PHYSICAL REVIEW C 84, 024321 (2011) Phase structure in a chiral model of nuclear matter Tran Huu Phat,1,2,* Nguyen Tuan Anh,3 and Dinh Thanh Tam4,1 1 Vietnam Atomic Energy Commission, 59 Ly Thuong Kiet, Hanoi, Vietnam 2 Dong Do University, 8 Nguyen Cong Hoan, Hanoi, Vietnam 3 Electric Power University, 235 Hoang Quoc Viet, Hanoi, Vietnam 4 University of Taybac, Sonla, Vietnam (Received 11 December 2010; revised manuscript received 30 May 2011; published 25 August 2011) The phase structure of symmetric nuclear matter in the extended Nambu-Jona-Lasinio (ENJL) model is studied by means of the effective potential in the one-loop approximation. It is found that chiral symmetry gets restored at high nuclear density and a typical first-order phase transition of the liquid-gas transition occurs at zero temperature, T = 0, which weakens as T grows and eventually ends up with a second-order critical point at T = 20 MeV. This phase transition scenario is confirmed by investigating the evolution of the effective potential versus the effective nucleon mass and the equation of state. DOI: 10.1103/PhysRevC.84.024321 PACS number(s): 11.30.Rd, 05.70.Fh, 21.65.−f I. INTRODUCTION Heavy-ion collisions at high energies presently are a powerful tool for generating hot and dense strongly interacting matter, and therefore they provide the opportunity to explore many interesting properties of matter under extreme conditions. In this connection, phase transitions in nuclear matter turn out to be a hot topic, attracting intense experimental and theoretical investigations. As was known, experiments reveal that with increasing excitation energy the behavior of excited nuclei can be described in terms of thermodynamics, and, consequently, in this regime statistical concepts turn out to be relevant. Numerous experimental analyses indicate that a dramatic change occurs in the reaction mechanism for excited energy per nucleon in the interval E/A ∼ 2–5 MeV, consistently corresponding to a first- or second-order phase transition of nuclear matter [1–9]. In addition, the critical exponents extracted from experimental data [10,11] are remarkably close to those of the liquid-gas phase transition and significantly different from the values derived from the mean-field treatment of the system concerned. In fact, the phase transition of nuclear matter has been often considered in numerous theoretical papers [12–22]. They are based on phenomenological models which are formulated directly in terms of nucleon degrees of freedom. Nonrelativistic nuclear models dealing with various types of nucleon-nucleon potentials are available only at low density, but they fail to reflect the physical characteristics of dense matter. The relativistic nuclear theory of Walecka [23] has successfully reproduced many physical properties of medium and heavy nuclei. Other relativistic nuclear models have been developed and have attained important results. However, all the preceding theories have a serious drawback: namely, they do not respect chiral symmetry, which is commonly accepted as one of the basic symmetries of the strong interaction. The chiral phase transition in a dense matter state plays a crucial role in studying physical properties of excited nuclei as well as the structure of compact stars and the evolution of the early universe. Nowadays, there has been considerable progress in exploring the chiral phase transition in quark matter within the framework of the lattice QCD simulation [24] and effective models of QCD [25,26]. The chiral phase transition has been one of the most important experimental realizations in relativistic heavy-ion collisions. In this regard, we are interested to those chiral models of nuclear matter which are able to explain spontaneous chiral-symmetry breaking in vacuum and its restoration at high densities. As was discussed in Ref. [21] the most suitable model for our goal might be the extended Nambu-Jona-Lasinio (ENJL) model. Therefore in this paper we start from the Lagrangian L = LNJL + μψ̄γ0 ψ, LNJL = ψ̄(i ∂ˆ − m0 )ψ Gv Gs + [(ψ̄ψ)2 + (ψ̄iγ5 τψ)2 ] − (ψ̄γμ ψ)2 2 2 Gsv (1) + [(ψ̄ψ)2 + (ψ̄iγ5 τψ)2 ](ψ̄γμ ψ)2 , 2 where μ = diag(μp , μn ), μp,n = μB ± μI /2; m0 is the “bare” mass of the nucleon, τ is the isospin Pauli matrices, and Gs , Gv , and Gsv are coupling constants. The main aim of this paper is to test the validity of the ENJL model (1): whether or not it reproduces the well-established results for symmetric nuclear matter, such as the value of the in-medium nucleon mass, the value of incompressibility, and the liquid-gas phase transition at subsaturation density, ρ < ρ0 ≈ 0.17 fm−3 . Therefore, we will deal only with symmetric nuclear matter (μI = 0) in what follows. In the mean-field approximation we replace (ψ̄Ŵi ψ)2 = 2ψ̄Ŵi ψψ̄Ŵi ψ − ψ̄Ŵi ψ2 , (ψ̄Ŵi ψ ψ̄Ŵj ψ)2 = ψ̄Ŵi ψ2 (2ψ̄Ŵj ψψ̄Ŵj ψ) + (2ψ̄Ŵi ψψ̄Ŵi ψ)ψ̄Ŵj ψ2 * − 3ψ̄Ŵi ψ2 ψ̄Ŵj ψ2 tranhuuphat41@gmail.com 0556-2813/2011/84(2)/024321(6) 024321-1 (2) ©2011 American Physical Society PHYSICAL REVIEW C 84, 024321 (2011) TRAN HUU PHAT, NGUYEN TUAN ANH, AND DINH THANH TAM with Ŵ = {1, iγ5 τ, γμ , γ5 γμ }, with averaging at finite density and temperature denoted by the angular brackets. Equations (2) combine with the bosonization σ = ψ̄ψ, π = ψ̄iγ5 τψ, Based on (9) and (10) the effective potential is derived immediately:  = ln Z = U (ρB , ρs ) + iTr ln S −1  d 3k [Ek + T ln(n− n+ )], = U (ρB , u) + 2Nf (2π )3 ωμ = ψ̄γμ ψ, yielding L = ψ̄(i ∂ˆ − m0 + γ0 μ)ψ + (Gs + Gsv ω2 )ψ̄(σ + iγ5 τπ )ψ 2 2 − [Gv − Gsv (σ + π )]ψ̄ ω̂ψ Gs 2 Gv 2 Gsv 2 − (σ + π 2 ) + ω −3 (σ + π 2 )ω2 . 2 2 2 where n∓ = [eE∓ /T + 1]−1 and Nf = 2 for nuclear matter and Nf = 1 for neutron matter. The pressure P is defined as (3) P = −taken at minimum . The structure of this paper is as follows. In Sec. II we present the equations of state of chiral nuclear matter for later use. Section III is devoted to determining static properties of symmetric nuclear matter at zero temperature. In Sec. IV the phase transition of chiral nuclear matter is studied. The conclusion and outlook are given in Sec. V. The energy density is obtained by the Legendre transform of P : E =  + T ς + μB ρB = U (ρB , u) + 2Nf + 2Nf II. EQUATIONS OF STATE OF CHIRAL NUCLEAR MATTER At μI = 0, the σ , π , and ω fields develop the ground-state expectation values σ  = u, π i  = 0, ωμ  = ρB δ0μ (4) in cold nuclear matter. Inserting (4) into (3) we arrive at LMFT = ψ̄(i ∂ˆ − M ∗ + γ0 μ∗ )ψ − U (ρB , u),  with μ∗ = μB − v = μB − (Gv − Gsv u2 )ρB ,   U (ρB , u) = 12 Gs u2 − Gv ρB2 + 3Gsv u2 ρB2 , G̃s = Gs + Gsv ρB2 = Gs [1 + α(ρB /ρ0 )2 ], (7) (8) or u = ρs = 2Nf S −1 (k) = k̂ − M ∗ + γ0 μ∗ , giving  k 2 + M ∗2 . (13) ∂ = 0, ∂u (10)  d 3k M ∗ (n− + n+ − 1), (2π )3 Ek which is usually called the gap equation. In terms of the baryon density  ∂P d 3k ρB = = 2Nf (n− − n+ ), ∂μB (2π )3 (14a) (14b) the expression for P reads (m0 − M ∗ )2 Gv 2 ρ + (μB − μ∗ )ρB − 2 B 2G̃s  d 3k [Ek + T ln(n− n+ )], −2Nf (2π )3 P =− with Ek = (12) (6) Integrating out the nucleon degrees of freedom it is found that   VU det S −1 , Z = exp − (9) T E∓ = Ek ∓ μ∗ , − n+ ) with the entropy density defined by  d 3k [n− ln n− + (1 − n− ) ln(1 − n− ) ς = −2Nf (2π )3 +n+ ln n+ + (1 − n+ ) ln(1 − n+ )]. V in which v (n− The ground state of nuclear matter is determined by the minimum condition α = ρ02 Gsv /Gs . det S −1 (k) = (k0 − E− )(k0 + E+ ), d 3k (2π )3 (5) The solution M ∗ of Eq. (6) is the nucleon effective mass, which reduces to the nucleon mass in vacuum. Starting from (5) we establish the partition function    β Z = D ψ̄DψDσ D πDω  d 3 x LMFT . dτ μ exp 0  d 3k Ek (n− + n+ − 1), (2π )3 where M ∗ = m0 − G̃s u, (11) (15) and the energy density takes the form  (m0 − M ∗ )2 Gv 2 d 3k E= Ek (n− +n+ −1). + ρB +2Nf 2 (2π )3 2G̃s (16) Equations (15) and (16) are the equations of state which govern all the phase transition processes of nuclear matter. 024321-2 PHYSICAL REVIEW C 84, 024321 (2011) PHASE STRUCTURE IN A CHIRAL MODEL OF NUCLEAR . . . III. STATIC PROPERTIES OF NUCLEAR MATTER 10 At zero temperature Eqs. (14a), (14b), (15), and (16) are simplified to  Nf  2 M∗ u=− 2 , (17) k dk 2 π kF (k + M ∗2 )1/2 bin MeV 5 kF3 , (18) 3π 2  (m0 − M ∗ )2 Gv 2 Nf  2 ρB − 2 k dk (k 2 + M ∗2 )1/2 . E= + 2 π kF 2G̃s ρB = Nf (i) It is evident that in vacuum, ρB = 0, kF = 0, we must have M ∗ = MN with MN = 939 MeV, the nucleon mass in vacuum. Consequently, Eq. (6) provides the first constraint MN = m0 − G̃s uvacuum , (20) with uvacuum satisfying the gap equation (17) taken at ρB = 0 and kF = 0. (ii) Another constraint is read out from applying the soft pion phenomenology and the partial conservation of the axial current (PCAC) equation for the axial-vector current of nucleons in vacuum, m2π fπ2 = m0 |uvacuum |, (21) where mπ = 138 MeV is the pion mass and fπ = 93 MeV is the pion decay constant in vacuum. (iii) The saturation mechanism requires that at normal density ρB = ρ0 ≃ 0.17 fm−3 the binding energy Ebin = −MN + E/ρB (22) attains its minimum value (Ebin )ρ0 ≃ −15.8 MeV, where E is given by Eq. (19). With the aid of MATHEMATICA [27] the numerical computation is achieved. It is found that m0 = 41.264 MeV, Gs = 8.507 fm2 , Gv /Gs = 0.933, α = 0.032, and  = 400 MeV. Figure 1 shows the graphs of binding energy against baryon density. Next, these values are used to calculate the in-medium mass of the nucleon, M ∗ /MN ≃ 0.684, and the incompressibility  ∂ 2 Ebin K0 = 9ρ0 ∂ρB2 5 10 15 0.0 (19) Now we follow the method developed in Ref. [21] to determine the five parameters m0 , Gs , Gv , α, and  for symmetric nuclear matter based on two constraints and the saturation condition: 0 0.5 1.0 1.5 2.0 2.5 ρB ρ0 FIG. 1. (Color online) Nuclear binding energy as a function of baryon density. Thus we have successfully obtained the values for two key nuclear quantities, which are in good agreement with those widely expected in the literature [30]. Table I lists all the calculated values for parameters and physical quantities concerned. Collecting the above calculated values for the model parameters we are ready to carry out the numerical study of phase transition. IV. PHASE STRUCTURE OF SYMMETRIC NUCLEAR MATTER In this section we proceed to the study of chiral restoration at high nuclear density and the well-known gas-liquid phase transition at subsaturation of nuclear matter, because they belong to the fundamental issues of modern nuclear physics. To this end, we first solve numerically the gap equation for a chiral condensate (14a) and obtain in Fig. 2 the evolution of a chiral condensate versus μB at various values of T . It is clear that chiral symmetry gets restored at large μB and, moreover, for T > 20 MeV the order parameter u is a single-valued function of the baryon chemical potential μB and smoothly tends to zero. This kind of behavior is usually defined as a crossover transition. Meanwhile, for lower T , 0 < T < 20 MeV, the order parameter turns out to be a multi-valued function of μB , where, due to Asakawa and Yazaki [25], a first-order chiral phase transition emerges. Then applying the method proposed in [25], which essentially identifies the multi-value region of a chiral condensate with the region of a first-order phase transition, we obtain the corresponding phase diagram for the chiral condensate in the (T , μB ) plane shown in Fig. 3. Here the solid line denotes a first-order phase transition, which ends up with a second-order critical end point, CEP (T = 20 MeV, μB = 907 MeV). This is exactly the liquid-gas ≃ 285.91 MeV. TABLE I. Values of parameters and physical quantities. ρ0 An analysis [28] shows that this value of incompressibility may be more compatible with the data than the one previously reported [29], K = 180–240 MeV.  (MeV) Gs (fm2 ) Gv /Gs 400 024321-3 8.507 0.933 m0 α 41.264 0.032 M ∗ /MN K0 0.684 285.91 PHYSICAL REVIEW C 84, 024321 (2011) TRAN HUU PHAT, NGUYEN TUAN ANH, AND DINH THANH TAM 1.00 1.0 1 0.95 T T T T T 0 5 MeV 10 MeV 15 MeV 20 MeV 0.8 0.9 0.90 6 0.6 1.0 0.85 3 0.8 0.80 860 880 900 920 940 MeV fm u u0 1.5 1 3 2 5 4 7 0.4 0.5 0.2 0.0 0.0 0 500 1000 1500 2000 0.6 µ B MeV 0.7 1.0 M MN FIG. 2. (Color online) The evolution of a chiral condensate u vs μB at various values of T . From the left the graphs correspond to T = 0, 100, 175 MeV, respectively. The inset shows u(T , μB ) at the low values of T , T = 0 (line 1), 5 MeV (line 2), 10 MeV (line 3), 15 MeV (line 4), and 20 MeV (line 5). phase transition at subsaturation of symmetric nuclear matter. Figure 3 is in good agreement with the conjectured QCD phase diagram [31]. Low-energy heavy-ion collisions experiments indicate that μCEP ∼ 923 MeV, TCEP = 15–20 MeV [32]. The above result can be tested by observing the evolution of the effective potential versus M ∗ for several values of T and μB belonging to the multi-valued region of a chiral condensate, 0 < T < 20 MeV and 907 < μB < 923 MeV. This is plotted in Fig. 4. For T < 20 MeV the first-order phase transition is represented by a graph with two minima corresponding to phases of restored and broken symmetry separated by a barrier. As T increases further these minima smear out and at T = 20 MeV the barrier between them disappears, signaling the onset of a second-order phase transition. This phenomenon is displayed clearly by means of the equation of state (EoS). Indeed, starting from the relation P = −taken at minimum . 250 FIG. 4. (Color online) The evolution of effective potential vs M ∗ at several values of T and μB . let us plot in Fig. 5 a set of isotherms. These bear the typical structure of the van der Waals equations of state. This structure resembles those derived from various nuclear theories [33,34] depending on the effective forces chosen, such as the Skyrme effective interaction and finite-temperature Hartree-Fock theory [33]. The line AB is exactly a part of the spinodal one which delimits the instability condition,   ∂P < 0. ∂ρB T The segments of those isotherms below the abscissa axis, which satisfy this condition, correspond to metastable states with negative pressure [35]. We show in Fig. 6 the μB dependence of nuclear density at several values of temperature. Combining Figs. 5 and 6 together indicates that the phase transition takes place at a critical point ρc ∼ 0.3–0.4ρ0 for Tc ∼ 16–18 MeV, which is consistent with other considerations [2]. It is worth pointing out that a first-order phase transition is characterized by the latent heat emitted during transition. Let us focus on the behavior of the energy density given by Eq. (16) for T varying around T = 20 MeV. As is seen in Fig. 7, for T < 20 MeV the energy density has a jump and an 2.0 1.5 3 150 100 P MeV fm T MeV 200 50 T 0 0 gas CEP liquid 800 1000 20 MeV 200 400 600 T 1.0 30 T 15 T 0.0 C T 10 B 0.5 1200 20 0.5 T A µ B MeV 1.0 0.0 FIG. 3. (Color online) The phase diagram of a chiral condensate in the (T , μB ) plane. The solid line means a first-order phase transition. CEP (T = 20 MeV, μB = 907 MeV) is the critical end point. The dashed line denotes a crossover transition. 0.2 0.4 0.6 5 T 0.8 0 1.0 1.2 ρB ρ0 FIG. 5. (Color online) The EoS for several temperature steps. ABC is the spinodal line. 024321-4 PHYSICAL REVIEW C 84, 024321 (2011) PHASE STRUCTURE IN A CHIRAL MODEL OF NUCLEAR . . . 500 P MeV fm 3 Symmetric Nuclear Matter 100 50 10 5 Expt. 1 1 2 3 4 5 ρB ρ0 FIG. 6. (Color online) The μB dependence of nuclear density at several values of temperature. amount of latent heat L = E(Tc + 0) − E(Tc − 0) is generated. If we take T = 10 MeV then L = 7.941 MeV. This value is interestingly compared with L = 8.1(±0.4)stat (+1.2 − 0.9)syst A MeV, which, for the first time, was experimentally estimated [8] for hot, heavy Z ∼ 70 nuclei. Finally, the theory is also highlighted by studying EoS at high densities and zero temperature. In Figs. 8 and 9 we show, respectively, the zero-temperature EoS for symmetric nuclear matter and neutron matter. In each figure the shaded region corresponds to the constraint on high-density behavior of the pressure from simulations of flow data from heavy-ion collision experiments [36]. V. CONCLUSION AND OUTLOOK In this paper we investigated systematically the phase transition in chiral nuclear matter described by the ENJL model. Our main results are summarized as follows. FIG. 8. (Color online) The EoS of cold symmetric nuclear matter at high baryon density. The shaded area denotes a constraint on the high-density behavior of the pressure consistent with the experimental flow data [36]. (i) Utilizing the effective potential in the one-loop approximation we reproduced the expected values for the inmedium mass of the nucleon and the incompressibility. (ii) The nuclear system manifests the following scenario of phase transitions: in the (T , μB ) plane a first-order phase transition of the liquid-gas type occurs at T = 0, μB = 923 MeV and extends to a second-order critical end point CEP at T = 20 MeV, μB = 907 MeV. This phenomenon is confirmed by considering the evolution of the effective potential versus effective nucleon mass, the EoS,and the T dependence of energy density. (iii) Chiral symmetry is restored at high baryon density for fixed T . The first-order phase transition of symmetric nuclear matter is the only one to be discovered in all phenomenological models. At present, all experimental data related to multifragmentations are consistent with the concept of a nuclear liquidgas phase transition, which shows that the phase transition is first order up to E ∗ /A = 3.8 ± 0.3 MeV, where E ∗ is the excited energy of hot nuclei, and it becomes continuous for 300 500 850 MeV 900 MeV 908 MeV 916 MeV Neutron Matter 3 200 P MeV fm MeV fm 3 µB µ 250 µB B µB 150 100 100 50 10 5 50 0 0 Expt.Asy_soft Expt.Asy_stiff 20 40 60 1 1 80 T MeV 2 3 4 5 ρB ρ0 FIG. 7. (Color online) The temperature dependence of the energy density at several values of μB . FIG. 9. (Color online) The EoS and constraint (shaded region) on the high-density behavior of the EoS for neutron matter [36]. 024321-5 TRAN HUU PHAT, NGUYEN TUAN ANH, AND DINH THANH TAM excited energy greater than this value. In this regard, it is of great significance to carefully test the energy scale on which the nuclear continuous phase transition could be observed. 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