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Deconfinement and Freezeout Boundaries in Equilibrium Thermal Models

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Deconfinement and Freezeout Boundaries in Equilibrium Thermal Models Hindawi Advances in High Energy Physics Volume 2020, Article ID 2453476, 8 pages https://doi.org/10.1155/2020/2453476 Research Article Deconfinement and Freezeout Boundaries in Equilibrium Thermal Models Abdel Nasser Tawfik ,1,2 Muhammad Maher,3 A. H. El-Kateb,3 and Sara Abdelaziz3 1Nile University, Egyptian Center for Theoretical Physics (ECTP), Juhayna Square of 26th-July-Corridor, 12588 Giza, Egypt 2Goethe University, Institute for Theoretical Physics (ITP), Max-von-Laue-Str. 1, D-60438 Frankfurt am Main, Germany 3Faculty of Science, Physics Department, Helwan University, 11795 Ain Helwan, Egypt Correspondence should be addressed to Abdel Nasser Tawfik; atawfi[email protected] Received 9 October 2019; Accepted 28 January 2020; Published 24 February 2020 Guest Editor: Jajati K. Nayak Copyright © 2020 Abdel Nasser Tawfik et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The publication of this article was funded by SCOAP3. In different approaches, the temperature-baryon density plane of QCD matter is studied for deconfinement and chemical freezeout boundaries. Results from various heavy-ion experiments are compared with the recent lattice simulations, the effective QCD-like Polyakov linear-sigma model, and the equilibrium thermal models. Along the entire freezeout boundary, there is an excellent agreement between the thermal model calculations and the experiments. Also, the thermal model calculations agree well with the estimations deduced from the Polyakov linear-sigma model (PLSM). At low baryonic density or high energies, both deconfinement and chemical freezeout boundaries are likely coincident, and therefore, the agreement with the lattice simulations becomes excellent as well, while at large baryonic density, the two boundaries become distinguishable forming a phase where hadrons and quark-gluon plasma likely coexist. 1. Introduction The program of heavy-ion collision experiments dates back to early 1980s. Past (AGS, SIS, and SPS), current (RHIC Strongly interacting matter under extreme conditions is and LHC), and future facilities (FAIR and NICA) help in characterized by different phases and different types of phase answering essential questions about the thermodynamics of transitions [1]. The hadronic phase, where stable baryons the strongly interacting matter and in mapping out the build up a great part of the Universe and the entire everyday temperature-baryon density plane [2]. The unambiguous evi- life, is a well known phase. At high temperatures and/or den- dence on the formation of QGP is an example of a great sities, other phases appear. For instance, at temperatures of a empirical achievement [5, 6]. The colliding nuclei are conjec- few MeV, chiral symmetry restoration and deconfinement tured to form a fireball that cools down by rapid expansion transition take place, where quarks and gluons are conjec- and finally hadronizes into individual uncorrelated hadrons. tured to move almost freely within a colored phase known The present script focuses on the temperature-baryon den- as the quark-gluon plasma (QGP) [2]. At low temperatures sity plane, concretely near the hadron-QGP boundaries, in but large densities, hadronic (baryonic) matter forming com- the framework of the equilibrium thermal model [7]. To this pact interstellar objects such as neutron stars is indubitably end, we put forward a basic assumption that both directions, observed in a conventional way, and very recently, gravita- the hadron-QGP and QGP-hadron phases are quantum- tional waves from neutron star mergers have been detected, mechanically allowed [8]. In other words, the picture drawn as well [3]. At higher densities, extreme interstellar objects so far seems in fundamental conflict with the time arrow. such as quark stars are also speculated [4]. In lattice quantum The concept of an arrow of time prevents the reverse direc- chromodynamics (QCD), different orders of chiral and tion, especially if the change in the degrees of freedom or deconfinement transitions have be characterized, especially entropies aren’t following the causality principle, the second at low baryon densities. law of thermodynamics. The statistical thermal approaches 2 Advances in High Energy Physics fi work well near both decon nement and chemical freezeout potential µb canp beffiffiffiffiffiffiffi related to the nucleon-nucleon center- boundaries [2, 9]. This could be understood in the light of of-mass energy sNN [20]: the thermal nature of an arbitrary small part of the highly entangled fireball states. Following the Eigenstate Thermali- μ apffiffiffiffiffiffiffi ð Þ zation Hypothesis [8, 10], the corresponding probability dis- b = , 4 1+b sNN tribution of the projection of these states remains thermal. We follow the line that the thermal models reproduce well : : : : −1 the particle yields and the thermodynamic properties of the where a =1245 ± 0 049 GeV and b =0244 ± 0 028 GeV . hadronic matter including the chiral and freezeout tempera- The number and energy density, respectively, can be tures. We compare our calculations with reliable lattice QCD derived as simulations, an effective QCD-like approach, and available ∂ ðÞμ experimental results. ðÞμ 〠 ln Zi T, V, ni T, = ∂μ The present script is organized as follows. In Section 2, i i fi ð∞ approaches for decon nement and freezeout boundaries in g 1 = 〠 i p2dp , equilibrium thermal models are introduced. The results are π2 ½ðÞμ − ε ðÞ i 2 0 exp i i p /T ±1 discussed in Section 3. Section 4 is devoted to the conclusions ð5Þ and outlook. ∂ ln Z ðÞT, V, μ ρ ðÞT, μ = 〠 i i ∂ðÞ1/T 2. Equilibrium Thermal Models i ð g ∞ −ε ðÞp ± μ = 〠 i p2dp i i : It was conjectured that the formation of the hadron reso- π2 ½ðÞμ − ε ðÞ i 2 0 exp i i p /T ±1 nances follows the bootstrap picture, i.e., the hadron reso- nances or the fireballs are composed of further resonances Likewise, the entropy and other thermodynamic quan- or fireballs, which in turn are consistent of lighter resonances tities can be derived straightforwardly. or smaller fireballs, and so on [11, 12]. The thermodynamic Both temperature T and the chemical potentialpffiffiffiffiffiffiffi µ = quantities of such a system can be deduced from the partition ⋯ function ZðT, µ, VÞ of an ideal gas. In a grand canonical Biµb + SiµS+ are related to each other and to sNN [2]. ensemble, this reads [2, 13–17] As an overall thermal equilibrium is assumed, µS is taken as a dependent variable to be estimated due to the strangeness μN − H conservation, i.e., at given T and µb, the value assigned to ZTðÞ, V, μ = Tr exp , ð1Þ µ is the one assuring hn i − hni =0. Only then is µ com- T S S S S bined with T and µb in determining the thermodynamic quantities, such as the particle number, energy, and entropy. where H is the Hamiltonian combining all relevant degrees of The chemical potentials related to other quantum charges, freedom and N is the number of constituents of the statistical such as the electric charge and the third-component isospin, ensemble. Equation (1) can be expressed as a sum over all can also be determined as functions of T, µb, and µS, and each hadron resonances taken from a recent particle data group of them must fulfill the corresponding laws of conversation. (PDG) [18] with masses up to 2.5 GeV [19]: This research intends to distinguish between deconfine- ment and freezeout boundaries in equilibrium thermal ðÞμ 〠 ðÞμ models. The latter is characterized by Tχ and µb, which are lnZT, V, = lnZi T, V, i conditioned to one of the universal freezeout conditions ð∞ 3 g −ε ðÞp [21], such as constant entropy density normalized to Tχ = V i ± p2dp ln 1 ± λ exp i , π2 i 2 0 T [22, 23], constant higher-order moments of the particle mul- tiplicity [24, 25], constant trace anomaly [26], or an analogy ð2Þ of the Hawking-Unruh radiation [27]. The experimental esti- mation for Tχ and µb, as shown in Figure 1, proceeds through where the pressure reads T∂ ln ZðT, V, µÞ/∂V and ± stands statistical fits for various particle ratios calculated in statisti- ε ð 2 2Þ1/2 cal thermal models. The former, the deconfinement transi- for fermions and bosons, respectively. i = p + mi is the dispersion relation and λ is the fugacity factor of ith tion, is conditioned to line-of-constant-physics, such as i ρ particle [2]. constant energy density, [28]. The inclusion of the strange quarks seems to affect the critical temperatures, as these μ μ μ come up with extra hadron resonances and their thermody- Bi b + Si S + Qi Q λ ðÞT, μ = exp , ð3Þ namic contributions, where the mass of strange quarks is of i T the order of the critical temperature. ð Þ ð Þ ð Þ where Bi µb , Si µS , and Qi µQ are baryon strangeness, and 3. Results electric charge quantum numbers (their corresponding chemical potentials) of the ith hadron, respectively. From Figure 1 depicts the freezeout and deconfinement parameters a phenomenological point of view, the baryon chemical Tχ and µb as determined from the measurements gained Advances in High Energy Physics 3 200 150 (MeV) 100 Tx 50 0 0 200 400 600 800 1000 �b (MeV) Freezeout: ql HADES (Ar+KCl) NA61/SHINE (p + p) Freezeout: ql + qs HADES (Au+Au) ALICE Deconfnement: ql NA49 (Pb+Pb) RHIC/SPS/AGS Deconfnement: ql + qs NA61/SHINE (Pb+Pb) RHIC/SPS/AGS/SIS Figure fi ff 1: The freezeout and decon nement parameters Tχ and µb as deduced from di erent experimental results (symbols with error bars) are depicted and confronted with the thermal model calculations for the chemical freezeout (dot-dashed and dashed curves) and deconfinement parameters (solid and long-dashed curves) with and without strange quarks.
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