Schwinger Pair Production in QCD from Flavor-Dependent Contact Interaction Model of Quarks

Historically, the phenomenon of electron-positron pair production from a non-perturbative vacuum in the presence of electric field strength $eE$ (where $e$ is an electric charge) was first introduced by Fritz Sauter [1], Heisenberg and Euler [2]. However, the complete theoretical framework provided by Julian Schwinger in the field of quantum electrodynamics (QED) [3], and named after him as the Schwinger effect or Schwinger mechanism. Later on, this phenomenon has been widely studied in the field of quantum chromodynamics (QCD), see for example, Refs. [4, 5, 6, 7, 8]. As we know that the QCD is a theory of strong color force among the quarks and gluons possess two special properties: The asymptotic freedom (i.e., the quarks interacts weakly at a short distancesor high energy scale) and the quark confinement (i.e., the quarks interacts strongly at a large distance or low energy scale). The Schwinger effect in QCD is thus, the production of quark-antiquark pairs in the presence of a strong electric field $eE$. The quark-antiquark pair production can be calculated from the Schwinger pair production rate $\Gamma$ and is, defined as the probability, per unit time and per unit volume that a quark-antiquark pair is created by the constant electric field $eE$.
Another important property of low-energy QCD is the dynamical chiral symmetry breaking, which is related to the dynamical mass generation of the quarks. The strong electric field tends to restore the dynamical chiral symmetry and as a result, the dynamically generated quark mass suppresses with the increase of $eE$. It can be understood by realizing that being closer together, the quark and antiquark pairs are reaching the asymptotic freedom regime faster by reducing the interaction strength as the intensity of the electric field $eE$ increases. Such a phenomenon is sometimes referred to as the chiral electric inhibition effect [9, 6, 10, 11, 12, 13, 14], or the chiral electric rotation effect [15] or inverse electric catalysis(IEC) [16, 17, 18]. The nature of the dynamical chiral symmetry breaking–restoration and confinement–deconfinement phase transition is of second-order when the bare quark mass $m=0$ (i.e., in the chiral limit) while cross-over when $m\neq 0$. It has been argued in Refs.  [13, 14, 17] that the pair production rate $\Gamma$, increases quickly near some pseudo-critical electric field $eE_{c}$, where the chiral symmetry is restored.
The study of the effect of the electric field on the chiral phase transition plays a significant role in Heavy-Ion Collision experiments. In such experiments, the magnitude of the electric and magnetic fields produced with the same order of magnitude ($\sim 10^{18}$ to $10^{20}$Gauss) [19, 20, 21, 22] 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. Besides, experiments with asymmetric Cu $+$ Au collisions, it is believed that the strong electric field is supposed to be created in the overlapping region [23, 24, 25]. It happens because there are different numbers of electric charges in each nuclei, which may be due to the charge dipole formed in the early stage of the collision. Some other phenomena like the chiral electric separation effect [26, 27], the particle polarization effect [28, 29, 30], etc., which may emerge due to the generation of vector and/or axial current in the presence of strong electromagnetic fields.
It is illustrative to approximate the created number of charged quark-antiquark pairs in the QGP produces in Heavy Ion collision, because in the QGP phase the dynamical chirl symmetry restores and the deconfinement occurs. According to advance numerical simulations, the electric fields created in Au $+$ Au collisions at center-of-mass energy $\sqrt{s}=200$ GeV due to the fluctuation is of the order $eE\sim m^{2}_{\pi}$ while in Pb$+$Pb collisions at $\sqrt{s}=2.76$ TeV is of the order $eE\sim 20m^{2}_{\pi}$ [20]. Then by assuming the space-time volume of the QGP is of the order $\sim(5{\rm fm})^{4}$, the total pair creation number is $N_{\rm RHIC}=3.5$, and $N_{\rm LHC}=1400$ [31], gives us a clear indication of the importance of Schwinger pair production of quark and antiquark in Heavy Ion Collisions.
It is well understood that the QCD exhibits confinement and chiral symmetry breaking with the small number of light quark flavors $N_{f}$. However, for larger $N_{f}$, Lattice QCD simulation [32, 33, 34, 35, 36], as well as the continuum methods of QCD  [37, 38, 39, 40, 41, 42, 18, 43], predicted that there is a critical value $N^{c}_{f}\approx 8$ above which the chiral symmetry is restored and quarks become unconfined. It has been discussed in detail in Ref. [18] that the critical number of flavors $N^{c}_{f}$ suppresses with the increase of temperature $T$ and enhances with the increasing magnetic field $eB$. Even the QCD phase diagram at finite temperature $T$ and density $\mu$ suppresses with the increase of light quark flavors, see for an instance Ref. [42]. Besides the higher number of light quark flavors, QCD with a larger number of colors $N_{c}$ in the fundamental $SU(N_{c})$ representation also plays a significant role. It has been demonstrated in Ref. [18, 42] that the chiral symmetry is dynamically broken above a critical value $N^{c}_{c}\approx 2.2$, as a result, the dynamically generated mass increases near and above $N^{c}_{c}$. Increasing the number of colors also enhances the critical temperature $T_{c}$ and the critical chemical potential $\mu_{c}$ of the chiral phase transition in the QCD phase diagram  [18]. Both $N_{c}$ and magnetic field $eB$ strengthen the generation of the dynamical masses of the quarks  [42].
It will be more significant to study the dynamical chiral symmetry breaking and Schwinger effect in pure electric field background for a higher number of light quark flavors $N_{f}$ and for a large number of colors $N_{c}$, which has not yet been studied so far, as far as we know. It may have a stronger impact not only on the theoretical ground but also in Heavy-Ion Collision experiments where a large number of light flavors of quark-antiquark pairs are produced. Our main objective of this work is to study the quark-antiquark pair production rate in the presence of a pure electric field $eE$ background for a higher number of light quark flavors $N_{f}$ and colors $N_{f}$. For this purpose, we use the Schwinger-Dyson equation in the rainbow-ladder truncation, in the Landau gauge, the symmetry preserving flavour-dependent confining vector-vector contact interaction model of quarks [18], and the Schwinger optimal time regularization scheme [3]. The pseudo-critical electrical field strength $eE_{c}$, the critical number of flavors $N^{c}_{f}$ and the critical number of colors $N^{c}_{c}$ for chiral symmetry breaking-restoration can be obtained from the peak of the correspondence gradient of dynamical quark mass, whereas the confinement-deconfinement can be triggered from the peaks of correspondence gradient of the confining length scale [44, 18, 42]. It should be noted that the chiral symmetry restoration and deconfinement occur simultaneously in this model [45, 44, 18, 42].
This manuscript is organized as follows: In Sec. 2, we present the general formalism for the flavor-dependent contact interaction model and QCD gap equation. In Sec. 3, We discuss the gap equation and the the Schwinger pair production rate in the presence of electric field $eE$ for a large number of flavors $N_{f}$ and colors $N_{c}$. In Sec. 4, we present the numerical solution of the gap equation and the Schwinger pair production rate in the presence of $eE$ for a higher number of $N_{f}$ and $N_{c}$. In the last Sec. 5, we present the summary and future perspective of our work.

In this section, we discuss the gap equation in the presence of a uniform and homogeneous pure electric field $eE$. In QCD Lagrangian, the interaction with pure electric field $A^{ext}_{\mu}$ embedded in the covariant derivative,

with $Q_{f}=(q_{u}=+2/3,q_{d}=-1/3)e$ is refers to the electric charges of $u$ and $d$-quark, respectively. We choose the symmetric gauge vector potential $A^{ext}_{\mu}=-\delta_{\mu 0}x_{3}E$, to obtain the resulting electric field along the z-axis. The gap equation in the presence of pure electric field continues to form Eq. (6), where $S_{f}(k)$ is now dressed with electric field $eE$, that is, ${S_{f}}(k)\rightarrow\tilde{S_{f}}(k)$ [3, 11, 13, 15, 57], and is given as

Where $\gamma$’s are the Dirac gamma matrices. Taking the trace “Tr” of the of the propagator Eq. (15), inserting it in Eq. (6) and after carrying out the the integration over $k$’s, the gap equation in the electric field $eE$ and with the flavor-dependent contact interaction model of quark [42] is given by

Next, we separate the vacuum and electric field dependent part by using the vacuum subtraction scheme[11, 13], as given by

The vacuum integral can be regularized with the two regulators as in Eq. (11), and thus the Eq. (17) can be reduced to:

The Eq. (18) can also be written as

As we have already regularized the integral in Eq. (18), using the vacuum subtraction scheme but we still have poles associated with the tangent term in the denominator of our gap equation when $|Q_{f}E|\tau=n\pi$ with $n=1,2,3.....,$. Upon taking the principle value (real value) of the integral [11], given in Eq. (19), we have

with

The effective potential $\Omega$, can be obtained by integrating the Eq. (19) over dynamical mass $M_{f}$:

We noted that there are infinite poles in the effective potential $\Omega$ too which are due to the contribution of the Schwinger pair production in the presence of the tangent (electric field-related) term . These poles yields the effective potential to be complex with its imaginary part giving the Schwinger pair production rate [3, 16, 13, 14]. The third term in the Eq. (22) can be further simplified by using the following trigonometric relation:

Using Eq. (23) in Eq. (23), we have

The Schwinger pair production rate $\Gamma$, corresponds to the imaginary part of the effective potential Eq. (24), and is thus, given by

which does not depend on the number of color $N_{c}$ or the number of flavors $N_{f}$ explicitly. However, since the dynamical quark mass $M_{f}$ depends on both $N_{f}$ and $N_{c}$ as it is obvious from the gap equation Eq. (17), they will affect the quark-antiquark pair production rate $\Gamma$ implicitly through $M_{f}$. The numerical solution of the gap equation and the Schwinger pair production rate will be discussed in the next section.

In this section, we present the numerical results of the gap equation at zero electric field and in the presence of electric field for higher $N_{c}$ and $N_{f}$. We use the following set of flavor-dependent contact interaction model parameters [42] i.e., $\tau_{{ir}}=(0.24~{}\mathrm{GeV})^{-1}$, $\tau_{uv}=(0.905~{}\mathrm{GeV})^{-1}$, and bare quark mass $m_{u}=m_{d}=0.007$ GeV. For fixed $N_{c}=3$ and $N_{f}=2$, we have obtained the dynamical mass $M_{u,d}=0.367$ GeV and the condensate $-\langle\bar{q}q\rangle^{1/3}_{u,d}=0.243$ GeV³. It should be noted that these parameters were fitted to reproduce the light mesons properties [49].
Next, we solve the gap equation Eq.(11) for various $N_{f}$ and at fix $N_{c}=3$ as shown in the Fig. 1. The dynamical mass monotonically decreases as we increase the $N_{f}$ as depicted in Fig. 1(a). The plot inside the Fig. 1(a), represents the flavor-gradient of the dynamical mass $\partial_{N_{f}}M_{f}$, whose peak is at $N^{c}_{f}\approx 8$, which is a critical number of flavors where near and above the dynamical chiral symmetry restored. In Fig. 1(b), we show the dynamical mass as a function of the number of colors $N_{c}$ for fix $N_{f}=2$. This plot represents that the dynamical chiral symmetry broken near or above some critical value of $N^{c}_{c}\approx 2.2$. The $N^{c}_{c}\approx 2.2$ is obtained from the peak of the color-gradient of the mass function $\partial_{N_{c}}M_{f}$. These findings are consistent with results obtained in [42, 43].

The inverse of the confining length scale $\tilde{\tau}^{-1}_{ir}$ and its flavor-gradient plotted for various $N_{f}$ and for fixed $N_{c}=3$ are plotted in Fig. 2(a). The confinement can be triggered from flavor-gradient $\partial_{N_{f}}\tilde{\tau}^{-1}_{ir}$, whose peak is at $N^{c}_{f}\approx 8$, and is approximated as a critical number of flavors above which the quark become unconfined. In Fig. 2(b), we show the inverse confining length scale $\tilde{\tau}^{-1}_{ir}$ as a function of the various number of colors $N_{c}$ for fix $N_{f}=2$. This plot represents that the confinement occurs near and above at some critical value of $N^{c}_{c}\approx 2.2$. The critical $N^{c}_{c}\approx 2.2$ for the confinement is obtained from the peak of the color gradient $\partial_{N_{c}}\tilde{\tau}^{-1}_{ir}$.

Next, we solve the gap equation Eq. (17), in the presence of electric field $eE$ and plotted the dynamical mass as a function of $eE$, for the various number of flavors $N_{f}$ and for fix $N_{c}=3$ as depicted in Fig. 3(a). The dynamical mass decreases as we increase the electric field $eE$ as we expected. Upon increasing the $N_{f}$, the plateau of dynamical quark mass as a function of $eE$ suppresses for larger values of $N_{f}$. There is no dynamical mass generation above $N^{c}_{f}=8$, this is because both electric field $eE$ and $N_{f}$ restore the dynamical chiral symmetry and quarks becomes unconfined.
In Fig. 3(b), we plotted the dynamical quark mass as a function of electric field strength $eE$ for various $N_{c}$ and for fixed $N_{f}=2$. The increasing $N_{c}$ enhances the plateau of the dynamical mass as a function of $eE$. For $N_{c}\geq 4$, the mass plots show the discontinuities in the dynamical symmetry restoration region where the nature of smooth cross-over phase transition changes to the first order. This may be due to the strong competition that occurs between $eE$ and $N_{c}$ (i.e. strong electric field tends to restore the dynamical chiral symmetry whereas larger $N_{c}$ enhance the dynamical chiral symmetry breaking.

The Schwinger pair production rate “$\Gamma$” Eq. (25), as a function of electric field strength $eE$, for various $N_{f}$ and for fix $N_{c}=3$ is shown in the Fig. 4(a). This figure clearly demonstrate that after some critical value $eE_{c}$, where near at and above the dynamical symmetry restored and the deconfinement occurred, the pair production rate “$\Gamma$” grows faster due to the weakening of the chiral condensates. In this situation, the QCD vacuum becomes more unstable and the pair of quark-antiquark becomes more likely to produces. Upon increasing the number of flavors $N_{f}$, the pair production rate grows faster and shifted towards the lower values of electric field $eE$. This is because, both the $eE$ and $N_{f}$ restore the dynamical chiral symmetry. Although, there is an unstable slow enhancement of pair production rate for small $N_{f}$ but become stable and faster for higher values of $N_{f}$ In Fig. 4(b), we plotted the quark-antiquark pair production rate“$\Gamma$” as a function of $eE$ for various $N_{c}$ but at this time we fix the number of flavors $N_{f}=2$. Upon increasing the number of colors $N_{c}$, the production rate grows slowly and higher values of $eE$ are needed for a stable and faster production rate. This is because, $N_{c}$ enhances the dynamical chiral symmetry breaking. For $N_{c}\geq 4$, the discontinuity occurs in the production rate near and above the chiral symmetry restoration and deconfinement region. This may be due to the strong competition between $N_{c}$ and $eE$, i.e., on one hand, the strong electric field $eE$ tends to restore the dynamical chiral symmetry whereas the $N_{c}$ tends to break it.

We then obtained the pseudo-critical electric field $eE_{c}$ for the chiral symmetry breaking -restoration from the electric field-gradient $\partial_{eE}M_{f}$ as a function of $eE$ for various $N_{f}$ and at fixed $N_{c}=3$, as depicted in Fig. 5(a). The peaks of the  $\partial_{eE}M_{f}$ shift towards lower values of the electric field $eE$. In Fig. 5(b), we plotted the electric field gradient  $\partial_{eE}M_{f}$ as a function of $eE$ for various $N_{c}$ and at fixed $N_{f}=2$. Upon increasing $N_{c}$, the peaks shift towards higher values of $eE$.

We then obtained the pseudo-critical electric field $eE_{c}$ for the chiral symmetry restoration/deconfinement different $N_{f}$, and draw a phase diagram in $N_{f}-eE$ plane a shown in the Fig. 6(a). The pseudo-critical electric field $eE_{c}$ decreases as we increase $N_{f}$, the nature of the chiral phase transition is of smooth cross-over. In Fig. 6(b), we sketch the phase diagram in $N_{c}-eE_{c}$ plane, the critical $eE_{c}$ grows upon increasing the number of colors $N_{c}$. The transition is of smooth cross-over until the critical endpoint $(N_{c,p}=4,eE_{c,p}=0.54GeV^{2})$ where it suddenly changes to the first-order. The phase diagram also shows that the quark-antiquark pair production grows quickly after pseudo-critical electric field $eE_{c}$ and how it varies with the flavors and colors. Our finding for $N_{c}=3$ and $N_{f}=2$, agrees well with the growth of the quark-antiquark production rate studied through another effective model of low energy QCD [31, 14].

In the next section, we discuss the summery and future perspectives of the this work.

In this work, we have studied the impact of pure electric field on the color-flavor chiral phase transitions and investigated the Schwinger quark-antiquark pair production rate $\Gamma$ for the higher number of colors $N_{c}$ and number flavors $N_{f}$. For this purpose, we implemented the Schwinger-Dyson formulation of QCD, with a gap equation kernel comprising a symmetry-preserving vector-vector flavor-dependent contact interaction model of quarks in a rainbow-ladder truncation. Subsequently, we adopted the well-known Schwinger proper-time regularization procedure. The outcome of this study is presented as follows:
First, we reproduced the results of dynamical chiral symmetry restoration for large $N_{f}$ with fixed $N_{f}=3$, where we evaluated the critical number of flavors $N^{c}_{f}\approx 8$. Further, we explore the dynamical symmetry breaking for a higher number of colors $N_{c}$ but kept $N_{f}=2$ and found the critical number of colors $N^{c}_{c}\approx 2.2$. Second, we consider the influence of the pure electric field $eE$, where we have explored the dynamical chiral symmetry restoration and deconfinement for various numbers of flavors $N_{f}$ and colors $N_{c}$. The plateau of the mass as a function of the electric field is noted to suppress upon increasing the number of flavors. Whereas upon increasing the number of colors, the dynamical mass as a function of $eE$ enhances and at some $N_{c}\geq 4$, we found the discontinuity in the chiral symmetry restoration region. Next, We obtained the Schwinger pair production (quark-antiquark) rate $\Gamma$ as a function of the pure electric field $eE$ for various $N_{f}$ and $N_{c}$. For fixed $N_{c}=3$, and upon varying the $N_{f}$, we found that quark-antiquark production rate $\Gamma$ grows quickly when we cross a pseudo-critical electric field $eE_{c}$. As a result, the pair production rate tends to initiates at lower values of $eE$ for larger values of $N_{f}$. Hence, the pseudo-critical electric field $eE_{c}$ reduced in magnitude upon increasing the number of flavors $N_{f}$. This is because both $N_{f}$ and $eE$ restored the dynamical chiral symmetry. While for fixed $N_{f}=2$ and upon increasing $N_{c}$, the Schwinger pair production tends to initiate at larger values of $eE$ for higher values of $N_{c}$. It is interesting to note that for $N\geq 4$, the discontinuity occurred in the production rate in the region where the chiral symmetry is restored. This may be due to the strong competition that occurred between $eE$ and $N_{c}$. Thus, the pseudo-critical $eE_{c}$ enhances as the number of colors $N_{c}$ increases. The transition is of smooth cross-over until the critical endpoint $(N_{c,p}=4,eE_{c,p}=0.54GeV^{2})$, where it suddenly changes to the first order. Our finding for $N_{c}=3$ and $N_{f}=2$ agrees well with the behaviour of the pair production rate studied through other effective models of low energy QCD. Qualitatively and quantitatively, the predictions of the presented flavor-dependent contact interaction model for fixed $N_{=}3$ and $N_{f}=2$ agree well with results obtained from other effective QCD models. Soon, we plan to extend this work to study the Schwinger pair production rate in hot and dense matter QCD in the presence of background fields. We are also interested in extending this work to study the properties of light hadrons in the background fields etc.

We acknowledge A. Bashir and A. Raya for their guidance and valuable suggestions in the process of completion of this work. We also thank to the colleagues of the Institute of Physics, Gomal University (Pakistan).