Chinese Physics C Vol. 43, No. 3 (2019) 034103

Chiral phase structure of the sixteen meson states in the SU(3) Polyakov linear-sigma model for finite temperature and chemical potential in a strong magnetic field

Abdel Nasser Tawfik1,2;1) Abdel Magied Diab3;2) M.T. Hussein4;3) 1Nile University, Juhayna sq. of 26th-July-Corridor, 16453 Giza, Egypt 2Frankfurt Institute for Advanced Studies, Ruth Moufang Str. 1, 60438 Frankfurt, Germany and World Laboratory for Cosmology And Particle Phys- ics (WLCAPP), 11571 Cairo, Egypt 3Egyptian Center for Theoretical Physics (ECTP), MTI University, 11571 Cairo, Egypt and World Laboratory for Cosmology And Particle Physics (WLCAPP), 11571 Cairo, Egypt 4Physics Department, Faculty of Science, Cairo University, 12613 Giza, Egypt

Abstract: In characterizing the chiral phase-structure of pseudoscalar (J pc = 0−+), scalar (J pc = 0++), vector (J pc = 1−−) and axial-vector (J pc = 1++t) meson states and their dependence on temperature, chemical potential, and magnetic field, we utilize the SU(3) Polyakov linear-sigma model (PLSM) in the mean-field approximation. We first

determine the chiral (non)strange quark condensates, σl and σs, and the corresponding deconfinement order paramet- ers, ϕ and ϕ∗, in thermal and dense (finite chemical potential) medium and finite magnetic field. The temperature and the chemical potential characteristics of nonet meson states normalized to the lowest bosonic Matsubara frequency are analyzed. We note that all normalized meson masses become temperature independent at different critical temper- atures. We observe that the chiral and deconfinement phase transitions are shifted to lower quasicritical temperatures with increasing chemical potential and magnetic field. Thus, we conclude that the magnetic field seems to have al- most the same effect as the chemical potential, especially on accelerating the phase transition, i.e. inverse magnetic catalysis. We also find that increasing the chemical potential enhances the mass degeneracy of the various meson masses, while increasing the magnetic field seems to reduce the critical chemical potential, at which the chiral phase transition takes place. Our mass spectrum calculations agree well with the recent PDG compilations and PNJL, lat- tice QCD calculations, and QMD/UrQMD simulations. Keywords: Chiral transition, magnetic fields, magnetic catalysis, critical temperature, viscous properties of QGP PACS: 11.10.Wx, 25.75.Nq, 98.62.En CPCDOI: 10.1088/1674-1137/43/3/034103 Accepted

1 Introduction characterizing the properties of this new-state-of-matter, for instance, STAR at the Relativistic Heavy-Ion Col- lider (RHIC) at BNL, ALICE at the Large Hadron Col- Quantum Chromodynamics (QCD) predicts that the lider (LHC) at CERN, the future CBM at the Facility for hadron-quark phase transition, where the hadronic matter Antiproton and Ion Research (FAIR) at GSI, and the fu- goes through a partonic phase transition and generates a ture MPD at the Nuclotron based Ion Collider fAcility new-state-of-matter, colored quark-gluon plasma (QGP), (NICA) at JINR. takes place under extreme conditions of density (large The explicit chiral-symmetry breaking in QCD is as- chemical potential4)) and temperature. Accordingly, at sumed to contribute to the masses of the elementary high temperature and/or chemical potential, the confined particles [2-4]. In QCD-like models, which incorporate hadrons are conjectured to dissolve into free colored some features of QCD so that they are able to give an ap- quarks and gluons. Various heavy-ion experiments aim at proximate picture of what the first-principle lattice QCD

Received 25 March 2018, Revised 5 December 2018, Published online 15 February 2019 1) E-mail: [email protected] 2) E-mail: [email protected] 3) E-mail: [email protected] 4) In cosmological context, low baryon density used to be related to low baryon chemical potential. But both quantities are apparently distinguishable, for instance, the phase diagrams T-μ and T-ρ aren't the same [1]. ©2019 Chinese Physical Society and the Institute of High Energy Physics of the Chinese Academy of Sciences and the Institute of Modern Physics of the Chinese Academy of Sciences and IOP Publishing Ltd

034103-1 Chinese Physics C Vol. 43, No. 3 (2019) 034103

simulations can lead to, such as the Polyakov linear- likely to be present in high-energy collisions, and the pos- sigma model (PLSM), or the Polyakov quark-meson sible modifications on the chiral phase-structure of vari- (PQM) and the Polyakov Nambu-Jona Lasinio (PNJL) ous (sixteen) mesonic states, including (pseudo) scalar models, some properties of the QCD meson sectors can and (axial) vector meson sectors, as function of temperat- be studied numerically. In the present paper, we study the ure and chemical potential in the presence of a finite chiral phase-structure of sixteen meson states in a finite magnetic field. The temperature and chemical potential magnetic field and its dependence on temperature and dependence of the chiral nonstrange and strange quark chemical potential. To characterize the dependence of condensates, σl and σs , and the corresponding decon- ∗ various meson masses for pseudoscalars (J pc = 0−+), scal- finement order-parameters, ϕ and ϕ , which determine the ars (J pc = 0++), vectors (J pc = 1−−) and axial-vectors quark-hadron phase transitions in a finite magnetic field, (J pc = 1++) on temperature, chemical potential and mag- shall be characterized for finite magnetic fields. We then netic field, we utilize SU(3) PLSM in the mean-field ap- investigate the various nonet meson states. Last but not proximation. The magnetic field is a significant ingredi- least, we estimate how the normalization of these meson ent to be taken into consideration, as it can be as high as states relative to the lowest bosonic Matsubara frequency 1019 Gauss in relativistic heavy-ion collisions. It is also behaves with temperature, chemical potential and mag- conjectured that the deconfinement phase transition gives netic field. This allows to approximately determine the considerable contributions to the meson mass spectra. We dissolving temperature and chemical potential for each shall discuss the conditions under which certain meson meson state in a varying magnetic field. states dissolve into free colored quarks and gluons, and An extensive comparison of our calculations of the which temperature and chemical potential would be able meson nonets with the latest compilation of the Particle to modify the chiral phase-structure of each of the nonet Data Group (PDG), lattice QCD calculations, and (sixteen) meson states. QMD/UrQMD simulations, shall be presented. It allows It is not only the peripheral heavy-ion collisions that not only to characterize the meson spectra in thermal and dense medium, but also to describe their vacuum phe- generate enormous magnetic fields, but also the central nomenology in finite magnetic fields. We conclude that ones as well. Due to the opposite direction of relativistic the results obtained are remarkably precise, especially for charges (spectators), especially in non-central collisions some light mesons, if these are extrapolated to vanishing and/or due to local momentum imbalance of the parti- temperatures. cipants, a huge short lived magnetic field can be created. The paper is organized as follows. We briefly de- At SPS, RHIC and LHC energies, for instance, the mean scribe the chiral LSM Lagrangian with three quark fla- values of such magnetic field have been estimated as vors in Section 2.1. We then give a short reminder of the 2 2 ≃ 18 ~0.1, ~1, and ~15mπ, respectively [5-7], where mπ 10 PLSM model in the mean-field approximation in Section Gauss. It should be emphasized that such estimations 2.2. In a vanishing or a finite magnetic field, the depend- were made with the Ultrarelativistic Quantum MolecularCPC Accepted ence of chiral nonstrange and strange quark condensates, Dynamics model (UrQMD) for various impact paramet- and the corresponding deconfinement order-parameters ers, i.e. different centralities. characterizing the quark-hadron phase transition, on tem- It is noteworthy to highlight the remarkable influence perature and chemical potential, calculated form the of the Polyakov-loop potentials on scalar and pseudoscal- SU(3) PLSM model, is presented in Section 3.1. The ar meson sectors. With the in (ex) clusion of the characterization of the magnetic catalysis is discussed as Polyakov-loop potentials in the LSM Lagrangian, the well. The chiral phase-structure of the various meson chiral phase-structure of the nonet meson states for finite states in a finite magnetic field is outlined in Section 3.2. temperatures has been evaluated [8]. The same study was The temperature dependence of the chiral phase-structure carried out with(out) axial anomaly term [9, 10]. A sys- of the sixteen meson states in a finite magnetic field is tematic study of the thermal (temperature) and dense analyzed in Section 3.2.1. The chemical potential depend- (chemical potential) influence on the chiral phase-struc- ence is introduced in Section 3.2.2. Section 3.2.3 is de- tures of the nonet (sixteen) meson states with (out) axial voted to the temperature dependence of these meson anomaly term, and also with in (ex) clusion of the states normalized to the lowest Matsubara frequency. The Polyakov-loop potentials, was published in [11]. For the conclusions are given in Section 4. sake of completeness, we recall that for finite chemical potential the SU(3) NJL model [12] and the SU(3) PNJL 2 A short reminder of the SU(3) Polyakov lin- model [13-15] have been applied to characterize the non- et meson states as well. ear-sigma model The present work represents an extension of Ref. [11] to a finite magnetic field. We present a systematic study To study QCD thermodynamics and characterize the of the influence of the finite magnetic field, which is various nonet mesons for finite temperatures and chemic-

034103-2 Chinese Physics C Vol. 43, No. 3 (2019) 034103 al potentials, the linear-sigma model shall be utilized as be generated increase as well. The full LSM Lagrangian an effective QCD-like model. In this section, we briefly in U(N f ) with the color degrees of freedom Nc is defined describe the chiral phase-structure of the LSM Lagrangi- as Lchiral = L f + Lm, i.e. fermionic and mesonic sectors. In an with three quark flavors N f = 3, which can be estab- other words lished for fermionic and mesonic potentials. Concretely, ● the first term represents quark (q) and antiquark the structure of this Lagrangian provides the tools to in- fields (q¯) [11], [ ( )] vestigate the chiral phase-structure of nonet meson states. ζ ζ L = q i∂/ − gT σ + iγ π + γζ V + γζ γ A q, (1) As we have done in a serious of previous publications, we f ¯ a a 5 a a 5 a treat SU(3) PLSM by means of the mean-field approxim- where g is the Yukawa coupling constant and ζ is an ad- ation, despite its well-know constrains. This approach en- ditional Lorentz index [16] ables to characterize the respective influence of finite ● the second term gives the mesonic Lagrangian, magnetic field on the quark-hadron phase transition, in which consists of the various contributions of the nonet both thermal and dense medium. By a dense medium we states, interactions and possible anomalies mean, even if it might not be exactly equivalent with, a fi- Lm = L + L + L + LU , (2) nite chemical potential. The intention is to examine the SP VA Int (1)A L pc = ++ chiral phase-structure of the sixteen meson states. By do- where SP stands for scalar (J 0 ) and pseudoscalar pc = −+ L pc = −− ing this, we are able to compare their mass spectra with (J 0 ), while AV represent vector (J 1 ) and pc = ++ L the experimental results, with the first-principle lattice axial-vector (J 1 ) mesons. Int represents interac- QCD calculations, and with the corresponding estima- tions that take place in the system of interest. The last tions from other QCD-like models. term is called an anomaly term U(1)A. For more details about the mesonic Lagrangian, interested readers are 2.1 Lagrangian of the linear-sigma model kindly advised to consult Refs. [11, 17-24]. Further de- tails about U(N ) × U(N )ℓ can be obtained from [17-24] As introduced in previous sections, LSM seems to [ f r f ] give a remarkably good description of various meson µ † µ 2 † † 2 LSP =Tr (D Φ) (D Φ) − m Φ Φ − λ1[Tr(Φ Φ)] states. With the increasing number of degrees of freedom, − λ Φ†Φ 2 + Φ + Φ† , the quark flavors N f , the number of meson states that can 2Tr( ) Tr[H( )] (3)    ( )  2 ( ) 1 2 2 m1  2 2  g2 µ ν µ ν µ ν L = − Tr Lµν + Rµν + Tr + ∆ Lµ + Rµ  + i (Tr{Lµν[L , L ]} + Tr{Rµν[R ,R ]}) + g [Tr(LµLνL L ) AV 4 2 2 3 ( ) ( ) ( ) µ ν µ ν µ ν µ ( ν) µ ν + Tr(RµRνR R )] + g4[Tr LµL LνL + Tr RµR RνR ] + g5 Tr LµL Tr RνR + g6[Tr(LµL )Tr(LνL ) µ ν + Tr(RµR )Tr(RνR )], CPC Accepted (4) h1 † 2 2 2 2 µ † L = Tr(Φ Φ)Tr(Lµ + Rµ) + h Tr[|LµΦ| + |ΦRµ| ] + 2h Tr(LµΦR Φ ), (5) Int 2 2 3 L = Φ + Φ† + Φ − Φ† 2 + Φ + Φ† ΦΦ† . U(1)A c[Det( ) Det( )] c0[Det( ) Det( )] c1[Det( ) Det( )]Tr[ ] (6)

PC ++ The complex matrices for scalar σa, i.e. J = 0 , freedom for (pseudo-)scalar and (axial-)vector and π PC = −+ µ PC = −− pseudoscalar a, i.e. J 0 , vector Va , i.e. J 1 couples them through g1, the coupling constant. For in- µ PC = ++ and axial-vector Aa, i.e. J 1 meson states can be stance, for N f = 3, the unitary matrices for SU(3)r× constructed as SU(3)ℓ can be obtained from Ref. [11], while Ta, the gen- 2− = λˆ / N∑f 1 erators of U(3) , can be expressed as Ta a 2, with = ... λˆ Φ = Ta(σa + iπa), a 0 8 , and a are the Gell-Mann matrices. a=0 Different PLSM parameters have been fixed in previ- 2− N∑f 1 ous studies [8, 25]. It should be noted that these two mod- µ µ µ L = Ta(Va + Aa), els are inconsistent and so is any parameterization based a=0 on them. The problem was solved in Ref. [4], where the 2− N∑f 1 authors introduced( )C1 combining various parameters, µ = µ − µ . = 2 + λ σ2 + σ2 R Ta(Va Aa) (7) C1 m0 1 l s . While C1 can only be estimated a=0 through parametrization, its individual parameters can 2 λ The various generators are defined according to N f . It not. While both m0 and 1 were taken from Ref. [8], the is obvious that the covariant derivative, DµΦ = ∂µΦ− remaining ones have been determined (see Table 3 in µ µ ig1(L Φ − ΦR ), is to be associated with the degrees of Ref. [25]).

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2.2 Polyakov linear-sigma model in the mean-field ap- external magnetic field, and is the focus of the present work, proximation ∫ ∑ | | ∑∞ ∞ = q f BT The LSM Lagrangian with N f 3, which are coupled Ωqq¯ (T,µ f , B) = − 2 (2 − δ0ν) dpz ∗ (2π)2 to Nc = 3 , can be written as L = Lchiral − U(ϕ,ϕ ,T). The f ν=0 0 ∗ { [ ( ) ] U ϕ,ϕ , E −µ E −µ E −µ potential ( T) should be adjusted from recent lat- ∗ − B, f f − B, f f −3 B, f f × ln 1 + 3 ϕ + ϕ e T e T + e T tice QCD simulations. Accordingly, various LSM para- [ ( ) ]} E +µ E +µ E +µ meters can be determined. It should be noted that this ∗ − B, f f − B, f f −3 B, f f +ln 1 + 3 ϕ + ϕe T e T + e T . Lagrangian has a Z(3) center symmetry [26-29]. Through the thermal expectation values of the color traced Wilson- (13) loop in the temporal direction, the dynamics of color EB, f is the dispersion relation for each of the quark fla- charges and gluons can be taken into consideration vors in a finite external magnetic field, B , 0 √ ϕ = ⟨ P⟩/ , ϕ∗ = ⟨ P†⟩/ . Trc Nc Trc Nc (8) = 2 + 2 + | | + − Σ , EB, f pz m f q f (2n 1 )B (14) The Polyakov-loop potential U(ϕ,ϕ∗,T) can be intro- σ σ =  / duced in the pure gauge limit as the temperature and where represents the spin quantum number, S 2 , density dependent quantity. There are various proposals and m f are the masses of quark flavors, which are dir- how to do so. ectly coupled to the corresponding sigma-fields, σ σ In previous works, we have implemented the polyno- ¯l ¯s ml = g , ms = g √ . (15) mial potential [11, 30-32]. In the present work, we intro- 2 2 duce calculations based on an alternatively-improved ex- For the sake of completeness, we recall that the light and ∗ tension to ϕ and ϕ [27, 29, 33], the logarithmic potential, strange condensates are determined from the partially U (ϕ,ϕ∗,T) − conserved axial-vector current (PCAC) relation [8], Log = a(T)ϕ∗ϕ 4 T 2 [ ] σ = , σ 1 − . ∗ ∗ ∗ ¯l fπ ¯s √ (2 fK fπ) (16) + b(T)ln 1 − 6ϕ ϕ + 4(ϕ 3 + ϕ3) − 3(ϕ ϕ)2 . 2 (9) where fπ and fK are pion and kaon decay constants, re- where T0 is the critical temperature for the deconfine- spectively. The quantity 2 − δ0ν is related to the degener- ment phase-transition in the pure-gauge sector, ate Landau levels ν. We would like to highlight that the 2 3 quantity 2n + 1 − σ can be replaced by a summation over a(T) = a0 + a1 (T0/T) + a2 (T0/T) and b(T) = b3 (T0/T) , (10) the Landau levels (ν). In nonzero magnetic field (eB , 0) but finite T and µ , both the Landau quantization and The remaining parameters a , a , a , and b can be de- f 0 1 2 3 magnetic catalysis, where the magnetic field is assumed termined by comparison with lattice QCD simulations. CPC Acceptedto be oriented along the z-direction, have been implemen- These are listed in Table 2 in Ref. [34]. ted. The quantization number (n) is known as the Landau In thermal equilibrium, the mean-field approximation quantum number. For details about Landau levels and of the Polyakov linear-sigma model can be implemented how they are occupied, interested readers are kindly ad- in the grand-canonical partition function (Z) for finite vised to consult Ref. [35]. temperature (T) and finite chemical potential (µ ). Here, f Regarding the fermionic vacuum term, let us now re- the subscript f refers to the quark flavors. For finite view the status of its role in various QCD-like models. In volume (V), the free energy reads F = −T · log[Z]/V. For Ref. [36], it was concluded that the inclusion of the fer- instance, for SU(3) PLSM, mionic vacuum term, commonly known as the no-sea ap- F = σ ,σ + U ϕ,ϕ∗, + Ω ,µ , . U( l s) ( T) qq¯ (T f B) (11) proximation, seems to cause a second-order phase trans- The three terms (potentials) can be described as follows. ition in the chiral limit. Accordingly, it was widely be- ● The purely mesonic potential, the first term, stands lieved that its inclusion is accompanied with first or for the strange (σs) and nonstrange (σl) condensates, i.e. second-order phase transition, depending on the baryon m2 c chemical potential, but also on the choice of the coupling σ ,σ = − σ − σ + σ2 + σ2 − √ σ2σ U( l s) hl l hs s ( l s ) l s constants [36]. For further investigation of its role, vari- 2 2 2 ous thermodynamic observables have been evaluated in λ λ + λ λ + λ + 1 σ2σ2 + (2 1 2)σ4 + ( 1 2)σ4. two different effective models, the NJL and quark-meson s s (12) 2 l 8 l 4 (QM), or LSM model. To this end, the adiabatic trajector- ● The Polyakov-loop potential, the second term, was ies in the QM model are found to exhibit a kink at the already detailed in Eq. (9). chiral phase transition, while in the NJL model, there is a ● The quark and antiquark potential, the third term, is smooth transition everywhere [37]. In light of this, it was obviously subject to important modifications in the finite suggested that the fermionic vacuum term can be dropped

034103-4 Chinese Physics C Vol. 43, No. 3 (2019) 034103 from the QM model, similar to the approach utilized in 3.1 Order parameters and magnetic catalysis the present work, while in the NJL model, this term must The chiral condensates (σl and σs) and the deconfine- be included. This effect underlies a first-order phase ment order-parameters (ϕ and ϕ∗) are analyzed for a wide transition in the LSM model in the chiral limit, especially range of temperatures, baryon chemical potentials, and when the fermionic vacuum fluctuations can be neg- magnetic field strengths. The various PLSM parameters lected [38]. The direct method of removing the ultravi- are estimated for the vacuum mass sigma meson σ = 800 olet divergences is the one in which the fermionic vacu- MeV, where the vacuum nonstrange and strange chiral um term is included [39], σ = 92.5 σ = 94.2 ∫ condensates are lo MeV and so MeV, re- ∑ 3 Ωvac = d p spectively. qq¯ 2NcN f E f Λ (2π)3 In Fig. 1, the temperature dependence of the different f ( [ ] ) order parameters are shown for different magnetic fields, − ∑ Λ + ϵ [ ] NcN f 4 Λ 2 2 = 2 = 2 = m ln − ϵΛ Λ + ϵ , (17) eB mπ (top panels) and eB 10mπ (bottom panels), and 8π2 f m Λ f f for different baryon chemical potentials, µ = 0 (solid / curves), 100 (dashed curves) and 200 MeV (dotted where ϵΛ = (Λ2 + m2 )1 2 and m is the flavor mass. Eq. f f curves). In the left-hand panels (a) and (c), the normal- (13) gives the fermionic contributions to the medium in ized chiral condensates (σl/σl and σs/σs ) are depicted. finite magnetic field, Therefore, many authors consider o o We note that the chiral condensates are slightly shifted to that the fermionic vacuum term has a negligible effect on lower critical temperatures when the baryon chemical po- the PLSM results, which we also assume. tential increases. In other words, the critical chiral tem- Coming back to our approach to PLSM and its mean perature (Tχ) decreases with the increase of the magnetic field approximation, we recall that the mean values of the field (eB), as well as with increasing µ. The procedure PLSM order parameters, such as the chiral condensates, utilized in determining Tχ is elaborated below. Such a be- σ and σ , and deconfinement phase transitions, ϕ and ϕ∗, l s havior is known as the inverse magnetic catalysis. We are evaluated such that one minimizes the free energy F bear in mind that this kind of inverse magnetic catalysis in a finite volume (V) with respect to the corresponding is related to the baryon chemical potential. This might be field, where σ = σ¯ , σ = σ¯ , ϕ = ϕ¯ and ϕ∗ = ϕ¯∗, l l s s seen as a novel discovery to be credited to the present ∂F ∂F ∂F ∂F = = = = . work. Furthermore, we conclude that the effect of the ∂σ ∂σ ∂ϕ ∂ϕ∗ 0 (18) l s min strong magnetic field is in the same direction as the bary- Their dependence on T, µ and eB can then be determined. on chemical potential, especially concerning the start of For a vanishing chemical potential (µ = 0), it is obvious the phase transition, i.e. that the phase transition takes that the order parameters of the deconfinement phase place at lower temperatures. In other words, similar to the transitions, ϕ and ϕ∗, are identical, but atCPC µ , 0, they are Acceptedinverse magnetic catalysis corresponding to temperature, distinguishable. The PLSM free energy for a finite V, Eq. we also observe an inverse magnetic catalysis related to (11), becomes complex. Therefore, the analysis of the the magnetic field. PLSM order parameters should be made by minimizing Right-hand panels (b) and (d) show the Polyakov- ∗ the real part of the free energy, Re(F ), at the saddle point. loop order parameters (ϕ and its conjugate ϕ ) as func- tion of temperature for the same values of baryon chemic- al potential and magnetic field as in the left-hand panels. 3 Results ∗ It is obvious that for µ = 0 (solid curves), ϕ = ϕ , i.e. the order parameters are indistinguishable. However, they be- We study the dependence of the quark-hadron phase come different for finite µ, µ = 100 MeV (dashed curves) transition on temperature and baryon chemical potential and µ = 200 MeV (dotted curves). The effects of the mag- in presence of a finite magnetic field. To this end, we first netic field are obvious. We observe that the deconfine- estimate the respective influence of the magnetic field on ment critical temperature (Tϕ) is shifted to lower values as the various chiral condensates and on the deconfinement the magnetic field increases, and also as the baryon order-parameters, Then, in the second part of this Section, chemical potential increases. This is similar to the left- we estimate the chiral phase-structure of sixteen meson hand panels. On the other hand, we conclude that ϕ and states in a thermal and dense medium. By a dense medi- ϕ∗ have different dependence on temperature. While ϕ in- um, we mean a medium with finite chemical potential. creases as the magnetic field increases, and as the baryon Last but not least, we shall present as well the temperat- chemical potential increases, we find that ϕ∗ decreases in ure and chemical potential dependence of the sixteen both cases. Accordingly, the magnetic catalysis related to meson states normalized to the lowest Matsubara fre- temperature is respectively a direct or an inverse effect. quency. Again, the determination of Tϕ is discussed below.

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Figure 2 illustrates the temperature dependence of the eB = 0 (solid curves), 0.1 (dashed curves), 0.2 (dotted normalized chiral quark-condensates (a) and deconfine- curves) and 0.4 GeV2 (dot-dashed curves). It is obvious ment order parameters (b) for different magnetic fields, that the critical deconfinement temperature (Tϕ) de- eB = 0 (solid curves), 0.1 (dashed curves), 0.2 (dotted creases very slightly as the magnetic field increases. curves) and 0.4 GeV2 (dot-dashed curves), at vanishing In both figures, we assure that the maximum Landau chemical potential (µ = 0). Relative to the previous figure, levels are fully occupied with quark states. The details this one presents a systematic study of the influence of about the Landau quantization as a crucial consequence magnetic fields. In the left-hand panel (a), we find that of the magnetic field, and how the corresponding levels the critical chiral temperature deceases with increasing are occupied with quark states, can be obtained from magnetic field. This means that the phase transition, Refs. [35, 40]. which is a crossover, becomes slightly sharper with in- Furthermore, we have calculated the chiral condens- creasing magnetic field. In the right-hand panel (b), the ates (σl and σs) and the deconfinement order-parameters temperature dependence of the deconfinement order para- (ϕ and ϕ∗) and their dependence on the baryon chemical meters is depicted for a vanishing baryon chemical poten- potential (µ) for different temperatures and magnetic field tial, i.e. ϕ = ϕ∗, but different magnetic field strengths, strengths. This is illustrated in Fig. 3, for temperatures

1.50 µ = 0.0 MeV µ = 0.0 MeV 1.0 µ =100 MeV 1.25 µ =100 MeV µ = 200 MeV µ = 200 MeV 0.75 1.0 f0 *

σ σs/σs0 φ / φ

f 0.75 ,

σ * 0.50 σl/σl0 φ φ 0.50 (b) 2 0.25 eB = mπ 2 (a) 0.25 eB = mπ 0 0 0 50 100 150 200 250 300 350 0 50 100 150 200 250 300 350 T (MeV) T (MeV)

1.50 µ = 0.0 MeV µ = 0.0 MeV 1.0 µ =100 MeV 1.25 µ =100 MeV µ = 200 MeV µ = 200 MeV 1.0 0.75 f0

* φ σ φ / 0.75 f σs/σs0 ,

σ * 0.50 σl/σl0 φ φ 0.50 2 (d) eB = 10 mπ 0.25 CPC Accepted0.25 (c) 2 eB = 10 mπ 0 0 0 50 100 150 200 250 300 350 0 50 100 150 200 250 300 350 T (MeV) T (MeV) Fig. 1. (color online) Left-hand panels (a) and (c) show the normalized chiral-condensate with respect to the vacuum value as func- tion of temperature. Right-hand panels (b) and (d) show the expectation value of the Polyakov-loop fields (ϕ and ϕ∗) as function of 2 2 temperature. The top panels present the results for magnetic field eB = mπ , while the bottom panels are for eB = 10mπ. The calcula- tions are performed for different baryon chemical potentials; µ = 0 (solid curves), 100 (dashed curves), 200 MeV (dotted curves).

1.25 1.25 eB = 0 GeV2 eB = 0 GeV2 2 M. L. L. = ∞ eB = 0.1 GeV 2 eB = 0.2 GeV2 eB = 0.1 GeV 1.0 eB = 0.4 GeV2 1.0 eB = 0.2 GeV2 eB = 0.4 GeV2 σs/σs0 0.75

f0 0.75 * *

σ = φ φ φ

/ / σl σl0 , f φ σ 0.50 0.50 (b) 0.25 µ = 0.0 MeV 0.25 (a) µ = 0.0 MeV 0 0 0 50 100 150 200 250 300 350 0 50 100 150 200 250 300 350 T (MeV) T (MeV) Fig. 2. (color online) Left-hand panel: the normalized chiral condensate as function of temperature for a vanishing baryon chemical 2 potential µ = 0 but different magnetic fields, eB = 0 (solid curves), 0.1 (dashed curves), 0.2 (dotted curves) and 0.4 GeV (dot-dashed curves). Right-hand panel: the same as in the left-hand panel but for the deconfinement order parameters ϕ and ϕ∗.

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1.2 0.8 T = 50.0 MeV σs/σs0 2 0.7 eB = m π (b) T = 50.0 MeV 1.0 2 eB = 10 m π 0.6 eB = 15 m2 0.8 π 0.5 eB = 20 m2 σl/σl0 π * f0 2 σ φ eB = 25 m π , / 0.6 0.4 f φ

σ 2 eB = m π 2 0.3 φ 0.4 eB = 10 m π 2 * eB = 15 m π (a) 0.2 φ 2 0.2 eB = 20 m π 2 0.1 eB = 25 m π 0 0 0 100 200 300 400 500 0 100 200 300 400 500 µ (MeV) µ (MeV)

1.2 1.4 2 (d) T = 100.0 MeV T = 100.0 MeV eB = m π 1.0 1.2 2 σs/σs0 eB = 10 m π 2 1.0 eB = 15 m π φ 0.8 2 eB = 20 m π

* 2 f0 0.8 2 eB = 25 m

σ π

/ 0.6 eB = m f π 2 φ,φ σ eB = 10 m 0.6 * eB = 15 m2π φ 0.4 2π eB = 20 m π σl/σl0 2 0.4 eB = 25 m π 0.2 (c) 0.2 0 0 0 100 200 300 400 500 0 100 200 300 400 500 µ (MeV) µ (MeV)

Fig. 3. (color online) Left-hand panels (a) and (c) show the chiral condensates (σl and σs) normalized to the vacuum value as func- tion of baryon chemical potential for different values of magnetic fields, eB = 1 (solid curves) 10 (dashed curves), 15 (dotted curves), 2 20 (dot dashed curves), and 25mπ (double dotted curves). Right-hand panels (b) and (d) show the same as in the left-hand panels but for the expectation value of the Polyakov-loop potentials, i.e. deconfinement order parameters (ϕ and ϕ∗). Top panels show the res- ults for T = 50 MeV, and bottom panels for T = 100 MeV.

T = 50 MeV (top panels) and T = 100 MeV (bottom pan- densates. The difference between σl and σs was also ob- els). In the left-hand panels (a) and (c), the density de- served in the previous two figures. pendence of the chiral condensates is given as function of The right-hand panels (b) and (d) show the depend- different magnetic field strengths, eB = 1 (solid curves) 10 ence of the deconfinement order parameter on the baryon (dashed curves), 15 (dotted curves), 20 (dot dashed chemical potential for temperatures T = 50 (top panel) 2 curves), and 25mπ (double dotted curves). We observe and T = 100 MeV (bottom panel), and fixed magnetic that increasing the temperature causes a rapid decrease inCPC Acceptedfield strengths eB = 1 (solid curves) 10 (dashed curves), 2 the chiral condensates around the chiral phase transition, 15 (dotted curves), 20 (dot dashed curves), and 25mπ similar to that observed in a previous study with PLSM (double dotted curves). The increase in temperature in- without the magnetic field [11] , and in the previous two creases the deconfinement order parameters for a larger figures as well. Such a decrease is likely related to a rap- baryon chemical potential. id crossover. It is obvious that the slope of ϕ and ϕ∗ with respect to We observe that increasing the magnetic field acceler- µ can be approximately estimated. We observe that the ates the rapid decrease in the chiral phase-structure. This magnetic field decreases both slopes, i.e. increases the leads to a decrease in the corresponding critical chemical corresponding critical chemical potentials, while increas- potential, which is determined in a similar manner as the ing the temperature increases the slopes and thus de- critical chiral temperature. Increasing the magnetic field creases the corresponding critical temperatures. Both tends to accelerate the phase transition, i.e. to accelerate thermal and magnetic effects of the hadronic medium on the formation of a metastable phase characterizing the re- the evolution of Polyakov-loop parameters seem to be gion of crossover. This phenomenon is known as magnet- very smooth. ic catalysis, and is actually an inverse one. In light of our study of the phase structure of PLSM There is a gap difference between the light and for finite T, µ, and eB, we can now speculate about the strange chiral condensates at high densities. This can be three-dimensional QCD phase-diagram. This was already understood because of the inclusion of the anomaly term analyzed in great detail in Ref. [35, 40], where the influ- in Eq. (12), which was discussed in Section 2.2, and is ence of a finite magnetic field on the QCD phase-dia- known as the c term. The resulting fit parameters are ac- gram, i.e. temperature vs. baryon chemical potential, was cordingly modified [8, 11]. This was conjectured as an analyzed. Interested readers are kindly advised to consult evidence of the numerical estimation of the chiral con- Refs. [35, 40], where T-eB, µ-eB, and T-µ QCD phase-

034103-7 Chinese Physics C Vol. 43, No. 3 (2019) 034103 diagrams are shown. The three-dimensional phase-dia- contribution of the valence quarks, the second term, gram was depicted as well in [35, 40]. [U(ϕ,ϕ∗,T)] gives the gluonic potential contribution, In the present work, we concentrate on a detailed ver- while the last term [Ωqq¯ (T,µ f , B)] represents the contribu- sion of the T-eB phase-diagram, Fig. 4. We distinguish tion of the sea quarks. It is well-known that the physical between the critical chiral and critical deconfinement mechanism for magnetic catalysis relies on a competition temperatures. Also, we confront our calculations to the between valence and sea quarks [43-45]. The mesonic po- recent 2 + 1 lattice QCD results [41, 42] , and observe ex- tential has a remarkable effect at very low temperature cellent agreement. [30]. In our calculations, we noted that the small value of U(σl,σs) leads to an increase in the contribution of the 200 sea quarks. Thus, applying a finite magnetic field results LQCD1 Tc: Deconfinement PT in the suppression of the chiral condensates relevant for 180 LQCD2 restoration of chiral symmetry breaking. This would ex- plain the "inverse magnetic catalysis", which is character- 160 ized by a decrease in Tc with increasing magnetic field, as illustrated in Fig. 4, where the critical temperatures of T (MeV) T 140 T : Chiral PT χ both types of phase transitions decrease with increasing 120 magnetic field. It should be noted that this phenomenon is RHIC LHC also observed in the lattice QCD simulations [41, 42]. 100 It is noteworthy to recall that Ref. [46] reported an 0 0.1 0.2 0.3 0.4 0.5 0.6 eB (GeV2) opposite results, i.e. a direct magnetic catalysis. The first

term in the free energy, Eq. (11), U(σl,σs), refers to the Fig. 4. (color online) Dependence of the critical chiral and contribution of the valance quarks. In a previous work critical deconfinement temperatures on a finite magnetic [30], we have analyzed the temperature dependence of field determined from PLSM (closed symbols). The results this specific potential for a vanishing chemical potential. are compared to the recent 2 + 1 lattice QCD simulations We found that it has a small effect at high temperatures. [curve (deconfinement) and open symbols (chiral)] [41, 42]. However, it plays an important role in assuring spontan- The vertical bands refer to the estimated averaged magnetic eous chiral symmetry breaking. PLSM describes the field at RHIC and LHC energies. quark-hadron phase structure, where the valence and sea Figure 4 illustrates our PLSM estimations of the de- quarks are simultaneously implemented, Eq. (11). We pendence of the critical chiral and critical deconfinement propose that the physical mechanism for magnetic cata- temperatures, Tχ and Tc, on the finite magnetic field. Both lysis relies on an interplay between the contributions of chiral (solid circles) and deconfinement (solid rectangles) sea and valance quarks [43, 45]. We have observed that calculations are confronted to the recentCPC 2 + 1 lattice QCD Acceptedthe magnetic field increasingly suppresses the chiral con- simulations [41, 42]. The open symbols stand for the lat- densates. Thus, the restoration of chiral symmetry break- tice estimations for Tχ, which are determined at the in- ing takes place at lower temperatures. Although it is un- flection point of the normalized entropy, labeled as likely that the number of the quark flavors alone is suffi- LQCD1 [42]. The curved band represents the lattice es- cient to explain the inverse or usual magnetic catalysis, we emphasize that our calculations assume + quarks timations of the critical deconfinement temperature (Tc) 2 1 at the inflection point of the strange quark-number sus- flavors. For the sake of completeness, we intend in a fu- ceptibility, labeled as LQCD2 [41]. ture work to investigate the magnetic catalysis in SU(4) We recall that there are at least two different methods PLSM, as well. that can be utilized to determine the critical temperature. 3.2 Chiral phase-structure of meson states in a finite In the first one, Tχ is calculated using the thermodynamic quantity s/T 3, where s is the entropy density. The second magnetic field one determines Tc as the position where the strange In a previous work, sixteen meson states in thermal quark-number susceptibility reaches a maximum. It is and dense medium were studied with SU(3) PLSM [11] clear that the PLSM results are in excellent agreement for a vanishing magnetic field. For a finite magnetic field, with both lattice QCD predictions [41, 42]. In light of this an estimation of the magnetic effects on four meson states study, we conclude that both PLSM and lattice QCD sim- was presented in Ref. [31]. In order to conduct a system- ulations clearly indicate that the two types of critical tem- atic study of the effects of the magnetic field strength on peratures (chiral and deconfinement) decrease with in- nonet (sixteen) meson states, the chiral phase-structure of creasing eB, i.e. show the inverse magnetic catalysis. each meson-state is analyzed depending on temperature, A few remarks concerning the magnetic catalysis are baryon chemical potential and finite magnetic field. in order. In Eq. (11), the first term, U(σl,σs), refers to the ● We start with an approximate range for the magnet-

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2 ic field strength that may be created in heavy-ion colli- (bq, f,B = 3(nq, f ) − cq, f,B) and antiquark read 2 2 sions at RHIC and LHC energies [5-7], eB = 1 − 25mπ. bq¯, f,B = 3(nq¯, f,B) − cq¯, f,B,

−E , /T ∗ − E , /T − E , /T ● We then construct the temperature dependence of Φe q f + 2Φ e 2 q f + e 3 q f nq, f,B = , (22) all meson states corresponding to the lowest Matsubara ∗ −E , /T −E , /T − E , /T 1 + 3(ϕ + ϕ e q f )e q f + e 3 q¯ f frequency. ∗ −E , /T − E , /T − E , /T ● Last but not least, the chiral phase-structure of the Φ e q¯ f + 2Φe 2 q¯ f + e 3 q¯ f nq¯, f,B = , (23) ∗ −E , /T −E , /T − E , /T meson states is then given in a wide range of baryon 1 + 3(ϕ + ϕe q¯ f )e q¯ f + e 3 q¯ f chemical potentials. −E , /T ∗ −2E , /T −3E , /T In quantum field theory, the hadron mass can be de- Φe q f + 4Φ e q f + 3e q f cq, f,B = , (24) ∗ −E , /T −E , /T − E , /T termined from the second derivative of the equation of 1 + 3(ϕ + ϕ e q f )e q f + e 3 q¯ f motion relative to the hadron field of interest. The free ∗ − / − / − / Φ Eq¯, f T + Φ 2Eq¯, f T + 3Eq¯, f T F T ,µ,⌉B,β = e 4 e 3e . energy ( ) apparently encodes details of the cq¯, f,B ∗ − / − / − / (25) + ϕ + ϕ Eq¯, f T Eq¯, f T + 3Eq¯, f T equation of motion, where β represents the correspond- 1 3( e )e e ing meson field. Assuming that the contribution of the - The quark and antiquark dispersion relations, quark-antiquark potential to the Lagrangian vanishes in Eq¯, f (T,eB,µ) and Eq, f (T,eB,−µ), are equivalent to EB, f in vacuum, the meson potential determines the mass matrix, Eq. (14). It should noted that due to the mixing between the given as ∂2F ,µ, ,β (pseudo)scalar and (axial)vector sectors through the cov- 2 = (T eB ) , mi,ab (19) ariant derivative, the tree-level expressions of the pseudo- ∂β , ∂β , i a i b min scalars and some scalars are not mass eigenstates. When where i stands for (pseudo)scalar and (axial)vector the corresponding wave functions are renormalized to the mesons, and a and b are integers in between 0 and 8. constants Z, such a mixing can be resolved. Details can Equation (19) can be split into two different terms. be found in Ref. [25]. ● The first one is related to vacuum, where the meson σ 3.2.1 Dependence of meson states on temperature masses are derived from the nonstrange ( l) and strange Figure 5 shows the (pseudo)scalar and (axial)vector (σs) sigma fields. More details about the estimation of meson states with U(1)A-anomaly as function of temper- masses in vacuum are given in Appendix A. In this term, ature for a fixed magnetic field strength and fixed baryon the effect of the magnetic field requires numerical estima- chemical potential. The left-hand[ panels] give the tions for nonstrange and strange sigma-fields. ,σ η∗ ,π [a0(980) (800)] scalar and[ (957) (134)] pseudoscalar ● The in-medium term, in which the magnetic field is ,κ states, while the scalar[ ] f0(1200) (1425) and pseudoscal- included, reads η , ∫ ar (547) K(497) meson states are presented in the ∑ | | ∑∞ ∞ 2 q f BT [middle panels. The right-hand panels illustrate the vector] m = ν (2 − δ ν) dp ∗ i,ab c 2 0 CPCz Acceptedρ , ,ω ,ϕ (2π) [ (775) K (891) (782) (1019) and] axial vector f ν=0 0 a (1030),f (1281),K∗(1270),f∗(1420) meson states. The [ ( 2 2 ) 1 1 1 1 m , m , 2 × 1 + 2 − f a f b solid curves represent the results for eB = mπ. The results (nq, f,B nq¯, f,B) m f,ab 2E , 2 µ = . B f 2EB, f for 0 0 are depicted in the top panels. The middle and bottom panels show the calculations for µ = 100 MeV and (m2 m2 )] f,a f,b µ = 200 MeV, respectively. The results for eB = m2 are + (bq, f,B + bq¯, f,B) , (20) π 2EB, f T given as dashed curves. To distinguish between such where νc = 2Nc. This expression gives the meson-mass curves, one should note that the curves corresponding to modifications, which can be estimated from PLSM, Eq. one meson state are usually bundled at low temperatures. (19), and the diagonalization of the resulting quark-mass At very high temperatures, many, if not all, curves ap- matrix. Details about other quantities in Eq. (20) can be proach asymptotic limits. In the left-hand panels, we ob- obtained in Ref. [9, 11]: serve that increasing magnetic field slightly reduces the - The quark mass derivative with respect to the meson meson masses, especially at the critical temperatures. In the middle and right-hand panels, the opposite is ob- fields (λ , ); m2 ≡ ∂m2 /∂λ , . i a f,a f i a tained, with one exception for Kaons. This might be a - The quark mass with respect to meson fields subject of an experimental study, especially in the future (λ , ∂λ , ), m2 ≡ ∂m2 /∂λ , ∂λ , . i a i b f,ab f i a i b facilities such as the Facility for Antiproton and Ion Re- - Correspondingly, the antiquark function search (FAIR) and the Nuclotron-based Ion Collider , ,µ = , ,−µ bq¯, f,B(T eB f ) bq, f,B(T eB f ), where fAсility (NICA). , ,µ = , ,µ − , ,µ . bq, f,B(T eB f ) nq, f,B(T eB f )(1 nq, f,B(T eB f )) The mass gap between the various meson states can (21) be estimated from the bosonic thermal contributions. The - The normalization factors for the quark masses of bosons are given as function of pure sigma

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2000 1600 2 2 * 1600 eB = m π eB = m π K 2 2 (c) µ = 0.0 MeV eB = 10 m eB = 10 m π π 1600 1400 * f f1 1200 o * ao 1200 1200 κ η* a1 = f1 800 κ σ 800 1000 φ M (MeV) M (MeV) η M (MeV) 400 ρ = ω 400 K (b) µ = 0.0 MeV 800 2 π (a) µ = 0 MeV eB = m π eB =10 m2 0 0 600 π 0 100 200 300 400 100 200 300 400 500 0 100 200 300 400 500 T (MeV) T (MeV) T (MeV)

1600 2000 1600 2 2 * eB = m π eB = m π K (f) µ = 100.0 MeV 2 2 eB = 10 m π 1600 eB = 10 m 1400 1200 π * f1 fo a * o 1200 1200 κ * 800 η κ a1 = f1 σ 800 1000 φ M (MeV) M (MeV) η M (MeV) 400 ρ = ω 400 K (e) µ = 100.0 MeV 800 eB = m2 π π (d) µ = 100.0 MeV eB =10 m2 0 0 600 π 0 100 200 300 400 100 200 300 400 500 0 100 200 300 400 500 T (MeV) T (MeV) T (MeV)

2000 1600 2 2 * 1600 eB = m π eB = m π K (i) µ = 200.0 MeV 2 2 eB = 10 m eB = 10 m π π 1600 1400 * fo f1 1200 a * o 1200 1200 κ * a = f η M (MeV) κ 1 1 800 σ 800 1000 φ M (MeV)

η M (MeV) ρ = ω 400 400 800 eB = m2 π K (h) µ = 200.0 MeV π (g) µ = 200.0 MeV eB =10 m2 0 0 600 π 0 100 200 300 400 100 200 300 400 500 0 100 200 300 400 500 T (MeV) T (MeV) T (MeV) Fig. 5. (color online) The temperature dependence of (pseduo)scalar and (axial)vector meson states calculated for finite magnetic 2 2 fields, eB = mπ (solid curves) and eB = 10mπ (dotted curve), at different baryon chemical potentials, µ = 0 MeV (top panels), µ = 100 MeV (middle panels) and µ = 200 MeV (bottom panels).

σ CPCσ Accepted fields, the nonstrange ( l) and the strange ( s). At low transition takes place at lower temperatures or chemical temperatures, the bosonic thermal contributions are dom- potentials. As a result, the chiral critical temperature de- inant and become stable and finite relative to the corres- creases with increasing magnetic field strength. ponding vacuum value of the meson state, until they ● The last (third) region represents the fermionic reach the chiral temperature (Tχ) related to the given thermal contributions. meson state. At high temperatures, the fermionic (quark) We conclude that the magnetic field has an evident thermal contributions complete the thermal behavior of effect on quarks and apparently accelerates and sharpens these states. As they increase with temperature, this leads the quark-hadron phase transition. to degenerate meson masses. It is obvious that the effects We observe that at temperatures exceeding the critic- of the fermionic (quark) thermal contributions are negli- al values (corresponding to each meson state), the meson gible at low temperatures. masses become degenerate. This can be understood as As depicted in Fig. 5, the effects of a finite magnetic due to the effect of fermionic fluctuations on chiral sym- field on the thermal contributions of the meson states can metry restoration [8], especially on the strange condens- be divided into three regions. ate (σs). Such an effect seems to melt the nonstrange con- ● The first region is dominated by the bosonic densate (σl) faster than the nonstrange one (σl), Fig. 1. At thermal contributions, and the magnetic field remains in- very high temperatures, the mass gap between mesons effective until the second region takes place. seems to disappear due to the melting of the strange con- ● In the second region, beyond the chiral phase-trans- densate (σs). This mass gap appears again at low temper- ition of a given meson state, the influence of the magnet- atures, where the nonstrange condensate remains finite. ic field becomes obvious. We observe that the increasing At temperatures higher than critical, only the strange con- magnetic field accelerates the chiral phase-transition of a densate remains finite. These thermal effects are strongly given meson. By accelerating, we mean that the phase related to the degenerate meson masses.

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∗ Also, Fig. 5 shows that the σ, π, a0, and η meson scription of their vacuum phenomenology in a finite mag- states become degenerate through a first-order phase netic field. In Table 1, an extensive comparison is given transition and apparently keep this state at higher temper- of the results for different scalar and vector meson (non- atures, assuring a completion of the chiral symmetry res- ets) from various effective models, PLSM (present work) toration. The middle panels depict the κ meson state and and PNJL [12-15, 47] , and from the very recent compila- illustrates that it reduces to the η and K meson states tion of PDG [48], lattice QCD calculations [49, 50] and through a first-order phase transition. f0 is degenerate at a QMD/UrQMD simulations [51]. We conclude that our higher temperature as well through a first-order phase PLSM calculations are remarkably precise, especially for ∗ some light mesons at a vanishing temperature. They are transition. In the right-hand panels, the meson states f1 , ϕ, a1, f1, ρ, and ω become degenerate through chiral comparable with measurements and lattice calculations, ∗ phase-transitions, whereas K∗ and κ become coincident as shall be elaborated in the following section. at very high temperatures. Only pseudoscalar and vector meson sectors are avail- Increasing the baryon chemical potential influences able from HotQCD [49] and PACSCS collaborations the behavior of the thermal contributions of different [50]. It is obvious that our estimations for meson masses meson states as well. We find that the increase of the ba- agree well with previous calculations [8, 9, 11, 25] for ryon chemical potential (from top to bottom panels) en- mixing strange with nonstrange scalar states, where it was hances various degenerate meson masses. For example, reported that one gets various states below 1 GeV and the σ and π states degenerate at the chiral temperature one above 1 GeV [25]. The agreement between PLSM Tχ ∼ 191.5 MeV and µ = 0 MeV. But for µ = 100 MeV, the and the available lattice-calculations is excellent. critical temperature drops to ∼ 186.5 MeV , and for µ = 0 3.2.2 Dependence of meson states on baryon chemical MeV, it decreases to ∼ 178.7 MeV. Thus, we conclude potential that the chiral temperature associated to the different Figure 6 shows the dependence of scalar and vector meson states decreases with increasing baryon chemical meson states (labeled curves) on the baryon chemical po- potential. This refers to a remarkable in-medium effect tentials in the presence of the U(1)A anomaly, for magnet- that could be verified in the future facilities with high- ic fields eB = 15mπ (solid curves) and eB = 20mπ (dashed density heavy-ion collision experiments. curves), and temperatures T = 50 (top panel), and 100 Our goal is not only the characterization of the meson MeV (bottom panel). As discussed in previous sections, spectra in thermal and dense medium, but also the de- the nonstrange (σl) and strange quark (σs) chiral condens- Table 1. Comparison of the (pseudo)scalar and (axial)vector meson states obtained in the present work (PLSM), and the corresponding results from PNJL [13-15], the latest compilation of PDG [48], QMD/UrQMD [51] and lattice QCD calculations [49, 50].

lattice QCD calculations sector meson states PDG [48] UrQMD model [51] present work PNJL model [12-15, 47] CPC Accepted Hot QCD[49] PACS-CS [50]

a0 a0(98020) 984.7 1015.73 837 κ ∗  K0(1425 50) 1429 1115 1013 scalar JPC = 0++ σ 400 − 1200 800 700

f0 f0(1281.0  0.5) 1200 − 1500 1102.8 1169

π π0(134.97700.0005) 139.57 150.4 126 1346 135.46.2

K K0 (497.6110.013) 493.68 509 490 422.611.3 49822 pseudoscalar JPC = 0−+ η η(547.8620.17) 517.85 553 505 5797.3 68832 ′ ′ η η (957.780.06) 957.78 949.7 949 ρ ρ .  . (775 26 0 25) 771.1 745 764.08 756.236 59786 ωX ω(782.650.12) 782.57 745 764.08 88418 86123 PC = −− vector J 1 ∗ K∗ K (891.760.25) 891.66 894 899.96 100593 1010.277 ω y ϕ(1019.4610.016) 1019.45 1005 1025.79

a1 a1(1230  40) 1230 980 1171.78

f1x f1(1281.90.5) 1281.9 980 1173.78 axial-vector JPC = 1++ ∗ ∗  . K1 K1(1272 7) 1273 1135 1035 21 ′ f .  . . 1512 1315 . 1y f1 (1426 4 71 30 9) 1531 55

034103-11 Chinese Physics C Vol. 43, No. 3 (2019) 034103 ates strongly depend on the temperature and baryon phase to parton phase, where quarks and gluons domin- chemical potential. The mass gap which distinguishes ate the underlying degrees of freedom. between the meson states results from dense bosonic con- ● The third region is characterized by large µ, where tributions. Such contributions remain independent on the the hadrons are conjectured to dissolve into their constitu- baryon chemical potential until they reach the Fermi sur- ent free quarks and gluons. face [11, 52]. The masses are then liberated through a As was obtained in the study of the temperature de- first-order phase transition. To assure that the Polyakov- pendence, the dependence of various meson masses on µ loop potentials include the deconfinement phase trans- is accompanied by a decrease of the bosonic contribu- ition in LSM, the dynamics of color charges was in- tions to the medium density, which become negligible at cluded as a new type of interaction. Apparently, the very large baryon chemical potentials. In this limit, the meson masses remain degenerate through the deconfine- dense fermionic contributions become dominant. Further- ment phase transition. When comparing these results with more, the latter lead to mass degeneration, which takes our previous ones [11], we conclude that the deconfine- place through dense fermionic contributions. Accord- ment phase transition in the present work, where a finite ingly, the meson states, if they pass through the phase magnetic field is included, seems to become sharper and transition as bound hadron states or survive the critical µ faster, i.e. the region of within which the masses of the density (µcrit) at which the phase transition takes place, meson states decrease, indicating a rapid change in the rely on dense fermonic contributions to overcome the en- underlying dynamics and degrees of freedom, becomes ergy gap of the Fermi surfaces. The latter differs from narrower as the magnetic field is increased. This finding one meson state to another. With increasing µ, the bound could be a subject of a future experimental verification. meson states are assumed to be free and entirely conver- Another finding of the present work is that for a fixed ted into deconfinment (partonic) phase [11, 52]. magnetic field, the finite baryon chemical potential is In the middle panels of Fig. 6, we note that the meson conjectured to contribute through baryon density to the masses reduce in the first-order phase transition. Con- masses of various meson states, and is dominant in three cretely, the mass of the κ state reduces to the masses of K regions: and η states at µcrit ∼ 350 MeV. The temperature is fixed µ ● The first region is characterized by low , where the at 50 MeV. It seems that the f0 meson state has a stronger gap difference between the meson masses seems to ori- Fermi surface because of the strange quarks. The mass of ginate in dense bosonic contributions. These are obtained this state seems to survive until µ ∼ 500 MeV. At T = 100 from the mesonic Lagrangian of LSM. MeV (bottom panel), the first-order phase transition ● The second region appears when the meson states seems to persist for small µ. The mass of κ reduces to the undergo a chiral phase-transition from confined hadron masses of K and η states at µcrit ∼ 298 MeV for the mag-

1500 1600 2 1600 2 f 2 CPCo AcceptedeB = 15 m π * eB = 15 m π eB = 15 m π K 2 2 2 eB = 20 m π eB = 20 m 1200 eB = 20 m π π 1400 * 1200 f1 ao κ 900 κ* η* 1200 a = f 800 1 1 σ 600 η φ M (MeV) M (MeV) M (MeV) 1000 K ρ = ω 400 300 π (a) T = 50.0 MeV (b) T = 50.0 (MeV) 800 (c) T = 50.0 (MeV) 0 0 0 100 200 300 400 500 600 700 800 100 200 300 400 500 600 700 800 100 200 300 400 500 600 700 800 µ (MeV) µ (MeV) µ (MeV)

1500 1600 1600 2 2 2 eB = 15 m fo eB = 15 m π eB = 15 m π K* π 2 2 eB = 20 m2 eB = 20 m π 1200 eB = 20 m π 1400 π 1200 * ao κ f1 900 κ* η* 1200 800 a1 = f1 σ 600 η φ M (MeV) M (MeV) M (MeV) 1000 K ρ = ω 400 300 π (d) T = 100.0 MeV (e) T = 100.0 (MeV) 800 (f) T = 100.0 (MeV) 0 0 0 100 200 300 400 500 600 700 800 100 200 300 400 500 600 700 800 100 200 300 400 500 600 700 800 µ (MeV) µ (MeV) µ (MeV) Fig. 6. (color online) Dependence of the masses of (pseudo)scalar and (axial)vector meson states on the baryon chemical potential for 2 2 magnetic field strengths eB = 15mπ (solid curves) and eB = 20mπ (dotted curve), and temperatures T = 50 MeV (top panels) and T = 100 MeV (bottom panels).

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2 2 netic field eB = 15mπ , and µcrit ∼ 200 MeV for eB = 20mπ. ter, we apply a normalization of the present meson sec- Also, the mass of f0 meson degenerates at lower µcrit than tors with respect to the lowest Matsubara frequency, at T = 50 MeV (top panel). which is associated to bosons 2πT [56]. This allows to In Fig. 6, two independent quantities (T and eB) have define the dissolving critical temperature at which the non-negligible effects on the dependence of meson various meson states are conjectured to liberate into the masses on µ. From the QCD phase-diagram [31, 32] and degrees of freedom of free quarks and free gluons. We the chemical freezeout boundary [53], we conclude that found that different meson states have different dissolv- increasing T reduces µcrit. Our results also show that in- ing temperatures, which means that different hadrons, in creasing T reduces µcrit, at which the bosonic contribu- this case mesons, have different chiral critical temperat- tion for a finite chemical potential becomes small. In- ures (Tχ). In other words, not all hadrons melt into a creasing µ has an obvious influence on scalar mesons, as quark-gluon plasma simultaneously. reaching stable levels of meson masses is accelerated. Figure 7 depicts the normalization of the different These are the levels at which the meson masses no longer meson states with respect to the lowest, in this case bo- µ depend on . Concretely, we observe that the masses of sonic, Matsubara frequency (2πT) for a vanishing (top vector mesons stay nearly constant until the phase trans- panel) and finite baryon chemical potential (bottom pan- ition takes place, while the masses of axial vector mesons 2 el), and different magnetic fields eB = mπ (solid curves) show stronger melting above the Fermi surface. Further- and eB = 10m2 (dotted curves). The legends are identical µ π more, increasing eB is assumed to reduce crit and thus to to those in previous figures. It is obvious that the masses enhance the chiral phase-transition, which explains the µ of almost all meson states become independent of tem- influences of finite eB on the dependence of various perature, i.e. construct a kind of a universal curve, espe- meson states. cially at high temperatures. This observation is a clear 3.2.3 Meson states normalized to the lowest Matsubara signature of dissociation of mesons, which are apparently frequency dissolving into free quarks and gluons. Also, it is found In thermal field theory, the Matsubara frequencies are that the characteristic critical temperature does not seem considered as a summation over all discrete imaginary to be universally valid for all meson states. As discussed frequencies ∑ in earlier sections, the two thermodynamics variables, temperature and magnetic field, seem to enhance the chir- S  =T g(iωn), (26) al phase-transition and therefore have a considerable in- iωn fluence on the corresponding critical temperature. This where  stands for bosons and fermions, respectively, and observation is also confirmed here. g(iω ) is a rational function, where ω = 2nπT and n n Table 2 summarizes the different meson states and ω = (2n + 1)πT represent bosons and fermions, respect- n their approximate dissolving critical temperatures (T d) ively. The integers n = 0,1,2,··· play an important role c CPC Acceptedfor the finite magnetic field eB = m2 and vanishing bary- in the quantization process. Using the Matsubara fre- π on chemical potential. The magnetic field strength quency, the weighting function Γη(z) has two poles at eB = m2 is likely to be generated at the top RHIC energy. z = iω . Then π n I The critical temperatures T d can roughly be read off from T c = Γ . the resulting graph. Statistical uncertainties (errors) have S  π g(z) η(z)dz (27) 2 i not yet been estimated. As mentioned, different meson Γ(z) can be chosen depending on which half plane the states seem to have different dissolving temperatures. convergence is to be controlled in,  This is a novel observation of the present work. To con-  1 + n(z) nect the latter with the critical temperatures, the first prin-  η  T ciple lattice calculations should be first reproduced by our Γ(z) = , (28) QCD-like calculations. The experimental results should   ηn(z) be verified as well. It should be noted that our calcula- T tions are based on PLSM, a QCD-like model in which not where the single particle distribution function is given as all aspects of QCD are taken into account. Nevertheless, z/T − n(z) = (1  e ) 1. PLSM as such seems to give a good picture of what lat- The boson masses are conjectured to get contribu- tice QCD simulations produce. As an outlook, we plan to tions from the Matsubara frequencies [54]. Furthermore, devote a future work to a systematic analysis of the res- at temperatures above the chiral phase-transition (T ⩾ Tc), ulting dissolving temperatures at varying magnetic fields. the thermal behavior of the thermodynamic quantities, The latter can be obtained at different beam energies and such as susceptibilities, and of the masses are affected by in different heavy-ion collision centralities, especially in the interplay between the lowest Matsubara frequency the future facilities which will enable to cover the gap at and the Polyakov-loop potentials [55]. In light of the lat- high densities.

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1000 800 2 1500 eB = m2 2 eB = mπ π eB = mπ 2 2 eB = 10 mπ eB = 10 m 700 eB =10 m2 750 1200 π π 600 T) T) π

2T) 900 π

π 500 500 M/(2 M/(

600 M/(2 400 250 (a) µ = 0 MeV 300 (b) µ = 0.0 MeV 300 (c) µ = 0.0 MeV 0 0 200 200 400 600 800 200 400 600 800 200 400 600 800 T (MeV) T (MeV) T (MeV)

1000 800 2 1500 2 2 eB = mπ eB = mπ eB = mπ 2 eB = 10 m2 eB = 10 mπ 700 eB =10 m2 750 π 1200 π 600 T) T) T) π π π 900 500 500

M/(2 600 M / (2 M / (2 400 250 (d) µ = 200.0 MeV 300 (e) µ = 200.0 MeV 300 (f) µ = 200.0 MeV 0 0 200 200 400 600 800 200 400 600 800 200 400 600 800 T (MeV) T (MeV) T (MeV) Fig. 7. (color online) Left-hand and middle panels depict the scalar and pseudoscalar meson sectors, respectively, as function of tem- perature for vanishing (top panels) and finite baryon chemical potential (bottom panels). The right-hand panel shows the same as in the left-hand panel but for vector and axial vector meson sectors.

d = 2 Table 2. Approximate dissolving critical temperatures, Tc , corresponding to different meson states at eB mπ . Statistical uncertainties (errors) have not yet been estimated.

comparison scalar mesons pseudoscalar mesons vector mesons axial vector mesons κ σ ′ ρ ∗ ω ϕ ∗ meson states a0 f0 π K η η K0 a1 K1 f1 f1 d Tc in MeV 430 450 470 450 320 230 335 240 495 495 495 495 495 495 495 495

2 ∼ 8 4 Conclusion spectively, where mπ 10 Gauss. In light of these estim- ates, studying the possible influence of the magnetic field CPC Acceptedon QCD matter is of great interest for our understanding The present study aims at a systematic investigation of heavy-ion collisions, and, therefore, it increasingly of the temperature and chemical potential dependence of gains popularity among particle physicists. pc = −+ sixteen meson states [pseudoscalars (J 0 ), scalars In the mean-field approximation of PLSM, a grand pc = ++ pc = −− (J 0 ), vectors (J 1 ) and axial-vectors canonical partition function can be constructed. In order pc = ++ (J 1 )] constructed from the SU(3) Polyakov linear- to prepare for the proposed calculations, we have to es- sigma model in the presence of finite magnetic fields. timate the temperature dependence of the deconfinement ∗ The introduction of magnetic fields in this QCD-like (ϕ and ϕ ) and chiral order parameters (σl and σs) in pres- model is accompanied by certain modifications in the ence of a finite magnetic field. We have found that, tak- phase space and in the dispersion relations, among others. ing into consideration the influence of finite magnetic Also, this requires Landau quantization to be implemen- fields, these parameters strongly affect the entire QCD ted, for which we use the Landau theory for quantized phase transition. cyclotron orbits of charged particles in an external mag- In a previous work, we have studied the distribution netic-field. Consequently, some restrictions are added to of Landau levels and showed how they are occupied for the color and electric charges of the quarks. finite magnetic field, temperature, and baryon chemical In relativistic heavy-ion collisions, a huge magnetic potential [35, 40]. We concluded that the occupation of field is expected. For instance, due to oppositely directed each Landau level varies with quark electric charge, as well relativistic motion of charges (spectators) in peripheral as with temperature and baryon chemical potential. This collisions and/or due to the local momentum-imbalance is assumed to be characterized by the QCD energy scale. of the participants in central collisions, a huge magnetic In PLSM with mean-field approximation, the masses field can be created. To get a picture of its magnitude, we can be determined from the second derivative of free en- recall that at SPS, RHIC and LHC energies, the strength ergy with respect to the corresponding hadron field, eval- 2 of such fields ranges between 0.1 to 1 to 10 − 15mπ, re- uated at its global minimum. We have presented the tem-

034103-14 Chinese Physics C Vol. 43, No. 3 (2019) 034103 perature and chemical potential dependence of sixteen between different meson states in thermal and dense me- SU(3) meson masses in finite magnetic fields. For a fi- dium. With increasing temperature, the fermionic contri- nite magnetic field, the broken chiral-symmetry is as- butions complete the thermal behavior of these states and sumed to contribute to the mass spectra. We utilize this lead to mass degeneracy at very high temperatures. assumption to define the chiral phase structure of nonet From the scalar and vector mesons normalized to the meson sectors as function of temperature and chemical lowest Matsubara frequency, we also conclude that a rap- potential. We found that the spectra of meson masses can id decrease of their masses is observed as the temperat- be divided into three regions. ure increases. Starting from the critical temperature cor- ● At small temperatures and baryon chemical poten- responding to each meson sector, we find that the temper- tials, the first region is related to the bosonic contribu- ature dependence almost vanishes. At high temperatures, tions which are apparently stable. we note that the masses of almost all meson states be- ● The second region is the chiral phase transition, come temperature independent, i.e. they construct a kind which seems to have a remarkable influence related to the of a universal line. This is to be seen as a signature of dis- finite magnetic field. It is obvious that the finite magnet- solving of confined mesons into colored quarks and ic field improves, i.e. sharpens and accelerates, the chiral gluons. In other words, the meson states undergo differ- phase transition of a given meson state. ent deconfinement phase transitions, i.e. the various had- ● The third region is characterized by fermionic con- rons very likely have different critical temperatures. This tributions, which increase with increasing temperature. is one of the essential findings of the present study, to be We conclude that the thermal bosonic (meson) mass confirmed by first-principle lattice simulations and ulti- contributions decrease with increasing temperature. The mately in future experiments. fermionic (quark) contributions considerably increases at We have compared our calculations for scalar and high temperatures. At low temperatures, bosonic contri- vector mesons with the latest compilation of PDG, lattice butions become dominant and reach a kind of a stable QCD calculations and QMD/UrQMD simulations, and plateaus relative to the vacuum mass of each meson sec- found that our results are remarkably precise, especially tor. In this temperature limit, the effects of fermionic con- for some light mesons at vanishing temperature. This im- tributions are negligibly small. It is noteworthy to high- plies that the parameters of the QCD-like model we have light that the bosonic contributions result in the mass gap utilized are reliable.

Appendix A: Masses of sixteen mesonic-states

In vacuum, the mesonic sectors are formulated independently ● For pseudoscalar meson states, the squared masses for the for nonstrange and strange fields: pseudoscalar sector (i = p) read CPC Accepted √ ( ) λ ● For scalar meson states, the squared masses for the scalar 2 2 2 2 2 2 2c mπ =m + λ1 σ¯ + σ¯ + σ¯ − σ¯ s, (A5) sector (i = S) are given as l s 2 l 2 √ ( ) λ ( √ ) ( ) 2 2 2 2 2 2 2 c 3λ2 2c m =m + λ σ¯ + σ¯ + σ¯ − 2σ ¯ σ¯ + 2σ ¯ − σ¯ , (A6) m2 =m2 + λ σ¯ 2 + σ¯ 2 + σ¯ 2 + σ¯ , (A1) K 1 l s 2 l l s s 2 l a0 1 l s 2 l 2 s 2 2 2 2 2 2 ( ) λ ( √ ) mη′ =m , cos θp + m , sin θp + 2m , sinθp cosθp, (A7) 2 = 2 + λ σ2 + σ2 + 2 σ2 + σ σ + σ2 + c σ , p 00 p 88 p 08 mκ m 1 ¯ l ¯ s ¯ l 2 ¯ l ¯ s 2 ¯ s ¯ l (A2) 2 2 2 2 2 2 2 2 mη =m , sin θp + m , cos θp − 2m , sinθp cosθp, (A8) 2 = 2 2 θ + 2 2 θ + 2 θ θ , p 00 p 88 p 08 mσ mS,00 cos S mS,88 sin S 2mS,08 sin S cos S (A3) 2 2 2 2 2 2 with m =m , sin θS + m , cos θS − 2m , sinθS cosθS , (A4) f0 S 00 S 88 S 08 ( ) λ ( ) c ( √ ) m2 =m2 + λ σ¯ 2 + σ¯ 2 + 2 σ¯ 2 + σ¯ 2 + 2σ ¯ + 2σ ¯ , with p,00 1 l s 3 l s 3 l s ( ) ( ) ( √ ) λ ( √ ) λ2 c 2 = 2 + 1 σ2 + σ σ + σ2 m2 =m2 + λ σ¯ 2 + σ¯ 2 + σ¯ 2 + 4σ ¯ 2 − 4σ ¯ − 2σ ¯ , m , m 7 ¯ 4 2 ¯ l ¯ s 5 ¯ s p,88 1 l s l s l s S 00 3 l √ 6 6 √ ( ) ( √ ) ( ) ( √ ) 2λ2 c 2 2 2c 2 = σ2 − σ2 − σ − σ , + λ σ¯ + σ¯ − 2σ ¯ + σ¯ , mp,08 ¯ l 2 ¯ s 2 ¯ l 2 ¯ s 2 l s 3 l s 6 6 λ ( √ ) 2 = 2 + 1 σ2 − σ σ + σ2 and the mixing angles are given by mS,88 m 5 ¯ l 4 2 ¯ l ¯ s 7 ¯ s 3  2 √ ( ) 2m ,  σ¯ 2  √ σ θ = i 08 , = , .  l 2 2c ¯ s tan2 i i S p (A9) + λ2  + 2σ ¯  + 2σ ¯ l − , m2 − m2 2 s 3 2 i,00 i,88 ( √ √ ) 2λ1 ● For vector meson states, the squared masses for the vector m2 = 2σ ¯ 2 − σ¯ σ¯ − 2σ ¯ 2 S,08 l l s s µ 3   sector (i = V ) can be expressed as √  σ2  ( √ )  ¯ l  c + λ  − σ2 + √ σ − σ . 2 2 1 2 h1 2 2 2  ¯ s  ¯ l 2 ¯ s mρ =m + (h + h + h )σ¯ + σ¯ + 2δ , (A10) 2 3 2 1 2 1 2 3 l 2 s l

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( ) σ¯ 2 ( ) 1 h1 2 2 l 2 m2 = m2 + 2g2 + h + h − h σ¯ 2 + σ¯ 2 + 2δ , (A14) m ⋆ =m + g + 2h + h a1 1 1 1 2 3 l s l K 1 4 1 1 2 2 2 2 ( ) ( ) ( ) σ¯ lσ¯ s σ¯ 1 1 + √ − 2 + s 2 + + + δ + δ , 2 = 2 + 2 + + σ2 − √ σ σ − 2 (h3 g1) g1 h1 h2 l s (A11) mK m1 g1 2h1 h2 ¯ l ¯ l ¯ s h3 g1 2 2 1 4 2 1 ( ) m2 =m2, (A12) + g2 + h + h σ¯ 2 + δ + δ , (A15) ωx ρ 2 1 1 2 s l s ( ) 2 2 h1 2 h1 2 m2 = m2 , (A16) mω = m + σ¯ + + h + h σ¯ + 2δ , (A13) f a1 y 1 2 l 2 2 3 s s 1x ( ) σ¯ 2 h ● And finally, for axial-vector meson states, the squared masses m2 = m2 + l h + 2g2 + 1 + h − h σ¯ 2 + 2δ . (A17) f1y 1 1 1 2 3 s s for the axial-vector sector (i = Aµ) are 2 2

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