter [30,31], finite nuclei and hypernuclei [32] at zero temperature. In this paper, we will apply the chiral SU(3) quark mean field model 2 to finite temperature and study the liquid-gas phase transition of strange hadronic matter. The paper is organized as follows. The model is introduced in section II. In section III we use the model to investigate strange hadronic matter at finite temperature. The numerical results are discussed in section IV and section V summarises our findings. II. THE MODEL Our considerations are based on the chiral SU(3) quark mean field model (for details see Refs. [30,32]), which contains quarks and mesons as basic degrees of freedom. Quarks are confined in baryons by an effective potential. The quark meson interaction and meson self-interaction are based on SU(3) chiral symmetry. Through the mechanism of spontaneous chiral symmetry breaking, the resulting constituent quarks and mesons (except for the pseudoscalars) obtain masses. The introduction of an explicit symmetry breaking term in the meson self-interaction generates the masses of the pseudoscalar mesons which satisfy partially conserved axial-vector current (PCAC) relations. The explicit symmetry breaking term of the quark meson interaction leads in turn to reasonable hyperon potentials in hadronic matter. For completeness, we introduce the main concepts of the model in this section. In the chiral limit, the quark field q can be split into left and right-handed parts q and L q : q = q + q . Under SU(3) SU(3) they transform as R L R L R × q = Lq , q = Rq . (1) L′ L R′ R The spin-0 mesons are written in the compact form 1 8 M(M+) = Σ iΠ = (σa iπa)λa, (2) ± √2 ± a=0 X where σa and πa are the nonets of scalar and pseudoscalar mesons, respectively, λa(a = 1,...,8) are the Gell-Mann matrices, and λ0 = 2 I. The alternative plus and minus signs 3 correspond to M and M+. Under chiral SU(3)qtransformations, M and M+ transform as M M = LMR+ and M+ M+′ = RM+L+. In a similar way, the spin-1 mesons are ′ → → introduced through: 1 1 8 l (r ) = (V A ) = va aa λa (3) µ µ 2 µ ± µ 2√2 µ ± µ aX=0(cid:16) (cid:17) with the transformation properties: l l = Ll L+, r r = Rr R+. The matrices Σ, µ → µ′ µ µ → µ′ µ Π, V and A can be written in a form where the physical states are explicit. For the scalar µ µ and vector nonets, we have the expressions 1 (σ +a0) a+ K + 1 8 √2 0 0 ∗ Σ = √2 σaλa =  a−0 √12 (σ −a00) K∗0 , (4) aX=0  K∗− K¯∗0 ζ      3 1 ω +ρ0 ρ+ K + 1 8 √2 µ µ µ µ∗ Vµ = √2 vµaλa =  ρ−µ (cid:16) (cid:17) √12 ωµ −ρ0µ Kµ∗0 . (5) aX=0  Kµ∗− (cid:16)K¯µ∗0 (cid:17) φµ    Pseudoscalar and pseudovector nonet mesons can be written in a similar fashion. The total effective Lagrangian has the form: = + + + + + + ,+ , (6) Leff Lq0 LqM LΣΣ LVV LχSB L∆ms Lh Lc where = q¯iγµ∂ q is the free part for massless quarks. The quark-meson interaction q0 µ qM L L can be written in a chiral SU(3) invariant way as = g Ψ¯ MΨ +Ψ¯ M+Ψ g Ψ¯ γµl Ψ +Ψ¯ γµr Ψ qM s L R R L v L µ L R µ R L − g(cid:16) 8 8 (cid:17) (cid:16) g 8 (cid:17) 8 = s Ψ¯ σ λ +i π λ γ5 Ψ v Ψ¯ γµvaλ γµγ5aaλ Ψ. (7) √2 a a a a ! − 2√2 µ a − µ a! a=0 a=0 a=0 a=0 X X X X In the mean field approximation, the chiral-invariant scalar meson and vector meson ΣΣ L self-interaction terms are written as [30,32] VV L 1 2 σ4 = k χ2 σ2 +ζ2 +k σ2 +ζ2 +k +ζ4 +k χσ2ζ ΣΣ 0 1 2 3 L −2 2 ! (cid:16) (cid:17) (cid:16) (cid:17) 1 χ4 δ σ2ζ k χ4 χ4ln + χ4ln , (8) − 4 − 4 χ4 3 σ2ζ 0 0 0 1 χ2 = m2ω2 +m2ρ2 +m2φ2 +g ω4 +6ω2ρ2 +ρ4 +2φ4 , (9) LVV 2 χ2 ω ρ φ 4 0 (cid:16) (cid:17) (cid:16) (cid:17) where δ = 6/33; σ , ζ and χ are the vacuum expectation values of the corresponding mean 0 0 0 fields σ, ζ and χ. The Lagrangian generates the nonvanishing masses of pseudoscalar χSB L mesons χ2 m2 = m2F σ + √2m2 F πF ζ , (10) LχSB χ20 " π π K K − √2 π! # leading to a nonvanishing divergence of the axial currents which in turn satisfy the relevant PCAC relations for π and K mesons. Pseudoscalar and scalar mesons as well as the dilaton field χ obtain mass terms by spontaneous breaking of chiral symmetry in the Lagrangian (8). The masses of the u, d and s quarks are generated by the vacuum expectation values of the two scalar mesons, σ and ζ. To obtain the correct constituent mass of the strange quark, an additional mass term has to be added: = ∆m q¯Sq, (11) L∆ms − s where S = 1 I λ √3 = diag(0,0,1), is the strangeness quark matrix. Through these 3 − 8 mechanisms, th(cid:16)e quark co(cid:17)nstituent masses are finally given by g s m = m = σ and m = g ζ +∆m , (12) u d 0 s s 0 s −√2 − 4 where g and ∆m are chosen to yield the constituent quark mass in vacuum – in our case, s s m = m = 313 MeV and m = 490 MeV. In order to obtain reasonable hyperon potentials u d s in hadronic matter, it has been found necessary to include an additional coupling between strange quarks and the scalar mesons σ and ζ [30]. This term is expressed as = (h σ + h ζ)s¯s. (13) h 1 2 L In the quark mean field model, quarks are confined in baryons by the Lagrangian = c Ψ¯ χ Ψ (with χ given in Eq. (14), below). The Dirac equation for the quark fieldLΨ , c c ij − under the additional influence of the meson mean fields, is given by iα~ ~ +χ (r)+βm Ψ = e Ψ , (14) − ·∇ c ∗i ij ∗i ij h i where α~ = γ0~γ, β = γ0, the subscripts i and j denote the quark i (i = u,d,s) in a baryon of type j (j = N,Λ,Σ,Ξ) and χ (r) is a confining potential – i.e. a static potential providing c confinement of quarks by meson mean-field configurations. The quark effective mass, m , ∗i and energy e are defined as ∗i m = giσ giζ +m (15) ∗i − σ − ζ i0 and e = e giω giφ, (16) ∗i i − ω − φ where e is the energy of the quark under the influence of the meson mean fields. Here i m = 0 for i = u,d (nonstrange quark) and m = ∆m = 29 MeV for i = s (strange i0 i0 s quark). Using the solution of the Dirac equation (14) for the quark energy e it has been ∗i common to define the effective mass of the baryon j through the ansätz: M = E 2 < p 2 >, (17) j∗ j∗ − ∗jcm q where E = n e +E is the baryon energy and < p 2 > is the subtraction of the j∗ i ij ∗i jspin ∗jcm contribution to the total energy associated with spurious center of mass motion. In the P expression for the baryon energy n is the number of quarks with flavor ”i” in a baryon ij with flavor j, with j = N p,n ,Σ Σ ,Σ0 ,Ξ Ξ0,Ξ ,Λ and E is the correction to ± − jspin { } { } { } the baryon energy which is determined from a fit to the data for baryon masses. There is an alternative way to remove the spurious c. m. motion and determine the effective baryon masses. In Ref. [33], the removal of the spurious c. m. motion for three quarks moving in a confining, relativistic oscillator potential was studied in some detail. It was found that when an external scalar potential was applied, the effective mass obtained from the interaction Lagrangian could be written as M = n e E0, (18) j∗ ij ∗i − j i X where E0 was found to be only very weakly dependent on the external field strength. We j therefore use Eq. (18), with E0 a constant, independent of the density, which is adjusted to j give a best fit to the free baryon masses. 5 Using the square root ansätz for the effective baryon mass, Eq. (17), the confining potential χ is chosen as a combination of scalar (S) and scalar-vector (SV) potentials as in c Ref. [30]: 1 χ (r) = [χS(r)+χSV(r)] (19) c 2 c c with 1 χS(r) = k r2, (20) c 4 c and 1 χSV(r) = k r2(1+γ0). (21) c 4 c Ontheotherhand,usingthelineardefinitionofeffectivebaryonmass, Eq.(18),theconfining potential χ is chosen to be the purely scalar potential χS(r). The coupling k is taken as c c c k = 1 (GeV fm 2), which yields baryon mean square charge radii (in the absence of a pion c − cloud [34]) around 0.6 fm. The properties of infinite nuclear matter and finite nuclei were calculated with these two treatments of effective baryon mass in Ref. [35]. As we have explained there, the linear definition of effective baryon mass has been derived using a systematic relativistic approach [33], while to the best of our knowledge no equivalent derivation exists for the square root case. For high baryon density, the predictions of these two treatments are quite different. Manyphysicalquantities changediscontinuously atsomecriticaldensityinthecaseofsquare root ansätz, while the linear definition of baryon mass yields continuous behavior for high density nuclear matter. Both treatments of the spurious c. m. motion fit the saturation properties of nuclear matter and therefore, for densities lower than the saturation density, these two treatments give reasonably similar results. In this paper, we will discuss the liquid-gas phase transition of strange hadronic matter with both treatments. We prefer the linear form because it has been derived. The square root case is reported here because it is widely used and in fact produces similar results in some regions. However, where they differ we believe that the linear form is the more reliable. III. STRANGE HADRONIC MATTER AT FINITE TEMPERATURE Based on the previously defined interaction, the Lagrangian density for strange hadronic matter is written as 1 1 1 1 1 = ψ¯ (iγµ∂ M )ψ + ∂ σ∂µσ + ∂ ζ∂µζ + ∂ χ∂µχ F Fµν S Sµν L B µ − B∗ B 2 µ 2 µ 2 µ − 4 µν − 4 µν gBψ¯ γ ψ ωµ gBψ¯ γ ψ φµ + , (22) − ω B µ B − φ B µ B LM where F = ∂ ω ∂ ω and S = ∂ φ ∂ φ . (23) µν µ ν ν µ µν µ ν ν µ − − 6 The term represents the interaction between mesons which includes the scalar meson M L self-interaction , the vector meson self-interaction and the explicit chiral symmetry ΣΣ VV L L breaking term , all defined previously. The Lagrangian includes the scalar mesons σ, ζ χSB L and χ, and the vector mesons ω and φ. The interactions between quarks and scalar mesons result in the effective baryon masses M , where subscript B labels the baryon B = N,Λ,Σ B∗ or Ξ. The interactions between quarks and vector mesons generate the baryon-vector meson interaction. The corresponding vector coupling constants gB and gB are baryon dependent ω φ and satisfy the relevant SU(3) relationships. In fact, we find the following relations for the vector coupling constants: 2 1 √2 gΛ = gΣ = 2gΞ = gN and gΛ = gΣ = gΞ = gN. (24) ω ω ω 3 ω φ φ 2 φ 3 ω At finite temperature and density, the thermodynamic potential for strange hadronic matter is defined as Ω = gjkBT ∞d3−→k ln 1+e−(Ej∗(k)−νj)/kBT (25) − (2π)3 j=NX,Λ,Σ,Ξ Z0 (cid:26) (cid:16) (cid:17) + ln 1+e−(Ej∗(k)+νj)/kBT M, −L (cid:16) (cid:17)(cid:27) 2 where E (k) = M 2 +−→k and g is the degeneracy of baryon j (g = 2, g = 1 and g = j∗ j∗ j N,Ξ Λ Σ r 3). The quantity ν is related to the usual chemical potential µ by ν = µ gjω gjφ. The j j j j− ω − φ energy per unit volume and the pressure of the system can be derived as ε = Ω 1 ∂Ω+ν ρ −T ∂T j j and p = Ω, where ρ is the baryon density. j − The mean field equation for meson φ is obtained by the formula ∂Ω/∂φ = 0. For i i example, the equations for σ, ζ are deduced as: 2δ χ2 k χ2σ 4k σ2 +ζ2 σ 2k σ3 2k χσζ χ4 + m2F 0 − 1 − 2 − 3 − 3σ χ2 π π 0 (cid:16) (cid:17) χ 2 ∂m ∂M m ω2 ω + j∗ < ψ¯ ψ >= 0, (26) ω j j − χ0! ∂σ j=N,Λ,Σ,Ξ ∂σ X δ χ2 1 k χ2ζ 4k σ2 +ζ2 ζ 4k ζ3 k χσ2 χ4 + √2m2F m2F 0 − 1 − 2 − 3 − 3ζ χ20 k k − √2 π π! (cid:16) (cid:17) χ 2 ∂m ∂M m φ2 φ + j∗ < ψ¯ ψ >= 0, (27) φ j j − χ0! ∂ζ j=Λ,Σ,Ξ ∂ζ X where g k2M < ψ¯ ψ >= j ∞dk j∗ [n (k)+n¯ (k)]. (28) j j π2 E (k) j j Z0 j∗ In the above equation, n (k) and n¯ (k) are the baryon and antibaryon distributions, respec- j j tively, expressed as 7 1 n (k) = (29) j exp E (k) ν /k T +1 j∗ − j B h(cid:16) (cid:17) i and 1 n¯ (k) = . (30) j exp E (k)+ν /k T +1 j∗ j B h(cid:16) (cid:17) i The equations for vector mesons ω and φ are expressed as χ2 m2ω +4g ω3 = gjρ , (31) χ2 ω 4 ω j 0 j=N,Λ,Σ,Ξ X χ2 m2φ+8g φ3 = gjρ , (32) χ2 φ 4 φ j 0 j=Λ,Σ,Ξ X where ρ is the density of baryons of type j, expressed as j g ρ = j ∞dkk2[n (k) n¯ (k)]. (33) j π2 j − j Z0 Let us now discuss the liquid-gas phase transition. For strange hadronic matter, we follow the thermodynamic approach of Refs. [12] and [13]. The system will be stable against separation into two phases if the free energy of a single phase is lower than the free energy in all two-phase configurations. This requirement can be formulated as [13] F(T,ρ) < (1 λ)F(T,ρ)+λF(T,ρ ), (34) ′ ′′ − where ρ = (1 λ)ρ +λρ , 0 < λ < 1, (35) ′ ′′ − and F is the Helmholtz free energy per unit volume. The two phases are denoted by a prime and a double prime. If the stability condition is violated, a system with two phases is energetically favorable. The phase coexistence is governed by the Gibbs conditions: µ (T,ρ) = µ (T,ρ ) (j = N,Λ,Σ,Ξ), (36) ′j ′ ′j′ ′′ p(T,ρ) = p (T,ρ ), (37) ′ ′ ′′ ′′ where thetemperature isthe same in the two phases. The chemical potentials of thebaryons satisfy the following relationship: µ = µ = (µ +µ )/2. (38) Λ Σ N Ξ Therefore, there are only two independent chemical potentials for the four kinds of baryons. They are determined by the total baryon density, ρ and the strangeness fraction, f , which B s are defined as ρ = (ρ +ρ +ρ +ρ ) and f = (ρ +ρ +2ρ )/ρ . B N Λ Σ Ξ s Λ Σ Ξ B 8 IV. NUMERICAL RESULTS AND DISCUSSIONS The parameters in this model were determined by the meson masses in vacuum and the properties of nuclear matter which were listed in table I of Ref. [35]. We now discuss the liquid-gas phase transition of strange hadronic matter. In Fig. 1, we plot the pressure of the system versus baryon density for various strangeness fractions, f , at temperature T = 15 s MeV for the square root ansätz of the effective baryon mass (Eq. (17)). For nonstrange hadronic matter, the p ρ isotherms exhibit the form of two phase coexistence with an B − unphysical region. The nuclear matter can be in a state of liquid-gas coexistence at this temperature. With increasing f , the pressure will increase. At a particular value of f , s s the pressure will increase monotonically with increasing density. As we will see later, the the strangeness fraction is different in the liquid and gas phases. Therefore, the system can still be in liquid-gas coexistence, even though the pressure increases monotonically with density. The unphysical region appears again in the range of 1.0 < f < 1.6. It is obvious s that the behavior of the pressure of strange hadronic matter is not monotonic with f . For s the linear definition of effective baryon mass (Eq. (18)), the results are plotted in Fig. 2. At small strangeness fraction, say f < 0.4, there are unphysical regions. In the range s 0.4 < f < 1.75, the pressure increases monotonically with increasing density, while for s f > 1.75, the unphysical regions appear again. s 0.5 0.4 -3m ) 0.3 Vf e M p ( 0.2 f=0 s f = 0 .5 s 0.1 f=1 s f=1.5 s f=2 s 0.0 0.00 0.02 0.04 0.06 0.08 0.10 0.12 r ( f m-3 ) B FIG. 1. The pressure of strange hadronic matter p versus baryon density ρ with different B strangeness fraction f at temperature T = 15 MeV in the case of square root ansa¨tz of effective s baryon mass. As we pointed out earlier, there are two independent chemical potentials for the baryons. We now show how the Gibbs conditions can be satisfied. As an example, we plot the chemical potentials of nucleon and Λ versus f at temperature T = 15 MeV and pressure s p = 0.23 MeV-fm 3 with the square root ansätz for the effective baryon mass in Fig. 3 (For − convenience, we use the reduced chemical potential which is defined as µ = µ M ). The j j j − solid and dashed lines are for nucleon and Λ, respectively. The Gibbs equations (36) and e 9 (37) for phase equilibrium demand equal pressure and chemical potentials for two phases with different concentrations. The desired solution canbefoundby means ofthe geometrical construction shown in Fig. 3, which guarantees the same pressure and chemical potentials of nucleon and Λ in the two phases with different f . Due to the chemical relationship between s the baryons, the chemical potentials of Σ and Ξ are also the same in the two phases. 0.7 f=0 s f=0.5 0.6 s f= 1 s f=1.5 s 0.5 f= 2 -3m ) s Vf 0.4 e M p ( 0.3 0.2 0.1 0.0 0.00 0.03 0.06 0.09 0.12 0.15 r ( fm -3 ) B FIG. 2. The pressure of strange hadronic matter p versus baryon density ρ with different B strangeness fraction f at temperature T = 15 MeV in the case of linear definition of effective s baryon mass. -27.75 -28.00 -28.25 V ) e -28.50 M ( ~~m, LN -28.75 m -47 -48 -49 -50 0.12 0.13 0.14 0.15 0.16 f s FIG. 3. Geometricalconstructionusedtoobtainthechemicalpotentialsandstrangenessfraction in the two-phase coexistence at temperature T = 15 MeV and p = 0.23 MeVfm 3 The solid and − dashed lines are for nucleon and Λ, respectively. 10