Magnetized QCD phase diagram
MMagnetized QCD phase diagram ∗ M´arcio Ferreira, Pedro Costa, and Constanc¸a Providˆencia,
CFisUC, Department of Physics, University of Coimbra, P-3004 - 516 Coimbra,PortugalUsing the 2+1 flavor Nambu-Jona-Lasinio (NJL) model with the Polyakovloop, we determine the structure of the QCD phase diagram in an externalmagnetic field. Beyond the usual NJL model with constant couplings, wealso consider a variant with a magnetic field dependent scalar coupling,which reproduces the Inverse Magnetic Catalysis (IMC) at zero chemicalpotential. We conclude that the IMC affects the location of the Critical-End-Point, and found indications that, for high enough magnetic fields,the chiral phase transition at zero chemical potential might change froman analytic to a first-order phase transition.PACS numbers: 24.10.Jv, 11.10.-z, 25.75.Nq
Introduction : The properties of hadronic matter in a magnetized envi-ronment is attracting the attention of the physics community. The effectof an external magnetic field on the chiral and deconfinement transitions isan active field of research with possible relevance in multiple physical sys-tems. From heavy-ion collisions at very high energies, to the early stages ofthe Universe and astrophysical objects like magnetized neutron stars, themagnetic field may play an important role.The catalyzing effect of an external magnetic field on dynamical chiralsymmetry breaking, known as Magnetic Catalysis (MC) effect, is well un-derstood [1]. However, Lattice QCD (LQCD) studies show an additionaleffect [2, 3, 4], the Inverse Magnetic Catalysis (IMC): instead of catalyzing,the magnetic field weakens the dynamical chiral symmetry breaking in thecrossover transition region. The chiral pseudo-critical transition tempera-ture turns out to be a decreasing function of the magnetic field strength.Different theoretical approaches have been applied in studying the mag-netized QCD phase diagram, and specifically the IMC effect. Several low-energy effective models, including the Nambu–Jona-Lasinio (NJL)-type mod-els, have been used to investigate the impact of external magnetic fields on ∗ Presented at
Excited QCD , 7-13 May 2017, Sintra, Portugal (1) a r X i v : . [ h e p - ph ] D ec eQCD2017˙Ferreira printed on December 25, 2017 quark matter (for a recent review see [5]). Model : We perform our calculations in the framework of the Polyakov–Nambu–Jona-Lasinio (PNJL) model. The Lagrangian in the presence of anexternal magnetic field is given by L = ¯ q [ iγ µ D µ − ˆ m f ] q + G s (cid:88) a =0 (cid:2) (¯ qλ a q ) + (¯ qiγ λ a q ) (cid:3) − F µν F µν − K { det [¯ q (1 + γ ) q ] + det [¯ q (1 − γ ) q ] } + U (cid:0) Φ , ¯Φ; T (cid:1) , where q = ( u, d, s ) T represents a quark field with three flavors, ˆ m f =diag f ( m u , m d , m s ) is the corresponding (current) mass matrix, and F µν = ∂ µ A EMν − ∂ ν A EMµ is the (electro)magnetic tensor. The covariant derivative D µ = ∂ µ − iq f A µEM − iA µ couples the quarks to both the magnetic field B , via A µEM , and to the effective gluon field, via A µ ( x ) = g A µa ( x ) λ a , where A µa is the SU c (3) gauge field. The q f represents the quark electric charge( q d = q s = − q u / − e/ z direction, A EMµ = δ µ x B . We employ the logarithmic effectivepotential U (cid:0) Φ , ¯Φ; T (cid:1) [6], fitted to reproduce lattice calculations.We use a sharp cutoff (Λ) in three-momentum space as a model regu-larization procedure. The parameters of the model are [7]: Λ = 602 . m u = m d = 5 . m s = 140 . G s Λ = 1 .
835 and K Λ = 12 . G s = G s and a magnetic field dependent coupling G s = G s ( eB ) [8]. In the latter, the magnetic field dependence is deter-mined phenomenologically, by reproducing the decrease ratio of the chiralpseudo-critical temperature obtained in LQCD calculations [2]. Its func-tional dependence is G s ( ζ ) = G s (cid:16) a ζ + b ζ c ζ + d ζ (cid:17) , where ζ = eB/ Λ QCD (withΛ
QCD = 300 MeV). The parameters are a = 0 . b = − . × − , c = 0 . d = 1 . × − [8]. Results (zero chemical potential) : Let us first compare both models atzero chemical potential. The up-quark condensate (all quarks show similarresults), normalized by its vacuum value, and the Polyakov loop value are inFig. 1. The presence of the IMC effect in the G s ( eB ) model its clear in Fig.1 (right top panel), by the suppression effect of the magnetic field on thequark condensate around the transition temperature region. Furthermore,the G s ( eB ) model still leads to Magnetic Catalysis at low and high tem-peratures: the magnetic field enhances the quark condensate away from thetransition temperature region, i.e., at low and high temperatures. The chiralpseudo-critical transition temperature, defined as the inflection point of the QCD2017˙Ferreira printed on December 25, 2017 G s (left) and G s ( eB ) (right). quark condensate, decreases for G s ( eB ) and increases for G s . The G s ( eB )makes possible not only the decreasing transition temperature, but also pre-serves the analytic nature of the chiral transition, in accordance with LQCDresults. The G s ( eB ) dependence also affects the Polyakov loop value (bot-tom panel). A decreasing pseudo-critical temperature for the deconfinementtransition with increasing magnetic field is obtained for G s ( eB ), contrast-ing with the increasing pseudo-critical temperature for G s . The G s ( eB )dependence induces a reduction of the Polyakov loop value in the transitiontemperature region (also seen in LQCD results [4]). Results (finite chemical potential) : Now, by introducing a finite chem-ical potential, we analyze the impact of the G s ( eB ) on the entire phasediagram. The results are displayed in Figs. 2, 3, and 4, where the respec-tive quantities are presented for two magnetic field intensities (0 . and 0 . ) within both models. From Fig. 2, we see that the (partial)chiral restoration is accomplished via an analytic transition (crossover) atlow chemical potentials, and through a first-order phase transition at higherchemical potentials. The region on which the chiral phase is broken (blueregion) shrinks as the magnetic field increases for the G s ( eB ) model, andthe opposite occurs for G s . Similar plots are shown in Fig. 3, but now eQCD2017˙Ferreira printed on December 25, 2017 for the strange quark. The general pattern shows a smoothly decrease of Baryonic Chemical Potential (MeV) T e m pe r a t u r e ( M e V ) T e m pe r a t u r e ( M e V ) T e m pe r a t u r e ( M e V ) T e m pe r a t u r e ( M e V ) Fig. 2. Up-quark condensate (normalized by its vacuum value) with G s (top) and G s ( eB ) (bottom) for eB = 0 . (left) and eB = 0 . (right). The colorscale represents the magnitude of the vacuum normalized condensate. the strange quark condensate over the whole phase diagram, though somediscontinuities appear, which are induced by the first-order phase transitionof the light quarks. An interesting result is seen for the G s ( eB ) model at eB = 0 . (bottom right panel of Fig. 3): a first-order phase transi-tion shows up for the strange quark at low temperatures which ends up ina Critical-End-Point (CEP) at a temperature around 50 MeV. Finally, werepresent the Polyakov value in Fig. 4. The general pattern is maintainedwithin both models. We see that the transition from confined quark matter(Φ ≈
0) to deconfinement quark matter (Φ ≈
1) is accomplished via an an-alytic transition, reflected in the continuous increase of the Polyakov loopvalue (there is a discontinuity induced by the chiral first-order phase tran-sition, on which the variation of the Polyakov loop value is small). Becausethe chiral broken phase region gets smaller with increasing magnetic field,the region on which the chiral phase is (approximately) restored but stillconfined (at low temperatures and high chemical potentials) enlarges withincreasing magnetic field strength. The opposite occurs for the model withconstant coupling.As a final step, we focus on the CEP’s location of the chiral transitionas a function of the magnetic field [9, 10]. The result is shown in Fig. 5. Animportant result shows up that clearly differentiates both models. Despite
QCD2017˙Ferreira printed on December 25, 2017 Baryonic Chemical Potential (MeV) T e m pe r a t u r e ( M e V ) T e m pe r a t u r e ( M e V ) T e m pe r a t u r e ( M e V ) T e m pe r a t u r e ( M e V ) Fig. 3. Strange-quark condensate (normalized by its vacuum value) with G s (top)and G s ( eB ) (bottom) for eB = 0 . (left) and eB = 0 . (right). Thecolor scale represents the magnitude of the vacuum normalized condensate. Baryonic Chemical Potential (MeV) T e m pe r a t u r e ( M e V ) T e m pe r a t u r e ( M e V ) T e m pe r a t u r e ( M e V ) T e m pe r a t u r e ( M e V ) Fig. 4. Polyakov loop value Φ with G s (top) and G s ( eB ) (bottom) for eB = 0 . (left) and eB = 0 . (right). The color scale represents the Polyakovloop magnitude. eQCD2017˙Ferreira printed on December 25, 2017 the agreement at low magnetic field strengths ( eB < .
1) between bothmodels on how the CEP reacts to the B presence, for higher magneticfields the CEP moves towards lower chemical potentials for G s ( eB ), whileit moves for higher chemical potentials for G s . This might indicate thatfor high enough magnetic fields, the chiral phase transition might changefrom an analytic to a first-order phase transition at zero chemical potential(there are some indications for this scenario [11]). S = G S ( e B ) G S = G e Be B T ( MeV ) m B ( M e V ) Fig. 5. The CEP position with increasing B field for G s (black) and G s ( eB ) (red). 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 Grants No. SFRH/BD/51717/2011and No. SFRH/BPD/1022 73/2014 from F.C.T., Portugal.REFERENCES [1] V. A. Miransky and I. A. Shovkovy, Phys. Rept. , 1 (2015).[2] G. S. Bali, et al., JHEP , 044 (2012).[3] G. S. Bali, et al., Phys. Rev. D , 071502 (2012).[4] F. Bruckmann, G. Endr¨odi and T. G. Kov´acs, JHEP , 112 (2013).[5] J. O. Andersen, W. R. Naylor and A. Tranberg, Rev. Mod. Phys. , 025001(2016).[6] S. Roessner, C. Ratti and W. Weise, Phys. Rev. D , 034007 (2007).[7] P. Rehberg, S. P. Klevansky and J. Hufner, Phys. Rev. C , 410 (1996).[8] M. Ferreira, P. Costa, O. Loureno, T. Frederico and C. Providˆencia, Phys.Rev. D , no. 11, 116011 (2014).[9] P. Costa, M. Ferreira, H. Hansen, D. P. Menezes and C. Providˆencia, Phys.Rev. D , no. 5, 056013 (2014).[10] P. Costa, M. Ferreira, D. P. Menezes, J. Moreira and C. Providˆencia, Phys.Rev. D , no. 3, 036012 (2015).[11] G. Endr¨odi, JHEP1507