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. to the nonperturbative approximation, such as the mean-field We notice that that Polyakov Lagrangian, Eq. (5), is not directly impacted by the OPT approach, for instance, η or δ are approximation (MFA), of the PLSM. This is the primary L L goal set forward for the present paper. In general, an OPT not present, while ψ¯ ψ and m are impacted. We also notice L η L η2 procedure aims at optimizing a linear δ expansion to the that ψ¯ ψ is given in terms of , while m of . The reason Lagrangian density. The basic idea of OPT becomes obvious for this can be understood due the OPT approach and the role when expanding the chiral Lagrangian, Eq. (1), from which we realize that even analytic nonperturbative calculations be- yond what MFA would reach become accessible. Concretely, 1This parameter is different from the PLSM parameters.

035210-2 POLYAKOV LINEAR-σ MODEL IN MEAN-FIELD … PHYSICAL REVIEW C 101, 035210 (2020) of η,Eq.(1). It is obvious that η as a modified mass parameter goes with the masses in the PLSM. Last but not least, when δ → 0, the standard PLSM Lagrangian [18] can be obtained. Up to now, construction of PLSM Lagrangian, MFA does not apply. It should be noticed that the Lagrangian density of the ψ¯ − δ ηψ fermions is deformed by adding a gaussian term (1 ) FIG. 1. Diagrams illustrating the corrections to the free energy to the original Lagrangian density. All terms, in which the up to δ2 expansion. The fermionic contributions are depicted as solid δ coupling constant g is included, are multiplied by .As lines. The propagators of σ fields are represented by the dashed lines. discussed earlier, at δ → 1, the original Lagrangian density of the fermionic and mesonic contributions can entirely be recovered. The generator operator is a complex matrix for nonet Then, the free energy density can be deduced as N2−1 = f σ + π T meson states, a=0 Ta(¯a i¯a ). In U(3) algebra, the ∗ FOPT(T,μf ) =− ln [Z] = (σl ,σs ) + U(φ,φ , T ) generator operators Ta = λˆ a/2 are related to the Gell-Mann V 2 + ,μ . matrices λˆ a [30]. The parameters m , hl , hs, λ1, λ2, and c ψψ¯ (T f ) (9) are determined, at σ meson mσ = 800 MeV [31]. It should δ2 be emphasized that the estimation of all these parameters is When recalling Feynman graphs up to , the OPT ap- not affected by MFA or OPT approaches. They are depending proach can be depicted. Figure 1 shows the contributions in δ on masses and decay constants as partly elaborated in the orders of and the color degrees of freedom Nc.Theleft δ0 / 0 O δ0 0 well-known PCAC relations [31,32]. panel (a) shows the zero order, , with 1 Nc or ( Nc ), When using OPT instead of MFA to evaluate the free which is shown as thick solid lines representing the fermionic energy F of the PLSM, analytic nonperturbative calculations contributions, for which the system is composed of quarks become possible through a prescription known as the principle and antiquarks. The free energy density of these contributions of minimal sensitivity (PMS) [33,34]. PMS states that F, have been evaluated by different approximations including η δ = MFA [12,14–21,36–38] and Hartree-Fock method (HFM) Eq. (9), can be minimized to the variations of ,at 1: ∂F [39]. The middle panel (b) of Fig. 1 shows the first-order OPT = . corrections of δ with 1/N0 or O(δ N0 ) and how this comes ∂η 0 (6) c c η,δ¯ =1 up with an additional contribution rather than those of both Accordingly, we emphasize that the expectation value ofη ¯ MFA and HFM. The dashed line represents the propagator of σ is related to the sigma fields σ f ; η ∼ σ f [35]. Consequently, the field. This is the version we are utilizing in the present the grand-canonical partition function2 Z, which is given in calculations. dependence on the temperature T and the chemical potentials The right panel (c) of Fig. 1 shows the second-order δ / 0 O δ2 0 of f th quark flavor μ f is defined by the path integral over all corrections of with 1 Nc or ( Nc ) belonging to the next- to-leading order (NLO) expansions [40,41]. We notice that the bosons, fermions, antifermions: first two diagrams are the propagators up to O(δ2 ). These = , , μ f Nˆ f − Hˆ Z = Tr exp f u d s are considered to be an OPT approach for a gas with free T fermions whose masses are converged by the modified mass parameter η → η + δ(σ f − η)[29,35]. The last diagram is 2 = DσaDπa DψDψ¯ already included in the second order of O(δ ) and written with a the usual mass parameter η [29,35]. Let us now consider the ⎡ ⎤ first two diagrams to determine the PLSM free energy density F ,μ δ × exp ⎣ d4x(L + μ ψ¯ γ 0ψ )⎦, (7) in finite volume, OPT(T f ), Eq. (9), the “optimized” f f f expansion. f =u,d,s As discussed, the full construction of the OPT free energy where Hˆ the chiral Hamiltonian density. The chemical poten- density in finite volume is given in Eq. (9). In the right-hand μ tials f are related to the conserved charge numbers of baryon side (rhs), the first term, (σl ,σs ), stands for the potential number (B), electric charge (Q), and strangeness (S) for each of the mesonic contributions in pure nonstrange (σl ) and pure = , , of the quark flavors, f [u d s]: strange (σs) condensates μ μ μ = B + 2 Q , σ 2 + σ 2 u 2 2 l s 3 3 (σl ,σs ) = (m + (1 − δ) η ) μ μ 2 μ = B − Q , d λ (2λ + λ ) (λ + λ ) 3 3 + δ 1 σ 2σ 2 + 1 2 σ 4 + 1 2 σ 4 l s l s μB μQ 2 8 4 μs = − − μS. (8) 3 3 c 2 − √ σ σs − hl σl − hs σs . (10) 2 2 l 2MFA also enters the play but only when constructing the partition We now explicitly show that the last term in rhs of Eq. (9), function Z as detailed in Refs. [12,14–21,36]. ψψ¯ (T,μf ), which represents the potential of the quarks

035210-3 ABDEL NASSER TAWFIK et al. PHYSICAL REVIEW C 101, 035210 (2020) and antiquarks contributions and the chiral condensates in the potentials in Eq. (9) are determined at δ → 1 and η = mean-ﬁeld limit reads 0 (or vanishing arbitrary mass parameter).

ψ¯ ψ (T,μf ) Having the free energy estimated weather in OPT or in 3 MFA, the thermodynamic quantities which are thought to d P + − =−2 T [I ( )(T,μ) + I ( )(T,μ)] describe the chiral structure of the QCD matter, at finite (2π )3 f f f f f =u,d,s temperatures and finite chemical potentials can be determined. These quantities play the role of the thermodynamic order pa- d3P (m + η) + 2 δ N f (η − σ ) rameters characterizing whether the system undergoes phase c (2π )3 E f transition. Concretely, the order parameters are the mean f =u,d,s f σ fields (σ ¯l , andσ ¯s) and the Polyakov-loop variables (φ,¯ ∗ (+) (−) and φ¯ ), which are analytically estimated from the global × [1 − n (T,μ) − n (T,μ)] . (11) f f f f minimizing of the real part of the PLSM free energy density in finite volume, Re [F (T,μ)], with respect to the associated I (+) ,μ OPT The Fermi-Dirac distribution functions f (T f ) and thermodynamic order parameter, at a saddle point (+) ,μ n f (T f ) are defined—in widely used notations—as ∂F ∂F ∂F OPT = , OPT = , OPT = , (+) (+) (+) 0 0 0 E E E ∂σ ∂σ ∂φ¯ f f f ¯l σ ¯s σ φ¯ (+) ∗ − − −3 ¯l ¯s I (T,μ) = ln 1 + 3 φ + φ e T e T + e T , f f ∂F OPT = . (12) ∗ 0 (14) ∂φ¯ φ¯∗ (+) (+) (+) E E E ∗ − f − f −3 f φ + 2φe T e T + e T These expressions can be solved, numerically. For MFA, these (+) ,μ = , n (T f ) (+) (+) (+) expressions are the ones assuring global minimization and f E E E ∗ − f − f −3 f 1 + 3 φ + φ e T e T + e T consequently determine the various order parameters. For OPT, we have to assure the principle of minimal sensitivity (13) (PMS), Eq. (6), in addition to Eqs. (14). Only when all ± expressions outlined in Eqs. (14) and (6) are fulfilled, T and μ where E ( ) = E ∓ μ are the energy-momentum dispersion f f f can be determined and then enter further OPT calculations. To relations, in which the upper sign is applied for quarks and summarize, the differences between OPT and MFA become the lower sign for antiquarks. These relations are subject to / now obvious, namely for MFA, F → F and Eq. (14)are = | |2 + + η 2 1 2 OPT modifications due to OPT, E f [ P (m f ) ] .By the only global minimization to be fulfilled. For OPT, Eq. (6) (+) (−) replacing E f with E f and the order parameter of the must be fulfilled, as well. Polyakov-loop variable φ with its conjugate φ∗ or vice versa, In the section that follows we present results on the I (−) ,μ (−) ,μ we find that the terms f (T f ) and n f (T f )are chiral condensates and the deconfinement order parameters, I (+) ,μ (+) ,μ Sec. III A. Section III C is devoted to results for the ther- identical to f (T f ) and n f (T f ), respectively. It is worth highlighting that the second term in the rhs of Eq. (11) modynamic pressure. The fluctuations and the correlations of approximately equals the derivative of the first term in the various conserved charges shall be discussed in Sec. III D.The mean-field of the averaged mass parameterη ¯. second-order moments shall be outlined in Sec. III D 1,while The second line in Eq. (11) is approximately the gap the higher-order moments are the topics of Sec. III D 2. equation of the quark condensate [42,43]. By minimizing the thermodynamic potential with respect to the quark condensate III. RESULTS AND DISCUSSION q¯ f q f , this definition becomes obvious, which shall play an essential role in clarifying the impacts of OPT compared In the present section, we study the quark-hadron phase to MFA, as illustrated in Fig. 5. The modified Fermi-Dirac structure of the QCD matter at finite temperature and chemical distribution functions for quarks and antiquarks, Eq. (13), potential in MFA and OPT in PLSM. To this end, we first σ σ with finite Polyakov-loop variables are derived explicitly by calculate the PLSM chiral condensates, l and s, and the φ φ∗ the summation over the Matsubara frequencies [42]. These deconfinement order parameters and . To judge about distribution functions straightforwardly lead to the standard the reliability of both types of approximations, we compare σ form, especially in the limit that φ, φ∗ → 1, i.e., within the the PLSM subtracted condensate for pure mesonic fields deconfined phase. On the contrary, within the confined phase, with the lattice QCD (LQCD) calculations. In doing this, φ, φ∗ → 0, the exponential term grows by a factor of 3. we can determine the pseudocritical temperatures at different For the free energy density in PLSM, OPT and MFA chemical potentials in MFA and then in OPT. The results procedures work as follows: in both approximations shall be summarized in the QCD phase diagram compared with recent lattice QCD simulations. (i) Optimized perturbation theory (OPT): Eq. (9)istobe As a further check, we discuss some PLSM features of the estimated, where Eqs. (10), (11), and (5) shall be taken thermodynamics. We limit the discussion on the pressure. into account, and Then, we move to the fluctuations and the correlations of (ii) Mean-field approximation (MFA): the chiral limit of various conserved charges, which are deduced from the sec- the mean-field is obtained when the thermodynamic ond derivatives of the pressure, for instance, with respect to

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σ σ σ σ MFA f / f0 OPT f / f0 φ φ φ0 φ0 μ μ μ 1 (a) f=0 MeV (b) f=50 MeV (c) f=100 MeV 0.8

0.6 σ σ l / l0 0.4 σ / σ 0.2 s s0

0 μ μ μ 1 (d) f=150 MeV (e) f=200 MeV (f) f=250 MeV

0.8

0.6

0.4

0.2

0 0 0.1 0.2 0.3 0.4 0 0.1 0.2 0.3 0.4 0 0.1 0.2 0.3 0.4 T[GeV] T[GeV] T[GeV]

FIG. 2. (a)–(f) The temperature dependence of the normalized condensates for nonstrange σl /σl0 and strange σs/σs0 chiral condensates in MFA (solid curves) and in OPT (dashed curves) and that of order parameter of Polyakov-loop variables φ (dashed curves) and φ∗ in MFA ∗ (dotted-dash curves) and φ (dotted curves) and φ in OPT (double dotted dash curves) are given at μ f = 0, 50, 100, 150, 200, and 250 MeV, respectively.

the corresponding chemical potential. Last but not least, we (i) The thermal behavior of σl and σs starts at almost the compare between the PLSM results on high-order moments same value; the one corresponding to their vacuum of particle numbers. condensates. In this region, the effect of the chemical potential is apparently negligible. The increase in chemical potential seems to repair the increase in the A. Chiral condensates and deconﬁnement order parameters phase transition. With this regard, we notice that OPT

As discussed in Sec. II, the chiral condensates (σl and is more sensitive than MFA. This is a fundamental σ ) and the Polyakov-loop variables (φ and φ∗) can be difference which might favor OPT against MFA. s σ evaluated by solving the gap equations, Eqs. (14) and (6), (ii) We also notice that in OPT l starts its prompt phase Re [F (T,μ)], at the subtle point. The vacuum values of transition more rapidly than MFA. While in MFA OPT σ μ = nonstrange and strange chiral condensates, at vanishing chem- l begins a likely first-order phase transition, at ical potential μ = 0MeV, are σ = 92.5 MeV and σ = 200 MeV, panel (e) of Fig. 2. In OPT, the first-order f l0 s0 σ μ = 94.2 MeV, respectively. The various PLSM parameters are phase transition is indicated for l ,at 100 MeV. σ estimated, at mσ = 800 MeV. As introduced, for MFA, the (iii) For s, there is always a smooth phase transition, gap equations to be solved for the ones given in Eq. (14). although, OPT at T Tχ is accompanied with a faster σ In panels (a), (b), (c), (d), (e), and (f) of Fig. 2 the drop in s than MFA. The corresponding Tχ remains temperature dependence of the order parameters which are nearly identical, Fig. 4. calculated in MFA and OPT at different chemical potentials μ = With this regard, it is in order now to summarize the pro- f 0, 50, 100, 150, 200, and 250 MeV, respectively, are cedures utilized in determining the pseudocritical temperature depicted. The temperature dependence of the order parameter Tχ [20]. of the Polyakov-loop variables φ (dashed curves) and φ∗ (dotted-dash curves) in MFA and φ (dotted curves) and φ∗ (i) The first one utilizes the temperature dependence of (double dotted dash curves) and in OPT are also illustrated in the chiral susceptibility. This is the second derivative the same panels, at the given chemical potentials. with respect to the chemical potential or the first ∗ Between the MFA and OPT calculations for φ and φ , derivative of the subtracted condensate ls,Eq.(15), there are small differences below and above Tχ . While, almost with respect to the chemical potential, Eq. (17) and no difference exists between the PLSM calculations for σl Fig. 6. Tχ is precisely positioned at the inflection point, and σs in MFA and OPT. In the hadronic phase, at T < Tχ , at which a maximum in the chiral susceptibility takes both approximations become distinguishable, especially in the place. region of the phase transition. A larger difference appears in (ii) The second procedure utilizes the intersection the quark-gluon phase. point of the normalized chiral condensate and the

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FIG. 3. Left panel (a) illustrates the temperature dependence of the subtracted condensate ls in MFA (solid curve) and in OPT (dashed curve) and compares these with recent lattice QCD simulations [48] (solid band). Middle panel (b) shows the MFA subtracted condensate, at μ = 0.0 (solid curve), 50.0 (dashed curve), 100.0 (dotted curve), 150.0 (dotted-dash curve), 200.0 (double dotted-dash curve), 250.0, (long-dash curve), and 300.0 MeV. (dotted long-dash curve). Right panel (c) depicts the same as in the middle panel but here for OPT.

Polyakov-loop variables, as dictated by the lattice observe that at low temperatures, ls keep their values (unity) QCD simulations, Fig. 2 [44]. unchanged for a while, i.e., until the temperatures reach some values. Again, the further increase in the temperature gives With this regard, we believe that the first procedure leads almost the same behavior as the one obtained in the left panel to a precise estimation for Tχ , at different chemical potentials. (a). We notice that the increase in the chemical potentials The pseudocritical temperature Tχ is an essential thermody- tends to increase the rate of the rapid drop in . The larger namic quantity for the QCD phase structure. In the section that ls chemical potential, the narrower is the temperature region, follows, we confront our calculations on σ and σ to recent l s within which decline to low values. It is apparent that lattice QCD calculations. ls hadron-quark phase transition of first order seems to start taking place, at μ f = 200–250 MeV. B. Pseudocritical temperatures The results obtained when comparing ls with recent lat- Another thermodynamic quantity which plays the role of tice QCD simulations, encourage mapping out the QCD phase an order parameter is the normalized net-difference between diagram. This is another confrontation of the PLSM results the nonstrange and strange chiral condensate, known as the in MFA and in OPT with the first-principle calculations. As subtracted condensates ls [45], this quantity was directly discussed earlier, the pseudocritical temperatures in PLSM, simulated in lattice QCD, Tχ is to be estimated in different procedures. Having this done, σ − / σ | we can now map out Tχ at different baryon chemical potential l (hl hs ) s T μ = μ ls = , (15) B 3 f . σl − (hl /hs ) σs|T =0 Figure 4 depicts the Tχ -μB plane. The PLSM results in where hl (hs) are nonstrange (strange) explicit symme- MFA (solid curve) and in OPT (dashed curve) are compared try breaking parameters which are to be estimated from with the lattice QCD simulations [49] (dashed band), [50] the Dashen-Gell-Mann-Oakes-Renner (DGMOR) relations (grid band), and also with the experimental estimations in the [46,47]. Figure 3 shows the temperature dependence of the STAR experiment at the BNL Relativistic Heavy Ion Collider subtracted condensate, at different chemical potentials. The (RHIC) (closed symbols) [51] and the ALICE experiment left panel (a) depicts the PLSM results in MFA (solid curve) at the CERN Large Hadron Collider (LHC) (open symbols) and in OPT (dashed curve). These are compared with the con- [52]. For the lattice QCD simulations [49], the dashed band tinuum extrapolation of recent lattice QCD simulations [48]. indicates the type of the phase transition. The grid band shows There is a good agreement, especially within the region of the the same as the dashed band but here for the first-principle phase transition, where the condensates decline very rapidly. lattice QCD calculations [50]. While the solid band refers It is obvious that at T < Tχ , ls first remains unchanged hav- to the boundary of the critical temperature, the dashed band ing unity as value. With the increase in temperature, a rapidly illustrates the boundary of Tχ , which was obtained from the decrease takes place within the region of the phase transition. baryon susceptibility, Fig. 6. With further increase in the temperature, ls becomes almost The PLSM results in MFA (solid curve) and in OPT temperature independent. In this region, the colored quark- (dashed curve) agree well with both lattice QCD calculations. gluon phase, ls keeps its low value almost unchanged, which The agreement looks very convincing, especially when focus- apparently means that the quark and gluon are deconfined and ing on the lattice QCD simulations [49] (solid and dashed their related degrees of freedom are liberated. bands), which are recently refined [50] (grid band). The The middle and right panels of Fig. 3 show the temperature inside box zooms out the region, within which the lattice dependence of the subtracted condensates at various chemical QCD calculations are reliable. When focusing on the possible potentials for MFA and OPT, respectively. Here, we also differences between MFA and OPT, we would report on

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200 of freedom and Nf quark degrees of freedom as LQCD: PLB 751 ,559 (2015) LQCD: PLB 751 ,559 (2015) . μ 2 pSB 0 8 Gg 2 G f 2 f HOTQCD: PLB 795 , 15 (2019) = π + 0.7 π + 3 4 STAR: EPJ Web Conf. , 08007 (2015) T 72 72 T 150 90 f ALICE: Nature 561, 321 (2018). PLSM in MFA 3 μ f 4 PLSM in OPT + , (16) 2π 2 T 100 170 [MeV] χ 160 where Gg and G f are the degeneracy factors for gluons T = 150 and quarks, respectively. These are defined as Gg 50 , × 2 − = 140 spin polarization[0 1] (Nc 1), G f spin polarization [+1/2, −1/2] × parity[ψ, ψ¯ ] × NcNf , i.e., the degeneracy 0 100 200 300 factors, Gg = 16 and G f = 36. In this limit, i.e., taking into 0 π 2 . × + . × account the first two terms 72 [(0 8 16) (0 7 36)], the 0 200 400 600 800 1000 / 4 = . μ SB limit for thermodynamic pressure reads pSB T 5 209 B [MeV] for Nf = 2 + 1 quark flavors. FIG. 4. The chiral phase diagram of the PLSM resulted in MFA The upper panels of Fig. 5 depict the temperature depen- / 4 (solid curve) and in OPT (dashed curve) is compared with recent dence of the normalized thermodynamic pressure p T de- lattice QCD simulations [49] (solid and dashed bands) and [50](grid duced form the PLSM in MFA and OPT, at different chemical band). Furthermore, these are confronted to the experimental results potentials. The left-hand panel (a) shows the PLSM pressure of the STAR [51] (closed symbols) and ALICE experiments [52] in MFA and in OPT, at μ f = 0 MeV and compares these (open symbols). with recent lattice QCD calculations [53] (closed circles). We observe that the PLSM results agree well with the lattice QCD calculations, as reported in Refs. [12,14–21,36]. At an almost identical result. Apart from the observation that low temperatures, the PLSM results are slightly lower than OPT is accompanied with a slightly higher Tχ , especially the lattice QCD calculations. The comparison seems to be at large μB, both approximations result in identical Tχ .We improved in the region of the quark-hadron phase transition. notice that the phase transitions in MFA and in OPT cross At high temperatures, there is a good agreement, at least over. Both result in identical Tχ ,atμB 450 MeV. At larger up to T 2.5Tχ . It is worth emphasizing that the PLSM μB, they start being distinguishable; OPT has a higher Tχ curves—similar to the lattice QCD—seem to saturate below than MFA. the SB limit. At T 2.5Tχ , the gap with the SB limit is about So far, we conclude that either in the order parameters or 31.8% for MFA (solid curve) and a little bit more for OPT in the subtracted condensates or in the pseudocritical temper- (dashed curve), 34.9%. In light of this, one concludes that the atures both MFA and OPT are almost identical, especially at phase transition in both approaches has the same transition small μB. At higher μB, OPT becomes slightly more sensitive (crossover), very similar to the lattice QCD calculations and than MFA, which represents a corner milestone in preparing the deconfined phase seems strongly correlated. PLSM for an extensive scan for critical phenomena, such as Middle (b) and right (c) panels of Fig. 5 show p/T 4 QCD critical endpoint. as functions of T ,atμ f = 0, 50, 100, 150, 200, 250, and In the section that follows, we compare PLSM results in 300 MeV in MFA and in OPT, respectively. We find that the OPT and in MFA with recent lattice QCD calculations for the increase in μ f apparently drives the phase transition form thermodynamic pressure, which can be derived directly from being a smooth crossover to a prompt first-order phase tran- the total free energy density, Eq. (9). sition. In light of this, the pseudocritical temperature Tχ (μB ) seems not to be a universal constant. At least, increasing μB decreases Tχ , similar to the behavior depicted in Fig. 4.Not C. Thermodynamic pressure shown here, we report on other dependences, namely that The thermodynamic pressure p(T,μf ) =−F(T,μf ) Tχ depends—as well—on different variables, for instance, plays an important role in deriving various thermodynamic the type of the approximation used in the PLSM, the input quantities. After addressing the PLSM in MFA and in OPT parameters, and the Polyakov potential which is there to and estimating the corresponding order parameters [chiral (σ integrate the dynamics of gluon interaction to the PLSM chiral fields) and deconfinement (Polyakov)], it is now comprehen- model. sible to analyze the thermodynamic pressure. As done in the The bottom panel of Fig. 5(d) shows differences between previous section, we aim at confronting the PLSM results on OPT and MFA in a semilog scale. The difference in the the pressure with recent lattice QCD calculations [53]. The scaled pressure p/T 4 (double-dot-dashed line) is compared earlier have been estimated in MFA and in OPT, separately. with the difference in second term of Eq. (11) (dot-dashed We aim at determining which approach agrees well with the curve). The latter represents main contributions added by the lattice QCD calculations. OPT approximation, i.e., an extra term relative to MFA. We With this regard, we calculate the Stefan-Boltzmann (SB) conclude that both approximations MFA and OPT seem not limit which can be defined from the partition function of an to be giving the same thermodynamic pressure. This would ideal gas of free quarks and gluons [54] with Nc color degrees not be obvious from a blind comparison as in the top panels.

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Nature 539, 69 (2016) MFA OPT MFA 6 OPT μ (a) f=0.0 MeV (b) (c) 5 Ideal gas Nf = 2+1 4 4 3

P/T μ f =0.0 MeV. μ =50.0 MeV. μ f 2 f =100.0 MeV. μ =150.0 MeV. μf 1 f =200.0 MeV. μ =250.0 MeV. μf f =300.0 MeV. 0

0.5 1 1.5 2 2.5 0.5 1 1.5 2 2.5 0.5 1 1.5 2 2.5 T/Tχ T/Tχ T/Tχ

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0.1

0.01

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FIG. 5. The same as in Fig. 3 but here for normalized pressure P/T 4 (a)–(c). The PLSM results are compared with lattice QCD simulations [53]. The bottom panel (d) illustrates the temperature dependence of the PLSM in MFA and OPT on the normalized pressure (double-dot- dashed line) and the second term in Eq. (11), the subtracted normalized pressure (dot-dashed curve).

A great portion of such a difference is to be credited to the derivative of the partition function Z(T,μX ) with respect to second term of Eq. (11). corresponding chemical potential μX as In the section that follows, we move to high-order mo- ∂ Z ,μ ments. In doing this, we aim at highlighting whether the ln[ (T X )] NX = T . (18) higher-order moments would enhance the difference between ∂μX MFA and OPT, as observed in Fig. 5 or eventually not. We first present calculations for quadratic fluctuations for the The cumulants of the quantum number distributions are given same quantum charges and correlations, i.e., mixed quantum as charges in Sec. III D 1. The higher-order moments shall be CX = VT3χ X , (19) elaborated in Sec. III D 2. n n σ 2 =δ 2 = 2χ X κ = where ( N ) VT 2 is the variance and X / σ 2 2 D. Fluctuations and correlations of conserved charges C4 ( ) is the kurtosis. One can construct products of The fluctuations and the correlations of the various con- moment, which can be related to the measured multiplicities served quantum charges, X = [B, Q, S], can be estimated of the produced particles [55], for instance, from the derivatives of the total free energy density of the CX χ X system of interest, Eq. (9), with respect to the associated κσ2 = 4 = 4 , (20) X χ X chemical potential, Eq. (8), C2 2 ∂i+ j+k , μ / 4 i.e., ratios of quartic to quadratic cumulants of the net- BQS (p(T ˆ X ) T ) χ = , (17) quantum number fluctuations. The fluctuations of conserved ijk (∂μˆ )i (∂μˆ ) j (∂μˆ )k B Q S quantum charges can be determined in the SB limit, i.e., whereμ ˆ X = μX /T , the superscripts i, j, and k run over for an ideal gas with free constituents. These are listed in integers giving the orders of the derivatives. With this regard, Table I, where only up to the fourth order cumulants are finite. it is informative to estimate the fluctuations (diagonal) and Comparing these values with our calculations indicates how correlations (off-diagonal) of PLSM in MFA and in OPT. far is the deconfined system from the SB limit. The thermal expectation values of the conserved charges In the next section, we focus on the second-order fluctua- X = [B, Q, S], first-order moments, are estimated from the tions and the correlations of various conserved charges.

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TABLE I. Cumulants for baryon, B, electric charge, Q,and curve). Figure 6 shows in panels (a), (b), and (c) the suscep- strangeness quantum numbers, S, in an ideal gas, i.e., SB limit. tibilities for the net-baryon numbers, the net-electric charge, and the net-strangeness, respectively. The phase transition Cumulants in SB limit BQS(crossover) is defined where the mesonic degrees of freedom χ X 1/32/31start to be liberated from the confined phase (bound mesons) 2 and the system is converted to the deconfined (free quarks and χ X 2/9π 2 4/3π 2 6/π 2 4 gluons) phase [59,60]. It is obvious that the susceptibilities seem to have small values, at low temperatures, where in this region the mesonic 1. Second-order moments contributions become dominant. Accordingly, the chiral con- With this regard, the PLSM results for the second-order densates have maximum values due to the relevant degrees fluctuations are deduced at i + j + k = 2. In the calculations of freedom responsible for the small fluctuations, Fig. 2.Itis presented in Fig. 6, we consider the MFA and OPT ap- apparent that in the region of crossover the fluctuations rapidly proaches, at μ f = 0. As done in almost all figures, the PLSM arise with the increase in temperature. It should be noticed that results are compared with recent lattice QCD calculations. the chiral structure of the mesonic states plays an essential Figure 6 depicts the temperature dependence of the role in the temperature dependence of the fluctuations and the χ X quark number multiplicities. It is believed that a new state of quadratic fluctuations 2 (left-hand panel) and the quadratic χ XY matter, i.e., massless quark and gluons, is produced. correlations 11 (right-hand panel) as calculated in the PLSM in MFA (solid curves) and in OPT (dashed curves) as func- At high temperatures, the system reaches the deconfined tions of T , at vanishing chemical potential. Our calculations, state and therefore the fluctuations raise from fixed values χ X to maxima, even if remains slightly below the SB limit. The the diagonal fluctuations, 2 , are compared with recent lattice QCD calculations; Wuppertal-Budapest (grid-filled bands) explicit calculations from PLSM, as well, reach ≈94.3% and [56] and HotQCD [57] (solid-filled bands) and [58] (solid ≈88.8% of their respective ideal gas limits in MFA and in

0.5 HOTQCD: PRD 86, 034509 (2012) LQCD: PRD 95, 054504 (2017) 0.03 0.4 PLSM in MFA PLSM in OPT 0.3 (a) 0.02 (d) BQ B 2 11 χ 0.2 χ 0.01 0.1

0 0 0.8 WB Group: JHEP 01, 138 (2012) 0.3 0.6 (b) (e) QS Q

2 0.2 11 χ 0.4 χ

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0 0 1 0.3 0.8 (c) (f) BS

S 0.6 11

2 0.2 χ χ χ - 0.4 0.1 0.2

0 0 0 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0.6 0.8 1 1.2 1.4 1.6 1.8 2 T/T χ T/T χ

χ B χ Q χ S FIG. 6. Left-hand panels illustrate the temperature dependence of the second-order ﬂuctuations of the conserved charges 2 , 2 ,and 2 in (a), (b), and (c), respectively. The PLSM results in MFA (solid curve) and in OPT (dashed curve) are confronted to recent lattice QCD calculations [56] (grid bands), [57] (solid bands), and [58] (dash band). Right-hand panels show the same as in the left-hand panels but here χ BQ χ QS χ BS for the correlations of the conserved charges 11 , 11 ,and 11 in (d), (e), (f), respectively.

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2 3 1.2 HOQCD: PRD 95, 054504 (2017) LQCD: PRD 79, 074505 (2009) PLSM in MFA 2.5 1 PLSM in OPT 1.5 2

0.8 2 2 2 σ σ σ 0.6 (a) Net-Baryonic 1 (b) Net-Charge 1.5 (c) Net-Strangeness κ κ κ 0.4 1 0.5 0.2 0.5 0 0 0 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0.6 0.8 1 1.2 1.4 1.6 1.8 2 T/Tχ T/Tχ T/Tχ

FIG. 7. The temperature dependence of the moment products κσ2 computed in PLSM in MFA (solid curve) and in OPT (dashed curve) for net-baryon (a), net-charge (b), and net-strangeness (c) multiplicities are compared with the lattice QCD calculations by HotQCD collaborations (solid band) [58]and[60] (closed symbols). The arrows refers to the respective SB limits. OPT, respectively. We notice that the differences between the signatures for the phase transition [63]. The PLSM results PLSM results and the corresponding SB limits vary according on the higher-order moments of different quantum charges in to different scenarios, for instance, the several mechanisms MFA are compared with the ones in OPT. We also compare of the inclusion dynamics and the degrees of freedom of these with the lattice QCD simulations and the available quark flavors and gluons. Furthermore, the differentiation with experimental results. We start first with the PLSM results μ κσ2 χ X /χ X respect to f and the various types of statistical errors in on the product of the higher-order moments or 4 2 , the numerical simulations would contribute to the potential Eq. (20), for which measurements at various beam energies, differences. for instance Ref. [64], and the lattice QCD calculations, at The right-hand panel of Fig. 6 illustrates the same as in finite temperatures [58,60] are available, Fig. 7. the left-hand panel but here for the off-diagonal fluctuations The susceptibility of the net-baryon number is proposed χ XY χ BQ (correlations) of the conserved quantum charges 11 ; 11 as a signature for deconfining hadrons into colored massless χ QS −χ BS quarks and gluons [65]. It was suggested that the particle (top panel), 11 (middle panel), and 11 (bottom panel). We notice that the PLSM correlations agree well with the number correlations are directly coupled to the variance of the lattice QCD calculations with continuum extrapolation [57]. order parameter and thus becomes sensitive to the correlation n(5−η)/2−3 The variation in the temperature behavior of the correlations length ξ in a universal manner that χn ∼ ξ , where η χ BQ is a critical exponent to be defined from the universality class. of the baryon number with the electric charge, 11 , panel (d), is dominated by the mesonic contributions, at low tempera- With this regard, we highlight that the higher-order moments χ BQ of conserved charges, which are proportional to powers of tures. 11 arises almost exponentially with the increase in the χ BQ the correlation length and expected to diverge at the critical temperature. Again 11 vanishes, at high temperatures, because in this limit the quarks become massless [59,60]. While endpoint of the QCD phase boundary, play an essential role the correlations of the strangeness S with the electric charge in locating CEP and thus providing an important signature for Q and that of the strangeness S with the baryon number B; its experimental characteristics. 2 χ QS −χ BS In Fig. 7, κσ for net-baryon number (left panel), for elec- 11 and 11 , respectively, clearly manifest the effect of the strangeness degrees of freedom [61,62]. These correlations tric charge (middle panel), and for strangeness (right panel) μ = are related to the quark-flavor fluctuations: χ BS =−(χ s + are depicted as functions of T ,at B 0 MeV. The results 11 2 are compared with the lattice QCD calculations [60] (closed 2χ us )/3 and χ QS = (χ s − χ us )/3[57]. As emphasized when 11 11 2 11 symbols) and HotQCD collaboration [58]. We find that at discussing the quadratic fluctuations, the correlations here low temperatures χ B/χ B starts from unity, where the mesonic are also sensitive to the quark-hadron contributions. We ob- 4 2 contributions are dominant within the hadron phase. With the serve that the PLSM results agree well with the continuum increase in T , a rapid drop in χ B/χ B = κσ2 takes place, due extrapolation of the lattice QCD calculations, especially at 4 2 to the rapid increase in χ B within the region of crossover, low temperature and within the region of crossover. At high 2 left panel of Fig. 7. At high temperatures, the temperature temperatures, we find that the PLSM results on χ QS and −χ BS 11 11 dependence of χ B/χ B reaches a saturated plateau, even with are below the SB limit. Such a difference is to be estimated as 4 2 much lower values than that in the confined phase. We notice ≈6% and ≈12% for MFA and OPT, respectively. that the PLSM results in MFA and OPT seem to nearly We conclude that the comparison between the PLSM approach the SB limit, χ B/χ B| = 2/(3π 2 ). results and the lattice QCD calculations shows that OPT 4 2 SB Middle and right-hand panels of Fig. 7 draw the moment reproduces the first-principle fluctuations relatively better than product κσ2 for the net-electric charge and the net-strangeness MFA, especially within the region of the phase transition. multiplicity, respectively, as calculated from the PLSM and Also, OPT seems to reproduce well the available lattice QCD compared with the lattice QCD calculations (closed symbols) correlations better that MFA. [60]. It is apparent that the κσ2 approaches unity, at low temperatures and also the SB limit, at high temperatures, while 2. Higher-order moments at T ∼ Tχ , there is an obvious peak observed in both quan- χ B/χ B 2 The ratios of the higher-order moments, such as 4 2 , tum charges [60]. At higher temperatures, κσ for the net- χ Q/χ Q χ S/χ S χ B/χ B χ B 4 2 , 4 2 , 6 2 , and 8 , are proposed as experimental charge agrees with the lattice QCD simulations, while that of

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2 0.4 HOTQCD: PRD 95, 054504 (2017) 1.5 0.3 LQCD: JHEP 10, 205 (2018) 1 PLSM in MFA 0.2 PLSM in OPT

B 0.5 0.1 2 B χ 8 / 0 0 B χ 6

χ -0.5 -0.1 -1 -0.2 -1.5 (a) -0.3 (b) -2 -0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0.6 0.8 1 1.2 1.4 1.6 1.8 2 T/Tχ T/Tχ

χ B/χ B χ B FIG. 8. The temperature dependence of susceptibilities of the conserved charges 6 2 and 8 computed in PLSM in MFA (solid curve) and in OPT (dashed curve) is graphically compared with the lattice QCD calculations [66] (solid symbols) and [58] (bands). net-strangeness multiplicity seems to have a different depen- and 200 GeV within the rapidity |y| < 0.5 and the transverse dence, at least within the region of temperatures covered by momentum range 0.4 < pT < 2.0GeV[67–69]. In a future the lattice calculations we compare with. These lattice calcu- study, a comprehensive comparison with the STAR results is lations are about 10 years old. To the authors’ best knowl- intended. edge, so far no recent calculations are available to compare with. We observe a qualitative agreement, especially at high temperatures so that the results approach the SB limits 2/π 2 IV. CONCLUSIONS and 6/π 2 for the net-electric charge and the net-strangeness We have generalized PLSM calculations in order to in- multiplicities, respectively. corporate zero- and higher-order δ expansions, i.e., MFA and The comparison between the PLSM results in MFA and in OPT. While MFA is originated in statistical physics and was OPT is qualitatively possible when confronting both results applied to a wide spectrum of numerical implications [12,14– with the continuum extrapolation of the lattice QCD results 21,36], OPT was developed in O(N) φ4 theory in order [58]. In Fig. 8, the temperature dependence of the susceptibil- to resume the higher-order terms of the naive perturbation χ B/χ B χ B ities of the conserved charges 6 2 (left panel) and 8 (right approach. In MFA the time evolution of fields are averaged panel) computed in PLSM in MFA (solid curve) and in OPT at a fixed time, while in OPT the time evolution of fields (dashed curve) is compared with the lattice QCD calculations are estimated, perturbatively. We have intended to point out [66] (solid symbols) and [58] (bands). whether the MFA generalization, OPT, reproduces identi- χ B/χ B / . For 6 2 (left panel), at T Tχ 1 2, although with the cal results or eventually improves the PLSM calculations vanishing temperature dependence, the PLSM results agree as reported in MFA. The present article is the first which excellently with the lattice QCD calculations. At lower tem- introduces a comprehensive comparison between MFA and peratures, we find a minimum at Tχ followedbyarapidarise. OPT. If improvements are achieved, we aim at highlighting χ B/χ B ≈ This continues to a maximum at 4 2 1. The lattice QCD how the high-order moments are affected by OPT. calculations are limited to 0.8Tχ . The temperature dependence At finite temperatures and chemical potentials, we have of these quantities are significant for the quark-hadron phase compared the chiral condensates and the decofinement or- transition [59,60]. Within their large error bars, we conclude der parameters, the thermodynamic pressure, the subtracted that both PLSM approaches, MFA and OPT, at least qualita- condensates, the pseudocritical temperatures, the second- and tively, reproduce well the recent lattice QCD calculations. high-order moments of various conserved charges (cumu- χ B / For 8 (right panel), the excellent agreement, at T Tχ lants) obtained in MFA with the ones obtained in OPT. These 1.2 exists here, as well. Decreasing the temperature unveils an calculations are then confronted to available lattice QCD interesting structure. It seems that a sinusoidal dependence is simulations. In general, we conclude that when moving to obtained. The sinusoidal oscillation is also supported by the lower- to higher-order moments, the OPT approach becomes lattice QCD calculations [66]. The most remarkable features more and more closer to lattice QCD simulations than the of such PLSM results in both MFA and OPT likely indicate a MFA. Concretely, we have found that OPT is more sensitive smooth crossover between the hadronic and partonic phases. to χ B/χ B and χ B than the MFA. The higher-order moments / . χ B 6 2 8 At T Tχ 0 8, 8 vanishes. The structure produced by OPT of conserved charges is conjectured to play an essential role fits well with the lattice QCD simulations. in positioning CEP. That OPT is found slightly more sensitive So far, we conclude that OPT is more sensitive to the than MFA furnishes PLSM with slight more capability to pre- χ B/χ B χ B higher-order moment products 6 2 and 8 than the MFA. dict signature for CEP and its possible experimental detection. The off-diagonal cumulants of the net-proton, the net-charge, The convincing OPT results outlined in this study encour- and the net-kaon multiplicity distributions are also measured age the trial to compare with the measurements that are and in the STAR experiment, at energies ranging between 7.7 shall be available in the near future. Accordingly, some PLSM

035210-11 ABDEL NASSER TAWFIK et al. PHYSICAL REVIEW C 101, 035210 (2020) parameters can be adjusted, properly, on one hand. On the ACKNOWLEDGMENTS other hand, the critical phenomena that might be detected The work of A.T. was supported by the ExtreMe Matter in the experimental results and are sensitive the higher-order Institute (EMMI) at the GSI Helmholtz Centre for Heavy Ion moments could likely be predicted. Research, visiting researcher.

[1] J. S. Schwinger, Phys. Rev. 82, 914 (1951). [35] J.-L. Kneur, M. B. Pinto, and R. O. Ramos, Phys. Rev. C 81, [2] J. S. Schwinger, Phys. Rev. 91, 713 (1953). 065205 (2010). [3] J. Schwinger, Phys. Rev. 91, 728 (1953). [36] A. M. Abdel Aal Diab, A. N. Tawfik, and M. T. Hussein, [4] J. Schwinger, Phys. Rev. 92, 1283 (1953). Electromagnetic effects on strongly interacting QCD-Matter, [5] J. Schwinger, Phys. Rev. 93, 615 (1954). arXiv:1611.06926 (2016). [6] J. Schwinger, Phys. Rev. 94, 1362 (1954). [37] B.-J. Schaefer, M. Wagner, and J. Wambach, Phys. Rev. D 81, [7] M. Gell-Mann and M. Levy, Nuovo Cim. 16, 705 (1960). 074013 (2010). [8] J. S. Schwinger, Ann. Phys. (NY) 2, 407 (1957). [38] H. Mao, J. Jin, and M. Huang, J. Phys. G 37, 035001 (2010). [9] M. C. Birse, J. Phys. G 20, 1537 (1994). [39] S. P. Klevansky, Rev. Mod. Phys. 64, 649 (1992). [10] D. Roder, J. Ruppert, and D. H. Rischke, Phys. Rev. D 68, [40] V. Dmitrasinovic, H. J. Schulze, R. Tegen, and R. H. Lemmer, 016003 (2003). Ann. Phys. (NY) 238, 332 (1995). [11] S. Gallas, F. Giacosa, and D. H. Rischke, Phys.Rev.D82, [41] M. Oertel, M. Buballa, and J. Wambach, Phys. Lett. B 477, 77 014004 (2010). (2000). [12] A. N. Tawfik and A. M. Diab, Phys.Rev.C91, 015204 (2015). [42] H. Hansen, W. M. Alberico, A. Beraudo, A. Molinari, M. Nardi, [13] C. Wesp, H. van Hees, A. Meistrenko, and C. Greiner, Eur. and C. Ratti, Phys. Rev. D 75, 065004 (2007). Phys. J. A 54, 24 (2018). [43] P. Costa, M. C. Ruivo, C. A. de Sousa, H. Hansen, and W. M. [14] A. N. Tawfik, A. M. Diab, and M. T. Hussein, J. Phys. G 45, Alberico, Phys. Rev. D 79, 116003 (2009). 055008 (2018). [44] P. Cea, L. Cosmai, and M. D’Elia, Nucl. Phys. B - Proc. Suppl. [15] A. M. Abdel Aal Diab and A. N. Tawfik, EPJ Web Conf. 177, 129, 751 (2004). 09005 (2018). [45] L. M. Haas, R. Stiele, J. Braun, J. M. Pawlowski, and J. [16] A. N. Tawfik, A. M. Diab, and M. T. Hussein, Chin. Phys. C 43, Schaffner-Bielich, Phys.Rev.D87, 076004 (2013). 034103 (2019). [46] M. Gell-Mann, R. J. Oakes, and B. Renner, Phys. Rev. 175, [17] A. N. Tawfik, A. M. Diab, N. Ezzelarab, and A. G. Shalaby, 2195 (1968). Adv. High Energy Phys. 2016, 1381479 (2016). [47] R. F. Dashen, Phys. Rev. 183, 1245 (1969). [18] A. N. Tawfik, A. M. Diab, and T. M. Hussein, Int. J. Adv. Res. [48] S. Borsanyi et al., J. High Energy Phys. 09 (2010) 073. Phys. Sci. 3, 4 (2016). [49] R. Bellwied et al., Phys. Lett. B 751, 559 (2015). [19] A. N. Tawfik, A. M. Diab, and M. T. Hussein, J. Exp. Theor. [50] A. Bazavov et al., Phys. Lett. B 795, 15 (2019). Phys. 126, 620 (2018). [51] S. Das, EPJ Web Conf. 90, 08007 (2015). [20] A. N. Tawfik, A. M. Diab, and M. T. Hussein, Int. J. Mod. Phys. [52] A. Andronic, P. Braun-Munzinger, K. Redlich, and J. Stachel, A 31, 1650175 (2016). Nature 561, 321 (2018). [21] A. N. Tawfik, A. M. Diab, M. T. Ghoneim, and H. Anwer, Int. [53] S. Borsanyi et al., Nature 539, 69 (2016). J. Mod. Phys. A 34, 1950199 (2019). [54] J. I. Kapusta and C. Gale, Finite-Temperature Field Theory: [22] J. Randrup, Nucl. Phys. A 616, 531 (1997). Principles and Applications (Cambridge University Press, Cam- [23] T. Kunihiro and T. Hatsuda, Prog. Theor. Phys. 71, 1332 (1984). bridge, 2006). [24] M. H. Thoma and H. J. Mang, Z. Phys. C 44, 349 (1989). [55] X. Luo and N. Xu, Nucl. Sci. Tech. 28, 112 (2017). [25] M. H. Thoma, Z. Phys. C 44, 343 (1989). [56] S. Borsanyi et al., J. High Energy Phys. 01 (2012) 138. [26] R. L. S. Farias, G. Krein, and R. O. Ramos, Phys.Rev.D78, [57] A. Bazavov et al., Phys.Rev.D86, 034509 (2012). 065046 (2008). [58] A. Bazavov et al., Phys.Rev.D95, 054504 (2017). [27] D. C. Duarte, R. L. S. Farias, P. H. A. Manso, and R. O. Ramos, [59] S. Borsanyi et al., J. Phys. G 38, 124060 (2011). Phys.Rev.D96, 056009 (2017). [60] M. Cheng et al., Phys.Rev.D79, 074505 (2009). [28]D.S.Rosa,R.L.S.Farias,andR.O.Ramos,Physica A 464, [61] V. Koch, A. Majumder, and J. Randrup, Phys. Rev. Lett. 95, 11 (2016). 182301 (2005). [29] T. E. Restrepo, J. C. Macias, M. B. Pinto, and G. N. Ferrari, [62] S. Jeon and V. Koch, Phys. Rev. Lett. 85, 2076 (2000). Phys.Rev.D91, 065017 (2015). [63] B. Friman, F. Karsch, K. Redlich, and V. Skokov, Eur. Phys. J. [30] S. Weinberg, The Quantum Theory of Fields, Vol.1: Foundations C 71, 1694 (2011). (Cambridge University Press, Cambridge, 2005). [64] L. Adamczyk et al., Phys. Lett. B 785, 551 (2018). [31] B.-J. Schaefer and M. Wagner, Phys. Rev. D 79, 014018 (2009). [65] M. A. Stephanov, Phys.Rev.Lett.102, 032301 (2009). [32] J. T. Lenaghan, D. H. Rischke, and J. Schaffner-Bielich, [66] S. Borsanyi et al., J. High Energy Phys. 10 (2018) 205. Phys.Rev.D 62, 085008 (2000). [67] R. Esha, Nucl. Phys. A 967, 457 (2017). [33] P. M. Stevenson, Phys.Rev.D23, 2916 (1981). [68] R. Esha, PoS CPOD2017, 003 (2018). [34] P. M. Stevenson, Phys.Rev.D24, 1622 (1981). [69] L. Chen, Z. Li, F. Cui, and Y. Wu, Nucl. Phys. A 957, 60 (2017).

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