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Polyakov Linear-Σ Model in Mean-Field Approximation and Optimized

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Polyakov Linear-Σ Model in Mean-Field Approximation and Optimized PHYSICAL REVIEW C 101, 035210 (2020) Polyakov linear-σ model in mean-field approximation and optimized perturbation theory Abdel Nasser Tawfik * Nile University, Egyptian Center for Theoretical Physics (ECTP), Juhayna Square of 26th-July-Corridor, 12588 Giza, Egypt and Goethe University, Institute for Theoretical Physics (ITP), Max-von-Laue-Str. 1, D-60438 Frankfurt am Main, Germany Carsten Greiner Goethe University, Institute for Theoretical Physics (ITP), Max-von-Laue-Str. 1, D-60438 Frankfurt am Main, Germany Abdel Magied Diab Modern University for Technology and Information (MTI), 11571 Cairo, Egypt and World Laboratory for Cosmology And Particle Physics (WLCAPP), 11571 Cairo, Egypt M. T. Ghoneim and H. Anwer Physics Department, Faculty of Science, Cairo University, 12613 Giza, Egypt (Received 21 August 2019; accepted 10 February 2020; published 30 March 2020) The optimized perturbation theory (OPT) is confronted to first-principle lattice simulations. We compare results from the Polyakov linear-sigma model (PLSM) in OPT with the conventional mean-field approximation (MFA). At finite temperatures and chemical potentials, the chiral condensates and the decofinement order parameters, the thermodynamic pressure, the pseudocritical temperatures, the subtracted condensates, the second- and high-order moments of various conserved charges (cumulants) obtained in MFA are compared with OPT and also confronted to available lattice quantum chromodynamics (QCD) simulations. We conclude that when moving from lower- to higher-order moments of different quantum charges, OPT becomes more closer to lattice QCD simulations. The higher-order moments of conserved charges, such as baryon, strange, and electric charge, are proportional to powers of the correlation length and thus expected to diverge at the critical endpoint of the QCD phase boundary. Making sure that one approximate approach is more sensitive than another, even slightly, is crucial for implementing PLSM, for instance, in positioning critical endpoint and providing an important signature for the possible experimental detection. DOI: 10.1103/PhysRevC.101.035210 I. INTRODUCTION SU(2)L× SU(2)R down to SU(2)L+R. This results in massive σ particle and light or even nearly massless Goldstone bosons. In quantized field theory [1–6], the linear-σ model Accordingly, the constituent quarks gain masses, as well, [7] (LSM) with a spinless scalar field σa [8] and triplet m = gfπ , where g is the coupling and fπ is the pion decay pseudoscalar fields π was introduced in order to describe q a constant. In the same matter, fermions can be inserted in this the pion-nucleon interactions and the chiral degrees of model either as nucleons or as quarks. It has been shown that freedom. This low-energy effective model has the generators the σ field under chiral transformations exhibits the same T = λ /2, where λ are Gell-Mann matrices, and a real a a a behavior as that of the quark condensates, i.e., σ can be taken classical field forming an O(4) vector; = T (σ , iπ ). The a a a as an order parameter for the quantum chromodynamics chiral symmetry is explicitly broken by the 3 × 3 matrix field (QCD) chiral phase transition [9–13]. Accordingly, the H = T h , where h are the external fields. Under SU(2) × a a a L phase structure [12,14–16], the properties of QCD in finite SU(2) chiral transformation, for instance, → L+R, σ R a magnetic fields [16–19], and also various thermodynamic acquires finite vacuum expectation values, which in turn break quantities have been estimated and reported, at finite baryon density [14,18,20] and isospin asymmetry [21]. In almost all these studies, a comprehensive confrontation with the *atawfi[email protected] first-principle lattice calculations was the main part. The LSM with its various extensions as examples on QCD-like effective Published by the American Physical Society under the terms of the models put forward a major goal to—as perfect as possible— Creative Commons Attribution 4.0 International license. Further reproduce various quantities simulated in lattice QCD. distribution of this work must maintain attribution to the author(s) Quantum mechanically, the spontaneous symmetry break- and the published article’s title, journal citation, and DOI. Funded ing could be achieved by introducing a coherent state and by SCOAP3. minimizing the free energy density. Assuming that the system 2469-9985/2020/101(3)/035210(12) 035210-1 Published by the American Physical Society ABDEL NASSER TAWFIK et al. PHYSICAL REVIEW C 101, 035210 (2020) of interest is enclosed in a cubic box and imposing periodic the OPT procedure in PLSM goes as follows: boundary conditions, the fields can be uniquely decomposed Lδ = − δ L η + δ L = L η + δ L − L η , φ , =φ +δφ , = φ · (1 ) 0( ) 0( ) [ 0( )] [22]into (r t ) (t ) (r t ) k k (t )exp(ik r), where φ(t ) is the spacial expectation value over the box (1) volume and δφ(r, t ) stands for the remaining fluctuations where η is an arbitrary mass parameter, which can be fixed relative to a constant background field. It is obvious that φ = k through an appropriate variational method [29],1 and L (η) φ exp(−ik · r) and the Fourier coefficients satisfy the sym- 0 ∗ is the free Lagrangian density, in which η is included. The metry relation, φ = φ− because the fields, themselves, are k k parameter η is equivalent to mass. Thus, the implementation real. Having all these estimated, we can now apply the mean- of OPT to PLSM is apparently accompanied by an expansion field approximation (MFA), which is originated in statistical in terms of the arbitrary parameter δ. Accordingly, Eq. (1) physics. In the LSM, the meson fields (or fermion/quark shows that the underlying symmetries in the chiral limit seems fields) are replaced by their spacial averaged values and all to not be modified. The term which is proportional to η is vacuum and thermal fluctuations are neglected. All quarks added to the Lagrangian, while the same term multiplied and antiquarks are retained as quantum fields, but averaged by δ is then subtracted. At δ = 1, the original Lagrangian as well. L can be recovered, straightforwardly. At δ = 0, a solvable A generalization of MFA could be the optimized perturba- Lagrangian L is obtained. In general, the parameter δ is also tion theory (OPT), also known as δ expansion or variational 0 used as a dummy constant, in order to label the orders of the perturbation theory [23]. OPT was developed in O(N) φ4 the- perturbative calculations. Thus, δ is initially taken as a small ory and resums higher-order terms of the naive perturbation value and afterwards fixed to unity. approach [24,25]. Implications in the symmetric and broken A short review on the LSM is now in order. LSM has chiral phases [26], Bose-Einstein condensation - Bardeen Cooper Lagrangian of N quark flavors including the structure of Schrieffer theory of superconductivity (BEC-BCS) [27], and f mesons and quarks. With the incorporation of the Polyakov- zero-dimensional O(N) scalar field model [28], are examples loop potential, of the reliability and applicability of OPT. Here, we apply ∗ OPT to O(4) σ model and then compare the results with the L = Lψ¯ ψ + Lm − U(φ,φ , T ), (2) ones obtained from MFA. We aim at determining the sensi- where the first term stands for the Lagrangian density for bary- tivity of OPT relative to MFA. This is the first time, where onic (fermionic) fields with N color degrees of freedom, the both approaches are compared with each other with reference c second term gives the contributions of the mesonic (bosonic) to the first-principle lattice calculations. Assuring that one fields, and finally the third term represents the Polyakov-loops approximate approach is more sensitive than another, even potential incorporating the gluonic degrees of freedom and the slightly, is very crucial for the preparation of the Polyakov dynamics of the quark-gluon interactions. linear-σ model (PLSM) for a comprehensive implementation When implementing the OPT approach, Eq. (1), on the in mapping out the QCD critical boundary and eventually PLSM Lagrangian, Eq. (2), we get finding the critical endpoint (CEP) and thus providing an crucial signature for the possible experimental detection. L = ψ γ μ − δ σ + γ π ψψ¯ f [i Dμ gTa( a i 5 a ) The present paper is organized as follows. A short review f on the optimized perturbation theory and the mean-field ap- − − δ η ψ , proximation will be given in Sec. II. Section III is devoted to (1 ) ] f (3) † μ 2 2 † the results and discussion. This includes chiral condensates Lm = Tr[∂μ ∂ − (m + (1 − δ) η ) ] and deconfinement order parameters in Sec. III A, pseudo- + δ { + † − λ † 2 critical temperatures in Sec. III B, thermodynamic pressure in c(Det[ ] Det[ ]) 1 (Tr [ ]) † 2 † Sec. III C, fluctuations and correlations of conserved charges, − λ2 Tr[ ] + Tr[H( + )]}, (4) in Sec. III D. The latter are detailed to second-order in U φ,φ∗, =− φφ∗ −a/T + − φφ∗ Sec. III D 1, and higher-order moments in Sec. III D 2.The ( T ) bT[54 e ln(1 6 final conclusions are outlined in Sec. IV. − 3(φφ∗)2 + 4(φ3 + φ∗3))], (5) where ψ are Dirac spinor fields for the quark flavors f = μ II. OPTIMIZED PERTURBATION THEORY [u, d, s], while Dμ,μ,γ , and g are covariant derivative, AND MEAN-FIELD APPROXIMATION Lorentz index, chiral spinors, and Yukawa coupling con- φ φ∗ We intend to check whether the optimized perturbation stant, respectively. and are the order parameter of the theory (OPT) would be able to play the role of an alternative Polyakov-loop variables and their conjugates, respectively.
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