# Magnetized QCD phase diagram^{1}

^{1}

## Abstract

Using the 2+1 flavor Nambuâ-Jona-Lasinio (NJL) model with the Polyakov loop, we determine the structure of the QCD phase diagram in an external magnetic field. Beyond the usual NJL model with constant couplings, we also consider a variant with a magnetic field dependent scalar coupling, which reproduces the Inverse Magnetic Catalysis (IMC) at zero chemical potential. 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 from an analytic to a first-order phase transition.

24.10.Jv, 11.10.-z, 25.75.Nq

Introduction: The properties of hadronic matter in a magnetized environment is attracting the attention of the physics community. The effect of an external magnetic field on the chiral and deconfinement transitions is an active field of research with possible relevance in multiple physical systems. From heavy-ion collisions at very high energies, to the early stages of the Universe and astrophysical objects like magnetized neutron stars, the magnetic field may play an important role.

The catalyzing effect of an external magnetic field on dynamical chiral symmetry breaking, known as Magnetic Catalysis (MC) effect, is well understood [1]. However, Lattice QCD (LQCD) studies show an additional effect [2, 3, 4], the Inverse Magnetic Catalysis (IMC): instead of catalyzing, the magnetic field weakens the dynamical chiral symmetry breaking in the crossover transition region. The chiral pseudo-critical transition temperature turns out to be a decreasing function of the magnetic field strength.

Different theoretical approaches have been applied in studying the magnetized QCD phase diagram,
and specifically the IMC effect.
Several low-energy effective models, including the Nambu–Jona-Lasinio (NJL)-type models,
have been used to investigate the impact of external magnetic fields on 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 an external magnetic field is given by

where represents a quark field with three flavors, is the corresponding (current) mass matrix, and is the (electro)magnetic tensor. The covariant derivative couples the quarks to both the magnetic field , via , and to the effective gluon field, via , where is the SU gauge field. The represents the quark electric charge (). We consider a static and constant magnetic field in the direction, . We employ the logarithmic effective potential [6], fitted to reproduce lattice calculations.

We use a sharp cutoff () in three-momentum space as a model regularization procedure. The parameters of the model are [7]: MeV, MeV, MeV, and .

We analyze two model variants with distinct
scalar interaction coupling: a constant coupling
and a magnetic field dependent coupling [8].
In the latter, the magnetic field dependence
is determined phenomenologically, by reproducing the decrease ratio of the
chiral pseudo-critical temperature obtained in LQCD calculations [2].
Its functional dependence is
,
where (with MeV).
The parameters are , , , and [8].

Results (zero chemical potential): Let us first compare both models at zero chemical potential. The up-quark condensate (all quarks show similar results), normalized by its vacuum value, and the Polyakov loop value are in Fig. 1.

The presence of the IMC effect in the model its clear
in Fig. 1 (right top panel), by the suppression effect of the magnetic field on
the quark condensate around the transition temperature region.
Furthermore, the model still leads to Magnetic Catalysis at low and high temperatures:
the magnetic field enhances the quark condensate away from the transition temperature
region, i.e., at low and high temperatures.
The chiral pseudo-critical transition temperature, defined as the inflection point of the quark condensate,
decreases for and increases for .
The makes possible not only the decreasing transition temperature, but also preserves the analytic nature
of the chiral transition, in accordance with LQCD results.
The dependence also affects the Polyakov loop value (bottom panel).
A decreasing pseudo-critical temperature for the deconfinement transition with increasing magnetic field
is obtained
for , contrasting with the increasing pseudo-critical temperature
for . The dependence induces a reduction of the Polyakov loop
value in the transition temperature region (also seen in LQCD results [4]).

Results (finite chemical potential): Now, by introducing a finite chemical potential, we analyze the impact of the on the entire phase diagram. The results are displayed in Figs. 2, 3, and 4, where the respective quantities are presented for two magnetic field intensities ( GeV and GeV) within both models. From Fig. 2, we see that the (partial) chiral restoration is accomplished via an analytic transition (crossover) at low chemical potentials, and through a first-order phase transition at higher chemical potentials. The region on which the chiral phase is broken (blue region) shrinks as the magnetic field increases for the model, and the opposite occurs for . Similar plots are shown in Fig. 3, but now for the strange quark.

The general pattern shows a smoothly decrease of the strange quark condensate over the whole phase diagram, though some discontinuities appear, which are induced by the first-order phase transition of the light quarks. An interesting result is seen for the model at GeV (bottom right panel of Fig. 3): a first-order phase transition shows up for the strange quark at low temperatures which ends up in a Critical-End-Point (CEP) at a temperature around MeV. Finally, we represent the Polyakov value in Fig. 4.

The general pattern is maintained within both models. We see that the transition from confined quark matter () to deconfinement quark matter () is accomplished via an analytic transition, reflected in the continuous increase of the Polyakov loop value (there is a discontinuity induced by the chiral first-order phase transition, on which the variation of the Polyakov loop value is small). Because the chiral broken phase region gets smaller with increasing magnetic field, the region on which the chiral phase is (approximately) restored but still confined (at low temperatures and high chemical potentials) enlarges with increasing magnetic field strength. The opposite occurs for the model with constant coupling.

As a final step, we focus on the CEP’s location of the chiral transition as a function of the magnetic field [9, 10]. The result is shown in Fig. 5. An important result shows up that clearly differentiates both models. Despite the agreement at low magnetic field strengths () between both models on how the CEP reacts to the presence, for higher magnetic fields the CEP moves towards lower chemical potentials for , while it moves for higher chemical potentials for . This might indicate that for high enough magnetic fields, the chiral phase transition might change from an analytic to a first-order phase transition at zero chemical potential (there are some indications for this scenario [11]).

Acknowledgments: This work was partly supported by Project PEst-OE/FIS/UI0405/2014 developed under the initiative QREN financed by the UE/FEDER through the program COMPETE “Programa Operacional Factores de Competitividade”, and by Grants No. SFRH/BD/51717/2011 and No. SFRH/BPD/1022 73/2014 from F.C.T., Portugal.

### Footnotes

- thanks: Presented at Excited QCD, 7-13 May 2017, Sintra, Portugal

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