Inverse Magnetic Catalysis in the three-flavor NJL model with axial-vector interaction
IInverse Magnetic Catalysis in the three-flavor NJL model with axial-vector interaction
Lang Yu , Jos Van Doorsselaere , and Mei Huang , Institute of High Energy Physics, Chinese Academy of Sciences, Beijing 100049, China Laboratoire de Mathemathique et Physique Theorique, Universite de Tours, 37000 Tours, France and Theoretical Physics Center for Science Facilities,Chinese Academy of Sciences, Beijing 100049, China (Dated: September 8, 2018)In this paper we explore the chiral phase transition in QCD within the three-flavor Nambu-Jona-Lasinio (NJL) model with a negative coupling constant in the isoscalar axial-vector channel,which is associated with a polarized instanton–anti-instanton molecule background. The QCD phasediagram described in this scenario shows a new first order phase transition around the transitiontemperature T c toward a phase without chiral condensates, but with nontrivial dynamic chiralchemical potentials for the light quarks, spontaneously giving rise to local CP violation and localchirality imbalance. The corresponding critical temperature T c for this phase transition decreaseswith the magnetic field and it gives a natural explanation to the inverse magnetic catalysis effectfor light quarks when incorporating a reasonable value of the coupling constant in the isoscalaraxial-vector channel. Furthermore, when the isoscalar axial-vector interaction is dominant in lightquark sector and suppressed in strange quark sector, it is found that there is no inverse magneticcatalysis for strange quark condensate, which agrees with lattice results. PACS numbers: 12.38.Aw,12.38.Mh
I. INTRODUCTION
Investigation of the QCD phase structure in the pres-ence of strong external magnetic fields has become amajor topic in both theoretical and experimental re-search into the physics of strongly interacting matter.This topic is of paramount importance to understand thephenomenology of noncentral heavy ion collisions at theRelativistic Heavy Ion Collider (RHIC) and the LargeHadron Collider (LHC), in which a strong magnetic fieldreaching up to √ eB ∼ (0 . − .
0) GeV [1–4] can begenerated. In addition, strong magnetic fields could alsohave existed in the strong and electroweak phase tran-sition [5, 6] of the early Universe, and exist in compactstars like magnetars [7].The breaking and restoration of chiral symmetry,which is described by the behavior of the quark con-densate, is one of the most intriguing nonperturbativeaspects of QCD. Therefore, it is of great interest to spec-ulate the effect of magnetic fields on the behavior of thechiral condensate in QCD at zero and finite tempera-tures. Since the 1990’s, a related phenomenon known asmagnetic catalysis has been recognized [8–11]. It refersto an enhancement of the quark condensate and thus anincrease of the chiral transition temperature T c underthe magnetic field. This is agreed by most of earlier low-energy effective models and approximations to QCD [8–10, 12–23] as well as lattice QCD simulations [24–28]in the past twenty years. However, recently, a latticegroup [29, 30] revealed surprising results that the tran-sition temperature T c decreases as a function of the ex-ternal magnetic field, and the chiral condensate showsa nonmonotonic behavior as a function of the externalmagnetic field in the crossover region. This prediction isin contrast to the majority of previous calculations, and the partly decreasing behavior of the chiral condensatewith the increasing B near T c , causing a decreasing de-pendence of T c on B , is called inverse magnetic catalysis.There are several recent studies [31–41] discussing theorigin of the phenomenon of the decreasing behavior ofthe chiral critical temperature with increasing B and theinverse magnetic catalysis around T c . For example, themagnetic inhibition [31], the mass gap in the large N c limit [32], the contribution of sea quarks [33], a run-ning scalar coupling parameter dependent on the mag-netic field intensity [36, 37] and an antiscreening effect ofthe color charges for quarks [40, 42], etc are proposed tounderstand this puzzle. Particularly, a very natural andcompetitive mechanism attributes the inverse magneticcatalysis to the local chirality imbalance induced by thenontrivial topological gluon configuration, arising from asphaleron transition [34] or the instanton–anti-instantonmolecule pairing [38].As discussed in Ref. [38], the chirality imbalance, whichis associated with the violation of the P and CP sym-metry, is induced by the nonzero topological charge Q T through the axial anomaly of QCD∆ N = (cid:90) d x∂ µ j µ = − N f Q T , (1)where N f is the number of flavors, ∆ N = N ( t =+ ∞ ) − N ( t = −∞ ), with N = N R − N L denoting thenumber difference between right- and left-hand quarks,and j µ = ¯ ψγ µ γ ψ denotes the isospin singlet axial vec-tor current. A consequence of this violation is the exis-tence of two kinds of local domains with the same quan-tum numbers but opposite net topological charges, whichwill lead to the generation of local chirality imbalancesbut zero average chirality, as well as the local P and CP violation. The modification of the QCD phase dia-gram by the chirality imbalance has been studied in some a r X i v : . [ h e p - ph ] D ec Refs. [43–45]. Especially, the recent observation of chargeazimuthal correlations at RHIC and LHC [46–48] may beresulting from the chiral magnetic effect (CME) with lo-cal P and CP violation, which is an interesting combinedeffect of both the strong magnetic field and the non-trivial topological gluon configuration of the quark-gluonplasma. Based on above discussions, the enhancementof the chirality imbalance by magnetic fields, which co-incides with the lattice results in Ref. [49], will naturallyresult in a decreasing chiral critical temperature, since itdestroys the pairing between the left-handed quark (an-tiquark) and the right-handed antiquark (quark).In Ref. [38], a mechanism has been presented to gen-erate local chirality imbalance based on the instanton–anti-instanton ( I ¯ I ) molecule picture [50–54], which isregarded as one effective mechanism responsible for non-perturbative properties of QCD in the region T (cid:39) T c − T c [52–54]. By using an unconventional repulsive iso-scalar axial-vector interaction in a two flavor Nambu-Jona-Lasinio (NJL) model [52], we find that a dynami-cal chiral chemical potential related to the local chiralityimbalance is induced spontaneously at the temperaturesnear T c . It is also found that the increasing magneticfield helps to lower the critical temperature due to theappearance of the local chirality. Moreover, since the lo-cal chirality imbalance can only be produced around T c ,it gives a reasonable explanation for why inverse mag-netic catalysis only appears at the temperatures around T c , while magnetic catalysis still occurs at zero and lowtemperatures.However, the lattice result in Ref.[30] shows that thestrange quark condensate does not exhibit inverse mag-netic catalysis, but simply increases with magnetic fieldat all temperatures. In this paper, we extend our analysisand calculations to the 2+1 flavors, and investigate thecorresponding effects of the axial-vector interaction onboth the local chirality imbalance and the chiral criticaltemperatures for u, d and s quarks. The paper is orga-nized as follows. In Sec. II, we give a general descriptionof the 2+1 flavor NJL model and formalism with con-sidering the repulsive axial-vector interactions stemmingfrom the interacting I ¯ I molecule model (IIMM). In Sec.III, we will discuss the main results of the numerical cal-culations. Finally, our conclusions and perspectives arepresented In Sec. IV. II. MODEL AND FORMALISM
In this section, we present the three flavor NJLmodel [55–64] with adding the vector and axial-vectorinteraction terms. The Lagrangian density of our modelin the presence of an external magnetic field is given by L = ¯ ψ ( iγ µ D µ − ˆ m ) ψ + L sym + L det − L V A , (2)where ψ = ( u, d, s ) T corresponds to the quark field ofthree flavors, and ˆ m = diag( m u , m d , m s ) is the corre-sponding current mass matrix. The covariant derivative, D µ = ∂ µ − iq f A µ , couples quarks to an external magneticfield B = (0 , , B ) along the positive z direction, via abackground Abelian gauge field A µ = (0 , , Bx, q f is defined as the electric charge of the quark field withflavor f . L sym and L det are given by L sym = G S (cid:88) a =0 (cid:104)(cid:0) ¯ ψλ a ψ (cid:1) + (cid:0) ¯ ψλ a iγ ψ (cid:1) (cid:105) , (3) L det = − K (cid:8) det (cid:2) ¯ ψ (1 + γ ) ψ (cid:3) + det (cid:2) ¯ ψ (1 − γ ) ψ (cid:3)(cid:9) , (4)where λ a are the Gell-Mann matrices in flavor space( λ = (cid:112) / I ) and the determinant is in flavor space also.The L sym term corresponds to the usual four-fermion in-teractions of scalar and pseudoscalar channels which re-spect SU (3) V ⊗ SU (3) A ⊗ U (1) V ⊗ U (1) A symmetry. The L det term corresponds to the ’t Hooft six-fermion deter-minant interactions [65] which break U (1) A symmetry.As for the L V A term, it represents the four-fermion in-teractions of vector and axial-vector channels under theinvariance of SU (3) V ⊗ SU (3) A ⊗ U (1) V ⊗ U (1) A sym-metry. The last two terms are added to introduce chiralinteractions, which are chosen such that they correspondto effective chirally asymmetric interactions with an in-stanton background.The connection between instantons and chiral symme-try breaking is well known. Essentially it is a consequenceof an index theorem that shows how a topologically non-trivial gauge configuration –the instanton– gives rise toan asymmetry in occupation of left- and right chiraleigenmodes, observed through the existence of a fermioncondensate. A feature of this index theorem is that allnon-trivial physics happens in the zero-mode space. In asense, the price we pay for those modes not to contributeto the action, is the violation of chiral symmetry. At zeroas well as low temperatures, the ’t Hooft interaction isdominant, the random instantons play an important rolein chiral symmetry breaking. However, at high tempera-tures and near the chiral phase transition, the instantonsare no longer random, but become correlated. Therefore,it was suggested in Refs. [50–52] that the growing cor-relations between instantons and anti-instantons near T c will lead to the decrease of random instantons but the in-crease of instanton–anti-instanton molecule pairs. Thismeans that the random instantons and anti-instantonsare not annihilated but paired up into the correlated I ¯ I molecules when chiral phase transition happens. Asshown in [52], in the temperature region of T (cid:38) T c , the I ¯ I molecules pairing induces a repulsive effective localfour-quark interactions in the isoscalar axial-vector chan-nel. This unconventional repulsive axial-vector interac-tion leads to a repulsive axial-vector mean field in thespace-like components but an attractive one in the time-like components, which naturally induces a spontaneouslocal CP violation and local chirality imbalance as shownin [38].In Refs. [52] and [38], the instanton background onlycouples to light u,d quarks, and the L V A in the isoscalarvector and axial-vector takes the form of L u,dV A = (cid:88) f = u,d (cid:104) G V (cid:0) ¯ ψ f γ µ ψ f (cid:1) + G A (cid:0) ¯ ψ f γ µ γ ψ f (cid:1) (cid:105) . (5)We may generalize this term to three-flavor case as fol-lowing: L u,d,sV A = (cid:88) f = u,d,s (cid:104) G V (cid:0) ¯ ψ f γ µ ψ f (cid:1) + G A (cid:0) ¯ ψ f γ µ γ ψ f (cid:1) (cid:105) . (6)In reality, considering the strange quark mass is heav-ier than light u,d quarks, the isoscalar vector andaxial-vector interaction induced by instanton backgroundmight take the form in between Eq.(5) and Eq.(6). Forexample, the quark propagator in a single instanton back-ground is associated with the quark zero-modes [65] ψ ± ( x ) = ρπ ± γ ( r + ρ ) / / rr U, , (7)where the superscript ± corresponds to an (anti-) in-stanton centered at x and with size ρ . The spin-colormatrix U satisfies ( (cid:126)σ + (cid:126)τ ) U = 0 and r = x − x . Thezero mode contributions will enter into the calculation ofthe correlators through the leading term in the spectralrepresentation of the quark background field propagator S ± q ( x, y ) = ψ ± ( x ) ψ ± † ( y ) m ∗ q ( ρ ) + O ( ρm ∗ q ) . (8)Here, the flavor dependent effective quark mass m ∗ q ( ρ ) = m q − π ρ (cid:104) qq (cid:105) (where q stands for up, down and strangequarks) in the denominator is generated by interactionswith long-wavelength QCD vacuum fields as shown inRef. [66]. It can be seen that the strange quark propaga-tor in the instanton background is suppressed by 1 /m ∗ s .Extending this to the instanton–anti-instanton moleculebackground, we can assume the actual interaction in theisoscalar vector and axial-vector channel can be writtenas: L V A = (cid:88) f = u,d (cid:104) G V (cid:0) ¯ ψ f γ µ ψ f (cid:1) + G A (cid:0) ¯ ψ f γ µ γ ψ f (cid:1) (cid:105) + (cid:104) G (cid:48) V (¯ sγ µ s ) + G (cid:48) A (cid:0) ¯ sγ µ γ s (cid:1) (cid:105) . (9)with G (cid:48) V (cid:28) G V and G (cid:48) A (cid:28) G A ( G (cid:48) V /G V or G (cid:48) A /G A ∼ ( m u /m s ) in the chirally symmetric phase by Eq. 8,where m u and m s represent the current masses of u and squarks, respectively). A compelling consequence of thisargument is that the L V A -term, which is the essentialsource of the inverse magnetic catalysis in our approach,distinguishes between the isoscalar channel with lightquarks and the one with strange quarks. It was foundon the lattice QCD [30] that the inverse magnetic catal-ysis effect appears only for light quarks not for strange quarks, an implicit feature in our approach. In the fol-lowing calculations, we take two cases for the isoscalarvector and axial-vector interaction:Case I : take L u,dV A (10)Case II : take L u,d,sV A . (11)Note that it has been discussed in Ref. [38] that G S and G V , are expected to be positive at the whole temperatureregion [59, 60, 62, 63] and are assumed to be keeping theconstants fixed by the mesonic properties in QCD vac-uum for simplicity, whereas G A is expected to be positiveat zero and low temperatures [59, 60] and to be negativeat the temperatures above T c as a result of the interact-ing instanton–anti-instanton molecule model [50–52]. Ithas been shown in Ref. [52] that a negative G A interac-tion induced by the correlated instanton–anti-instantonmolecule pairs near T c , will give rise to nontrivial influ-ence on the chiral phase transition. Therefore, we willtreat G A as a free parameter in the NJL model of SU(3)flavor version as we did in Ref. [38] with two light flavors.Working at the mean field level, one gets the thermo-dynamical potential per unit volume Ω by integrating outthe quark fields ψ of the Lagrangian density of Eq. (2),Ω = 14 G S (cid:88) f = u,d,s σ f + K G S σ u σ d σ s − ˜ µ G A + (cid:88) f = u,d,s Ω f , (12)where σ f = − G S (cid:104) ¯ ψ f ψ f (cid:105) ( f = u, d, s ) and˜ µ = − G A (cid:88) f = u,d (cid:104) ¯ ψ f γ γ ψ f (cid:105) , for Case I , ˜ µ = − G A (cid:88) f = u,d,s (cid:104) ¯ ψ f γ γ ψ f (cid:105) , for Case II . (13)The contributions from the fermion loop for each flavoris given byΩ f = − N c | q f B | π (cid:88) s z ,k α s z k × (cid:20) (cid:90) ∞−∞ dp z π f ( p ) ω fk ( p )+2 T ln (cid:0) e − ω fk /T (cid:1)(cid:21) , (14)where ω fk = (cid:113) M f + (cid:2) | p | + s z ˜ µ sgn( p z ) (cid:3) for f = u, d (cid:113) M f + | p | for f = s, (15)are dispersion relation for the thermal eigenfrequencieswith spin factors s z = ± ω fk = (cid:113) M f + (cid:2) | p | + s z ˜ µ sgn( p z ) (cid:3) f = u, d, s, (16)are dispersion relation for the thermal eigenfrequencieswith spin factors s z = ± M u = m u + σ u + K G S σ s σ d ,M d = m d + σ d + K G S σ s σ u ,M s = m s + σ s + K G S σ u σ d . (17)The 3-momentum p in a magnetic field is given by p = p z + 2 | q f B | k , (18)and k = 0 , , , . . . is a non-negative integer number la-beling the Landau levels. The spin degeneracy factor isexpresses as α s z k = δ s z , +1 for k = 0 , qB > ,δ s z , − for k = 0 , qB < , k (cid:54) = 0 . (19)Following Ref. [43] we use a smooth regularization formfactor f Λ ( p ) = (cid:115) Λ N Λ N + | p | N , (20)where we take N = 5. Now, by making use of Eq. (12), σ f and ˜ µ can be determined self-consistently as solutionsto the saddle point equations ∂ Ω ∂σ f = ∂ Ω ∂ ˜ µ = 0 . (21)Numerically one can obtain these solutions by a minimi-sation and moreover find the true vacuum by looking atthe global minimum. This will prove essential for ourmodel which has multiple minima breaking chiral sym-metry spontaneously.The parameters of our model, the cutoff Λ, the cou-pling constants G S and K , and the current quark masses m u = m d and m s are determined by fitting f π , m π , m K and m η (cid:48) to their empirical values by using the smoothregularization method. We obtain Λ = 604 . m u = m d = 5 . m s = 133 . G S Λ = 3 .
250 and K Λ = 10 . G A will be treated as a free parameter, and we willperform our calculations over a limited range of ratios r A = G A /G S . (22)It is discussed in Ref. [38] that when r A (cid:62) r A < III. NUMERICAL RESULTS AND DISCUSSION
Considering the interaction in strange quark channelis suppressed, for our numerical calculations, we mostly (a)(b)(c)FIG. 1. (color online). The quark condensate σ f (f=u,d,s)and dynamical chiral chemical potential ˜ µ as a function of T at r A = 0 for several different values of eB . (a) σ f and ˜ µ for eB = 0 . GeV at r A = 0. (b) σ f and ˜ µ for eB = 0 . GeV at r A = 0. (c) σ f and ˜ µ for eB = 0 . GeV at r A = 0. take the isoscalar axial-vector interaction of Case I, i.e.take the form in Eq.(5).We study the chiral phase transition at finite tem-perature by using Eq. (21) for several different valuesof the parameters r A and eB , representing different in-tensity for the instanton–anti-instanton molecule back-ground and the magnetic field background, respectively.Consequently, the diagrams of σ − T for different quarkflavors as well as the diagrams of ˜ µ − T can be obtained,which allow us to efficiently find the transition tempera- (a)(b)(c)FIG. 2. A 2D minimal surface of the potential Ω for Case I asa function of σ d and ˜ µ at r A = − . eB = 0 . forseveral different values of temperature. (a) Ω at r A = − . eB = 0 . for T = 100 MeV (below T c ). (b) Ω at r A = − . eB = 0 . for T = 150 MeV (around T c ). (c) Ω at r A = − . eB = 0 . for T = 200MeV (above T c ). tures as functions of the parameters r A and eB . Actually,each of the condensates has thus its own specific transi-tion temperature, which is defined by the temperature atthe inflection point of the σ − T diagram for each flavor,i.e., the maximum point of the quantity ∂σ/∂T . Here wewill use σ u + σ d to determine the transition temperature T c for the chiral phase transition of QCD.In Fig. 1, we display the quark condensates of σ u , σ d and σ s as well as dynamical chiral chemical potential ˜ µ as functions of T for several different values of eB withoutconsidering additional axial-vector couplings, i.e., r A = 0and ˜ µ ≡
0, and the ordinary magnetic catalysis effectcan be exactly found from the plots. In fact, even ifwe choose positive values for the paramter r A , fitted bythe conventional SU(3)-flavor NJL model, we will acquirethe same results. This is because of the fact that when r A >
0, the potential energy density Ω can only has alocal maximum at nonzero ˜ µ , which forces (cid:104) ¯ ψγ γ ψ (cid:105) tobe zero so that ˜ µ = 0 accordingly.Next, as discussed in Ref. [38], by switching on anegative r A , we introduce a non-trivial dependence ofthe thermodynamical potential Ω on the dynamic chiralchemical potential ˜ µ , which adds an extra dimension tothe mean field surface of Ω. As is shown in Fig. 2, thereare two local minima for the potential Ω. When the tem-perature is low, the original local minimum representingnonzero quark condensates is dominant, since it is theglobal minimum. However, sufficient heating of the QCDsystem makes the local minimum for non-trivial ˜ µ en-ergetically more favourable while the chiral condensatesis weakened to the trivial state. As a result, the vacuumtunnels to a state with local chirality imbalance betweenright- and left-handed quarks as this metastable state be-comes the new global minimum. It is important to pointout that the potential Ω in Eq. (12) is even in ˜ µ so onecan get separated local domains with chiral densities ofboth signs. Moreover, another important consequence ofthe competition between these two local minima is thatno ‘mixed’ state appears, so one has either nonzero quarkcondensates but no chirality imbalance or an instabilitytowards the formation of nonzero dynamic chiral chem-ical potential but no presence of condensates. This isclear from the fact that the minima in Fig. 2 appear oneither of the axes and is a consistent feature of all oursimulations.In fact, the magnitude of the unconventional negative G A , which leads to an attractive mean field in the time-like components of the axial-vector channel, reflects thecoupling strength of the attraction for ˜ µ . In Figs. 3, 4and 5, we compare our numerical results of quark conden-sates σ f ( f = u, d, and s) and dynamic chiral chemicalpotential ˜ µ as functions of T for several magnetic fieldsat r A = − . − . − .
7. One can distinguish twodistinct cases as a result of the magnitude of negative r A : (i) T c ( eB = 0; r A = 0) < T c ( eB = 0) and (ii) T c ( eB = 0; r A = 0) > T c ( eB = 0).If magnitude of G A is small, approximately − . 3. (b) σ f and ˜ µ for eB = 0 . GeV at r A = − . 3. (c) σ f and ˜ µ for eB = 0 . GeV at r A = − . the case (i), i.e. T c > T c ( r A = 0) at eB = 0. Forexample, as shown by Fig. 3 of r A = − . r A = − . 5, when the external magnetic field is not strongenough, the ordinary phase transition into the chirally re-stored phase takes place at a lower temperature and isthe dominant effect in destroying the quark condensates;whereas a local CP -odd first order phase transition for˜ µ is spontaneously generated at a higher critical tem-perature T c > T c . As the magnetic field grows, both (a)(b)(c)FIG. 4. (color online). For Case I, the quark condensate σ f (f=u,d,s) and dynamical chiral chemical potential ˜ µ asa function of T at r A = − . eB . (a) σ f and ˜ µ for eB = 0 . GeV at r A = − . 5. (b) σ f and ˜ µ for eB = 0 . GeV at r A = − . 5. (c) σ f and ˜ µ for eB = 0 . GeV at r A = − . critical temperatures, T c and T c , approach each otherand two local minima in the thermodynamic potential Ωco-exist like in the example of Fig. 2. At some criticalvalue of magnetic field B c for a given r A , these two criti-cal temperatures meet with each other and the first ordertransition for nonzero dynamical chiral chemical poten-tial ˜ µ becomes dominant effect, which makes σ u and σ d drop to zero at T c = T c . Therefore, we find that, for uand d quarks in the case (i), the critical temperature T c (a)(b)(c)FIG. 5. (color online). For Case I, the quark condensate σ f (f=u,d,s) and dynamical chiral chemical potential ˜ µ asa function of T at r A = − . eB . (a) σ f and ˜ µ for eB = 0 . GeV at r A = − . 7. (b) σ f and ˜ µ for eB = 0 . GeV at r A = − . 7. (c) σ f and ˜ µ for eB = 0 . GeV at r A = − . decreases with eB , while T c increases at first and thendecreases as the magnetic field grows (see Fig. 7). Asfor s quark condensate, it shows a slight jump becauseof ˜ µ background and then continues to dissolve with in-creasing temperature. The critical temperature T c ( σ s )will increase with eB always, depicted by Fig. 6, whichis consistent with the lattice results in Ref. [30] in somesense.If magnitude of G A is large enough ( r A < − . (a)(b)(c)FIG. 6. (color online). For Case I, the quark condensates σ s , σ u and σ d as a function of T at r A = − . eB . (a) σ s at r A = − . eB . (b) σ u at r A = − . eB .(c) σ d at r A = − . eB . in the case (ii), the light quark condensates are destroyedat T c = T c because the QCD ground state with chiral-ity imbalanced density becomes more favorable aroundthe critical temperature for any values of eB , before σ u and σ d are dissolved at their original critical tempera-ture T c without considering ˜ µ , e.g., shown by Fig. 5 at r A = − . 7. Hence, the critical temperatures both T c (a)(b)(c)FIG. 7. (a) The critical temperature T c as a function of eB for several different values of r A , comparing with T c as afunction of eB at r A = 0. (b) The critical temperature T c asa function of eB for several different values of r A . (c) Thecritical temperature T c ( σ s ) as a function of eB for severaldifferent values of r A . The results are for Case I. and T c for u and d quark condensates decreases with eB starting from eB = 0, which is just the decreasing T c dependence on B predicted by Ref. [29]. On the otherhand, the condensates σ u and σ d increase with the mag-netic field at zero and low temperatures still, which is theordinary magnetic catalysis effect validated in Ref. [30].The behavior of both strange quark condensate σ s andthe critical temperature T c ( σ s ) is similar to that in thecase (i).We can now put the data about the critical temper-atures obtained from Figs. 3, 4 and 5 into one T c − eB phase diagram of QCD. Adding more data points fromidentical simulations with other values of the backgroundparameters r A and eB , we can find that the middle dia-gram of Fig. 7 shows two different possible types of de-pendence of T c on the magnetic field as a result of the freeparameter r A , which bas been discussed explicitly above.As a consequence, a reasonable strength of r A , approxi-mately between − . − . 55 by the simulations of ourmodel, will naturally explain the decreasing dependenceof T c on eB obtained in a recent lattice QCD study [29].If the magnitude of r A is too small, less than 0 . 5, we cannot find a monotonously decreasing dependence of T c on eB ; If the magnitude of r A is too big, more than 0 . 55, thevalue of T c at eB = 0 will deviate from the lattice QCDresult greatly. Furthermore, we can distinguish case (i)and case (ii) easily by the T c ( B ) function from the topdiagram of Fig. 7, the separation given by the thick blackline for the critical temperature T c at r A = 0. As for thecritical temperature T c ( σ s ), depicted by the bottom dia-gram of Fig. 7, one can find that its behavior at differentnegative values of r A is similar to that at r A = 0, showinga slightly increasing dependence on eB .The above calculations are based on Case I, where theisoscalar axial-vector interaction only involves light u,dquarks. The isoscalar nature of the interaction L V A is es-sential for the nature of the phase transition. With littleextra effort we were able to simulate the Case II where thefour-fermion chiral attraction treats all flavors equally asgiven by Eq.(6). It can be seen that the results of σ u and σ d are very similar to what we found before, but ratherthan a small shift in the value of the heavy strange quarkcondensate, σ s undergoes the same first order phase tran-sition as the other two light flavors and vanishes at thetransition temperature T c , as shown in Fig. 8. As weargued in the previous section, this kind of equal cou-pling with negative G A to all three quark flavors is not tobe expected for an axial-vector coupling induced by aninstanton–anti-instanton molecule background, and un-surprisingly it does not reproduce the qualitative latticeresult.Before drawing our final conclusions, it is important torealize we can only trust our results qualitatively. Sincethe new minimum and the corresponding phase transitionshown in Fig.2 are in fact at a scale well beyond the cut-off of our theory, exact quantitative prediction are beyondthe scope of the NJL framework. Qualitatively, however,we can be sure that the instability will emerge, and a newvacuum state will appear that is more favored than thechiral condensate when increasing magnetic fields around T c and thus give rise to inverse magnetic catalysis ef-fect. In that sense we think that our model is a goodrepresentation of the effect of instanton–anti-instantonmolecule background on the chiral condensates, but wecannot produce accurate predictions for the large chiraldensities involved. FIG. 8. For Case II, the quark condensate σ f (f=u,d,s) anddynamical chiral chemical potential ˜ µ as a function of T at r A = − . eB = 0 . when introducing a negativeaxial-vector coupling to all three flavor quarks equally. IV. CONCLUSIONS In this paper we extend our study to the QCD phasediagram as well as the behavior of quark condensatesat finite temperature under an external magnetic fieldwithin the three-flavor NJL model including additionalisoscalar vector and axial-vector channels. Note that animportant and unconventional feature of our model is theisoscalar axial-vector interaction with a negative couplingconstant depending mostly on the up and down quarks,while the interaction in the strange quark sector is sup-pressed due to its heavier mass, which can be derivedfrom the instanton–anti-instanton molecule model [52].In this scenario, we have shown that a new way of de-stroying chiral condensates appears around T c , replacingthem by dynamical chiral chemical potential ˜ µ in a firstorder phase transition, which corresponds to a sponta-neous generation of local P and CP violation and localchirality imbalance. Moreover, the critical temperatureof this first order phase transition shows inverse mag-netic catalysis, meaning that it decreases with increasingmagnetic field.The dominant features of the phase transition with re-spect to destroying the light quark condensates dependson the parameters of the model, a tunable axial-vectorcoupling constant G A and the background magnetic field B . When increasing the magnitude of G A , we can findthat it will decrease the critical temperature for the firstorder phase transition of ˜ µ . And the increase of the magnetic field at a given G A will also decrease the criti-cal temperature T c . It means that, in the generic case,the increase of the magnitude of both parameters, G A and eB , will catalyze the appearance of the local chiralityimbalance. Therefore, a reasonable value of G A , makingthe ordinary chiral phase transition meet with the newlyfound first order phase transition at eB = 0 (however,this is not in agreement with previous lattice results atfinite temperature, and possible reasons are discussed inRef. [38]), the inverse magnetic catalysis effect can benaturally explained and a phase diagram is reproducedin Fig. 7, consistent with lattice QCD results [29].On the other hand, since the lattice results of Ref. [30]indicated that strange quark condensate experience mag-netic catalysis only, we investigate the behavior of strangequark condensate in our model also. When we intro-duce a negative axial-vector interaction channel includ-ing light-quark currents only, it is found that the criticaltemperature T c ( σ s ) exhibits little modification as a resultof the instanton–anti-instanton molecule background, insome sense consistent with lattice results, although thestrange condensate shows a slight jump arising from theappearance of ˜ µ in the σ − T diagrams. This mightbe improved by considering the spatial structure of thetopological density distribution, which is one of our fu-ture plans. In reality, the interaction in the isoscalarvector and axial-vector channel might look like Eq.(9),with a dominant isoscalar axial-vector interaction in thelight quark sector and with a suppressed interaction inthe strange quark sector. In this case, if the interactionin the strange sector is small enough, the result of chiralphase transitions will be the same as that in Case I, andwe can get the inverse magnetic catalysis for light quarkcondensate but magnetic catalysis for strange quark con-densate around critical temperature, which is in agree-ment with lattice results. ACKNOWLEDGEMENT We thank valuable discussions with M. Chernodub,J.Y. Chao and I. Shovkovy. This work is sup-ported by the NSFC under Grant No. 11275213, and11261130311(CRC 110 by DFG and NSFC), CAS keyproject KJCX2-EW-N01, and Youth Innovation Promo-tion Association of CAS. L.Yu is partially supported byChina Postdoctoral Science Foundation under Grant No.2014M550841. The work of JVD was supported by agrant from La Region Centre (France) and the Chinese-French Cai Yuanpei 2013 grant. [1] V. Skokov, A. Y. .Illarionov and V. Toneev, Int. J. Mod.Phys. A , 5925 (2009) [arXiv:0907.1396 [nucl-th]].[2] V. Voronyuk, V. D. Toneev, W. Cassing,E. L. 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