PoS(ICHEP2016)634 http://pos.sissa.it/ ma model (PLSM) is constructed in charm condensates are likely not af- wide range of temperatures and den- ng the chemical potentials decreases P), 11571 Cairo, Egypt. P), 11571 Cairo, Egypt. nding to charm condensates are greater versity for Technology and versity for Technology and logy and Information, 11671 Cairo, Z. Khalilov st. 23, AZ-1148, Baku, shall be analysed at finite temperatures and densi- ∗ for light, strange and charm quarks, respectively, and φ c σ and φ and s ive Commons σ , l σ onal License (CC BY-NC-ND 4.0). ∗ ) Polyakov linear-sigma model at finite [email protected] 4 sities. The chiral condensates In mean-field approximation, the SU(4) Polyakov linear-sig order to characterize the quark-hadron phase structure in a fected by the QCD phase-transition.the corresponding Furthermore, critical increasi temperatures. Speaker. than that to strange and light ones, respectively. Thus, the ties. We conclude that, the critical temperatures correspo the deconfinement order-parameters ∗ Copyright owned by the author(s) under the terms of the Creat c Attribution-NonCommercial-NoDerivatives 4.0 Internati temperature and density Information (MTI), 11571 Cairo, Egypt, World Laboratory for Cosmology And Particle Physics (WLCAP Abdel Magied DIAB Egyptian Center for Theoretical Physics (ECTP), Modern Uni Azar I. Ahmadov Department of Theoretical Physics, Baku State University, E-mail: Eiman Abou El Dahab Faculty of Computer Sciences, Modern University for Techno SU( Egyptian Center for Theoretical Physics (ECTP), Modern Uni Azerbaijan. Abdel Nasser Tawfik Information (MTI), 11571 Cairo, Egypt, World Laboratory for Cosmology And Particle Physics (WLCAP Egypt. 38th International Conference on High Energy3-10 Physics August 2016 Chicago, USA

PoS(ICHEP2016)634 , c = σ (2.3) (2.1) (2.2) 2 with chiral / , we use  , and a , L s 2 λ )] ) σ † 4 matrix for φ , = l ∗ Φ × a σ φ T + ( 3 Φ ( Abdel Nasser Tawfik − , the Ployakov-loop ∗ ) H with tentials in section 2. [ 3 φ , ) φ Tr a 3 matrices and the chiral + π + and i 5 GeV. The corresponding × 3 2 . ∗ ) φ + , 3 is a complex 4 φ f a are Gell-Mann matrices, while Φ tions between quarks and glu- ( q ed approximation to the PLSM † ∼ Φ σ a 4 stance Refs. [1, 2]. At vanish- ( i λ Φ ) a ( to + chiral-condensates, a T nio (PNJL) model [6, 7, 8] and the d various QCD-like approaches. In re as that of SU(3) [9], for instance. , 16, 17, 18, 19, 20, 21, 22, 23] have aim at describing the quark-hadron s. By imposing chiral symmetries, π φ Tr itly broken [3] and the QCD phase- . -freedom and gluon dynamics, where 5 ∗ = 2 arks and gluons become confined and 2 γ perties of the QCD Lagrangian, while o the final conclusions. ) i φ λ arameters, analysed. Φ φ 6 + 4 instead of 3 − Polyakov-loop fields, [27] ∗ a 2 − φ × σ )] 1 ( )(  a Φ T † ln ( gT is an additional Lorentz index. The pure mesonic Φ d ) 1 ( − ζ T ( ζ Tr )+ [ b 3 1 D symmetry group [9]. , respectively, are Polyakov-loop fields. Details about the chiral LSM ζ φ λ + ) a γ ) , is usually added, where 4 π i φ + − ( h ∗ T )] ) 3 , f † ∗ U φ ∗ Φ q Φ φ and ) † φ ( ( f , a T Φ ∑ ) ( φ 2 2 σ ( T a 2 Det = m ( , c − q ) U − c )+ , L − + = s Φ symmetries, the LSM Lagrangian [9] can be constructed as Φ , µ ( ) R d ∂ , T † chiral u , Det 2 and 3 can be found in Refs. [9, 12, 26, 27]. In the present work ( ∗ [ Φ L c µ φ = ∈ , SU(4) ∂ = f ( q 4 φ ( N × 1 are the generators of T Tr L -term, − PLSM = c f 2 4 quarks flavors at finite temperatures and baryon chemical po L N m PolyLog 2 [25]. , where , , ) Polyakov linear-sigma model = ∗ m L U 4 √ f φ / ··· is the flavor-blind Yukawa coupling [24]. L ˆ I N , ) PLSM at finite temperature and density In SU(4) Quantum chromodynamics (QCD) describes the strong interac The Polyakov-loop potentials introduces color degrees-of The present work is organized as follows. We utilize mean-fil g 0 4 + = and q = 0 phase-structures can be studied for mesons having masses up a fields, shall be discussed in section 3. Section 4 is devoted2. t SU( respectively, and the corresponding deconfinement order-p λ φ SU( 1. Introduction ons. As the temperature decreasesthe or at physical low degrees-of-freedom densities, can the qu bevarious dominated approaches besides by lattice hadron QCDphase simulations transitions (LQCD) in thermaling and quark dense masses, QCD the medium, chiralat symmetries see represent finite for basic current in pro quark masses,diagram can the be chiral described symmetry by the ischaractering first-principle SU(2) explic LQCD and [4, SU(3), 5] the an Polyakov Polyakov linear-sigma Nambu-Jona model Lasi (PLSM) [9, 10, 11,been 12, implemented. 13, The 14, SU(4) 15 Lagrangian hasAll the meson same fields structu should be parametrized in terms of 4 The thermal dependence of the light, strange and charm quark part is given as quark number susceptibilities and correlations can also be with Lagrangian for polynomial logarithmic parametrisation potential for the and to which an L scalar and pseudoscalar mesons PoS(ICHEP2016)634 . ) d 0 , φ c )+ , 2. → (2.6) (2.5) (2.4) a T , and c 2 √ / . σ / λ 0 = ( c . , and the 15 , T is related x ) σ 1 ( . σ s g c λ 5 MeV and tively. The 1 4 c  3  . σ h x 2 , s 2 ( µ µ σ = s √ ) − + σ 94 T T + h 2. In order to f f c 2 2 l , with E E 0 l density can be , 2 ] − λ m . The thermody- √ = l 3 4 σ 3 x 3 0 b / h 1 − − V ) 2 + Abdel Nasser Tawfik σ π 2 e e , λ / T 1 2 2 √ 1 / m λ 0 + + ) = [ / + . The orthogonal basis Z π m ( ) T f µ µ = ( T c e expressed as function π s 2 − ln + ( T L T f + c f σ f b x − , the strange s E E e σ 4 T s ) σ − l 2 D − 4 − 2 [14], and σ − 2 − l e e σ ) K m being the flavor mass of light, 1 f ( √ 2 σ from pion, kaon and D-meson [ ddition parameter , D × f × / λ c f 2 1 4 s b )= 15 tial can be obtained when 2 m nts (averages) in space and imag-   ) σ c + − σ µ µ grangian, g = ( σ 1 √ T − ) + 3 T T , 0. f f 0 / λ 2 s c s = E E 1 ( or current relation (PCAC) [9, 13]. At 0 √ = σ σ = s σ − 2 − , T , with c + l e e ( m 2 + h ∗ / σ + 4 φ l min 0 2 ), should be minimized at the saddle point; s 1 ( | and the deconfinement order-parameters φ 8 2, , to the light b ) σ ∗ σ + Ω / f c ) σ 2 U 2 l + 15 2 ∗ σ 2 3 + m σ σ 4 MeV, λ ∂φ φ )= φ . 2 g l 8 / )+ + T r   + 2 σ ( T 92 Ω 2 = ( 3 3 1 , b ), for instance. Accordingly, grand-canonical par- and − l ∂ 2 P ~ , and λ ∗ 0 = µ + + s 8 2 m m φ σ = ( 1 σ 1 π , was introduced in Ref. [7, 28], σ , and f = ( ) , 2 1   , + φ ] 0 l f + µ ( 2 c ∂φ ln ln σ σ , = ) = 2 E c σ [24]; / symmetries [9], the purely mesonic potential can be driven 0 T  s c T l + σ ( g Ω h R 2 / σ 3 s σ have been determined in Ref. [27], 2 ¯ ) 0 qq 2 is subtracted. The latter is stemming from the anomaly term ∂ P ~ σ − PolyLog d 3 π 1 T Ω s √ = 2 d ( λ U , 2 σ ( 4 c s / SU(4) x 15 s + ∞ h 87 MeV. 0 )+ σ , and . σ ∂σ 2 c × Z − c 2 l 6 / µ )+ refer to degenerate light, strange and charm quarks, respec L σ c l , 1 , , σ 2 s T l Ω √ σ c a 293 , 2 T c l l / σ ∂ ( ∑ h 0 = 1 + = f ¯ qq λ = 8 T − 2 ( T s Ω σ , and ) can be constructed, from which the thermodynamic potentia 3 ) and chemical potential ( 2 √ + 3 s x / 1 T ∂σ − , ) = Z ) √ l + / )= c π f 1 σ µ Ω + [ , . The exchanges between particles and antiparticles shall b , -term) added to mesonic potential, Eq. (2.2). All parameter / s quark flavor basis 0 ∂ ) = − τ ] c T σ σ ) 2 ( µ D , c = ) , f 2 l Ω l σ 1 T T 2 σ , the real part of thermodynamic potential, Re( ( √ ( / ( ∗ 0 ∂σ ¯ 0, the quark condensates reads ) PLSM at finite temperature and density It is noteworthy highlighting that SU(3) pure mesonic poten The values of the vacuum chiral-condensates are determined By introducing SU(4) In mean-field approximation, all fields are treated as consta qq φ U = = ( / 4 T l Ω ( 0 = have been determined at different sigma masses [9, 12]. The a Ω c σ 2 ∂ and σ SU( The various coefficients Furthermore, one can obtain that charm (known as tition function ( namic antiquark-quark potential, decay widths by means of theT partially conserved axial-vect evaluate the PLSM chiral condensates of temperature ( The subscripts and the additional term c to the pion- and D-meson masses by Ward identities; in basis of light, strange and charm quark from the mesonic La x energy-momentum relation is given as strange, and charm quark coupled to inary time transformation from the original ones: derived PoS(ICHEP2016)634 , 4. 4, µ . and = 92 g f N = with the l estimated 0.4 ∗ σ and decon- ∗ φ φ c 575 MeV for σ = mical potentials 0.35 and c and χ * T φ φ Abdel Nasser Tawfik φ = * , and φ φ s 0.3 σ ass is coupled to , l 4, and . σ 0.25 mization the real part of the T [GeV] φ 283 ates refer to varying critical temper- tential is formulated for = 0.2 r-parameters ∗ ven as functions of temperatures at s χ φ T el (a) depicts the normalized chiral e to charge quark chiral condensate; seems to decrease with increasing ers added to the SU(4) model, such as ply other methods, such as, peaks of 6, ing to their vacuum values modynamic antiquark-quark potential 0.15 and . tures coincide, especially at vanishing φ (solid, dotted and double-dotted curves) for ing this, we recall the well-know lattice e different values of the baryon chemical φ 0 (b) c 174 σ 0.1 temperatures corresponding to light, strange, t-hand panel (b): the same as in left-hand panel / 1 0 = c 0.8 0.6 0.4 0.2

l

σ

χ

,

φ φ * T 3 . ∗ , and φ 1 0 s σ / and s [left-hand panel (a)] and deconfinement order-parameters σ c φ 0 leads to , 0.8 σ = T 0 l µ at varying baryon chemical potentials. We notice the criti- = 0.0 = = T/2 µ µ σ 0 ∗ c / µ - l σ / φ c , and σ - σ s 0.6 σ , and l σ T [GeV] φ 0 0.4 s - σ / s . Right-hand panel (b) shows the Polyakov-loop fields - σ T . While the critical temperature from 295 MeV as functions of temperatures at different baryon che µ = 0.2 = 0 µ l c increases. - σ / (a) l σ ∗ - σ φ 0 0 1 0.6 0.4 0.2 0.8 Left-hand panel (a): normalized chiral-condensates are gi 2, and

are found identified. We have introduced how the charm quark m

[right-hand panel (b)] have been evaluated from global mini

0 /

f f

/

c 5, and σ σ - - ∗ . T h φ 94 0, ) PLSM at finite temperature and density We present the temperature dependence of the chiral condens Also, we notice the deconfinement order-parameters In determining various PLSM parameters, the pure mesonic po The chiral condensates 4 and = = and c s µ Accordingly, the extra degrees-of-freedom modifyand the the energy-momentum ther dispersion relations. The paramet finement order-parameters corresponding quark condensate at the charm quark condensate in vacuum. from the polynomial logarithmic parametrisation [27] at th and charm quarks. The intersection of the deconfinement orde potentials. It is possible to determine the chiral critical σ φ σ Figure 1: SU( cal temperatures increases when moving from light to strang atures with varying light, strange, and charm quarkbut for flavors, the respectively. deconfinement Righ order-parameters light, strange, and charm quark flavors,results respectively. that In both do chiral andbaryon deconfinement chemical critical potential. tempera Alternatively,chiral we condensates might susceptibilities, etc. also ap the one from 4. Conclusion different baryon chemical potentials 3. Results thermodynamic potential at thecondensates saddle of point. light, strange, Left-hand and pan charm quarks correspond PoS(ICHEP2016)634 , (Cambridge ions” Abdel Nasser Tawfik , 074502 (2013). 88 , 045202 (2001). and the dissociation of 64 χ T , 085008 (2000). 62 , 4 (2016). , 112 (2015). . The chiral phase-structure of 3 , 074023 (2007). χ 51 T thermal and dense medium are of 76 quark-hadron phase transition. Fur- , 1604.08174 [hep-lat]. ki, Phys. Rev. D s. Sci. d A. G. Shalaby, Adv. High Energy Phys. h]. e, Phys. Rev. C s. J. A ev. D ts. Phys. Rev. D , 531 (1969). , 055210 (2014). 41 89 4 , 014018 (2009). , 1343 (1973). 79 , 705 (1960). 30 , 015204 (2014). , 015206 (2015). 16 , 173 (1984). decreases the corresponding , 054019 (2010). , 015204 (2015). , 025015 (2007). 90 91 , 015004 (2015). 81 µ 234 91 75 42 575 MeV, respectively. This leads to draw a conclusion that , 1346 (1973). , 277 (2004). , 213 (1988). , 114028 (2008). 30 = , 014019 (2005). c χ 591 77 73 161 T , 074017 (2013). ”Finite-temperature field theory: Principles and applicat 88 -quarks obviously takes place at higher 4, and c . 283 , Phys. Rev. D = s χ et al. T 6, , 1381479 (2016). . 174 2016 University Press, UK, 2006). ) PLSM at finite temperature and density 4 = l χ [2] U. G. Meissner, Phys. Rept. [1] S. Gasiorowicz and D. A. Geffen, Rev. Mod. Phys. [3] C. Vafa and E. Witten, Nucl. Phys. B [5] H. D. Politzer, Phys. Rev. Lett. [4] D. J. Gross and F. Wilczek, Phys. Rev. Lett. [6] K. Fukushima, Phys. Lett. B [8] C. Ratti, [7] K. Fukushima, Phys. Rev. D [9] J. T. Lenaghan, D. H. Rischke, and J. Schaffner-Bielich, SU( T heavy mesons and the correlationsgreat importance including with charm respect quarks to recent in experimental resul References the charm condensate wouldn’t be affectedthermore, by the we well-known conclude that increasing hadrons consisting of [24] O. Scavenius, A. Mocsy, I. N. Mishustin, and D. H. Rischk [23] A. Tawfik and N. Magdy, Phys. Rev. C [22] A. Tawfik and N. Magdy, Phys. Rev. C [19] A. Tawfik, A. Diab, and M.[20] T. Hussein, Abdel 1610.06041 Nasser [nucl-t Tawfik, Abdel Magied Diab, N. Ezzelarab, an [21] A. Tawfik and N. Magdy, J. Phys. G [18] A. Tawfik, A. Diab, and M. T. Hussein, Int. J. Adv. Res. Phy [17] Abdel Nasser Tawfik, Abdel Magied Diab, and M.T. Hussein [16] A. Tawfik and A. Diab, Phys. Rev. C [15] A. Tawfik, N. Magdy, and A. Diab, Phys. Rev. C [14] P. Kovacs and Z. Szep, Phys. Rev. D [26] B.-J. Schaefer, J. M. Pawlowski, and J. Wambach, Phys. R [25] M. Gell-Mann and M. Levy, Nuovo Cim. [27] P. M. Lo, B. Friman, O. Kaczmarek, K. Redlich, and C. Sasa [13] Walaa I. Eshraim, F. Giacosa, and D. H. Rischke, Eur. Phy [12] B. J. Schaefer and M. Wagner, Phys. Rev. D [11] V. Tiwari, Phys. Rev. D [28] J. I. Kapusta and C. Gale, [10] U. Gupta and V. Tiwari, Phys. Rev. D