Inverse magnetic catalysis in the Polyakov-Nambu-Jona-Lasinio and entangled Polyakov-Nambu-Jona-Lasinio models
aa r X i v : . [ h e p - ph ] D ec Inverse magnetic catalysis in thePolyakov–Nambu–Jona-Lasinio and entangledPolyakov–Nambu–Jona-Lasinio models ∗ Constanc¸a Providˆencia, M´arcio Ferreira, Pedro Costa
Centro de F´ısica Computacional, Department of Physics, University of Coimbra,P-3004 - 516 Coimbra, PortugalWe investigate the QCD phase diagram at zero chemical potential andfinite temperature in the presence of an external magnetic field withinthe three flavor Polyakov–Nambu–Jona-Lasinio and entangled Polyakov–Nambu–Jona-Lasinio models looking for the inverse magnetic catalysis.Two scenarios for a scalar coupling parameter dependent on the magneticfield intensity are considered. These dependencies of the coupling allow toreproduce qualitatively lattice QCD results for the quark condensates andfor the Polyakov loop: due to the magnetic field the quark condensates areenhanced at low and high temperatures and suppressed for temperaturesclose to the transition temperatures while the Polyakov loop increases withthe increasing of the magnetic field.PACS numbers: 24.10.Jv, 11.10.-z, 25.75.Nq
1. Introduction
Presently, the investigation of magnetized quark matter is attracting theattention of the physics community due to its relevance for different regionsof the QCD phase diagram [1]: from the heavy ion collisions at very highenergies, to the understanding of the early stages of the Universe and forstudies involving compact objects like magnetars. In the presence of an ex-ternal magnetic field B , the competition between two different mechanismsdetermine the behavior of quark matter: on one hand, the increase of lowenergy contributions leads to an enhancement of the quark condensate; onthe other hand, the suppression of the quark condensate due to the partialrestoration of chiral symmetry. ∗ Presented at EEF70 (1)
CP˙EEF70 printed on May 16, 2018
At zero baryonic chemical potential, almost all low-energy effective mod-els, including the Nambu–Jona-Lasinio (NJL)-type models, find an enhance-ment of the condensate due to the magnetic field, the so-called magneticcatalysis (MC), and no reduction of the pseudocritical chiral transition tem-perature with the magnetic field [2]. However, the suppression of the quarkcondensate, also known as inverse magnetic catalysis (IMC), was obtainedin lattice QCD (LQCD) calculations with physical quark masses [3, 4, 5].Due to the IMC effect the pseudocritical chiral transition temperature de-creases and the Polyakov loop increases with increasing B . In Ref. [5] it isargued that the IMC may be a consequence of how the gluonic sector reactsto the presence of a magnetic field, and, it is shown that the magnetic fielddrives up the expectation value of the Polyakov field. The distribution ofgluon fields changes as a consequence of the distortion of the quark loops inthe magnetic field background. Therefore, the backreactions of the quarkson the gauge fields should be incorporated in effective models in order todescribe the IMC.It is also known that in the region of low momenta, relevant for chiralsymmetry breaking, there is a strong screening effect of the gluon inter-actions which suppresses the condensate [5, 6]. In this region, the gluonsacquire a mass M g of the order of p N f α s | eB | , due to the coupling of thegluon field to a quark-antiquark interacting state. In the presence of a strongenough magnetic field, this mass M g for gluons becomes larger. This, alongwith the property that the strong coupling α s decreases with increasing B ( α s ( eB ) ∼ [ b ln( | eB | / Λ QCD )] − with b = (11 N c − N f ) / π = 27 / π [6]), leads to an effective weakening of the interaction between the quarksin the presence of an external magnetic field, and damps the chiral con-densate. This suggests that the effective interaction between the quarksshould include the reaction of the gluon distribution to the magnetic fieldbackground. Having this in mind, the present work shows two different ap-proaches of taking into account the influence of the presence of an externalmagnetic field in the gluonic sector.We perform our calculations in the framework of the Polyakov–Nambu–Jona-Lasinio (PNJL) model. The Lagrangian in the presence of an externalmagnetic field is given by L = ¯ q [ iγ µ D µ − ˆ m f ] q + G s X a =0 (cid:2) (¯ qλ a q ) + (¯ qiγ λ a q ) (cid:3) − K { det [¯ q (1 + γ ) q ] + det [¯ q (1 − γ ) q ] } + U (cid:0) Φ , ¯Φ; T (cid:1) − F µν F µν , (1)where the quarks couple to a (spatially constant) temporal backgroundgauge field, represented in terms of the Polyakov loop. Besides the chi- P˙EEF70 printed on May 16, 2018 . . . . G s ( Φ ) / G s ; G s ( e B ) / G s . . . . . . eB [GeV ] T = 0210290 G s ( eB ) /G s T c [ M e V ] . . . . . . eB [GeV ] T χc T Φ c Fig. 1. Comparison between both cases: left panel − the scalar coupling G s versusthe magnetic field; right panel − the chiral and deconfinement temperatures as afunction of eB , being the full (dashed) lines for Case I ( Case II ). ral point-like coupling G s , that denotes the coupling of the scalar-type four-quark interaction in the NJL sector, in the PNJL model the gluon dynamicsis reduced to the chiral-point coupling between quarks together with a sim-ple static background field representing the Polyakov loop. The Polyakovpotential U (cid:0) Φ , ¯Φ; T (cid:1) is introduced and depends on the critical temperature T , that for pure gauge is 270 MeV. In addition to the PNJL model, we alsoconsider the effective vertex depending on the Polyakov loop [7] (EPNJLmodel), G s (Φ; ¯Φ) = G s [1 − α Φ ¯Φ − α (Φ + ¯Φ )], that generates an entan-glement interaction between the Polyakov loop and the chiral condensate.In Case I we adopt a running coupling of the chiral invariant quarticquark interaction in the PNJL model with the magnetic field [8, 9]. Thedamping of the strength of the effective interaction is build phenomenolog-ically: since there is no available LQCD data for α s ( eB ), we fit G s ( eB ) inorder to reproduce the chiral pseudocritical temperature T χc ( eB ) obtainedin LQCD calculations [3]. The G s ( eB ) coupling, that reproduces T χc ( eB ),is calculated in the NJL model and is shown in the left panel of Fig. 1 (solidblack line). Now, using this G s ( eB ) coupling in the PNJL model, both thedeconfinement transition and chiral transition pseudocritical temperaturesare decreasing functions with eB , up to eB ∼ . Due to the existingcoupling between the Polyakov loop field and quarks, the G s ( eB ) does notonly affect the chiral transition but also the deconfinement transition.In Case II we introduce an eB − dependence on the pure-gauge criticaltemperature T , reproducing the LQCD data for the deconfinement transi-tion [3], in order to mimic the reaction of the gluon sector to the presenceof an external magnetic field.We will use the EPNJL model because within the PNJL it is not possibleto implement the above scheme, since the chiral transition temperaturesincrease strongly with the external magnetic field. In order to bring these CP˙EEF70 printed on May 16, 2018 temperatures down, it would be necessary to use very small values of T ,for which the deconfinement phase transition becomes of first order.Nevertheless, within EPNJL the chiral condensates and the Polyakovloop are entangled. Thus, the chiral transition temperatures are pulleddown to temperatures close to the deconfinement transition temperature.This model, however, at moderate magnetic fields, still predicts a first ordertransition for both transitions, when a small T is needed. As a consequence,a too small value of T leads to a first order phase transition within bothPNJL and EPNJL models, and, therefore, the range of T values we areinterested in is limited to the values that maintain the crossover transition.A larger range of validity is obtained if the quark backreactions are nottaken into account at eB = 0, i.e. when T = 270 MeV as obtained inpure gauge. This gives T Φ c = 214 MeV, 40 MeV higher than the predictionof lattice QCD data in [3]. This parametrization reproduces the referredlattice QCD data for T Φ c ( eB ), shifted by 40 MeV, for magnetic fields upto 0 .
61 GeV . Above 0.61 GeV , a first order phase transition occurs. Wewill use the last scenario in Case II to illustrate our results because largermagnetic fields are achieved.Moreover, in the EPNJL the coupling G s depends on the Polyakov loop,thus, in the crossover region, where the Polyakov loop increases with tem-perature, the coupling G s becomes weaker. This is shown in Fig. 1 (leftpanel), where the coupling G s [Φ( T )] is plotted for several temperatures(dashed curves) [10]. Within the PNJL model with constant coupling G s ,no IMC effect was obtained even with T ( eB ), because T ( eB ) does notaffect the coupling G s .In Fig. 1 (right panel), the results for the pseudocritical temperaturesfor both cases are compared. In Case II , the pseudocritical temperatureshave a much flatter behavior at small values of eB than in Case I , reflectingthe softer decrease of the coupling G s at small magnetic field values as shownin Fig. 1 (left panel). Also, within the EPNJL with T ( eB ), the differencebetween the pseudocritical temperatures T χc and T Φ c is much smaller, due tothe strong coupling between the Polyakov loop and the quark condensates.For eB = 0 these temperatures are almost coincident, but a finite strongmagnetic field destroys this coincidence. The PNJL model with G s ( eB )does not have this feature and different temperatures for T χc and T Φ c arepredicted.Next, we discuss the effect of the magnetic field on the quark condensatesand on the Polyakov loop, for both cases.According to [4], we define the change of the light quark condensate dueto the magnetic field as ∆Σ f ( B, T ) = Σ f ( B, T ) − Σ f (0 , T ) , with Σ f ( B, T ) = M f m π f π [ h ¯ q f q f i ( B, T ) − h ¯ q f q f i (0 , , where the factor m π f π in the denomi- P˙EEF70 printed on May 16, 2018 − . . . . ∆ ( Σ u + Σ d ) / . . . . eB [GeV ]T=0160170180200270 − . . ∆ ( Σ u + Σ d ) / . . . . . . eB [GeV ] T = 0190200215220290 Fig. 2. The light chiral condensate, ∆(Σ u + Σ d ) /
2, as a function of eB for severalvalues of T in MeV: left panel − Case I ; right panel − Case II . nator contains the pion mass in the vacuum ( m π = 135 MeV) and (the chirallimit of the) pion decay constant ( f π = 87 . u +Σ d ) / eB < eB < .
61) GeV in Case I (Case II) ,at temperatures close to the respective T Φ c ( eB = 0). The main conclusionsare: i) for both cases the qualitative behavior shown in Fig. 2 of Ref. [3]and in Fig. 6 of Ref. [5] is reproduced, that is, the non-monotonic behaviorof the condensates as a function of the magnetic field, having the T = 0curves the highest ∆(Σ u + Σ d ) / ii) for temperatures close T χc ( eB = 0)the strong interplay between the partial restoration of chiral symmetry andthe condensate enhancement due to the magnetic field gives rise to curvesthat increase, for small values of eB , and as soon as the partial restorationof chiral symmetry becomes dominant the curve starts to decrease.Finally, the effect of the magnetic field on the Polyakov loop is seen inFig. 3, where the Polyakov loop value, Φ, is plotted as a function of thetemperature, for several magnetic field strengths. As can be seen, for bothcases, the Polyakov loop increases sharply with the magnetic field aroundthe transition temperature, and the transition temperature decreases withthe magnetic field with B , in close agreement with the LQCD results [5].Indeed, the suppression of the condensates achieved by the magnetic fielddependence of the coupling parameter results in an increase of the Polyakovloop, with this effect being stronger precisely for temperatures close to thetransition temperature. Acknowledgments : This work was partly supported by Project PEst-OE/FIS/UI0405/2014 developed under the initiative QREN financed by theUE/FEDER through the program COMPETE − “Programa OperacionalFactores de Competitividade”, and by Grant No. SFRH/BD/51717/2011. CP˙EEF70 printed on May 16, 2018 G s ( eB ) . . . Φ . . . . . . T/T Φ c ( eB = 0) eB = 00 . . . T ( eB ) . . . Φ . . . . . . T/T Φ c ( eB = 0) eB = 00 . . . Fig. 3. The Polyakov loop as a function of T for different values of eB (in GeV )renormalized by the deconfinement pseudocritical temperature at eB = 0: T Φ c =171 MeV for Case I (left panel) and T Φ c = 214 MeV for Case II , (right panel).
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