The influence of magnetic field on the pion superfluidity and phase structure in the NJL model
aa r X i v : . [ h e p - ph ] O c t The influence of magnetic field on the pion superfluidity and phase structure in theNJL model
Xiaohan Kang, Meng Jin, and Jiarong Li
Key Laboratory of Quark and Lepton Physics (MOE) and Institute of Particle Physics,Central China Normal University, Wuhan 430079, China
Juan Xiong
Department of Applied Physics, College of Sciences,Huanzhong Agriculture University, Wuhan 430070,China (Dated: September 23, 2018)The influence of the magnetic field on the pion superfluidity and the phase structure is analyzedin the framework of the two-flavor Nambu–Jona-Lasinio(NJL) model. To do this, we first derivethe thermodynamic potential from the Lagrangian density of the NJL model in the mean field ap-proximation. Using this thermodynamic potential, we get the gap equation of the chiral condensateand the pion condensate. The effect of external magnetic field on the pion condensate is not simplepromotion or suppression, which we will discuss in detail in the paper. It is shown that the tricriticalpoint on the pion superfluidity phase transition line moves to the space with smaller isospin chemicalpotential and higher temperature when the external magnetic field becomes stronger. The influenceof external magnetic field on the chiral condensate is also studied.
PACS numbers: 11.30.Qc, 12.39.-x, 21.65.+f
I. INTRODUCTION
Quantum Chromodynamics (QCD) is the gauge theoryof strong interaction. Recently, the QCD phase diagramincluding chiral symmetry breaking (restoration) [1–9],quark confinement (deconfinement) [10, 11], color super-conductivity [12–17] were investigated with finite tem-perature and finite chemical potential. Furthermore, thestudies are extended to finite isospin chemical poten-tial [18–25]. It is found that at a critical isospin chemicalpotential, which is about the pion mass in vacuum, thephase transition from normal phase to pion superfluidityphase will happen. The understanding of the propertiesof pion superfluidity and phase structure under the ex-treme conditions are important in conducting research ofthe physics of the compact objects and relativistic heavyion collisions.It is reported that very strong magnetic field may begenerated in the heavy-ion collision [26–28]. The mag-netic field magnitudes were estimated to be 5.3 m π /e at the Relativistic Heavy-Ion Collision(RHIC) and6 m π /e at the Large Hadron Collision(LHC), and evenhigher [29]. Such magnetic field can also exist in mag-netars [30] and in the early universe [31–33] despite theorigin of such strong field is not clear yet. Above phe-nomena lead us to think deeply about the influence ofthe magnetic field on the QCD phase diagram [34]. Re-cently, much work has been done about the influence ofthe magnetic field on the properties of quark matter [35],chiral transition [36] and color superconductivity [37–39].Normally, QCD is widely accepted as the correct the-ory describing strongly interacting matter at high tem-perature and high density. The low energy QCD vacuumis hard to be fully understood by using of perturbativemethods, because the characteristics of chiral symmetry breaking and color confinement have a non-perturbativeorigin. To solve the problem, two approaches are intro-duced, one is lattice QCD simulations [40] and the otheris the effective model method.The Nambu–Jona-Lasinio (NJL) model [1, 42–45] isused as a simple and practical chiral model, which sat-isfies the basic mechanism of spontaneous breaking ofchiral symmetry and key features of QCD at finite tem-perature and chemical potential. One of the basic prop-erties of this model is that it includes a gap equationwhich connect the chiral condensate to the dynamicalquark mass. It is well known that the hadronic massspectra and the static properties of meson, especiallythe chiral symmetry spontaneous breaking, can be ob-tained through the mean field approximation (RPA) ofmeson in the NJL model. This model and its extendedversion (PNJL, EPNJL, etc.) are also widely used tostudy the properties of deconfinement phase, the color su-perconductivity phase, and the pion superfluidity phaseand the related phase transitions under extreme condi-tions. Recently, the strong magnetic field effect on theproperties of quark matter [46–48], and the phase tran-sitions, including chiral restoration transition, deconfine-ment transition [49, 50] and color superconductivity tran-sition [39, 51], has been investigated in the NJL-typemodels by many groups.In this paper, we will mainly focus on the effect ofexternal magnetic fields on the pion condensate and thepion superfluidity phase structure at finite temperatureand finite isospin chemical potential. The influence of themagnetic field on the chiral condensate is also studied.This paper is organized as follows. In Sec. II, we willgive a simple introduction of our model and calculate thethermodynamics potential within mean field approxima-tion. Sec. III is devoted to the numerical results of thequark pair condensation and the phase diagram in the T − µ I − eB space. We summarize and conclude our jobin Sec. IV. II. THE MODEL
The Lagrangian density of two-flavor Nambu-Jona-Lasinio (NJL) model is defined as: L = ¯ ψ ( iγ µ D µ − ˆ m ) ψ + G [( ¯ ψψ ) + ( ¯ ψiγ ~τ ψ ) ] , (1)where ψ = ( ψ u , ψ d ) T is the quark field, ˆ m = diag ( m u , m d ) is the current quark mass matrix with m u = m d ≡ m (the isospin symmetry). D µ = ∂ µ + ieA µ isthe covariant derivative, A µ = δ µ A and A = − iA represents the gauge field. G is the four-quark couplingconstant with dimension GeV − . The Pauli matrices τ i ( i = 1 , ,
3) are defined in isospin space.With scalar and pseudoscalar interactions correspond-ing to σ and π excitation, the lagrangian density has thesymmetry of U B (1) N SU I (2) N SU A (2), correspondingto baryon number symmetry, isospin symmetry and chi-ral symmetry, respectively. The chiral symmetry SU A (2)breaks down to U A (1) global symmetry which is associ-ated with the chiral condensation of the σ meson. σ = < ¯ ψψ > = σ u + σ d , (2) σ u = < ¯ uu >, σ d = < ¯ dd > . (3)The isospin symmetry SU I (2) breaks down to U I (1)global symmetry with the generator I which is relatedto the condensate of charged pions, π + and π − , π + = < ¯ ψiγ τ + ψ > = √ < ¯ diγ u >, (4) π − = < ¯ ψiγ τ − ψ > = √ < ¯ uiγ d > . (5)With τ ± = ( τ ± τ ) / √
2. At extremely high µ I > u and anti- d quark is favored. At extremelyhigh µ I < d and anti- u quark is favored.The system is in the global thermal equilibrium, π + = π − , as is assumed in this paper, the whole superfluidityis electric charge neutral.By means of the mean field approximation, the ther-modynamic potential of two flavor NJL model at finiteisospin chemical potential, finite temperature and strongmagnetic field is given asΩ = G ( σ + π ) − N c X f = u,d X κ α κ Z + ∞−∞ dp z π | Q f eB | π [ ω πf + 2 T ln (1 + e − βω πf )] , (6)with ω πf on behalf of ω πu and ω πd , the quasi-particle energyof u and d quark. Q f means the electric charge of Q u and Q d . κ is non-negative integer which denotes the Landau level and α κ = 2 − δ κ is the corresponding degeneracy.In the equation (6), ω πf = q ( ω f ± µ I ) + 4 G π ,ω f = q p z + 2 | Q f eB | κ + M , (7)and M = m − Gσ. (8)If we use the replacement for the momentum integraland the quark energy,2 Z d p (2 π ) ↔ | Q f eB | π X κ α κ Z dp z π , (9) p p + M ↔ q p z + 2 | Q f eB | κ + M , (10)the thermodynamic potential Ω above will coincide withthat in the zero magnetic field case [24]. For the upperlimit of p z integral, we use the hard cut p Λ − κ | eB | .Then the upper limit of κ sum is κ max = Int [ Λ | Q f eB | ],for Q u = + e and Q d = − e .The gap equation of the mean field σ and π are derivedfrom ∂ Ω ∂σ = 0 , ∂ Ω ∂π = 0 . (11)When there exist multi-roots, only the solution whichsatisfies the minimum condition is physical. The gapequation of σ is σ + 2 N c Z + ∞−∞ dp z π X κ α κ M | eB | π { Q u [1 − f ( ω πu )] ω u + µ I ω πu ω u + Q d [1 − f ( ω πd )] ω d − µ I ω πd ω d } = 0 . (12)The gap equation of π is1 − N c G X f = u,d X κ α κ Z + ∞−∞ dp z π | eB | π Q f [1 − f ( ω πf )] ω πf = 0 . (13) π = 0 is the trival solution of this equation, which hasbeen ignored here. The Fermi function is f ( x ) = e βx ) .We choose the following values as numerical analysis pa-rameters: m = 5 M eV , a three-dimensional momen-tum cut-off Λ = 650 . M eV ( T = 0), G = 5 . GeV − , m π = 139 M eV and | < ¯ ψ u ψ u > | / = − M eV . III. NUMERICAL RESULTSA. the pion and chiral condensate
In Fig. 1 we plot the behavior of the chiral condensate σ and the pion condensate π , measured in units of the chi- T = eB=0eB=30 m Π eB=20 m Π Μ I H MeV L Π (cid:144) Σ T = eB=0eB=30 m Π eB=20 m Π Μ I H MeV L Σ (cid:144) Σ FIG. 1:
Upper panel:
Pion condensate π , shown as a functionof the isospin chemical potential µ I with respect to variousmagnetic field magnitudes. Lower panel:
Chiral condensate σ , shown as a function of the isospin chemical potential µ I with respect to various magnetic field magnitudes. ral condensate in vacuum σ , with varing isospin chem-ical potential µ I at fixed temperature T = 100 M eV forvarious magnitude of the magnetic field, eB = 0, 20 m π ,and 30 m π .In the upper panel, for eB = 0, we reproduced theresult of Ref. [24]. The π superfluidity begin at criticalisospin chemical potential µ cI = m π / ≈ M eV , thenthe pion condensate increases when µ I becomes larger,which says that the isospin chemical potential enhancethe pion condensate [54].When the external magnetic field is included, such as eB = 20 m π and 30 m π , the critical µ I for occurrence ofpion condensate increases with the increasing magneticfields, indicating the magnetic field suppresses the for-mation of the pion superfluidity. For different magneticfields, the pion condensate gaps remain the similar shapeas a function of µ I , the difference is that the maximumof gap is bigger with larger eB . For example, the max-imum of π gap is 1 . σ when eB = 0, but it reaches to1 . σ when eB = 30 m π . For µ I > M eV , there existthree solutions for the π condensation corresponding toa given isospin chemical potential (one of them is zero).Such feature of the order parameter normally indicatesthat the phase transition is of first order.In the lower panel, for eB = 0, the chiral conden-sation also coincides with the result in Ref. [24]. Thechiral condensate decreases with increasing µ I , which re- Μ I = T=130 MeVT=100 MeVT=0 MeV (cid:144) m Π Π (cid:144) Σ Μ I = T=0T=100 MeVT=130 MeV (cid:144) m Π Σ (cid:144) Σ FIG. 2:
Upper panel:
Pion condensate π , shown as a func-tion of the magnetic field strength eB with respect to varioustemperature magnitudes. Lower panel:
Chiral condensate σ ,shown as a function of the magnetic field strength eB withrespect to various temperature values. flects the restoration of the chiral symmetry. When theexternal magnetic fields are taken into account, such as eB = 20 m π and 30 m π , the σ condensate is enhanced withincreasing B , this is the so called chiral catalytic effectof the magnetic field [52, 53]. The restoration of chi-ral symmetry still happen with the increasing µ I . Fromthe change tendency of the σ condensate we can roughlyextract the critical isospin chemical potential µ cI . Thebigger the magnetic field, the bigger the µ cI . This impliesthat the external magnetic field hinders the restorationof chiral symmetry.In Fig. 2 we show the behavior of the chiral con-densate σ and pion condensate π , with varing mag-netic field strength at fixed isospin chemical potential µ I = 100 M eV for several values of temperature, T = 0,100 M eV and 130
M eV .In the upper panel, the π/σ almost remains stablewith small eB . When eB increases to certain value, the π condensation starts to decrease oscillately, and thendisappears when the external magnetic field is strongenough. This implies that π condensate is suppressedwith increasing eB . From the figure we can see the higherthe temperature, the narrower range of the magnetic fielddomain of π = 0 and the smaller the critical eB for pionvanishing. That is to say the pion condensate is sup-pressed by the temperature. An interesting phenomenonis that the pion condensate exhibits an oscillation behav- Μ I = T = T = T = (cid:144) m Π Π (cid:144) Σ FIG. 3: The Pion condensate π , shown as a function of themagnetic field strength B with respect to various temperaturemagnitude at fixed isospin chemical potential µ I = 400 MeV . ior as a function of eB . From our calculation, the oscil-lating behavior is only related to the external magneticfield but not the temperature or the isospin chemical po-tential, it is also verified in the lower panel.In the lower panel, with increasing eB , the chiral con-densate increases oscillately. That is coincide with thechiral catalytic effect. The oscillation is obvious evenwhen the temperature is very high, such phenomenonis the so-called Alfven-de Haas oscillation. The oscil-lating behavior related to the external magnetic field issimilar with the π . But the oscillating behavior van-ishes when eB > . m π and the σ condensate increasesmonotonously with increasing eB . The reason is thatwhen the eB is higher enough, the Lowest Landau Level(LLL) will chiefly contribute to the properties of the sys-tem and then the oscillation disappears. The influenceof temperature on chiral condensate is not such apparentas the pion condensation, the curves of σ/σ coincide ap-proximately for the three values of temperature we chose.We noticed in some eB regions, for example, between eB = 13 m π ∼ m π , the chiral condensate will decreaseas the increasing eB . The decrease is a consequence ofthe oscillating behavior of the σ condensate, which isdifferent from the inverse chiral magnetic catalysis effect.In Fig. 3, we show the pion condensation varies with eB for µ I = 400 M eV . Now the pion condensation in-crease oscillately with eB , impling the magnetic field canalso enhance the pion condensate. This contradicts thebehavior for the case of µ = 100 M eV in Fig. 2. Normallythe isospin chemical potential can promote the pion con-densate while the temperature depress it. However, theeffect of magnetic field is complicated. In some case itdepresses the π condensation and in the other case it re-inforce the π condensation. For some given temperatureand isospin chemical potential, such as T >
M eV , µ I = 400 M eV , the pion condensate does not exist at eB = 0, and then it occurs as the eB increases, showingthat the external magnetic field can promote the forma-tion of the pion superfluidity. This phenomenon is theresult of the coupled influence of µ I and eB . When we Μ I = eB=0eB=23 m Π eB=20 m Π H MeV L Π (cid:144) Σ Μ I = eB=0eB=23 m Π eB=20 m Π H MeV L Σ (cid:144) Σ FIG. 4:
Upper panel:
Pion condensate π , shown as a func-tion of the temperature with respect to various magnetic fieldmagnitudes. Lower panel:
Chiral condensate σ , shown as afunction of the temperature with respect to various magneticfield magnitudes. increase eB higher enough, the π condensate start to de-crease and will disappear at some value of eB (it is notshown in the Figure), that is similar with the pion con-densate in Fig.2.In Fig. 4 we show the behavior of the pion conden-sate π and the chiral condensate σ , with varing T forisospin chemical potential µ I = 100 M eV but for vari-ous magnetic field magnitudes, eB = 0, 20 m π and 23 m π ,respectively.In the upper panel, for eB = 0 the π condensatedecreases monotonously as the temperature increases.When T increases to the critical value, the pion con-densate disappears. This implies that the temperaturedepress the pion condensate, as is shown similarly in Fig.2 and Fig. 3. Taking the magnetic field into considera-tion, the greater the eB , the narrower range of temper-ature domain of the π = 0 and the lower the critical Tfor π superfluidity disappearing, reflecting that π con-densate is suppressed by the strong magnetic field for µ I = 100 M eV . This result is self-consistent with whatwe have obtained in Fig. 2.In the lower panel, the σ condensate rises slowlywith increasing temperature. The above phenomenononly happens in the pion superfluidity phase.In the high T region the π = 0, the σ condensation decreasesmonotonously as the T increases, meaning the chiral sym-metry will restore in this region. When magnetic field is Μ I = eB=0eB=30 m Π eB=20 m Π H MeV L Π (cid:144) Σ FIG. 5: The Pion condensate π , shown as a function of thetemperature with respect to various magnetic field magnitudeat fixed isospin chemical potential µ I = 400 MeV . included, the varying behaviors of the chiral condensa-tion are similar, but the chiral condensate increases with eB . That is to say that the magnetic field can improvethe σ condensate, which is of course the chiral magneticcatalysis effect again.The influence of the external magnetic field on the pioncondensation is different for different µ I . Fig. 5 showsthe change tendency of the pion condensation with µ I =400 M eV .For fixed eB , the π condensate decreases monotonouslywith increasing temperature, that coincides with the casewhen µ I = 100 M eV . This implies that the tempera-ture always depress the pion condensation. Taking themagnetic field into consideration, the greater the eB , thewider range of temperature domain of the π = 0 andthe larger the critical temperature for π superfluiditytransition,reflecting that π condensate is promoted withthe strong magnetic field for µ I = 400 M eV . Compar-ing the varying behaviors of the pion condensation for µ I = 100 M eV with that for µ I = 400 M eV , we proveagain the influence of the external magnetic field on thepion condensation is not simple promotion or suppres-sion. The effect of the external magnetic field on the pionsuperfluid is related to the value of the isospin chemicalpotential.
B. the pion superfluidity phase diagram
From the change in the π and σ order parameter asa function of T, eB, and µ I , we can calculate both thepion superfluidity and the chiral phase transition phasediagram. We confine our analysis to the pion phase dia-gram in this paper. Here we define the transition happenswhen the π condensation goes to zero, namely the pionsuperfluidity phase transform to the normal phase.In the upper panel of Fig. 6 we plot the pion super-fluidity phase diagram in T − µ I plane at eB = 0. Theregion inside of the curve with π = 0 is superfluid phase,and the region outside of the curve with π = 0 is normal eB=0 Π¹ Π =0 Μ I H MeV L T H M e V L T = = m Π
400 500 600 700 800 - - - - - - - Μ I H MeV L W H (cid:144) f m L FIG. 6:
Upper panel:
The Pion superfluidity phase diagramin T − µ I plane at eB = 0. Lower panel:
The thermodynamicpotential as a function of µ I at eB = 0 and T = 60 MeV . phase. When µ I is small, the phase transition is secondorder, and then it becomes first order at large µ I . Thepoint connects the first and the second order phase tran-sition lines is the so-called tricritical point. At eB = 0,this point is approximately at ( T, µ ) = (456 , µ I when the pioncondensation exists, and on the lower dashed line the µ I is the point where the pion condensate disappears. Thethick dashed line in the middle is the so-called Maxwellline. The points on the Maxwell line meet the conditionof phase equilibrium of the first order phase transitionwith p = p , µ I = µ I and T = T .The lower panel of Fig. 6 clearly shows how we de-termine this Maxwell line. We plot the behavior of thethermodynamic potential Ω(= − P ) with varying µ I , forexample, for T = 60 M eV and eB = 0. From the crosspoint on the figure we obtain the value of µ I at this tem-perature. By changing the temperature and doing thesame calculation, we then obtain the middle dashed lineon the upper panel of Fig. 6.Fig. 7 show the pion superfluidity phase diagram ofour model including the strong magnetic field effect. Inthe upper panel, we show the pion superfluidity phasediagram in T − µ I plane at different magnetic field, eB =0, 20 m π , and 30 m π , respectively. The regions outside ofthe curves are in normal phase. The regions inside the eB=30 m Π eB=20 m Π eB=0 Π¹ Π =0 Μ I H MeV L T H M e V L Π¹ Π =0 T = T = T = Μ I H MeV L e B (cid:144) m Π Μ I = Μ I = Μ I = Π ¹ Π = (cid:144) m Π T H M e V L FIG. 7:
Upper panel:
The pion superfluidity phase diagramin T − µ I plane. Middle panel:
The pion superfluidity phasediagram in eB − µ I plane. Lower panel:
The pion superfluidityphase diagram in T − eB plane. curves with lower T are the π superfluidity phases.For eB = 0 case, when µ I is larger than µ cI (about m π / π superfluidity phase forms. The pion super-fluidity vanishes with increasing T . When the externalmagnetic fields are included, such as eB = 20 m π and30 m π , the regions of π = 0 shrink as the eB increases for µ I < M eV , which can be verified by the upper panelin Fig.4. When µ I is very large, for µ I > M eV , theregions of π = 0 become larger as the eB increases, thisphenomenon is coincide with the lower panel in Fig.4. Allin all, the region of π = 0 is nonmonotonously affectedby the external eB . From our calculation, the tricriticalpoint moves to the space with smaller µ I and higher T when the external magnetic field becomes stronger.The middle panel shows us the pion superfluidity phasediagram in eB − µ I plane at different values of temper- ature, T = 0, 150 M eV , and 190
M eV , respectively. Theregions in the left-side of the curves are the normal phasesand the right-side of the curves with higher µ I are thepion superfluidity phase. The regions of π = 0 decreasewith increasing T because the temperature will suppressthe formation of the π superfluidity.In the upper panel, we can roughly conclude thatthe critical µ cI where the pion superfluidity occurs in-creases as the eB increases. If we investigate it in detail,shown in the middle panel, the µ cI actually does not risemonotonously but oscillatorily as the eB increases whenthe magnetic field is not so strong. This oscillation is re-lated to the splitting quark energy level in the surround-ing of the magnetic field. For eB > m π , the π super-fluid phase boundary becomes monotonous and smoothwith the increasing µ I . This is because the Lower Lan-dau Level (LLL) chiefly contributes to the behaviors ofphase transition boundaries for large eB .In the lower panel we show the pion superfluidityphase diagram in T − eB plane for different values ofthe isospin chemical potential, µ I = 80 M eV , 100
M eV ,and 150
M eV , respectively. The regions inside the curveswith low eB and low T are the π superfluidity phases.The larger the µ I , the wider the region of π = 0. Thismeans that the π superfluid is enhanced with increasing µ I . Here we still obtain the oscillated phase boundaryfor the pion superfluid phase and this results to a in-teresting phenomenon with the increasing eB . Taking µ I = 150 M eV and T = 185 M eV for an example, whenincreasing the eB the pion superfluidity appears occa-sionally. This phenomenon of the superfluid phase carryon alternatively. The oscillatory behavior of the pion su-perfluid phase boundary is also a typical subsequence ofthe magnetic field effect. This result is similar with theconclusion of the middle panel in this figure. IV. CONCLUSION AND DISCUSSION
The effect of the external magnetic field on the pioncondensate and chiral condensate is investigated in thetwo flavor NJL model at finite temperature and isospinpotential. The chiral and pion condensation are calcu-lated in the surrounding of the various magnetic field.The oscillating behavior of the chiral and pion condensa-tion, related to the so-called Alfven-de Haas oscillation,is shown by the numerical result. The phase boundary ofthe pion superfluid are also investigated with the consid-eration of the magnetic field. An oscillatory phase bound-ary is found with increasing the magnetic field. The firstorder phase transition in the high isospin chemical po-tential region still exists when the magnetic field effectis taken into account. With the magnetic field consid-ered, the tricritical point on the phase diagram of the T − µ I plane moves to the high T and low µ I region.Because of the discrete of Landau Level, the influence ofthe external magnetic field on the pion condensation andthe phase boundary is not simple promotion or suppres-sion, which is different from the effect of the temperatureand the isospin chemical potential. The σ condensate isenhanced with increasing magnetic field, which is sup-ported by the chiral magnetic catalysis effect. However,because of the oscillation of the chiral condensate, there exist some magnetic field region where the chiral conden-sation would decrease with the increasing eB . Acknowledgement: