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Dynamical chiral symmetry breaking and confinement are the two fundamental properties of non-perturbative quantum chromodynamics (QCD). At zero or low temperatures T, the fundamental degrees of freedom of quantum chromodynamics (QCD) are the low energy hadrons, whereas at high T, the interaction gets increasingly screened and thus becomes weak, causing hadrons to melt down to a new phase in which the predominant degrees of freedom are quarks and gluons. This phenomenon is referred to as the confinement–deconfinement phase transition. As the strength of the QCD interaction decreases with increasing T, only the current quark mass survives when the parameter T exceeds a critical value. This is termed as the chiral symmetry breaking–chiral symmetry restoration phase transition. Such a phase transition is expected to have occured in the early universe, a few microseconds after the Big Bang, and it is experimentally observed in heavy-ion collisions at the Large Hadron Collider (LHC) in CERN and Relativistic Heavy Ion Collider (RHIC) at Brook Heaven National Laboratory (BNL). In addition, when the hadronic matter is subjected to an external electromagnetic field background, it yields a significant impact on phase transition. It is well known that at
T=0 , in the pure magnetic case, the strong magnetic field tends to strengthen the formation of quark anti-quark condensate, and the system remains in the chiral symmetry broken phase, even at a high magnetic field strengtheB . This phenomenon is known as magnetic catalysis (MC) [1-10]. It was verified in earlier studies that at a finite T, the pseudo-critical temperatureTχ,Cc of the chiral symmetry restoration and deconfinement increases with an increase ineB ; hence, magnetic catalysis is also observed at finite T [1, 2, 5, 11, 12]. Recently, the lattice QCD simulation [13-15] predicted that at finite T, a magnetic field suppresses the formation of quark-antiquark condensates and tends to restore the chiral symmetry at approximately the pseudo-critical temperatureTχ,Cc . Consequently,Tχ,Cc decreases with an increase ineB , and such a phenomenon is known as the inverse magnetic catalysis (IMC). This phenomenon is validated and supported by effective models of low energy QCD [16-23], as well as in holographic QCD models [24].In a pure electric case, and at
T=0 , the situation is relatively different from that of a pure magnetic field background. The strong electric field suppresses the formation of a quark-antiquark condensate, and thus tends to restore the chiral symmetry, i.e., the electric field anti-screens the strong interaction. Such a phenomenon is known as the chiral electric inhibition effect [1, 2, 5, 11, 12, 25-28] or the chiral electric rotation effect [29]. The nature of chiral phase transitions is of the second order in the chiral limit but cross-over when the bare quark mass is considered. At finite temperature, it is well understood that the pseudo-critical temperatureTχ,Cc decreases with an increase in electric field strengtheE , and this phenomenon is known as the inverse electric catalysis (IEC) [28, 30]. The study on the influence of electric fields on the chiral phase transitions, as well as the effect of the magnetic field, is equally important from theoretical and experimental perspectives. Experimentally, in heavy-ion collisions, the electric and magnetic fields are generated with the same order of magnitude (∼1018 to1020 Gauss) [31-34] in the event-by-event collisions using Au+ Au at RHIC-BNL and in a non-central heavy-ion collision of Pb+ Pb in ALICE-LHC. Moreover, some interesting anomalous effects, such as the chiral magnetic [35, 36], chiral electric separation [37, 38], and particle polarization [39-41] effects, which may arise owing to the generation of vector and/or axial currents in the presence of strong electromagnetic fields, are also required for theoretical exploration. Recently, a special case of considering the electric field strength parallel to the magnetic field strength (eE∥eB ) significantly focused on exploring the above-mentioned phenomenon in the effective models of QCD [28, 29, 42, 43]. A major reason behind this approach is that the parallel electric and magnetic fields play important and prominent roles in several heavy-ion collision experiments [44, 45].Considering the aforementioned facts and findings, the objective of this study is to elucidate the traits of the dynamical chiral symmetry breaking-restoration transition in the presence of parallel electric and magnetic fields. The unified framework of this study is based on Schwinger-Dyson equations (SDE) in the rainbow-ladder truncation, Landau gauge, symmetry preserving the confining vector-vector contact interaction model (CI) [46], and Schwinger optimal time regularization scheme [47]. We adopt the quark-antiquark condensate
−⟨ˉqq⟩ as an order parameter for the chiral symmetry breaking-restoration, whereas for the confinement-deconfinement transition, we use the confinement length scale [22, 48]. It should be noted that the chiral symmetry restoration and deconfinement occur simultaneously in this model [22, 49].This article is organized as follows. In Sec. II, we present the general formalism and contact interaction model at zero temperature, in the absence of background fields. In Sec. III, we discuss the gap equation at zero temperature, in the presence of parallel electric and magnetic fields presented in Sec. IV. Next, in Sec. V, we present the phase diagram at finite temperature, in the presence of parallel electric and magnetic fields. Finally, in Sec. VI, we present the summary and perspectives of this study.
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We begin with the Schwinger-Dyson's equations (SDE) for a dressed-quark propagator
Sf , which is given byS−1f(p)=iγ⋅p+mf+∫d4k(2π)4g2Δμν(p−k)λa2γμSf(k)λa2Γν(p,k),
(1) the subscript f represents the two light quark flavors, i.e., up (u) and down (d) quarks, g is the coupling constant, and
mf is the current quark mass, which can be set to zero in the chiral limit.λa represents the conventional Gell-Mann matrices,Γν is the dressed quark-gluon vertex, andΔμν depicts the gluon propagator.In literature, it is well known that the properties of low energy hadrons can be reproduced by assuming that the interaction among the quarks occurs not via a mass-less vector boson exchange but by the symmetry preserving four-fermions vector-vector CI with a finite gluon mass [46, 50-53]:
g2Δμν(k)=δμν4παirm2G≡δμναeff,
(2) where
αir=0.93π represents the infrared enhanced interaction strength parameter, andmG=800 MeV is the gluon mass scale [54]. In the CI model, for a small value ofαir and a large value of the gluon mass scalemG , there must be a critical valueαeff ; above this critical value, the chiral symmetry is broken, and below this value, there is a lesser chance for the generation of the dynamical mass. The d-dimensional (arbitrary space-time dimensions) dependence of this effective coupling and its critical value on the chiral symmetry breaking, using an iterative method in the superstrong regime, has been comprehensively studied in Ref. [55].The CI model Eq. (2), together with the choice of rainbow-ladder truncation
Γν(p,k)=γν , forms the kernel of the quark SDE, Eq. (1), which brings the dressed-quark propagator into a very simple form [56]:S−1f(p)=iγ⋅p+Mf.
(3) This is possible because the wave function renormalization trivially tends to unity in this case, and the quark mass function
Mf becomes momentum independent:Mf=mf+4αeff3∫Λd4k(2π)4Tr[Sf(k)].
(4) In this truncation, the quark-anitquark condensate is given by
−⟨ˉqq⟩f=Nc∫Λd4k(2π)4Tr[Sf(k)],
(5) with
Nc=3 representing the number of colors. The form of the proposed gap equation, Eq. (4), is significantly similar to the NJL model gap equation, except the coupling parameterαeff=92G [49].Further simplification of Eq. (4) yields the following gap equation.
Mf=mf+16αeff3∫Λd4k(2π)4Mfk2+M2f,
(6) where
Mf represents the dynamical mass, and the symbol∫Λ stresses the need to regularize the integrals. Usingd4k=(1/2)k2dk2sin2θdθsinϕdϕdψ , and performing trivial regular integrations with the variables=k2 , the above expression reduces to:Mf=mf+αeffMf8π2∫∞0dsss+M2f.
(7) The integral in Eq. (7) is not convergent; therefore, we need to regularize it via the proper-time regularization scheme [47]. In this scheme, we take the exponent of the integrand's denominator and then introduce an additional infrared cutoff, in addition to the conventional ultraviolet cut-off, which is widely adopted in NJL model studies. Accordingly, the confinement is implemented using an infrared cut-off [57]. By adopting this scheme, the quadratic and logarithmic divergences are eliminated, and the axial-vector Ward-Takahashi identity [58, 59] is satisfied. From Eq. (7), the denominator of the integrand is given by
1s+M2f=∫∞0dτe−τ(s+M2f)→∫τ2irτ2uvdτe−τ(s+M2f)=e−τ2uv(s+M2f)−e−τ2ir(s+M2f)s+M2f.
(8) Here,
τ−1uv=Λuv is an ultra-violet regulator that plays the dynamical role and sets the scale for all dimensional quantities.τ−1ir=Λir represents the infrared regulator whose non zero value implements confinement by ensuring the absence of quarks production thresholds [60]. Hence,τir corresponds to the confinement scale [22]. From Eq. (8), it is clear that the location of the original pole is ats=−M2 , which is canceled by the numerator. Accordingly, we discarded the singularities, and the propagator is free from real and complex poles, which is consistent with the definition of confinement: "an excitation described by a pole-less propagator would never reach its mass-shell" [57].After performing integration over "s", the gap equation is given by
Mf=mf+M3fαeff8π2Γ(−1,τuvM2f,τirM2f),
(9) where
Γ(a,x1,x2)=Γ(a,x1)−Γ(a,x2),
(10) and
Γ(a,x)=∫∞xtα−1e−tdt , which is the incomplete Gamma function. By using the parameters of Ref. [50], i.e.,τir=(0.24GeV)−1 andτuv=(0.905GeV)−1 , with the bare quarkmu=md=0.007 GeV, we obtain the dynamical massMu=Md=0.367 GeV and the condensate⟨ˉqq⟩u=⟨ˉqq⟩d=−(0.0143) GeV3 . -
In this section, we study the gap equation in the presence of a uniform and homogeneous electromagnetic field with
eE∥eB and at zero temperature. In the QCD Lagrangian, the interaction with the parallel electromagnetic fieldAextμ embedded in the covariant derivative is expressed asDμ=∂μ−iQfAextμ,
(11) where
Qf=(qu=+2/3,qd=−1/3)e refers to the electric charges of u and d-quarks, respectively. We adopt the symmetric gauge vector potentialAextμ=(iEz,0,−Bx,0) in Euclidean space, where both the electric and magnetic fields are selected along the z-axis. The gap equation in the presence of the parallel electromagnetic field remains in the form of Eq. (4), whereSf(k) is dressed with parallel background fields, i.e.,Sf(k)→~Sf(k) .~Sf(k) , in the Schwinger proper time representation [1, 2, 5, 47], in the presence of the parallel magnetic field in Euclidean space [26, 29], is given by~Sf(k)=∫∞0dτe−τ(M2f+(k24+k23)tan(|QfE|τ)|QfE|τ+(k21+k22)tanh(|QfB|τ)|qfB|τ)×[−γ4k4+Mf+tan(|QfE|τ)(γ4k3−γ3k4)−itanh(|QfBτ|)(γ1k2−γ2k1)]×[1−itanh(|QfB|τ)tan(|QfE|τ)γ5−itanh(|QfB|τ)γ1γ2+tan(|QfE|τ)γ4γ3],
(12) where the magnetic field couples with the coordinate indices
1 and2 , while the electric field couples with3 and4 . Now, taking the trace of Eq. (12) and introducing both infrared and ultraviolet cut-offs, the gap equation in the presence of parallel electric and magnetic fields is given by~Mf=mf+16αeff3∑f=u,d∫˜τ2irτ2uvdτ˜Mfe−τ˜M2f×∫dk1(2π)dk2(2π)dk3(2π)dk4(2π)e−τ((k24+k23)tan(|QfE|τ)|QfE|τ+(k21+k22)tanh(|QfB|τ)|QfB|τ.
(13) The infrared cu-off
τir is introduced in the model to mimic confinement by ensuring the absence of quarks production thresholds. In the presence of botheE andeB , it is required to vary slightly with botheE andeB . Therefore, the entanglement between dynamical chiral symmetry breaking and confinement is expressed using an expliciteE andeB -dependent regulator in the infrared [22]:˜τir=τirMf˜Mf,
(14) where
Mf is the dynamical mass in the absence of background fields, and˜Mf represents theeE andeB dependent dynamical mass. For chiral quarks (i.e.,mf=0 ), the confining scale˜τir diverges at the chiral symmetry restoration parameter near its critical value, which ensures the simultaneity of deconfinement and chiral symmetry restoration [22]. After the integration over k, the gap equation, Eq. (13), can be written as˜Mf=mf+αeff3π2∑f=u,d∫ˉτ2irτ2uvdτ˜Mfe−τ˜M2f[|QfE|tan(|QfE|τ)|QfB|tanh(|QfB|τ)].
(15) When both fields
eE andeB→0 , Eq. (15) reduces to Eq. (9). The gap equation for the pure electric field can be obtained by settingeB→0 , while for pure magnetic field,eE→0 . The quark-antiquark condensate in the presence of background fields is in this form.In this study, we adopt two flavors
f=2 , i.e., u- and d-quarks. We use the same current quark mass for the up and down quarks, i.e.,mu=md=0.007 GeV, such that the iso-spin symmetry is preserved. As is well-known, the response to the electromagnetic field is different for u and d-quarks because they have different electric charges [1, 5].The numerical solution of the gap equation Eq. (15) as a function of
eB for fixed values ofeE , is plotted inFig. 1. In the pure magnetic case (eE→0 ), the dynamical mass~Mf monotonically increases with an increase ineB , which ensures the magnetic catalysis phenomenon. The increase in~Mf as a function ofeB reduces in magnitude upon varyingeE from its smaller to larger given values. We noticed that ateE⩾0.33 GeV2 , the dynamical mass~Mf exhibits the de Haas-van Alphen oscillatory type behavior [61]. In other words, it remains constant for smalleB values, then monotonically decreases with an increase ineB , and abruptly declines to lower values in the regioneB≈[0.3−0.54] GeV2 , where the chiral symmetry is partially restored via first-order phase transition; above it, the value increases again. This phenomenon can be attributed to the strong competition that occurs between paralleleB andeE , i.e., on the one hand,eB enhances the mass function, while on the other hand,eE suppresses it. We also sketch a zoom-in plot of the mass function, which shows the de Haas-van Alphen oscillatory behavior in the regioneB≈[0.3−0.54] GeV2 witheE=0.33 GeV2 , as illustrated in Fig. 2. Such a behavior type is also explored and discussed in the other effective model of QCD, for example, refer to Refs. [29, 62].Figure 1. (color online) Behavior of the dynamically generated quark mass, Eq. (15), as a function of the magnetic field strength
eB at several given values ofeE. Figure 2. Zoom-in plot of the mass function that shows the de Haas-van Alphen oscillatory behavior in the magnetic field region
eB=[0.3−0.5] GeV2 with the electric fieldeE=0.33 GeV2 The behavior of the quark-antiquark condensate, Eq. (16), as a function of
eB at various given values ofeE , is illustrated in the Fig. 3. For a pure magnetic case (eE→0 ), the magnetic field strengtheB facilitates the formation of the quark-antiquark condensate. For given non zero values ofeE , particularly, ateE≈0.33 GeV2 the evolution in the condensate is suppressed in the regioneB=[0.3− 0.54] GeV2 . The nature of the transition in this region suddenly changes to first-order; however, above this region, it is enhanced witheB . The pseudo-critical values of the fields at which the chiral symmetry partially is restored and first-order phase transition occurres areeBc≈0.3 GeV2 andeEc=0.33 GeV2 . Although such values of electric or magnetic field strength are sufficiently large in terms of what is typically generated during heavy-ion collisions, they may be relevant to astronomical objects, such as neutron stars, magnetars, etc. The confinement parameter˜τ−1ir as a function ofeB for different fixed values ofeE is presented in Fig. 4. We observed the same pseudo-critical fieldseBc≈0.3 GeV2 andeEc= 0.33 GeV2 for the confinement transition, similar to the case of chiral symmetry breaking.Figure 3. (color online) Quark-antiquark condensate, Eq. (16), as a function of
eB for several values ofeE. In the following, we discuss the variation of
~Mf and
˜τ−1ir as a function ofeE , in the pure electric field, as well as for several non-zero values ofeB . In Figs. 5, 6, and 7, we illustrate the behaviors of all three parameters as a function ofeE , for various values ofeB⩾0 .Figure 6. (color online) Quark-antiquark condensate, Eq. (16), as a function of
eE for several values ofeB. Figure 7. (color online) Confinement scale
˜τ−1ir as a function ofeE for a given several values ofeB. In the pure electric field case (
eB→0 ), the three-parameters monotonically decrease with an increase ineE , and at a pseudo-critical field strengtheEχ,Cc , the chiral symmetry is partially restored and deconfinement transitions occurred; hence, the nature of the transitions here becomes smooth cross-over. Accordingly, we observed the chiral rotation or chiral electric inhibition effect in the contact interaction model, as already predicted by other effective models of QCD [1, 2, 25-29]. For non-zeroeB values, we determined an interesting behavior for all the three parameters, as a function ofeE . All three parameters are enhanced for given smaller to larger values ofeB , except the chiral symmetry restoration and deconfinement regions, where all the parameters are suppressed by highereB values. The pseudo-critical field strengthEχ,Cc decreases with an increase ineB , and at a pseudo-criticaleBc≈0.3 GeV2 , the transition changes from cross-over to first order. The nontrivial behaviors of all three parameters represent the competition between the magnetic catalysis and electric inhibition effects, both induced byeB in the presence of paralleleE [29]. As elucidated and validated in Ref. [25], the cause of the chiral inhibition effect is the second Lorentz invariant of the electromagnetic field,E⋅B . The magnitude ofeEχ,Cc at which the chiral symmetry restoration and deconfinement occurred are triggered from the inflection point of the electric gradients ofand∂eE˜τ−1ir , as presented in Figs. 8 and 9, respectively. In the pure electric case, the chiral symmetry restoration and deconfinement transition occur ateEχ,Cc≈0.34 GeV2 . For several given values ofeB≠0 ,eEχ,Cc decreases with an increase ineB . We determined a smooth cross-over phase transition up to the pseudo-critical magnetic field strengtheBc≈0.3 GeV2 , where the transitions take the first-order nature. Here,eEχ,Cc remains constant in the regioneB≈[0.3−0.54] GeV2 and then increases with larger values ofeB , as shown in Fig. 10. A similar behavior has already been demonstrated in [29]. The boundary point where the cross-over phase transition ends and the first order phase transition starts is known as a critical end point, and its co-ordinates are at(eBp≈0.3,eEp≈0.33) GeV2 .Figure 8. (color online) Electric gradient of the quark-antiquark condensate
as a function of
eE for fixed values ofeB . At a particular fixed value ofeB=0.3 GeV2 , the electric gradientdiverges at
eEχc,≈0.33 GeV2 and above this value, the first order phase transition occurs. -
In this section, we elucidate the behaviors of the dynamical mass, condensate, and confinement length scale at a finite temperature and in the presence of parallel electric and magnetic fields. We also explore the IEC, MC, and IMC phenomena, as well as the competition among them. Finally, we sketch the QCD phase diagram.
The finite temperature version of the gap equation, Eq. (13), in the presence of parallel electric and magnetic fields can be obtained by adopting the standard convention for momentum integration:
∫d4k(2π)4→T∑n∫d3k(2π)3,
(17) and the four momenta
k→(ωn,→k) , withωn=(2n+1)πT , represent the fermionic Matsubara frequencies. The Lorentz structure does not preserve anymore at finite temperature. By making the following replacements in Eq. (13),∫dk1(2π)dk2(2π)dk3(2π)dk4(2π)→T∑n∫dk1(2π)dk2(2π)dk3(2π),
and
k4→ωn , we have^Mf=mf+16αeff3T∑n∑f=u,d∫ˆτ2irτ2uvdτ^Mfe−τˆM2f×∫dk1(2π)dk2(2π)dk3(2π)e−τ((ω2n+k23)tan(|QfE|τ)|QfE|τ+(k21+k22)tanh(|QfB|τ)|QfB|τ),
(18) with
ˆMf=Mf(eE,eB,T) . Performing sum over Matsubara frequencies and integrating overk′s , the gap equation can be written asˆMf=mf+αeff3π2∑f=u,d∫ˆτ2irτ2uvdτˆMfe−τˆM2fΘ3(π2,e−|QfE|4T2tan(|QfE|τ))×[|QfE|tan(|QfE|τ)|QfB|tanh(|QfB|τ)],
(19) where
Θ3(π2,e−x) is the third Jacobi's theta function. The confinement scaleˆτir here slightly varies with T,eE , andeB , and it takes the formˆτir=τirMfˆMf.
(20) Here,
Mf=Mf(0,0,0) is the dynamical mass ateB=eE=T=0 , whereasˆMf=Mf(eE,eB,T) is the variation of the dynamical mass at finiteeB ,eE , and T. The quark-antiquark condensate is given by the following.First, we consider the case of a pure electric field at a finite temperature. The numerical solution of Eq. (19) as a function of T, for different given values of
eE , is presented in Fig. 11. The dynamical massˆMf monotonically decreases with the increase of T until the dynamical chiral symmetry is partially restored. The response ofeE is to suppress the dynamical mass. The quark-antiquark condensate, Eq. (14), and confinement length scale, Eq. (15), as a function of T for various given values ofeE are depicted in Figs. 12 and 13, respectively. We infer that both parameters decrease with temperature T, and at a pseudo-critical temperatureTχ,Cc , the chiral symmetry is partially restored, and deconfinement occurs. We note that the electric fieldeE suppresses both parameters in the low-temperature regions and also reduces the pseudo-critical temperatureTχ,Cc . In Figs. 14 and 15, we plotted the thermal gradients of the condensate and the confinement length scale, respectively. The peaks in the thermal gradients of both parameters shift toward the low-temperature regions upon increasing the value ofeE . Consequently, the critical temperatureTχ,Cc decreases with an increase in electric field strengtheE ; hence, inverse electric catalysis is observed at a finite temperature in the proposed contact interaction model. Therefore, the observations of this study are consistent with other effective models of QCD [28-30]. The magnitude of the critical temperatureTχ,Cc≈0.22 GeV2 is obtained from the inflection points of the thermal gradients ofand
∂Tˆτ−1ir .Figure 11. (color online) Dynamical mass, Eq. (19), as a function of temperature for various given values of electric field strength
eE. Figure 12. (color online) Quark-antiquark condensate, Eq. (21), plotted as a function of temperature for various given values of electric field
eE. Figure 13. (color online) Behavior of confinement scale, Eq. (20), plotted as a function of temperature for different given values
eE. Figure 14. (color online) Thermal gradient of the quark-antiquark condensate
plotted as a function of temperature T for various values of the electric field
eE . The peaks in the derivatives shift toward low temperature regions from small to large values ofeE. Figure 15. (color online) Thermal gradient of the confinement scale
∂Tˆτ−1ir plotted as a function of temperature T for various values ofeE. Second, we consider the case of a pure magnetic field at finite temperature T. We plot the thermal gradient of the quark-antiquark condensate and the confinement length scale, as presented in Figs. 16 and 17, respectively. We note that the inflection points in both parameters shift toward the higher temperatures. In other words,
eB enhances the pseudo-critical temperatureTχ,Cc of chiral symmetry restoration and deconfinement. Hence, in a pure magnetic case and at a finite temperature, we observe the magnetic catalysis phenomenon, which has already been observed in the CI model [22, 63].Figure 16. (color online) Thermal gradient of the quark-antiquark condensate
plotted as a function of temperature T for various values of magnetic field
eB . From the plots, it is evident that the peaks shift toward higher temperatures.Figure 17. (color online) Thermal gradient of the confinement scale
∂Tˆτ−1ir plotted as a function of temperature for several values of the magnetic field strength fieldeB. In most effective model calculations of QCD, it is well demonstrated that to reproduce the inverse magnetic catalysis effect as predicted by the lattice QCD [13, 14], the effective coupling must be taken as magnetic field dependent [17, 22, 28, 63] or both temperature and magnetic field dependent [16, 19, 23, 63, 64]. In the case of this study, we just employed the following functional form of the
eB -dependent effective couplingαeff(eB) [22], where the coupling decreases with the magnetic field strength asαeff(κ)=αeff(1+aκ2+bκ31+cκ2+dκ4),
(22) here
κ=eB/Λ2QCD , withΛQCD=0.24 GeV. The parameters a, b, c, and d were extracted to reproduce the behavior of critical temperatureTχ,Cc for the chiral symmetry restoration and deconfinement in the presence of magnetic field strength, obtained by lattice QCD simulations [13, 14]. The thermal gradients of the condensate and the confinement length scale, with the magnetic field dependent coupling, Eq. (22), is plotted in Figs. 18 and 19, respectively. It can be observed that the criticalTχ,Cc decreases with an increase ineB ; hence, the inverse magnetic catalysis can be observed at a finite T.Figure 18. (color online) Thermal gradient of the condensate with magnetic field dependent coupling, Eq. (22), plotted as a function of temperature for several values of magnetic field strength. The IMC effect is depicted in this figure.
Figure 19. (color online) Behavior of
∂Tˆτ−1ir with the magnetic field dependent coupling, Eq. (22), as a function of temperature for several values of magnetic field strengtheB. In Fig. 20, we sketch the combined phase diagram in the
Tχ,Cc−eE,eB planes, where we demonstrated the inverse electric catalysis, magnetic catalysis, and inverse magnetic catalysis (with magnetic dependent coupling). In the pure electric background, the solid-black curve indicates that the critical temperatureTχ,Cc decreases with an increase ineE ; hence, the electric field strength inhibits the chiral symmetry breaking and confinement. In the pure magnetic limit (without magnetic field dependent coupling), the magnetic fieldeB enhances the critical temperatureTχ,Cc , and thuseB acts as a facilitator of chiral symmetry breaking and confinement. If we use the magnetic field dependent coupling, it can be observed that the critical temperatureTχ,Cc decreases as the magnetic field strengtheB increases (red dotted-dashed curve). In this case,eB acts as an inhibitor of the chiral symmetry and confinement.Figure 20. (color online) Combined phase diagram in the
Tχ,Cc vseE,eB planes of chiral symmetry breaking and confinement. The plot shows the evalution of IEC as well as MC without and IMC witheB -dependent coupling, Eq. (22). TheTχ,Cc values are obtained from the inflection points in∂Tˆσ and∂Tˆτ−1ir. Third, we adopt both the non-zero values of
eE andeB , and draw the phase diagram in theTχ,Cc−eE plane for various given values ofeB , as shown in Fig. 21. At this point, the competition between IEC vs MC starts: theeB tends to catalyze the chiral symmetry breaking and confinement; consequently,Tχ,Cc is enhanced. In contrast,eE tries to inhibit the chiral phase transition and tends to suppresTχ,Cc ; finally, it is concluded with a combined inverse electromagnetic catalysis (IEMC). This may be different for a very strongeB , where theeB dominates over theeE . Next, we consider theeB -dependent coupling, Eq. (22), and sketch the phase diagramTχ,Cc vseE for various given values ofeB in Fig. 22. It can be observed that the effect of paralleleE andeB witheB -dependent coupling tends to reduceTχ,Cc . Therefore, botheE andeB produce IEC and IMC simultaneously. -
In this work, we studied the influence of uniform, homogeneous, and external parallel electric and magnetic fields on the chiral phase transitions. In this context, we implemented the Schwinger-Dyson formulation of QCD, with a gap equation kernel comprising a symmetry preserving vector-vector contact interaction model of quarks in a rainbow-ladder truncation. Subsequently, we adopted the well-known Schwinger proper-time regularization procedure. The findings of this study are presented as follows.
At zero temperature, in the pure magnetic background, the magnetic field facilitated the dynamical chiral symmetry breaking and confinement; hence, we observed the magnetic catalysis phenomenon. However, the electric field tends to restore chiral symmetry and deconfinement above the pseudo-critical electric field
eEχ,Cc≈0.34 GeV2 , i.e., the chiral rotation effect demonstrated in the proposed model. We observed that the electric field acted as an inhibitor of the chiral symmetry breaking and confinement. When botheE andeB were considered, we determined the magnetic catalysis effect for small given values ofeE up to and above the critical electric field strengtheEc≈0.33 GeV2 , where the other parameters exhibited the de Haas-van Alphen oscillatory type behavior in the regioneB=[0.3−0.54] GeV2 . In this particular region,eE dominated overeB , and we observed the electric chiral inhibition effect, while above that region whereeB was superior toeE , we observed magnetic catalysis again. We also realized that the pseudo-critical strengtheEχ,Cc was suppressed with the increase ineB , and the transition nature was determined to be the smooth cross-over nature up to and above the pseudo-critical magnetic field strengtheBχ,Cc≈0.3 GeV2 , where the transition suddenly changed to the first order transition. Subsequently, we located the position of the critical endpoint at(eBp≈0.3,eEp≈0.33) GeV2 . We further determined thateEχ,Cc initially remains constant in the limited regioneB=[0.3−0.54] GeV2 and then increases with larger values ofeB .Finally, we sketched the phase diagram at a finite temperature and in the presence of parallel electric and magnetic fields. At a finite T, in the pure electric field limit, we determined that the pseudo-critical temperature decreased as we increased
eE ; hence, the inverse electric catalysis was observed. However, for the pure magnetic field background, we observed the magnetic catalysis effect in the mean-field approximation and inverse magnetic catalysis witheB -dependent coupling. The combined effect of botheE andeB onTχ,Cc yielded the inverse electromagnetic catalysis, with and withouteB− dependent effective coupling of the model.Qualitatively and quantitatively, the predictions of the presented CI-model agree well with results obtained from other effective QCD models, as well as modern Lattice QCD results. In the near future, we plan to extend this work to study the Schwinger pair production rate, including the dynamical chiral symmetry breaking for a higher number of colors, flavors, and in a parallel electromagnetic field. We are also interested in extending this work to study the IEMC phenomenon with the electric field and temperature-dependent coupling. Regarding the dependence of the model coupling on the considered electric field, there is no first-principle simulation for comparisons because lattice simulations in the presence of real electric background fields are not feasible to date. However, progress in this case has commenced, and results will be published somewhere else. The next strategy will be to study the properties of light hadrons in the background of parallel electric and magnetic fields.
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The author acknowledges A. Bashir, A. Raya, R.L.S. Farias, and E. V. Gorbar for their valuable suggestions in the process of completing this work. The author also thanks his colleagues at the Institute of Physics, Gomal University, Pakistan.
