1. Introduction
The relativistic heavy ion collider and the large hadron collider focus on the new state of quark gluon plasma (QGP) at sufficiently high temperature and/or density, which expands as a hydrodynamics expansion of an ideal fluid. The color screening of heavy quarks in a deconfined medium is suggested to lead to a signal of the formation of QGP [1]. To reflect the anisotropic structure with a rapid longitudinal expansion of QGP created in HICs, the anisotropic Coulomb potential can be produced through an angle-averaged screening mass [2]. The screening of electric fields in quantum chromodynamics (QCD) is initially studied by the leading order term depending on the temperature. And it is mainly modified by the nonperturbative characteristic when the temperature is not very larger. It is widely accepted that the important medium effects are entirely characterized by the Debye mass and screening scale in perturbative and nonperturbative contribution.
In literature, the phenomenological models overcome the difficulty of the QCD theory at finite chemical potentials. The quark quasiparticle model, as an extended bag model, has been developed in studying the bulk properties of the dense quark matter at finite density and temperature. The advantage of the quasiparticle model is the introduction of the medium-dependent quark mass scale to reflect the nonperturbative QCD properties [3]. In particular, the quark confinement and vacuum energy density can be well described by the density-dependent bag constant. To describe the strong interaction effects in terms of effective fugacities, the effective fugacity quasiparticle model was proposed by Chandra and Ravisankar [4, 5]. The transport properties of the QGP have been well investigated by the noninteracting/weakly interacting particles with effective masses [6]. These models are good at studying the collective excitation of hot QCD medium [7, 8], but behave poorly in investigating the chiral symmetry of QCD-like matter. In contrast, the Nambu−Jona-Lasinio (NJL) model has proved to be very successful in the description of the spontaneous breakdown of chiral symmetry exhibited by the true (nonperturbative) QCD vacuum [9]. It explains very well the spectrum of the low-lying mesons, which is intimately connected with chiral symmetry as well as many other low energy phenomena of strong interaction. The deconfinement transition related to the breaking of the Z(3) symmetry has also been produced at the finite temperature in Polyakov–NJL model. Recently, the confinement properties have been extended to be investigated at zero temperature in the PNJL0 model [10]. The confining potential of the Polyakov loop is essential for the Z(3) symmetry at extreme densities in the PNJL model [11–13] and the nonlinear σ model [14, 15]. In this paper, the NJL model is employed to investigate the medium effect on the chiral restoration of QCD in strong magnetic fields.
It is known that the interaction potential between quarks will be modified in a hot and dense QCD medium [16]. The medium modification effect due to the Debye screening mass is responsible for the quarkonium binding energy and dissociation chemical potential [17]. The original work on the Debye mass is the measurement of the Debye mass with gauge invariant operators in the high-temperature phase using lattice simulations [18]. Nonperturbative Debye mass in finite temperature QCD is an essential quantity to describe the coherent static interaction. The electric screening mass feels influenced by both the hot QCD medium effects and the magnetic field [19], which is important for phenomenological discussion of the formation of QGP. The presence of the strong magnetic field can drastically change the properties of matter [20]. Consequently, it promotes a change in the size and location of the chiral phase transition, and the magnetic field distribution would have an important effect on the neutron stars [21, 22]. Our aim is to study the behavior of the Debye mass in the chiral restoration and suggest that there exists a relation between the Debye mass or the susceptibility with respect to the temperature and the signal of the chiral crossover.
This paper is organized as follows. In section 2 , we present the thermodynamics of the magnetized quark matter in the two-flavor NJL model. In section 3 , the numerical results for the typical variation of the Debye mass are shown in the chiral restoration at finite temperatures. And the discussions are focused on the signals characterized by the Debye mass and the inverse magnetic catalysis (IMC) realized by the thermomagnetic running coupling constant G(B, T). The last section is a short summary.
2. Thermodynamics of NJL model in strong magnetic fields
In the presence of strong external magnetic fields, the Lagrangian density of the two-flavor NJL model in a strong magnetic field is given as
$\begin{eqnarray}{{ \mathcal L }}_{{\rm{NJL}}}=\bar{\psi }(i\,| \rlap{/}{D}-m)\psi +G[{\left(\bar{\psi }\psi \right)}^{2}+{\left(\bar{\psi }{\rm{i}}{\gamma }_{5}\vec{\tau }\psi \right)}^{2}],\end{eqnarray}$
where the covariant derivative Dμ = ∂μ + iQAμ represents the coupling of the quarks to the electromagnetic field. The Q is the electric charge matrix acting in the flavor space Q = diagf (qu, qd) = diagf (2e/3, −e/3) with e > 0. A sum over flavor and color degrees of freedom is implicit. In the mean-field approximation [9], the dynamical quark mass is $\begin{eqnarray}{M}_{i}=m-2G\langle \bar{\psi }\psi \rangle ,\end{eqnarray}$
where the quark condensates include u and d quark contributions as $\langle \bar{\psi }\psi \rangle \equiv \phi ={\sum }_{i=u,d}{\phi }_{i}$. The dynamical mass depends on both flavor condensates. Therefore, the same mass Mu = Md = M is available for u and d quarks. The contribution from the quark with flavor i includes the vacuum, the magnetic field, and the medium term as $\begin{eqnarray}{\phi }_{i}={\phi }_{i}^{\mathrm{vac}}+{\phi }_{i}^{\mathrm{mag}}+{\phi }_{i}^{\mathrm{med}}.\end{eqnarray}$
Since the model is not renormalizable at zero temperature, we should introduce a cutoff Λ in the 3 momentum space as in the usual way. The vacuum contribution ${\phi }_{i}^{\mathrm{vac}}$ is $\begin{eqnarray}\begin{array}{rcl}{\phi }_{i}^{\mathrm{vac}} & = & -\displaystyle \frac{{{MN}}_{c}}{2{\pi }^{2}}\left[{\rm{\Lambda }}\sqrt{{{\rm{\Lambda }}}^{2}+{M}^{2}}\right.\\ & & \left.-{M}^{2}\mathrm{ln}\left(\displaystyle \frac{{\rm{\Lambda }}+\sqrt{{{\rm{\Lambda }}}^{2}+{M}^{2}}}{M}\right)\right].\end{array}\end{eqnarray}$
The magnetic field and medium contributions to the quark condensation are [23, 24] $\begin{eqnarray}\begin{array}{rcl}{\phi }_{i}^{\mathrm{mag}} & = & -\displaystyle \frac{M| {q}_{i}| {{BN}}_{c}}{2{\pi }^{2}}\left\{\mathrm{ln}[{\rm{\Gamma }}({x}_{i})]-\displaystyle \frac{1}{2}\mathrm{ln}(2\pi )\right.\\ & & \left.+{x}_{i}-\displaystyle \frac{1}{2}(2{x}_{i}-1)\mathrm{ln}({x}_{i})\right\},\end{array}\end{eqnarray}$
$\begin{eqnarray}{\phi }_{i}^{\mathrm{med}}=\sum _{{k}_{i}=0}^{\infty }{a}_{{k}_{i}}\displaystyle \frac{M| {q}_{i}| {{BN}}_{c}}{2{\pi }^{2}}\int \displaystyle \frac{{f}_{i}({\omega }_{i})}{{\omega }_{i}}{dp},\end{eqnarray}$
where ${a}_{{k}_{i}}=2-{\delta }_{{k}_{i}0}$ and ki are the degeneracy label and the Landau quantum number, respectively. The dimensionless quantity xi is defined as xi = M2/(2∣qi∣B). In the second equation above, the medium contribution with finite chemical potential is introduced in the fermion distribution function as $\begin{eqnarray}{f}_{i}({\omega }_{i})=\displaystyle \frac{1}{1+\exp [({\omega }_{i}-{\mu }_{i})/T]}.\end{eqnarray}$
The effective quantity ${\omega }_{i}=\sqrt{{p}_{z}^{2}+{s}_{i}^{2}}$ sensitively depends on the magnetic field through ${s}_{i}=\sqrt{{M}^{2}+2{k}_{i}| {q}_{i}| B}$.The total thermodynamic potential density in the mean-field approximation reads
$\begin{eqnarray}{\rm{\Omega }}=\displaystyle \frac{{\left(M-{m}_{0}\right)}^{2}}{4G}+\sum _{i=u,d}{{\rm{\Omega }}}_{i},\end{eqnarray}$
where the first term is the interaction term. In the second term, Ωi is defined as ${{\rm{\Omega }}}_{i}={{\rm{\Omega }}}_{i}^{\mathrm{vac}}+{{\rm{\Omega }}}_{i}^{\mathrm{mag}}+{{\rm{\Omega }}}_{i}^{\mathrm{med}}$. The vacuum, the magnetic field, and the medium contributions to the thermodynamic potential are $\begin{eqnarray}{{\rm{\Omega }}}_{i}^{\mathrm{vac}}=\displaystyle \frac{{N}_{c}}{8{\pi }^{2}}\left[{M}^{4}\mathrm{ln}\left(\displaystyle \frac{{\rm{\Lambda }}+{\epsilon }_{{\rm{\Lambda }}}}{M}\right)-{\epsilon }_{{\rm{\Lambda }}}{\rm{\Lambda }}({{\rm{\Lambda }}}^{2}+{\epsilon }_{{\rm{\Lambda }}}^{2})\right],\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}{{\rm{\Omega }}}_{i}^{\mathrm{mag}}=-\displaystyle \frac{{N}_{c}{\left(| {q}_{i}| B\right)}^{2}}{2{\pi }^{2}}\\ \quad \times \left[{\zeta }^{{\prime} }(-1,{x}_{i})-\displaystyle \frac{1}{2}({x}_{i}^{2}-{x}_{i})\mathrm{ln}({x}_{i})+\displaystyle \frac{{x}_{i}^{2}}{4}\right],\end{array}\end{eqnarray}$
$\begin{eqnarray}{{\rm{\Omega }}}_{i}^{\mathrm{med}}=-T\sum _{{k}_{i}=0}^{\infty }{a}_{{k}_{i}}\displaystyle \frac{| {q}_{i}| {{BN}}_{c}}{2{\pi }^{2}}\int {\rm{d}}p\,\left\{\mathrm{ln}\left[1+\exp (-\displaystyle \frac{{\omega }_{i}}{T})\right]\,\right\},\end{eqnarray}$
where the quantity εΛ is defined as ${\epsilon }_{{\rm{\Lambda }}}=\sqrt{{{\rm{\Lambda }}}^{2}+{M}^{2}}$. The $\zeta (a,x)={\sum }_{n\,=\,0}^{\infty }\tfrac{1}{{\left(a+n\right)}^{x}}$ is the Riemann–Hurwitz zeta function. The $\zeta ^{\prime} (-1,x)=\tfrac{{\rm{d}}\zeta (z,x)}{{\rm{d}}z}{| }_{z=-1}$ is defined in the magnetic term.The electromagnetic Debye screening (in terms of the self-energy) has been influenced by both the hot QCD medium effects and the magnetic field. The Debye mass mD for electromagnetic screening can be obtained by taking the static limit of the 00 component of the photon polarization tensor [16, 19, 25].
$\begin{eqnarray}{m}_{D}^{2}=-4\pi {\alpha }_{s}{g}_{n}\int \displaystyle \frac{{{\rm{d}}}^{3}p}{{\left(2\pi \right)}^{3}}\displaystyle \frac{\partial }{\partial {\omega }_{k}}f({\omega }_{k}),\end{eqnarray}$
where gn = 4NcNf is the degeneracy factor for quark-antiquark, spin, and color freedom in the strong magnetic field. The electric screening mass encodes the medium effect in terms of the temperature and the baryon chemical potential. In the presence of a strong magnetic field, the Debye mass of the two-flavor quarks can be derived as $\begin{eqnarray}\begin{array}{rcl}{m}_{D}^{2} & = & G{{\rm{\Lambda }}}^{2}\displaystyle \sum _{i=u}^{d}\displaystyle \sum _{{k}_{i}=0}^{\infty }{g}_{{k}_{i}}\displaystyle \frac{| {q}_{i}B| }{2{\pi }^{2}}{\displaystyle \int }_{0}^{\infty }\displaystyle \frac{{{\rm{d}}p}_{z}}{T}f({\omega }_{{k}_{i}})\\ & & \times [1-f({\omega }_{{k}_{i}})]+({\mu }_{i}\to -{\mu }_{i})\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}=G{{\rm{\Lambda }}}^{2}\displaystyle \sum _{i=u}^{d}\displaystyle \sum _{{k}_{i}=0}^{\infty }{g}_{{k}_{i}}\displaystyle \frac{| {q}_{i}B| }{2{\pi }^{2}}{\displaystyle \int }_{0}^{\infty }\displaystyle \frac{{\rm{d}}p}{T}\\ \quad \times \,\displaystyle \frac{{{\rm{e}}}^{-(\sqrt{{p}^{2}+{M}^{2}+2{k}_{i}| {q}_{i}| B}-{\mu }_{i})/T}}{{\left(1+{{\rm{e}}}^{-(\sqrt{{p}^{2}+{M}^{2}+2{k}_{i}| {q}_{i}| B}-{\mu }_{i})/T}\right)}^{2}}+({\mu }_{i}\to -{\mu }_{i}).\end{array}\end{eqnarray}$
Accordingly, the remaining degeneracy factor changes to ${g}_{{k}_{i}}={N}_{c}\times (2-{\delta }_{0{k}_{i}})$ indicating the color and spin degeneracy for the landau levels of quarks.3. Numerical result and conclusion
In the present calculation with the SU(2) NJL model, the following parameters are adopted: mu = md = 5.5 MeV for the up and down quarks current masses, Λ = 650 MeV and G = 4.50373 GeV−2 for the three momentum cutoff and the coupling constant.
The Lattice QCD indicates that there is no genuine phase transition at the zero chemical potential but a crossover. The pseudocritical temperature for the crossover can be obtained by the behavior of the dynamical mass in the gap equation. In figure 1 at the magnetic field eB = 0.2 GeV2, the dynamical quark masses as the chiral order parameter are plotted as the temperature increases on the top-left panel. It is clear that the vacuum mass is almost independent of the temperature and the chemical potential. As expected, the pseudocritical temperature of the chiral crossover is indicated by the susceptibility of the dynamical mass with respect to the temperature on the low-left panel. The pseudocritical temperature marked by the peak is decreased as the chemical potential increases.
Figure 1. Variation of the dynamical mass (left panel) and the Debye mass of the quark (right panel) with temperature for different chemical potentials μ = 0 by solid line, μ = 50 MeV by dashed line, μ = 100 MeV by dotted line at the magnetic field eB = 0.2 GeV2. |
In fact, the role of the quark dynamical mass in the bulk matter to signal the chiral crossover can be replaced by the Debye mass to some extent. According to the definition in equation (12 ), the dynamically produced mass would affect the fermion distribution in turn, and consequently changes the Debye mass of quarks. On the top-right panel in figure 1, it is shown that the Debye mass increases with the temperature at different chemical potentials μ = 0, 50, 100 MeV. There is an inflection point on the monotonous ascending lines, which reflects the dependence of the fermion distribution on the dynamical quark mass. The susceptibility of the Debye mass with respect to the temperature, dmD/dT, would display a smooth peak to indicate the chiral crossover on the low-right panel. However, as the chemical potential increases up to the magnitude of μ = 100 MeV, there are apparently two peaks on the line due to the rapid increase of the distribution function. The two peaks are meaningless and indistinguishable for the determination of the chiral restoration. One can be taken as a signal for the chiral crossover by comparison with the chiral condensate. The other is only due to a mathematical property and could be understood by the fact that the occupation number of quarks with a given momentum would be enhanced mostly by the increasing chemical potential. In order to remove the additional peak due to the sudden increase of the distribution function, it is necessary to construct a quantity dependent on the Debye mass to signal the chiral transition at finite chemical potentials.
In figure 2, the magnetic fields eB = 0.2, 0.4, 0.6 GeV2 are marked by solid, dashed and dotted lines. On the top-left panel, the thermal mass in the vacuum at the fixed chemical potential μ = 100 MeV is shown to be apparently enhanced by the magnetic fields, which is consistent with the general statement that a constant magnetic field leads to the generation of a fermion dynamical mass. The movement of the peak of the susceptibility − dM/dT to higher temperatures on the low-left panel indicates the so-called magnetic catalysis (MC) effect.
Figure 2. Left panel: the dynamical mass and the susceptibility are shown as functions of temperature at the chemical potential μ = 100 MeV and different values of the magnetic fields eB = 0.2, 0.4, 0.6 GeV2. Right panel: the Debye mass and the quantity TdmD/dT are shown as functions of temperature. |
Accordingly, the Debye mass is increased as the temperature increases on the top-right panel. There is an inflection point to change the slope of the line at the pseudocritical temperature of the chiral crossover. Since the additional peak of the dmD/dT occurs at lower densities, we can try to construct the product of T and dmD/dT to eliminate the peak. The single peak location on the bottom-right panel can signal the chiral crossover. This can be understood that the magnetic field influences the Debye screening mass by entering through the quark loop in the gluon propagator [16]. The Debye mass is a fundamental quantity of plasma that can measure the color-electric screening effects in the medium. And it might get large modifications in the presence of the magnetic field and high temperature due to the fact that the dynamical mass affects the fermion distribution. In literature, the Debye mass was used to define the effective coupling constant in strong magnetic fields. The Debye mass susceptibility would be related to the discontinuous behavior in the chiral restoration to some extent. The peak of the TdmD/dT indicates the increase of the pseudocritical temperature with the increasing magnetic fields, which is the same result of the MC effect revealed by the susceptibility of the dynamical mass on the bottom-left panel. It can be understood that the screening effect characterized by the Debye mass shows a response to the phase structure and displays a cusp structure at the pseudocritical temperature. So it can be concluded that the MC effect can be equivalently indicated by typical variations of the dynamical mass and the quark Debye mass with the coupling constant G interaction.
In figure 3, we investigate the chiral restoration under the fixed coupling constant G at zero chemical potential (left panel) and the finite chemical potential μ = 100 MeV (right panel). The four values of magnetic fields eB = 0.2, 0.4, 0.6, 0.8 GeV2 are marked by the solid, the dashed, the dotted, and the dash-dotted lines respectively. At μ = 0, the peak of the dmD/dT shows the pseudocirtical temperature is increased by the increasing magnetic fields. At μ = 100 MeV, the same MC effect is perfectly demonstrated by the single peak of the quantity TdmD/dT.
Figure 3. The susceptibility dmD/dT at μ = 0 (left panel) and TdmD/dT at μ = 100 MeV (right panel) of Debye mass as functions of temperatures with the coupling constant G. |
In the original version of NJL model, the coupling constant G is used to describe the four-fermion interaction and is determined by the pion decay constant and quark condensate from QCD sum rules. In order to reflect the asymptotic freedom, the more reasonable parameterization by fitting the lattice QCD suggests that the coupling constant should run with the magnetic field strength. In literature, the coupling constant was taken as a function of chemical potential and temperature at the first order correction [26] and up to two loops [3]. In order to reflect the asymptotic freedom in the thermal region, we adopt the running G(B, T) for the comparison [27]
$\begin{eqnarray}G(B,T)=c(B)[1-\displaystyle \frac{1}{1+{{\rm{e}}}^{\beta (B)[{T}_{a}(B)-T]}}]+s(B)\end{eqnarray}$
where the parameters c, β, s, and Ta are adopted as the values of [27]. The parameterization was obtained by accurately fitting from lattice QCD results for the magnetized quark condensates. The quantitative changes of the running of the coupling G(B, T) have been expected to give rise to a physical implication in the signatures of the chiral transition.The dynamical mass of quarks is obtained under the running coupling constant G(B, T) at zero chemical potential in figure 4. The different magnetic fields eB = 0.2, 0.4, 0.6, and 0.8 GeV2 are marked by the solid, the dashed, the dotted, and the dashed-dotted lines respectively. On the left panel, the M in the vacuum is enhanced by the magnetic field. As the temperature increases, the M shows a decreasing slope indicating a chiral crossover. The corresponding pseudocritical temperature can be shown by the peak of the thermal susceptibility on the right panel. The movement towards the low temperature region indicates the IMC effect.
Figure 4. The dynamical mass and the thermal susceptibility as functions of temperatures with the running coupling G(B, T) at μ = 0. |
In figure 5, the running coupling constant G(B, T) is employed in the calculation of the Debye mass and the chiral restoration. Similar to figure 3, the two cases of μ = 0 and μ = 100 MeV are investigated at the different magnetic fields. As usual, the Debye mass increases as the temperature increases. Interestingly, it is found that there is a peak of the Debye mass itself. No matter how large the chemical potential is, the Debye mass of quarks under the thermomagnetic running coupling G(B, T) could directly signal the chiral restoration. Moreover, the IMC effect is realized in contract to the MC effect with the fixed coupling G. The quantity TdmD/dT is shown at μ = 0 (left panel) and μ = 100 MeV (right panel) as functions of temperatures in figure 6. It can be found that the chiral crossover can also be characterized by the discontinuous behavior of the Debye mass. As the temperature increases, the TdmD/dT increases until the chiral crossover happens, which is marked by a sudden peak. The peak position would move to a lower temperature as the magnetic field increases. It is emphasized again that the running coupling constant would play an important role in the IMC of the chiral restoration discovered by lattice QCD. Furthermore, the IMC can be equivalently demonstrated by the dynamical mass in figure 4 and the Debye mass in figures 5 and 6.
Figure 5. The Debye mass as functions of temperatures with the running coupling G(B, T) at μ = 0 and μ = 100 MeV. |
Figure 6. The TdmD/dT at μ = 0 (left panel) and μ = 100 MeV (right panel) as functions of temperatures with the running coupling G(B, T). |
4. Summary
In this paper, we have investigated the Debye mass of two flavor quark matter in the framework of the NJL model at strong magnetic fields. The electromagnetic screening Debye mass is obtained by the 00 component of the photon polarization tensor. In a uniform phase, the Debye mass increases with the temperature, the chemical potential and the magnetic field. The Debye screening mass is determined by the nonperturbative quark distribution function. It has been demonstrated to play a significant role in determining the pseudocritical temperature for the chiral crossover. Especially, the derivation of the quark Debye mass with respect to the temperature can signal the position of the chiral crossover at zero or very low chemical potentials. At finite chemical potentials, the two peaks of the susceptibility dmD/dT occur due to the rapid increase of the distribution function of the fermions. They are hard to distinguish from the chiral signal. Fortunately, it has been found that the product of the temperature and the Debye mass susceptibility, TdmD/dT, can remove the additional peak at finite chemical potential. Consequently, the single peak for the TdmD/dT could indicate the chiral crossover as the same MC result as the dynamical mass at the fixed coupling G.
In order to reflect the asymptotic freedom phenomenologically, the refined thermomagnetic running coupling constant G(B, T) by fitting the lattice QCD is employed in our investigation. Interestingly, the ordinary inflection point on the slope of the Debye mass becomes an evident peak at the pseudocritical temperature. Moreover, the pseudocritical temperature would decrease with the increasing strength of the magnetic field. The IMC effect is directly indicated by the behavior of the Debye mass itself. In conclusion, both the Debye mass and the quark condensate can equivalently trace out the signature of the chiral crossover. The running coupling constant plays an important role in reproducing the IMC effect of the chiral restoration. However, it seems to be impossible to directly test the Debye mass in experiments. Our aim is to display the possible behavior of the Debye mass related to the chiral restoration. We hope that the present study could inspire further investigation into the physical explanation of the Debye screening phenomenon in the chiral restoration process.