Deconfinement and chiral phase transitions in quark matter with a strong electric field

The deconfinement and chiral phase transitions are studied in the context of the electrized quark matter at finite temperature in the two-flavor Polyakov-Nambu--Jona-Lasinio model. Using the mean field approximation and an electric field independet regularization we show that the effect of temperature and/or electric fields is to partially restore the chiral symmetry. The deconfinement phase transition is slightly affected by the magnitude of the electric field. To this end we show how the effective quark masses and the expectation value of the Polyakov Loop are affected by the electric fields at finite temperatures. As a very interesting result, the pseudocritical temperatures for chiral symmetry restoration and deconfinement decrease as we increase the magnitude of the electric fields, however, both start to increase after some critical value of the electric field.


I. INTRODUCTION
Recent numerical simulations provide possibilities for strong electromagnetic fields to be present in noncentral heavy ion collisions (HIC) [1][2][3][4]. These indications suggest even more properties to be explored in the strongly interacting quark matter, besides the usual strong magnetic fields [5], that are supposed to be created in noncentral HIC or in magnetars [6,7]. The event-by-event fluctuations of the proton positions in the colliding nuclei in Au þ Au heavy-ion collisions at ffiffi ffi s p ¼ 200 GeV in RHIC@BNL and Pb þ Pb at ffiffi ffi s p ¼ 2.76 TeV in ALICE@LHC indicate the creation of strong electric fields that can be of the same order of magnitude of the already predicted magnetic fields [1][2][3]8]. In asymmetric Cu þ Au collisions [4,9,10] it is expected that a strong electric field is generated in the overlapping region. This happens because there is a different number of electric charges in each nuclei and it is argued that this is a fundamental property due to the charge dipole formed in the early stage of the collision. Also, different projectile-target combinations are studied from symmetric to asymmetric collisions systems, showing that the electric field is more significant in the former [11]. The time evolution for electric and magnetic fields in HIC can be estimated and the prediction depends on the conductivity of the medium [12]. In a complementary way, the study of strong electric fields can be very useful when searching for the chiral magnetic effect (CME) [5], given the speculated possibility of reversing the sign of some experimental observables related to the CME [4,13] if the lifetime of such fields is long enough. Such scenarios could give an opportunity to explore anomalous transport properties such as chiral electric separation [14][15][16][17], which is a generation of an axial current in a system with both vector and axial densities [18]. It is natural in this scenario to ask how some properties of quantum chromodynamics(QCD) under strong conditions (i.e., high temperatures and densities) can be affected by such fields. In the low energy limit we should use effective theories or lattice QCD (LQCD) techniques, once we are dealing with the nonperturbative behavior of QCD. One of the main aspects that should be explored is the chiral symmetry restoration of QCD, where several studies have explored a series of interesting phenomena, like the magnetic catalysis [19][20][21], the chiral magnetic effect [5], and the inverse magnetic catalysis [22][23][24][25][26][27] predicted by lattice QCD results [28,29], all of them studied in the context of a pure magnetic field. A natural extension of these works can be done by exploring the role of magnetic fields in the deconfinement transition as well [30][31][32][33][34][35]. The effective theories or LQCD investigations in general are dealing with the chiral condensate, which is an approximate order parameter for the chiral phase transition [30]. On the other hand, the deconfinement in a gauge theory is associated with the spontaneous symmetry breaking of the center symmetry, in which the approximate order parameter is the Polyakov loop [36,37]. The predictions of some effective models for the physics associated with the Polyakov loop have been widely studied in the literature [38][39][40][41][42].
The main goal of our work is to include for the first time the effects of a pure electric field on the Polyakov extended SU(2) Nambu-Jona-Lasinio model (PNJL) [43], to study how the deconfinement phase transition temperature T l pc is affected by these fields, and also to explore its connection with the chiral pseudocritical temperature T χ pc . To this end, we will implement the Schwinger proper-time quark propagators in a constant electric field, which have been explored in previous works [44][45][46][47] related to the study of the gap equation and thermodynamical properties of the model. We also implement in the Schwinger proper-time formalism the Polyakov loop, which should be useful for future evaluations even at eE ¼ 0. As the Nambu-Jona-Lasinio model SU(2) model (NJL) in 3 þ 1 D is nonrenormalizable, the regularization procedure is the same as that presented in [44], where the pure-electric field contribution was analytically solved and the finite thermoelectric part was numerically evaluated. Finally, the Schwinger pair production [48,49] will be presented. Previous studies have shown [44,45] that in the present approximation this quantity is mainly determined by the effective quark masses, which in turn incorporates the effects of temperature and electric field. Thus, there will be quantitative differences between the results for the pair production rate when calculated in the PNJL or NJL model.
The work is organized as follows. In Sec. II we introduce the standard formalism for the two-flavor NJL and PNJL models at finite temperature, including the effects due to the strong electric field. In Sec. III we present the regularization scheme adopted in this work. In Sec. IV we present our numerical results. Finally, in Sec. V we discuss our results and conclusions. We leave for the Appendix the explicit calculations of the trace of the Polyakov loop.

II. GENERAL FORMALISM
In the presence of an electromagnetic field the two-flavor NJL model Lagrangian can be written as where A μ , F μν ¼ ∂ μ A ν − ∂ ν A μ are the electromagnetic gauge potential and tensor fields, respectively; G is the coupling constant; and D μ ¼ ði∂ μ − QA μ Þ is the covariant derivative.
Also, ⃗ τ are the isospin Pauli matrices; Q is the diagonal quark charge matrix, Q ¼ diagðq u ¼ 2e=3; q d ¼ −e=3Þ; ψ ¼ ðψ u ; ψ d Þ T is the two-flavor quark fermion field; and m ¼ diagðm u ; m d Þ represents the current quark mass matrix. Here, we adopt the isospin approximation, i.e., m u ¼ m d ¼ m. We choose A μ ¼ −δ μ0 x 3 E in order to obtain a constant electric field in the z-direction.
In the mean field approximation the Lagrangian density reads where the constituent quark mass is defined by the following expression: where we have both a pure electric field, ϕ E f , and a thermoelectric field, ϕ E;T f , contribution. We will use the following definition of the quark condensate: where f ¼ u, d stands for the quark flavors. Following the procedure described in [44], the pure electric condensate contribution ϕ E f can be calculated using the full Schwinger proper-time quark propagator [48] in a constant electric field, resulting in the expression where E f ¼ jq f jE and N c ¼ 3 is the number of colors. As already derived in [44,45], the thermoelectric contribution can be written as The thermodynamical potential has been derived in [44,45] simply by integrating Eq. (3) with respect to the effective quark mass M, yielding where we have defined θ E f and θ E;T The Schwinger pair production rate is given by Γ ¼ −2ℑðΩÞ [45,48], where ℑðΩÞ corresponds to the imaginary part of the effective potential. Explicitly, one obtains for Γ where we need to perform the summation over the flavor indices f ¼ u, d, and as we will see later, the entire Schwinger pair production dependence on the external conditions is contained only in the effective mass M ≡ MðE; TÞ.

A. The electrized SU(2) PNJL
The extended version of the two-flavor NJL model Lagrangian in the presence of an electromagnetic field coupled with the Polyakov loop is given by now the covariant derivative is given by A μ a ðxÞ, λ a are the Gell-Mann matrices, and A μ a ðxÞ is the SUð3Þ gauge field. For the pure gauge sector, let us define the Polyakov line as where P is a path ordering and β ¼ 1 AÞ. The effective potential Uðl;l; TÞ for the Polyakov fields is parametrized in order to reproduce lattice results in the mean field approximation [38,39]. We adopt the Ansatz [39], where b 2 ðTÞ is given by The parameters of the potential Uðl;l; TÞ will be given in Sec. IV. We should also mention that the thermal expectation value of the Polyakov loop is given by [40] As we will see, our most interesting results occur for temperatures around the pseudocritical temperatures for deconfinement T l pc and for chiral symmetry restoration T χ pc . We verify that the different Ansätze for Uðl;l; TÞ [50] almost agree in this region. Therefore, our results remain essentially the same if we change the form of the effective potential for the Polyakov loop Uðl;l; TÞ.
Once we are working with the background field A 4 , we can obtain the condensate as a straightforward generalization of the expression given in Eq. (4) for zero temperature and density using the following symbolic replacements [41]: where w n ¼ ð2n þ 1ÞπT is the Matsubara frequency. In this work we consider only the zero baryon chemical potential case, and hence l ¼l. Once the traced Polyakov loop is given by l ¼ 1 N c Tr expði A 4 T Þ, we can write the background field in the Polyakov gauge [42,51] , and it is straightforward to see that A ð8Þ 4 ¼ 0 at μ ¼ 0. Therefore, we implement the Polyakov loop in the condensate ϕ E;T;l at μ ¼ 0 by using the prescriptions given by Eqs. (16) and (17). After calculating the trace in the color space, one obtains By making use of this expression in Eq. (3) the SU(2) PNJL gap equation reads Finally, the thermodynamical or effective potential is obtained following a standard procedure [41] using the prescriptions (16) and (17), yielding It is straightforward to show that The effective quark masses and the expectation value of the Polyakov loop are obtained by minimizing the thermodynamical potential, i.e., calculating ∂Ω ∂M ¼ 0 and ∂Ω ∂l ¼ 0. The derivative ∂Ω ∂M ¼ 0 was already obtained in Eq. (19) and ∂Ω ∂l ¼ 0 is given by

III. REGULARIZATION
In this work we use the vacuum-subtraction scheme [44][45][46][47]. We define for the vacuum-subtracted condensate the following quantity: The gap equation, Eq. (3) in the SU(2) NJL model and Eq. (19) in the SU(2) PNJL, should be calculated using the following regularized condensate ϕ E f : where we have adopted the 3D cutoff scheme [52,53] to regularize the infinite field independent vacuum contribution where In the same way, for the effective potential in the SU(2) NJL model, Eq. (7), and Eq. (20) in the SU(2) PNJL model, we should use the regularized θ E f given by where the finite field dependent term is and the infinite vacuum contribution θ vac f is given by The regularized θ vac f is given in the 3D cutoff scheme by It is important to note that the poles associated with the imaginary part of the effective potential can be associated to the zeros of the sinðE f sÞ which appear in the denominator of both our gap equation and the effective potential when E f s ¼ nπ for n ¼ 1; 2; 3;… . We interpret these integrals as the Cauchy principal value [44,46] and that the true ground state of the theory is given by the real part of the thermodynamical potential.
Using the techniques adopted in [44], we can use the principal value (or the real part) ofθ E f , given by where y f ¼ M 2 =ð2E f Þ. In the same manner, we obtain for the principal value of the vacuum-subtracted condensateφ E As discussed in [44], the quantities ϕ E;T f and θ E;T f depend on the temperature and following the procedure adopted in Ref. [54] we set the lower limits of the integration to zero, since theses integrals are ultraviolet finite.

IV. NUMERICAL RESULTS
In this section we show our numerical results. The parameter set for the SU(2) NJL model is [53] Λ ¼ 587.9 MeV, GΛ 2 ¼ 2.44, and m ¼ 5.6 MeV. The parameters associated with the pure gauge sector are the following: a 0 ¼ 6.75, a 1 ¼ −1.95, a 2 ¼ 2.625, a 3 ¼ −7.44, b 3 ¼ 0.75, and b 4 ¼ 7.5. In the pure gauge sector, the transition temperature is given by T 0 ¼ 270 MeV [39]. A lower value of T 0 is usually necessary in order to include the effects of N f ¼ 2 in the theory. Following the procedure adopted in Refs. [35,55], we use T 0 ¼ 208 MeV.
In Fig. 1 we show the effective quark masses as a function of the temperature for different values of electric fields in the two-flavor NJL model. As expected, for the NJL model at eE ¼ 0.01 GeV 2 , as we increase the temperature, the effective quark masses decrease, as a signature of the chiral symmetry restoration. If we increase the electric field to eE ¼ 0.10 GeV 2 , at low temperatures the effective quark masses slightly decrease as an effect of the restoration of the chiral symmetry guided by the electric field. This effect was already explored in previous works in the literature at finite temperature [44,45] and at T ¼ 0 [52,56]. As we increase the electric field it can be seen that the pseudocritical temperature has decreased with the increase of the electric field. In Fig. 2, we show how − dM dT changes with the increase of the electric field. The peak of each curve is interpreted as the pseudocritical temperature for chiral symmetry restoration T χ pc for the corresponding electric field. In Fig. 3 we can see the pseudocritical temperature as a function of electric field for the two-flavor NJL model, showing the behavior previously predicted, that as we grow the electric field the pseudocritical temperature decreases until we reach a critical value eE ¼ 0.31 GeV 2 . At this point, the pseudocritical temperature starts increasing as we increase the electric field.
Next, we discuss our results for the SU(2) PNJL model. We show in Fig. 4 the expectation value of the Polyakov loop as a function of the temperature for different values of electric fields. We can see that the effect of the strong electric fields slightly changes the Polyakov loop expectation value in comparison with the effect on the effective quark mass. These changes are more prominent at T ∼ 170 MeV, where the increase of the electric field tends to strengthen the deconfined phase. Also, we should mention that the effect of the electric field in the Polyakov loop is given in an indirect way though the gap equation (19) and Eq. (23) (that are coupled), once the electric fields do not couple directly with the Polyakov loop.
In Fig. 5 we show the effective quark masses as a function of the temperature for different values of electric fields in the two-flavor PNJL model; the behavior of M as a function of eE and T is in qualitative agreement with that predicted in the NJL model. The quantity − dM dT for the twoflavor PNJL model is in Fig. 6 and in Fig. 7 we show the pseudocritical temperature for the chiral symmetry restoration as a function of T, as we increase the electric field. The results obtained in the PNJL model are in quantitative The quantity dl dT as a function of the temperature for different values of electric fields is shown in Fig. 8; the pseudocritical temperature for deconfinement T l pc corresponds to the maximum of each curve for different values of electric fields. In Fig. 9 we show T l pc as a function of the electric field. The pseudocritical temperature for the deconfinement transition slightly decreases as we increase the electric field until eE ∼ 0.270 GeV 2 . At this point, the deconfinement temperature transition starts to increase in a similar way that we have predicted for the pseudocritical temperature for chiral symmetry restoration.
In Fig. 10 for the NJL SU(2) model the variation of the effective quark masses as a function of the electric field is shown for fixed temperatures: T ¼ 0, 170, 200, and 220 MeV. At low values of the electric fields eE ∼ 0, we can see the temperature effect on the partial restoration of the chiral symmetry. As we increase the magnitude of the electric field, the general aspect is the restoration of the chiral symmetry with the electric field.
For the PNJL SU(2) results we have almost the same analysis, but the quantitative results are different as we can see in Fig. 11. In Fig. 12, where we compare the two models at T ¼ 170 MeV and T ¼ 220 MeV, we can see quantitative differences on the numerical results.
At T ¼ 170 MeV the NJL SU(2) model partially restores the chiral symmetry with a lower electric field than the PNJL SU(2) model. In the lower panel of Fig. 12 we have the comparison of the models at T ¼ 200 MeV. We see that the PNJL SU(2) model has a much higher value of the effective quark mass at eE ∼ 0 than the NJL SU(2) model. Also, both models tends to partially restore the chiral symmetry to higher values of electric fields, following the previous analysis. At T ¼ 0 almost no difference is seen. All these results show quantitative differences between the two models and how the confinement can change the scenario of the restoration of the chiral symmetry.
It is interesting to point out the prediction of the differences of the effects of the electric fields on T l pc and T χ pc in the PNJL model. The electric fields tends to influence more easily the chiral transition than the deconfinement. The behavior of the chiral condensate and the Polyakov loop in an environment with constant magnetic field has been explored in [31].
We see that the difference between the pseudocritical temperatures for chiral symmetry restoration at eE ¼ 0.15 GeV 2 and eE ¼ 0.2 GeV 2 is about ΔT χ pc ∼8.7 MeV. In the same manner, the difference in the deconfinement temperature is given by ΔT l pc ∼ 1.43 MeV. As a conclusion, the quarks can be in a deconfined phase with chiral symmetry still not restored. This is a very interesting result, in the opposite direction of what happens when we have the quarkyonic phase [51,57] that occurs for some values of μ ≠ 0 and eE ¼ eB ¼ 0. At strong enough electric fields, where both T χ pc and T l pc increase as when we increase the electric field, the variation in the respective transition temperatures become larger and different. For example, if we take eE ¼ 0.25 GeV 2 and eE ¼ 0.30 GeV 2 we have, respectively, ΔT χ pc ∼ 15.88 MeV and ΔT l pc ∼ 2.18 MeV. We should pay attention to the fact that we are working with magnitudes of the electrical fields that are valid for the NJL and PNJL models eE ∼ Λ 2 . In Fig. 13 we show the Schwinger pair production as a function of the temperature at fixed electric fields eE ¼ 0.05 GeV 2 , eE ¼ 0.10 GeV 2 , eE ¼ 0.15 GeV 2 , eE ¼ 0.20 GeV 2 , eE ¼ 0.25 GeV 2 , and eE ¼ 0.30 GeV 2 . In the region where we have chiral symmetry restoration we can see that the production rate grows, and after some value of temperature the Schwinger pair production stabilizes. Another interesting aspect is that if you fix the electric field, in the NJL model the Schwinger pair production stabilizes for larger values and it happens at low temperatures if we compare with the PNJL model results.

V. CONCLUSIONS
In this work we have presented a study for strongly electrized quark matter within the SU(2) PNJL and NJL models at finite temperatures in the mean field approximation. We have shown that the constituent quark masses decrease as we grow both electric fields and temperatures, as a signature of the partial restoration of the chiral symmetry. In this scenario, as expected, the pseudocritical temperature for chiral symmetry restoration decreases as we grow the electric fields. The deconfinement is guided by the expectation value of the Polyakov loop, and our results show that the electric fields tend to anticipate the transition to the deconfined phase, and the effects due to the electric fields are more prominent in the chiral transition than the deconfinement one. On the other hand, we show that for strong enough electric fields, both pseudocritical temperatures for chiral symmetry and deconfinement temperatures start to increase after a critical value of the electric field, a very interesting effect that has been identified for the first time in the literature. This effect propagates to all quantities, as the Schwinger pair production. For comparison, we also show the results in the NJL model, where the same qualitative results are obtained, revealing that the main characteristics of this type of theory are model independent. Also, we observe that for some values of electric fields, the quarks can be in a deconfined phase with the chiral symmetry still not restored, in the opposite direction of the observed quarkyonic phase for systems at finite baryonic density and zero external fields. We expect to use this type of phenomenology to extend the analysis for electromagnetic fields and more general purposes in the future, e.g., in systems that present a chiral imbalance of right-handed and left-handed quarks, in finite baryonic densities, and in the physics behind the chiral magnetic effect [5,58,59]. Works in these directions are under way and we expect to report our results soon.

APPENDIX: THE TRACE OF THE POLYAKOV LOOP
The Polyakov loop can be expressed by a diagonal matrix L ¼ diagðe iφ ; e iφ 0 ; e −iðφþφ 0 Þ Þ. As discussed in [42,60], the perturbative vacuum has φ ¼ φ 0 ¼ 0, i.e., the T ¼ 0 limit, and the confining vacuum can be chosen to be φ ¼ 2π 3 and φ 0 ¼ 0. For simplicity, we can adopt φ 0 ¼ 0 from the beginning. The Polyakov loop with this assumption assumes l ¼ 1 N c Tr c L ¼ 1 N c Tr c L † . This is true for the limit μ ¼ 0 [61].
The simplification adopted here implies l ¼ 1 3 ð1 þ 2 cos φÞ. It is useful to invert the last relation for future evaluations: In this work we will implement the μ ¼ 0 case (where we havel ¼ l) and Tr c ; i.e., the trace over the color space is performed following the steps given in Eqs. (16) and (17). First we decompose the trace as Tr c ½ðLÞ n þ ðL † Þ n ¼ Tr c ðLÞ n þ Tr c ðL † Þ n : ðA2Þ To evaluate the trace of L n and ðL † Þ n we should note that the Ansatz is already in the Jordan form [62]; then we can use the following result: where the λ i are the eigenvalues of the matrix A. Applying this result directly to Eq. (A2), we obtain Tr c ðLÞ n þ Tr c ðL † Þ n ¼ 2ð1 þ e iφn þ e −iφn Þ; ¼ 2ð1 þ 2 cos nφÞ: ðA4Þ Using now Eq. (A1) in the last equation, we can reach Tr c ðLÞ n þ Tr c ðL † Þ n