Deformed QCD phase structure and entropy oscillation in the presence of a magnetic background
aa r X i v : . [ h e p - ph ] J u l Deformed QCD phase structure and entropy oscillation in the presence of a magneticbackground
Guo-yun Shao,
1, 2, ∗ Wei-bo He, ∗ and Xue-yan Gao School of Science, Xi’an Jiaotong University, Xi’an, 710049, China MOE Key Laboratory for Nonequilibrium Synthesis and Modulation of Condensed Matter,Xi’an Jiaotong University, Xi’an, 710049, China
The QCD phase transitions are investigated in the presence of an external magnetic field in thePolyakov improved Nambu–Jona-Lasinio (PNJL) model. We detailedly analyze that how the fillingof multiple Landau levels by light (up and down) quarks deforms the QCD phase structure underdifferent magnetic fields. In particular, we concentrate on the phase transition under a magnetic fieldpossibly reachable in the non-central heavy-ion collisions at RHIC. The numerical result shows thattwo first-order transitions or more complicate phase transition in the light quark sector can exist forsome magnetic fields, different from the phase structure under a very strong or zero magnetic field.These phenomena are very interesting and possibly relevant to the non-central heavy-ion collisionexperiments with colliding energies at several A GeV as well as the equation of state of magnetars.Besides, we investigate the entropy oscillation with the increase of baryon density in a magneticbackground.
PACS numbers: 12.38.Mh, 25.75.Nq
I. INTRODUCTION
Over the decades, intensive investigations have beenperformed to explore the structure of strongly interact-ing matter. At high temperature and small chemical po-tential, the heavy-ion collision experiments indicate thatthe transformation from quark-gluon plasma (QGP) tohadrons is a smooth crossover [1], which is consistentwith the lattice QCD (LQCD) calculations [2–7]. A first-order phase transition, with a critical endpoint (CEP)connecting with the crossover transition, is predicted atlarge chemical potentials by some popular quark modelswhich incorporate the symmetry of QCD (e.g.,[8–18]).Searching for the critical endpoint is one of the primarytasks of RHIC STAR [19, 20]. The second phase of thebeam energy scan (BES-II) at STAR is being performedwith enhanced luminosity, focusing on the energy range √ s NN = 7 . ∼
20 GeV where some possible indicationsrelated to critical phenomenon were reported based onthe preliminary result of BES-I [21–23].A more challenging question is how the properties ofstrongly interacting matter will be affected when an ex-ternal magnetic field emerges (for recent reviews, pleaserefer to [24–26] ). There are at least two areas related tostrong interaction where magnetic field plays a very im-portant role: magnetars and non-central heavy-ion col-lisions. In the core of magnetars, the magnetic fieldstrength possibly reach 10 ∼ G [27, 28], whichgives birth to a stiffer equation of state of neutron starmatter and thus can support a massive compact star. Inthe non-central heavy-ion collisions, the intensity of mag-netic field depends on the beam energy and centrality.The magnetic field, up to B = 10 G or eB ∼ m π , is ∗ These authors contributed equally to this work possibly created at RHIC [29, 30], while up to eB ∼ m π can possibly be reached at LHC [31–33]. In all the abovesituations, the magnetic field intensity can be the sameorder as or larger than Λ QCD scale, therefore it will defi-nitely produce a profound effect on the QCD phase tran-sition.There are two main aspects in the study of stronglyinteracting matter under an external magnetic field re-lated to heavy-ion collisions. One is the chiral magneticeffect (CME) [29, 34] and the related phenomena suchas the chiral separation effect (CSE) [35] and the chiralmagnetic wave (CMW) [36–38]. The essence of the CMEis the imbalance of the chirality. The possibility that theCME can be observed in heavy-ion collisions has stim-ulated the exploration of strong interactions in presenceof a chirality imbalance and a magnetic field [39–44].The other aspect is the QCD phase transition drivenby a strong magnetic field. At zero temperature, thelattice studies indicate that the chiral condensate tendsto increase with the increasing magnetic field, which isknowns as magnetic catalysis (MC) [45]. It means thatthe magnetic field contributes to the chiral symmetrybreaking. Correspondingly, it suggests that the criticaltemperature of chiral symmetry restoration should be en-hanced for a stronger magnetic field. Later lattice cal-culation with the pion mass in the range 200 −
480 MeVindeed shows that the critical temperature slightly in-creases with the enhancement of magnetic field [46, 47].However, when the physical pion mass m π = 145 MeV istaken, the inverse magnetic catalysis (IMC) occurs, i.e.,the increase of magnetic field tends to suppress the quarkcondensate near the critical temperature of chiral transi-tion, and lowers the phase transition temperature at zerochemical potential [48, 49]. The discovery of IMC effectat high temperature has motivated the improvements ofthe effective quark models to give a consistent result withLQCD calcualtion. Different mechanisms have been pro-posed in literatures to explore the IMC effect and QCDphase transitions ( see, e.g., [50–62]).In the present study, we are more interested in thecomplete QCD phase diagram under a background mag-netic field. Related studies with relatively larger mag-netic fields have be done in Ref. [63] and meaningful re-sults about the phase transition in strange quark sec-tor have been achieved. However, the phase diagramat low temperatures and densities where up and downquarks dominate strongly depends on the magnetic fieldintensity. How the QCD phase structure changes fromsmall to large magnetic fields needs to be explored. Weherein will focus on the QCD phase structure under rel-atively smaller magnetic field. In particular, we will de-tailedly analyze the relation between the deformed first-order transition and the filling of Landau levels by upand down quarks. In addition, we will study the entropyoscillation with the increasing baryon density under anexternal magnetic field. The study is in some degreerelevant to the non-central collisions at STAR where rel-atively smaller magnetic field can be generated.The paper is organized as follows. In Sec. II, we intro-duce the thermodynamics of quark matter in the pres-ence of an external magnetic field within the 2+1 flavorPNJL quark model. In Sec. III, we illustrate the numer-ical results of the deformed QCD phase structure underdifferent magnetic fields, and discuss the influence of thefilling of Landau levels on the phase transition as well asthe entropy oscillation along baryon density. A summaryis finally given in Sec. IV. II. THERMODYNAMICS OF MAGNETIZEDQUARK MATTER
We first briefly introduce the thermodynamics of mag-netized quark matter in the 2+1 flavor PNJL model.In the presence of an external magnetic field, the La-grangian density takes the form, L = ¯ q ( iγ µ D µ + γ ˆ µ − ˆ m ) q + G X k =0 (cid:2) (¯ qλ k q ) +(¯ qiγ λ k q ) (cid:3) − K (cid:2) det f (¯ q (1 + γ ) q ) + det f (¯ q (1 − γ ) q ) (cid:3) − U (Φ[ A ] , ¯Φ[ A ] , T ) − F µν F µν , where q denotes the quark fields with three flavors, u, d ,and s ; the current mass ˆ m = diag ( m u , m d , m s ) andthe quark chemical potential ˆ µ = diag ( µ u , µ d , µ s ) in fla-vor space; G and K are the four-point and six-point in-teracting constants, respectively. The covariant deriva-tive is defined as D µ = ∂ µ − iA µ − iq i A µEM , where A µ = g A aµ λ a , in which A aµ represents the SU(3) gaugefield and λ a are the Gell-Mann matrices. A µEM is theelectromagnetic vector potential, and A µEM = δ µ x B for a static and constant magnetic field in the z direc-tion. F µν = ∂ µ A νEM − ∂ ν A µEM are used to account forthe external magnetic field. The effective potential U (Φ[ A ] , ¯Φ[ A ] , T ) is expressedwith the traced Polyakov loop Φ = (Tr c L ) /N C and itsconjugate ¯Φ = (Tr c L † ) /N C . The Polyakov loop L is amatrix in color space L ( ~x ) = P exp (cid:20) i Z β dτ A ( ~x, τ ) (cid:21) , (1)where β = 1 /T is the inverse of temperature and A = iA . The Polyakov-loop effective potential in presentstudy takes the form U (Φ , ¯Φ , T ) T = − a ( T )2 ¯ΦΦ + b ( T )ln (cid:2) − + Φ ) −
3( ¯ΦΦ) (cid:3) , where a ( T ) = a + a ( T T ) + a ( T T ) and b ( T ) = b ( T T ) .The parameters a = 3 . a = − . a = 15 .
2, and b = − .
75 were derived in [64] by fitting the thermody-namics of pure gauge sector in LQCD. T = 210 MeV isimplemented when fermion fields are includedIn the mean field approximation, the thermodynami-cal potential of magnetized quark matter can be derivedas [43]Ω = X f = u,d,s (Ω f + Ω T f ) + 2 G ( φ u + φ d + φ s )+4 Kφ u φ d φ s + U (Φ , Φ , T ) , (3)where Ω f = − N c | q f | eB π ∞ X n =0 α n Z ∞−∞ dp z π E f,n , (4)andΩ T f = − T | q f | eB π ∞ X n =0 α n Z + ∞−∞ dp z π (cid:16) Z + f + Z − f (cid:17) . (5)In Eq. 5, Z + f = ln(1+3Φ e − Ef,n − µfT +3 ¯Φ e − Ef,n − µfT + e − Ef,n − µfT ) , (6)and Z − f = ln(1+3 ¯Φ e − Ef,n + µfT +3Φ e − Ef,n + µfT + e − Ef,n + µfT ) . (7)The quasi-particle energy is E f,n = (2 n | q f | B + p z + M f ) / , (8)where n (= 0 , , , ... ) represents the n th Landaulevel (LL).To deal with the divergence in the vaccum part Ω f , wetake a smooth cutoff regularization procedure introducedin [65]. A form factor f Λ ( p f ) multiplying the integralkernel of Ω f is taken to avoid cutoff artifact, thus wehaveΩ f = − N c | q f | eB π ∞ X n =0 α n Z ∞−∞ dp z π f ( p f ) E f,n , (9)where f Λ ( p ) = s Λ N Λ N + p N , (10) N = 10 is chosen in the numerical calculation. One cansee that f Λ ( p ) is reduced to the sharp cutoff function θ (Λ − | p | ) in the N → ∞ limit. Since the thermal partof Ω Tf is not divergent, it is unnecessary to introduce aregularization function.In fact, the original PNJL model with a backgroundmagnetic field can not describe well the IMC effect de-rived in LQCD. To solve this problem, a magnetic fielddependent coupling constant for four-fermion interactionis proposed in Ref. [63]. The coupling takes the form G ( B ) = G aζ + bζ cζ + dζ , (11)where ζ = eB Λ QCD , with Λ
QCD = 300 MeV. The parame-ters are a = 0 . b = − . × − , c = 0 . d = 1 . × − . We will take such a magneticfield dependent coupling in the present study.Other thermodynamic quantities relevant to the bulkproperties of magnetized quark matter can be obtainedfrom Ω. We take the model parameters obtained in [66]:Λ = 603 . G Λ = 1 . K Λ = 12 . m u,d = 5 . m s = 140 . f π = 92 . M π = 135 . m K = 497 . m η =957 . µ u = µ d = µ s is taken in the calculation. III. NUMERICAL RESULTS ANDDISCUSSIONS
In this section, we demonstrate the numerical results ofthe deformed QCD phase diagram in the presence of anexternal magnetic field, and discuss its relation with Lan-dau quantization. We mainly concentrate on the first-order transition region at finite temperature.
A. Deformed ρ B − µ B curves under magnetic field To illustrate how the magnetic field strength affect thefirst-order transition, we present, in Fig. 1, the ρ B − µ B re-lations at different temperatures with eB = 0 , . , . . , respectively. The upper panel shows that,at T = 100 MeV, the ρ B − µ B curve for each eB hasone single spinodal structure, which is a typical charac-teristic of a first-order transition. But with the decreaseof temperature, the ρ B − µ B curves become more andmore complicated, as shown in the middle and lower pan-els. Particularly, multiple inflections appear for relativelysmaller magnetic field at low temperatures.The complicated twist in the ρ B − µ B curves at lowtemperatures are closely related to the Landau quanti-zation. The filling of multiple Landau levels should be eB=0.155GeV eB=0.40GeV eB=0.155GeV eB=0.40GeV eB=0 eB=0.05GeV T=20MeV B / eB=0.155GeV eB=0.40GeV T=100MeV B / eB=0 eB=0.05GeV eB=0 eB=0.05GeV T=5MeV B / B (MeV) FIG. 1: (color online) ρ B − µ B curves with eB = 0 , . , . . at three different temperatures. responsible for the number and locations where the in-flection points appear in the ρ B − µ B curve. To see thisclearly, we start analysis from the quark number den-sity. For each quark flavor f , the number density can bederived as ρ f = ∂ Ω ∂µ f = ∞ X n =0 ρ f,n , (12)where ρ f,n is the number density of the n th Landau level.Specifically, ρ f,n = 3 q f B π α n Z ∞−∞ d p z π (cid:2) f + f,n − f − f,n (cid:3) , (13)where f + f,n = Φ e − Ef,n − µfT + 2 ¯Φ e − Ef,n − µfT + e − Ef,n − µfT (1+3Φ e − Ef,n − µfT +3 ¯Φ e − Ef,n − µfT + e − Ef,n − µfT ) , (14) f − f,n = ¯Φ e − Ef,n + µfT + 2Φ e − Ef,n + µfT + e − Ef,n + µfT (1+3Φ e − Ef,n + µfT +3 ¯Φ e − Ef,n + µfT + e − Ef,n + µfT ) . (15)In the following, we will take T = 5 MeV and eB =0 .
05 GeV as an example to discuss the relation between ρ B − µ B curve and the filling of Landau levels. The curveof ρ f,n for each Landau level as a function of ρ B is plottedin Fig. 2. In this and all subsequent figures, LL0 meansthe lowest Landau level, LL1 is the first Landau level,LL2 is the second, and so forth. The upper and lowerpanels in Fig. 2 respectively illustrate the number densityof different Landau levels of up and down quarks. In thedensity region of 0 < ρ B ≤ ρ , the lowest four Landaulevels for u quarks are sequentially occupied as the baryondensity increases, and the lowest eight Landau levels areoccupied for d quarks. u , n ( f m - ) LL0 LL1 LL2 LL3
LL4 LL5 LL6 LL7 d , n ( f m - ) B / LL0 LL1 LL2 LL3
FIG. 2: (color online) Curves of ρ f,n for the first few Landaulevels as functions of ρ B . The corresponding ρ B − µ B curve is demonstrated inFig. 3. The thresholds of different Landau levels aremarked with the horizontal lines. The subscript n of u n and d n in Fig. 3 means the n th Landau level of u (d)quark. A horizontal line marked with two Landau lev-els, such as u and d , means that the two Landau levelsare filled (almost) at the same baryon density. It can beseen that when quarks fill a new Landau level a twistwill appear in the ρ B − µ B curve. The twist induced bythe Landau level is more distinct at small baryon density.Besides, the zigzag at ρ B > ρ results from the strangequark filling the new Landau levels. Therefore, the mul-tiple Landau levels are responsible for the twisted ρ B − µ B relations when the magnetic field is considered.Since Φ ≈ ¯Φ ≈ ρ f,n ≈ q f B π α n Z ∞ d p z (2 π ) (cid:20)
11 + e ( Ef,n − µf ) T (cid:21) . (16)From Eq. (16), it is easy to know that the chemical po-tential of ρ f,n from zero to non-zero approximately satis-fies the condition q M f + 2 | q f | Bn = µ f . For the lowest d d d d d B / B (MeV) u u u eB=0.05 GeV T=5.0 MeV
FIG. 3: (color online) ρ B − µ B curve for eB = 0 .
05 GeV at T = 5 MeV. The subscript i of u i and d i means the i thLandau level of u (d) quark. A horizontal line marked withtwo Landau levels, such as u and d , means that the twoLandau levels are filled (almost) at the same baryon density. u / d / ( u d )/2/ T=5.0MeVeB=0.4GeV ud / B / FIG. 4: (color online) ρ u and ρ d as functions of ρ B . Thefluctuation at low density is induced by u quark filling thelowest Landau level LL0 and the fluctuation at high densityis due to the strange quarks filling the new Landau level. Landau level, the condition becomes M f = µ f . For alarge magnetic field, the isospin symmetry is clearly bro-ken since q u = q d . For example, M u = 431 MeV and M d =400 MeV are derived for eB = 0 . at zero den-sity for T=5 MeV. Therefore the threshold of ρ u, will belarger than that of ρ d, , as shown in Fig. 4. This figurealso shows that ρ d, decreases with the onset of ρ u, . Cor-respondingly, we can understand that the small zigzag atlow density in the ρ B − µ B curve of eB = 0 . in thelower panel of Fig. 1 is induced by that u quark beginsto occupy the lowest Landau energy (LL0).Eq. (16) also indicates that the maximum n of thefilled Landau level satisfies n max =Floor( µ f − M f | q f | B ). It isinversely proportional to the quark charge q f and themagnetic field strength eB , which is also indicated bythe numerical results in Figs. 1 and 2. At high den-sity (large chemical potential), the dynamic quark mass M f approaches the current quark mass after chiral sym-metry restoration. Since a u quark is charged 2/3 and a d quark is charged − /
3, when u quarks fill one Landaulevel d quarks will fill two. B. Entropy oscillation with the increase of density
Many studies have involved the de Haas-van Alpheneffect of magnetized matter, a phenomenon related tothe filling of Landau levels, in which a physical quantityoscillates as a function of magnetic field intensity [67–70].In this subsection, we discuss the entropy oscillation asa function of baryon number density. en t r op y den s i t y ( f m - ) s eB=0.0 s s u +s d +s s en t r op y den s i t y ( f m - ) B / s u s d s s FIG. 5: (color online) Entropy density without and with amagnetic field eB = 0 .
05 GeV at T=5 MeV. s is the totalentropy density; s i is the entropy density of quark flavor i . The total entropy density s as a function of baryondensity is plotted in Fig. 5 without and with a magneticfield eB = 0 .
05 GeV at T=5 MeV. A distinct oscillat-ing behavior appears when the external magnetic fieldis considered. The numerical results in the upper panelshow that the total entropy density s is approximatelyequal to s u + s d + s s . This can be understood since thecontribution from the gauge sector is very small at verylow temperatures.The lower panel of Fig. 5 describes the entropy densi-ties of different quark flavor. It shows that the entropydensity oscillation at low baryon densities mainly comes from the u and d quarks. The contribution of strangequarks appear at high density. It also shows that thenumber of the peaks of s d is almost twice of s u . Sincenearly half of the peaks of s d appear at the same baryondensities with the peaks of s u , the larger peaks of the to-tal entropy density s in the upper panel reflect the super-position of s u and s d before the strange quarks appear.The smaller peaks are only induced by s d . Recalling theprevious conclusion about the number of Landau levelsfilled by u and d quarks, it reminds us that the oscillationof entropy density may be also induced by quarks fillingmultiple Landau levels. LL3 s u , n ( f m - ) LL1 LL2
LL5 LL6 LL2 LL3 LL4 s d , n ( f m - ) B / FIG. 6: (color online) Entropy density s f,n of each Landaulevel as a function of baryon density. The upper (lower) paneldescribes the entropy densities of different Landau levels of u ( d ) quark. Considering the contribution of each Landau level tothe total entropy density, we can decompose the totalentropy density before the appearance of strange quarksas (contribution from gauge sector is neglected at lowtemperature) s ≈ s u + s d = X f = u,d ∞ X n =0 s f,n . (17)Fig. 6 illustrates the curves of s f,n with the increaseof baryon density. It shows that each s f,n varies non-monotonically as the density increases. The location ofthe peak of each s f,n corresponds to the density whered ρ f,n / d ρ B takes the maximum value. This can be seenby comparing Figs. 6 with 7.Fig. 7 describes the derivative of ρ f,n respect to ρ B .It shows that d ρ f,n / d ρ B has also several minima at thelocations where d ρ u,n ′ / d ρ B or d ρ d,n ′ / d ρ B of the subse-quent Landau levels take maxima. This can be under- d u , n / d B LL1 LL2 LL3 d d , n / d B B / LL2 LL3 LL4 LL5 LL6
FIG. 7: (color online) Differential of ρ f,n of the n th Landaulevel of quark flavor f respect to ρ B . eB=0.05 GeV eB=0.155GeV eB=0.40 GeV s / B B / eB=0.00 FIG. 8: (color online) Entropy per baryon as functions of ρ B for eB = 0 , . , . . . stood from the following relation (before strange quarksappear at high density) ρ B = 13 X f = u,d ∞ X n =0 ρ f,n . (18)For a fixed ρ B , any two d ρ f,n / d ρ B with different f or n are in a competitive relationship. The growth of one sidemust be accompanied by the decrease of the other side.In Fig. 8, we present the entropy per baryon ( s/ρ B ) asfunctions of density for eB = 0 , . , . . ,respectively. It can be seen that, for a smaller magneticfield such as eB = 0 .
05 GeV , the frequent oscillationsoccur because more Landau levels are filled. The oscilla- tions take place around the curve of s/ρ B for eB = 0.The oscillation frequency decreases with the enhance-ment of magnetic field strength. Furthermore, the nu-merical analysis indicates that each valley in the s/ρ B curves corresponds to the threshold of a new Landaulevel, and each peak corresponds to the global maximumof d ρ f,n / d ρ B for a Landau level. C. Magnetic field dependence of QCD phasestructure
In this subsection, we analyze the magnetic field de-pendence of the QCD phase diagram, in particular thedeformation of the first-order phase transition.We first discuss the first-order transition under dif-ferent magnetic field intensity for a fixed temperature T = 5 MeV. The ρ B − µ B curves without ( eB = 0)and with an external magnetic field for eB = 0 .
155 and0 . are plotted in Fig. 9. For the case of zero mag-netic field, the first-order transition occurs at ρ A and ρ D . The two regions of A - B and C - D are the metastablephases, in which nucleation and bubble formation possi-bly occur. The region of B - C is the mechanically unsta-ble phase because of ∂p/∂ρ <
0, which is known as thespinodal region. When the bulk uniform matter entersinto this region, a small fluctuation in density will lead tophase separation via the spinodal decomposition. Gener-ally, for an equilibrium transition, the unstable phase cannot be observed. But it is difficult to estimate the role itplays on observables such as the particle fluctuations ina rapid expanding system.For eB = 0 .
155 GeV , the phase structure is quitedifferent with that of zero magnetic field. Two first-order phase transitions, from A to D and from E to H , take place. The locations of the two transitions aredetermined according to the conditions for phase equi-librium: T A = T D , µ A = µ D and P A = P D as well as T E = T H , µ E = µ H and P E = P H . Moreover, betweenthe two first-order transitions, a stable phase of the mag-netized matter exists in the region of D - E . Such a spe-cial phase structure is driven by the Landau quantizationwith the filling of different Landau levels by quarks.With the increase of magnetic field intensity, for ex-ample eB = 0 . , the right panel of Fig. 9 showsthat the first-order transition occur at ρ A and ρ D , sim-ilar with the case of eB = 0. However, there exists aregion marked as O - P - Q - R , which has a ρ B − µ B struc-ture opposite to a standard first-order transition. At thelocations O and R , the conditions for two phase equi-librium are fulfilled, but it is not a first-order transitionbecause O and R lie in the unstable phase. On the otherhand, bulk matter in the interval of P - Q are metastable,which can not be observed for an equilibrium transition,since the phase transition from ρ D to ρ A will first takeplace.The dynamical mass of u quark as functions of baryonchemical potential are illustrated in Fig. 10 for eB = B (MeV) A BC D FG HE eB=0.155GeV eB=0.40GeV B (MeV) A B
OP QR
C D eB=0
DC BA B / B (MeV) FIG. 9: (color online) ρ B − µ B curves without ( eB = 0) and with an external magnetic field for eB = 0 .
155 and 0 . at T = 5 MeV. For eB = 0, the first-order transition takes place at the locations of A and D . For eB = 0 . A and D as well as E and H . For eB = 0 .
4, the first-order transition takes place at thelocations of A and D . , .
155 and 0 . . It can be seen that M u in thechiral breaking phase increases with the enhancementof magnetic field intensity, which reflects the magneticcatalysis effect at low temperature. It is consistent withLQCD calculation [45]. The solid dots with the samecolor on M u − µ B curves indicate the locations wherethe first-order transition takes place. The circles on thecurve of eB = 0 . does not mean a first-order tran-sition although the conditions for phase equilibrium arefulfilled. The curves of eB = 0 .
155 GeV indicates thattwo first-order transition can take place. Similar phe-nomenon was discovered at zero temperature [71–73]. M u ( M e V ) B (MeV) eB=0 eB=0.155GeV eB=0.40 GeV FIG. 10: (color online) Dynamical quark mass of u quark asa function of baryon chemical potential for eB = 0 , .
155 and0 . , respectively, at T=5 MeV. The complete phase diagram of the chiral phase transi-tion is demonstrated in Fig. 11. For the first-order transi-tion at low temperatures, the associated metastable and unstable regions are also included. This figure distinctlyillustrates the deformed QCD phase structures driven bymagnetic field with different field intensity. When the ex-ternal magnetic field is larger than eB = 0 . , themagnetized quark matter has a similar phase structureto eB = 0 . . Two first-order transitions exist inthe vicinity of eB = 0 .
155 GeV . For a smaller magneticfield such as eB = 0 .
05 GeV , more Landau levels will beoccupied and the corresponding phase diagram is morecomplicated, as shown in Fig. 3.The phase diagrams including the contribution ofstrange quark at high density (large chemical potential)are illustrated in Figs. 12 and 13. Compared with thephase structure without an external magnetic field, thetwo figures indicate that the first-order transitions inthe strange quark sector can also be induced by Landauquantization when s quarks fill the different Landau lev-els. The locations where the first-order transitions takeplace depend on the magnetic field intensity. One canalso refer to [63] for the related discussion about the first-order transition in strange quark sector.In this study, we find the phase structure of strongly in-teracting matter under an external magnetic field highlydepends on the field intensity. Experimently, the high-density region can possibly be reached in future heavy-ioncollisions at RHIC, NICA and FAIR. At the same time,the magnetic field with eB . .
155 GeV can possiblybe created in the non-central collisions, therefore, themulti-first-order transitions or more complicated phasestructure of light quarks may give birth to some observ-able effects in the beam energy scan with relatively lowercollision energies. B / B / B / T ( M e V ) FIG. 11: (color online) Phase diagrams in the T − ρ B plane for eB = 0 , .
155 and 0 . . The dashed line in each panelcorresponds to the chiral crossover transition at high temperatures. The first-order transitions are marked with the solid redlines. The blue lines indicate the spinodal regions. The orange lines and region for eB = 0 . indicate the range with astructure opposite to a standard first-order transition. B / B / B / T ( M e V ) FIG. 12: (color online) Phase diagrams in the T − ρ B plane including the contribution of strange quarks at high density for eB = 0 , .
155 and 0 . . The first-order transitions of strange quark are marked with the solid black lines. The purple linesindicate the corresponding spinodal regions
600 800 1000 1200 140050100150200 600 800 1000 1200 1400 400 600 800 1000 1200 1400 1600eB=0 B (MeV) T ( M e V ) eB=0.155GeV B (MeV) eB=0.4GeV B (MeV) FIG. 13: (color online) Phase diagrams in the T − µ B plane including the contribution of strange quarks at large chemicalpotential for eB = 0 , .
155 and 0 . . IV. SUMMARY
In this study, we investigated the chiral phase tran-sition in the presence of an external magnetic field inthe improved PNJL model. The calculations show thatthe phase structure of magnetized quark matter stronglydepends on the intensity of magnetic field. Differentfrom the first-order phase transition without a mag-netic field, two first-order transitions or more compli-cated phase transition in the light quark sector can oc-cur for eB . .
155 GeV . The study also indicates thatthe deformation of the phase structure under an externalmagnetic field is attributed to the Landau quantizationwith the filling of different Landau levels. Generally, fora relatively smaller magnetic field, more Landau levelswill be filled which leads to a twisted ρ B − µ B relationand then produce a complicated phase structure.We also found that the distribution of quarks at mul-tiple Landau levels causes the entropy density oscillationdue to the Landau quantization. The numerical resultsindicate that the entropy density as well as the entropy per baryon begin to increase at the threshold of a newLandau level. Each peak of the entropy density (en-tropy per baryon) corresponds to a maximum value of ∂ρ f,n /∂ρ B of a Landau level.If the high-density quark matter and eB ∼ .
155 GeV could be created in the non-central heavy-ion collisions,some signals different from the standard first-order tran-sition may manifest in future experiments. However, itis difficult in measurements due to the decay of magneticfield in the expansion. More simulations are needed tocatch the relevant signatures. This study is also refer-ential to investigate the magnetized neutron star matterwith a quark core or a magnetized quark star. Acknowledgments
This work is supported by the National Natural Sci-ence Foundation of China under Grant No. 11875213and the Natural Science Basic Research Plan in ShaanxiProvince of China (Program No. 2019JM-050). [1] S. Gupta, X. F. Luo, B. Mohanty, H. G. Ritter, and N.Xu, Science , 1525 (2011).[2] Y. Aoki, G. Endrodi, Z. Fodor, S. D. Katz, K. K. Szabo,Nature , 675 (2006).[3] A. Bazavov, et al., hotQCD Collaboration, Phys. Rev. D , 054503 (2012).[4] S. Bors´anyi, Z. Fodor, S. D. Katz, S. Krieg, C. Ratti andK. K. Szab´o, Phys. Rev. Lett. , 062005 (2013).[5] A. Bazavov, et al., hotQCD Collaboration, Phys. Rev.D. , 094503 (2014).[6] A. Bazavov, et al., hotQCD Collaboration, Phys. Rev. D , 074510 (2017).[7] S. Bors´anyi, Z. Fodor, C. Hoelbling, S. D. Katz, S. Krieg,and K. K. Sabz´o, Phys. Lett. B , 99 (2014).[8] K. Fukushima, Phys. Lett. B , (2004) 277; Phys. Rev.D , 114028 (2008).[9] C. Ratti, M. A. Thaler, and W. Weise, Phys. Rev. D ,014019 (2006).[10] P. Costa, M. C. Ruivo, C. A. de Sousa, and H. Hansen,Symmetry , 1338 (2010).[11] W. J. Fu, Z. Zhang, and Y. X. Liu, Phys. Rev. D ,014006 (2008).[12] T. Sasaki, J. Takahashi, Y. Sakai, H. Kouno, and M.Yahiro, Phys. Rev. D , 056009 (2012).[13] B. J. Schaefer, M. Wagner, and J. Wambach, Phys. Rev.D , 074013 (2010).[14] V. Skokov, B. Friman, and K. Redlich, Phys. Rev. C ,054904 (2011).[15] S. X. Qin, L. Chang, H. Chen, Y. X. Liu, and C. D.Roberts, Phys. Rev. Lett. , 172301 (2011).[16] F. Gao, J. Chen, Y. X. Liu, S. X. Qin, C. D. Roberts,and S. M. Schmidt, Phys. Rev. D , 094019 (2016).[17] C. S. Fischer, J. Luecker, and C. A. Welzbacher. Phys.Rev. D , 034022 (2014).[18] C. Shi, Y. L. Wang, Y. Jiang, Z. F. Cui, H. S. Zong,JHEP , 014 (2014). [19] M. M. Aggarwal, et al., STAR Collaboration, Phys. Rev.Lett. , 022302 (2010).[20] L. Adamczyk, et al., STAR Collaboration, Phys. Rev.Lett. , 032302 (2014).[21] X. Luo (for the STAR Collaboration), PoS(CPOD2014)(2015) 019.[22] X. Luo, Nucl. Phys. A , 75 (2016).[23] X. Luo and N. Xu, Nucl. Sci. Tech. , 112 (2017).[24] D. Kharzeev, K. Landsteiner, A. Schmitt, and Ho-UngYee, Lect. Notes Phys. , 1 (2013).[25] V. A. Miransky and I. A. Shovkovy, Phys. Rept. , 1(2015).[26] J. O. Aadersen and W. R. Naylor, A. Tranberg,Rev. Mod.Phys. , 025001, (2016).[27] M. Bocquet, S. Bonazzola, E. Gourgoulhon, and J. No-vak, Astron. Astrophys. , 757 (1995).[28] E. J. Ferrer, V. de la Incera, J. P. Keith, I. Portillo, P.L. Springsteen, Phys. Rev. C , 065802 (2010).[29] D. E. Kharzeev, L. D. McLerran, and H. J. Warringa,Nucl. Phys. A , 227 (2008).[30] V. Skokov, A. Y. Illarionov, and V. Toneev, Int. J. Mod.Phys. A , 5925 (2009).[31] V. Voronyuk, V. D. Toneev, W. Cassing, E. L.Bratkovskaya, V. P. Konchakovski, and S. A. Voloshin,Phys. Rev. C , 054911 (2011)[32] A. Bzdak, V. Skokov, Phys. Lett. B , 171 (2012)[33] W. T. Deng, X. G. Huang, Phys. Rev. C , 044907(2012)[34] P. V. Buividovich, M. N. Chernodub, E. V.Luschevskaya, and M. I. Polikarpov, Phys. Rev. D , 054503 (2009).[35] D. E. Kharzeev and H. U. Yee, Phys. Rev. D , 085007(2011).[36] M. A. Metlitski and A. R. Zhitnitsky, Phys. Rev. D ,045011 (2005).[37] Y. Burnier, D. E. Kharzeev, J. Liao, and H. U. Yee, Phys. Rev. Lett. , 052303 (2011).[38] E. V. Gorbar, V. A. Miransky, I. A. Shovkovy, Phys. Rev.D , 085003 (2011).[39] K. Fukushima, D. E. Kharzeev, H. J. Warringa, Phys.Rev. D , 074033 (2008).[40] K. Fukushima, M. Ruggieri, Phys. Rev. D , 054001(2010).[41] R. Gatto, M. Ruggieri, Phys. Rev. D , 054013 (2012).[42] M. Ruggieri, Phys. Rev. D , 014011 (2011).[43] W. J. Fu, Phys. Rev. D , 014009 (2013).[44] C. A. B. Bayona, K. Peeters, and M. Zamaklar, J. HighEnergy Phys. , 092 (2011).[45] P. Buividovich, M. N. Chernodub, E. V. Luschevskaya,and M. I. Polikarpov, Phys. Lett. B
484 (2010).[46] M. D’Elia, S. Mukherjee, F. Sanfilippo, Phys. Rev. D ,051501(R) (2010) .[47] E. M. Ilgenfritz, M. Kalinowski, M. M¨uller-Preussker, B.Petersson, A. Schreiber, Phys. Rev. D (2012) 114504.[48] G. S. Bali, F. Bruckmann, G. Endrodi, Z. Fodor,S. D. Katz, S. Krieg, A. Schafer, and K K. Szabo, JHEP , 044 (2012).[49] G. S. Bali, F. Bruckmann, G. Endrodi, Z. Fodor,S. D. Katz and A. Schafer, Phys. Rev. D , 071502(R)(2012).[50] K. Fukushima and J. M. Pawlowski, Phys. Rev. D ,076013 (2012).[51] J. Chao, P. Chu, and M. Huang, Phys. Rev. D , 054009(2013).[52] E. S. Fraga, J. Noronha, and L. F. Palhares, Phys. Rev.D ,031601 (2013).[54] K. Fukushima and Y. Hidaka, Phys. Rev. Lett. ,102301 (2016).[55] S. Fayazbakhsh, and N. Sadooghi, Phys. Rev. D ,105030 (2014).[56] M. Ferreira, P. Costa, and C. Providˆencia, Phys. Rev. D , 016012 (2014).[57] P. Costa, M. Ferreira, D. P. Menezes, J. Moreira, and C.Providˆencia, Phys. Rev. D , 036012 (2015).[58] E. S. Fraga, B. W. Mintz, and J. Schaffner-Bielich, Phys.Lett. B , 154 (2014).[59] L. Yu, J. Van Doorsselaere, and M. Huang, Phys. Rev.D , 074011 (2015).[60] A. Ayala, C. A. Dominguez, L. A. Hernandez, M. Loewe,and R. Zamora, Phys. Lett. B , 99 (2016).[61] V. P. Pagura, D. Gomez Dumm, S. Noguera, and N. N.Scoccola, Phys. Rev. D , 034013 (2017).[62] D. P. Menezes, M. B. Pinto, S. S. Avancini, A. P. Mar-tinez, and C. Providˆencia, Phys. Rev. C , 035807(2009).[63] M. Ferreira, P. Costa, and C. Providˆencia, Phys. Rev. D , 014014 (2018).[64] S. R¨oßner, C. Ratti, and W. Weise, Phys. Rev. D ,034007 (2007).[65] K. Fukushima, M. Ruggieri, and R. Gatto, Phys. Rev. D , 114031 (2010).[66] P. Rehberg, S. P. Klevansky, and J. H¨ufner, Phys. Rev.C , 410 (1996).[67] X. J. Wen and J. J. Liang, Phys. Rev. D , 014005(2016).[68] L. Wang and G. Cao, Phys. Rev. D , 034014 (2018)[69] K. I. Aoki, H. Uoi, and M. Yamada, Phys. Lett. B ,580 (2016)[70] G. Lugones and A. G. Grunfeld, Phys. Rev. C , 015804(2017)[71] P. G. Allen and N. N. Scoccola, Phys. Rev. D , 094005(2013).[72] R. Z. Denke and M. B. Pinto, Phys. Rev. D , 056008(2013).[73] A. G. Grunfeld, D. P. Menezes, M. B. Pinto, andN. N. Scoccola, Phys. Rev. D90