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Redalyc.Spinodal Instability in the Quark-Gluon Plasma Brazilian Journal of Physics ISSN: 0103-9733 [email protected] Sociedade Brasileira de Física Brasil Aguiar, C. E.; Fraga, E. S.; Kodama, T. Spinodal Instability in the Quark-Gluon Plasma Brazilian Journal of Physics, vol. 34, núm. 1A, march, 2004, pp. 307-309 Sociedade Brasileira de Física Sâo Paulo, Brasil Available in: http://www.redalyc.org/articulo.oa?id=46400135 How to cite Complete issue Scientific Information System More information about this article Network of Scientific Journals from Latin America, the Caribbean, Spain and Portugal Journal's homepage in redalyc.org Non-profit academic project, developed under the open access initiative Brazilian Journal of Physics, vol. 34, no. 1A, March, 2004 307 Spinodal Instability in the Quark-Gluon Plasma C. E. Aguiar, E. S. Fraga, and T. Kodama Instituto de F´ısica, Universidade Federal do Rio de Janeiro C.P. 68528, Rio de Janeiro, 21941-972, RJ, Brazil Received on 15 August, 2003. We investigate the onset of spinodal decomposition in a relativistic fluid of quarks coupled to a nonequilibrium chiral condensate. Studying small fluctuations around equilibrium, we identify the role played by sound and chiral waves in the generation of unstable modes. The hadronization of the quark-gluon plasma (QGP) Lagrangian: possibly produced in the early universe or in high-energy ¹ 0 1 ¹ heavy-ion collisions may proceed in a number of differ- L = q[iγ @¹ + ¹qγ ¡ W (Á)]q + @¹Á@ Á ¡ V (Á) : (1) ent ways, depending on the nature of the QCD phase tran- 2 sition. In the heavy-ion case, some results from CERN- Here q = (u; d) is the constituent-quark field and ¹q = ¹=3 SPS and BNL-RHIC suggest what has been called sudden is the quark chemical potential. The interaction between hadronization [1] or explosive behavior [2, 3]. From the the quarks and the chiral field is given by W (Á) = g(σ + theoretical side, this phenomenon has been associated as- 2 2 2 2 2 iγ5~¿ ¢ ~¼), and V (Á) = (¸ =4)(σ + ~¼ ¡ v ) ¡ hqσ is sociated with deep supercooling of the QGP followed by the self-interaction potential for Á. The parameters above spinodal decomposition, and also to rapid changes in the ef- are chosen such that chiral SUL(2) ­ SUR(2) symmetry is fective potential of QCD near the critical temperature, such spontaneously broken in the vacuum. The vacuum expecta- as predicted, for instance, by the Polyakov loop model [4]. tion values of the condensates are hσi = f¼ and h~¼i = 0, Clearly, an understanding of the interplay between the typi- where f¼ = 93 MeV is the pion decay constant. The ex- cal space and time scales of the expanding plasma is wel- plicit symmetry breaking term is due to the finite current- come. Some attempts in this direction can be found in quark masses and is determined by the PCAC relation, giv- Refs. [5, 6, 7, 8, 9, 10, 11, 12, 13]. 2 ing hq = f¼m¼, where m¼ = 138 MeV is the pion mass. 2 2 2 2 2 This yields v = f¼ ¡ m¼=¸ . The value of ¸ = 20 leads In this paper, we discuss the onset of spinodal instability 2 2 2 2 to a σ-mass, mσ = 2¸ f¼ + m¼, equal to 600 MeV. The in an expanding plasma. As a phenomenological model to quark-chiral field coupling constant is taken to be g = 3:3, mimic the case of the QGP, we use a relativistic plasma of for which the constituent quark mass is 307 MeV, about 1/3 quarks coupled to a chiral field. The later is not necessarily of the nucleon mass. in thermal equilibrium with the quark fluid. Although we In what follows, we treat the gas of quarks as a heat bath derive a phenomenological nonequilibrium chiral hydrody- for the chiral field, with temperature T and baryon-chemical namics from a variational principle, we do not focus on the potential ¹. Integrating over the fermionic degrees of free- numerical solution of the resulting hydrodynamic transport dom and using a classical approximation for the chiral field, equations. we can write the thermodynamic potential as To model the mechanism of chiral symmetry breaking T ­(T; ¹; Á) = V (Á) ¡ ln detf[G¡1 + W (Á)]=T g ; (2) present in QCD, we adopt a simple low-energy effective chi- V E ral model: the linear σ-model coupled to quarks [14], which in turn comprise the hydrodynamic degrees of freedom of where GE is the fermionic Euclidean propagator and V is the system. Similar approaches, relying on low-energy ef- the (infinite) volume of the system. The determinant that fective models for QCD and making use of a number of tech- results from the functional integration over the quark and niques to treat the expanding plasma, can be found in the lit- anti-quark fields can be calculated to one-loop order in the erature [5, 6, 7, 8, 9]. The gas of quarks provides a thermal standard fashion [15, 16]. As already mentioned, the chiral bath in which the long-wavelength modes of the chiral field field Á plays the role of an order parameter, its equilibrium evolve. The latter plays the role of an order parameter in a value corresponding to the minimum of the grand canonical Landau-Ginzburg description of the chiral phase transition potential (2) for given (T; ¹). [8, 9]. The collective modes of the quark heat bath will be treated within the framework of ideal relativistic hydrody- Let us consider a chiral field Á = (σ; ~¼), where σ is a namics. To obtain the equations of motion we generalize the scalar field and ¼i are pseudoscalar fields playing the role of variational principle of Ref. [17], including the coupling of the pions, coupled to two flavors of quarks according to the the hydrodynamical degrees of freedom to the chiral field 308 C.E. Aguiar, E.S. Fraga, and T. Kodama ¹ ~ dynamics. This approach provides a natural way of merging where à stands for n, s, ~v or Á, and K = (!; k). Ãeq cor- chiral and fluid dynamics in a unified Lagrangian system. responds to a situation of stable or metastable equilibrium, For a different treatment of the hydrodynamics of nuclear given by the equation R = 0. Keeping terms up to first matter in the chiral limit, see [18]. order in the perturbation Ã1, the field and hydrodynamical We describe the state of the fluid in terms of the four- equations become velocity u¹(x) = (γ; γ~v), where ~v(~r; t) is the flow velocity of matter, the proper baryon density, n(x), and the proper entropy density, s(x). The action of the fluid-chiral field 2 ~ 2 2 (! ¡ k ¡ m )~¼1 = 0 ; (10) system is then defined as ¼ 2 ~ 2 2 k 0 Z · ¸ (! ¡ k ¡ mσ)σ1 = weqR v1 ; (11) 4 1 ¹ ! S ´ d x @¹Á@ Á ¡ ²(n; s; Á) ; (3) 2 0~ 2 0 2 (! ¡ p k )v1 = !kR σ1 ; (12) where ² = ­ + T s + ¹n is the energy density, from which the temperature and chemical potential are obtained by the where the masses of the σ and ~¼ fields are m2 = usual thermodynamic relations: T = @²(n; s; Á)=@s and σ (@2²(n; s; Á)=@σ2) and m2 = (@2²(n; s; Á)=@¼2) . We ¹ = @²(n; s; Á)=@n. The variation of n, s and u¹ in the eq ¼ eq have also defined p0 ´ [@p=@²] and R0 ´ [@R =@²] , action principle is performed under constraints arising from eq σ eq ¹ evaluated at constant (s=n) and Á. The flow velocity ~v1 is baryon number conservation, @¹(nu ) = 0, entropy conser- ¹ parallel to the wave vector ~k (sound waves are longitudi- vation, @¹(su ) = 0, and normalization of the 4-velocity, ¹ nal), and its magnitude determines the baryon number and u u¹ = 1. With these conditions the variational principle, ±S = 0, leads to the equations of motion: entropy amplitudes: 2Á = ¡R; (4) n1 = (k=!)neqv1 ; (13) ¹ º ¹ º º º u @¹(wu ) = ¡(@¹u )wu + @ p + R@ Á; (5) s1 = (k=!)seqv1 : (14) where p = ¡­ is the pressure, w = ² + p = T s + ¹n is the enthalpy density, and R = @­(T; ¹; Á)=@Á = @²(n; s; Á)=@Á (6) We see from the linearized equations (10)–(12) that, to first order, only the σ field is coupled to the hydrodynamic gives the dynamical coupling between the fluid and the chi- degrees of freedom. The dispersion relation for the coupled ral field. Note that R has four components, and can be writ- modes reads ten as @V (Á) R = + g½(T; ¹; Á) ; (7) @Á 2 0~ 2 2 ~ 2 2 02~ 2 (! ¡ p k )(! ¡ k ¡ mσ) = weqR k : (15) where the (scalar/pseudoscalar) density ½ is Z d3k 1=E (Á) ½ = gÁº k + (¹ !¡¹ ) : For long wavelength fluctuations we can approximate the q 3 [E (Á)¡¹ ]=T q q (2¼) e k q + 1 roots of (15) by (8) Here ºq = 12 stands for the color-spin-isospin degeneracy of the quarks, E (Á) = (~k2 + m2(Á))1=2, and m (Á) = k q q M 2 (g2Á2)1=2 = g(σ2 + ~¼2)1=2 plays the role of an effective !2=~k2 = p0 ¡ + O(~k2) ; (16) s m2 mass for the quarks. σ 2 2 ~ 2 Equation (5) is the relativistic Euler equation for the !σ = mσ + O(k ) ; (17) quark fluid in the presence of the chiral field. Introducing the energy-momentum tensor of the fluid in the usual way, T ¹º = wu¹uº ¡ pg¹º , we can write the Euler equation as 2 02 where M ´ weqR . These modes can be identified as ¹º º sound waves (frequency !s) and chiral waves (frequency @¹T = R@ Á: (9) !σ). The total energy-momentum tensor of the fluid-field sys- The onset of instabilities takes place when !2 < 0, i.e., ¹º ¹º 1 ® ¹º ¹ º tem, T = T ¡ @®Á@ Ág + @ Á@ Á, is, of course, 0 2 2 2 2 either for (p ¡M =mσ) < 0 or for mσ < 0.
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