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Quark Matter in the Chiral Color Dielectric Model

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Quark Matter in the Chiral Color Dielectric Model View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by CERN Document Server Quark Matter in the Chiral Color Dielectric Mo del a b a Alessandro Drago , Manuel Fiolhais and Ubaldo Tambini a Dipartimento di Fisica, UniversitadiFerrara, and INFN, Sezione di Ferrara, Via Paradiso 12, Ferrara, Italy 44100 b Departamento de Fisica da Universidade and Centro de FisicaTeorica, P-3000 Coimbra, Portugal (March 30, 1995) Abstract We study the equation of state (EOS) of quark matter at zero temp erature, using the Color Dielectric Mo del (CDM) to describ e con nement. Sensible re- sults are obtained in the version of the CDM for which con nement is imp osed smoothly. The two{phases version of the mo del turns out to give unrealistic results for the EOS. Chiral symmetry plays a marginal r^ole and the quarks are massive till high densities. The decon nement phase transition is smo oth and unlikely to b e rst order. Instabilities of the quark matter and the gap equation are discussed. 24.10.Jv Relativistic mo dels, 24.85.+p Quarks, gluons and QCD in nuclei and nuclear pro cesses HEP-PH-9503462 Typ eset using REVT X E 1 I. INTRODUCTION The study of the Equation Of State (EOS) of Quark Matter (QM) has b ecome a fash- ionable topic in view of the next exp eriments using heavy ions in program at RHIC(BNL) and at LHC(CERN) [1]. Furthermore the inner structure of neutron stars is now under in- vestigation: the connection b etween the comp osition of the star and the co oling time, which can b e measured, allows to discriminate among the various mo dels, indicating the p ossible existence of a quark matter phase (see, e.g. [2]). The study of the equation of state of matter at high densities can also give usefull informations to traditional nuclear physics, since one can search heavy nuclei for precursor phenomena, b oth of the decon nement and/or of the chiral restoration phase transition (for a review see [3]). In many mo del calculations of the decon nement phase transition the frame of the MIT bag mo del has b een used [4,5]. In suchaway a rst order decon nement phase transition is obtained (apart from sp eci c, ad hoc choices of the mo del parameters) and the decon nement phase transition coincide with the chiral restauration one. At densities and temp eratures slightly bigger than the critical ones the right degrees of freedom are already quarks, having current masses, and p erturbative gluons. There are anyway indications, from lattice calcu- lations, that at temp eratures bigger than the critical one non p erturbative e ects are still present in the quark{gluon plasma [6]. In this pap er we study the EOS of QM using the Color Dielectric Mo del (CDM) to describ e con nement [7]. We shortly review CDM in sect. 2. This mo del has b een widely used to study b oth the static and the dynamical prop erties of the nucleon. Morover it can b e used to describ e many{nucleon systems: for a twonucleon system it allows to compute anucleon{nucleon p otential qualitatively similar to the ones used in nuclear physics [8]; in the case of a homogeneous, in nite system of nucleons the CDM can b e used to construct a nonlinear version of the Walecka mo del [9]. The aim of our work is to extend previous calculations of the decon nement phase transition, where the same mo del has b een used [11]. An imp ortant p oint of our calculation will b e to x the mo del parameters in order to repro duce the basic static prop erties of the single nucleon, as was already done in the study of the nucleon structure functions [12]. We will later use the same parameters to study the EOS of QM, whichwe de ne as a system of totally decon ned quarks. In suchaway the study of the QM's EOS will turn out to b e a severe test for the di erentversions of the CDM, and we will b e able to make some predictions of the prop erties of matter at high densities. Within CDM (with a double minimum p otential for the scalar eld), we will investigate the p ossibility of getting a scenario similar to the one describ ed by the MIT bag mo del, with two phases undergoing a sharp rst order phase transition (sect. 3). Our results show that such a description is incompatible with the CDM. We than study another version of the CDM (with a single minimum p otential for the scalar eld), where con nement is imp osed more smo othly, and we get a sensible EOS for QM, without a sharp decon nement transition. An imp ortant feature of this decon ned quark matter is that quark's masses are big till high densities. Chiral restauration and decon nement do not o ccur at the same density. The reasons whychiral simmetry is restaured so slowly are discussed in sect. 4. In the last sections we analyze the prop erties of QM, as describ ed in the CDM. In sect. 5 the stability of quark matter is studied, using the technology of the resp onse function. In 2 sect. 6 we consider a gap equation, trying to understand the formation of quarks' clusters. Finally sect. 7 is devoted to the concluding remarks. I I. THE COLOR DIELECTRIC MODEL A. The mo del lagrangian In this section we shortly summarize the main features of the CDM. For a comprehensive review see the ones by Pirner [7] and by Birse [10]. We will use a chiral invariantversion of the CDM, as the one used in Ref. [11] and also in Ref. [12]. The Lagrangian reads g L = i @ + ( + i ~ ~ ) 5 1 1 1 2 2 2 (@ ) U ()+ (@ ) + (@ ~) U (;~) ; (1) + 2 2 2 where U (;~) is the usual mexican-hat p otential, as in ref. [15]. L describ es a system of interacting quarks, pions, sigmas and a scalar-isoscalar chiral singlet eld , whose p otential U () has an absolute minimum for = 0. In suchaway, in the case of the single nucleon problem, the quarks' e ective mass g= diverges outside the nucleon. The simplest p otential for is a quadratic one: 1 2 2 U ()= M : (2) 2 Using this p otential, uctuates around zero, reaching this value asymptotically, at large distances. In the following we will refer to this version of the mo del as the Single Minimum (SM) one. Another p ossibility is to use a p otential for having a Double Minimum (DM): 2 3 ! ! ! 2 4 4 8 6 1 2 2 4 5 1+ M 2 + 1 ; (3) U ()= 2 2 2 M M where the absolute minimum is still in the origin and the relative one is in = M . Cho osing appropriately the parameters it is p ossible to obtain solutions for the single nucleon where the eld interp olates b etween the relative minimum at the center of the nucleon and the absolute minimum at big distances. In the following, we will consider for DM only solutions where has the ab ove discussed b ehaviour. A complete analysis of the various p ossible solutions of the mo del (in a non{chiral{invariantversion) can b e found in Ref. [13]. In Fig.1 we showtwotypical solutions of the mo del for the single nucleon problem. As it app ears, in the DM case the transition b etween the interior of the nucleon and the external region is sharp er than in the SM, where all the elds havea very smo oth b ehaviour. Corresp ondingly, the kinetic energy contribution (which comes mainly from the quarks) will b e bigger in DM than in SM. This p oint will b e relevant when studying the EOS of QM. The Lagrangian is chiral invariant, and it can b e considered a con ning version of the traditional {mo del. An imp ortant p oint concern the value of the chiral elds, the pion and the sigma, in this mo del, in the single nucleon case. These elds are always near their 3 vacuum value. This p oint has b een checked out numerically several times and some euristic explanations have b een prop osed [10,14]. As a consequence, the chiral elds cannot `wind' around the mexican hat p otential. The pion is thus just a p erturbation and cannot develop a non{trivial top ology. B. Fixing the parameters The parameters of the mo del are: the chiral meson masses m =0:14 GeV, m =1:2 GeV, the pion decay constant f =0:093 GeV, the coupling constant g , and the parameters app earing in the quadratic (2) or in quartic p otential (3). The free parameters are xed to repro duce the basic prop erties of the nucleon. At the mean eld level we use the hedgehog ansatz which is an eigenstate of the so called Grand ~ ~ ~ Spin G = S + I , and is a sup erp osition of various bare nucleon and delta states. In order to describ e the single nucleon state we p erformed a double pro jection on linear and angular momentum eigenstates from the hedgehog, whose details can b e found in Ref. [15]. For the DM version of the mo del we use the sets of parameters of Ref. [16], for whicha go o d description of the static prop erties of the nucleon was obtained.
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