The interquark static potential for heavy mesons described by a massive One Gluon Exchange interaction obtained from the propagator of the truncated Dyson-Schwinger equations does not reproduced the expected Cornell potential. I show that no formulation based on a finite propagator will lead to confinement of quenched QCD. I propose a mechanism based on a singular nonperturbative coupling constant which has the virtue of giving rise to a finite gluon propagator and (almost) linear confinement. The mechanism can be slightly modified to produce the screened potentials of unquenched QCD.
The interquark potential is constructed by making use of the new analytic running coupling in QCD. This running coupling arises under ``analytization'' of the renormalization group equation. The rising behavior of the interquark potential at large distances, which provides the quark confinement, is shown explicitly. The estimation of the parameter $Λ_{QCD}$ is obtained.
We calculate the lowest-order corrections to the static potential for both Yang-Mills theory and gluodynamics in curved space-time, in the presence of a quantum gravity induced minimal length. Our analysis is carried out within stationary perturbation theory. As a consequence, the potential energy is the sum of a Yukawa-like potential and a linear potential for gluodynamics in curved space-time, leading to the confinement of static charges. Interestingly enough, we obtain that the coefficient of the linear term ("string tension") is ultraviolet finite. We should highlight the role played by th…
Starting from the observation that colour charge is only well defined on gauge invariant states, we construct perturbatively gauge invariant, dynamical dressings for individual quarks. Explicit calculations show that an infra-red finite mass-shell renormalisation of the gauge invariant, dressed propagator is possible and, further, that operator product effects, which generate a running mass, may be included in a gauge invariant way in the propagator. We explain how these fields may be combined to form hadrons and show how the interquark potential can now be directly calculated. The onset of co…
We address the question of the Lorentz nature of the effective long-range interquark interaction generated by the QCD string with quarks at the ends. Studying the Dyson-Schwinger equation for a heavy-light quark-antiquark system, we demonstrate explicitly how a Lorentz-scalar interaction appears in the Diraclike equation for the light quark, as a consequence of chiral symmetry breaking. We argue that the effective interquark interaction in the Hamiltonian of the QCD string with quarks at the ends stems from this effective scalar interaction.
We consider the potential-model approach for obtaining the spectrum of charmonium and bottomonium, replacing the usual gluon propagator by one obtained from lattice simulations. The resulting spectra are compared to the corresponding ones in the Cornell-potential case. We also estimate the interquark distance in both cases.
We address the question of the Lorentz nature of the effective interquark interaction in QCD which leads to the formation of the QCD string between colour charges. In particular, we start from a manifestly vectorial fundamental interaction mediated by gluons and demonstrate that, as soon as chiral symmetry is broken spontaneously, the effective interquark interaction acquires a selfconsistently generated scalar part which is eventually responsible for the formation of the QCD string. We demonstrate this explicitly for a heavy-light quarkonium, using the approach of the Schwinger-Dyson-type equ…
The resolution of Dyson-Schwinger equations leads to the freezing of the QCD running coupling (effective charge) in the infrared, which is best understood as a dynamical generation of a gluon mass function, giving rise to a momentum dependence which is free from infrared divergences. We calculate the interquark potential for heavy mesons by assuming that it is given by a massive One Gluon Exchange potential and compare with phenomenologyical fits inspired by lattice QCD. We apply these potential forms to the description of quarkonia and conclude that, even though some aspects of the confinemen…
The exact nonperturbative confining solutions of the SU(3)-Yang-Mills equations recently obtained by author in Minkowski spacetime with the help of the black hole theory techniques are analysed and on the basis of them the gluon propagator corresponding to linear confinement at large distances (small momenta) is constructed in a nonperturbative way. At small distances (large momenta) the resulting propagator passes on to the standard (nonperturbative) gluon propagator used in the perturbative quantum chromodynamics (QCD). The results suggest some scenario of linear confinement for mesons and q…
Properties of single heavy flavor baryons in a non relativistic potential model with colour coulomb plus power law confinement potential have been studied. The ground state masses of single heavy baryons and the mass difference between the ($J^{P}={3/2}^{+}$ and $J^{P}={1/2}^{+}$) states are computed using a spin dependent two body potential. Using the spin-flavour structure of the constituting quarks and by defining an effective confined mass of the constituent quarks within the baryons, the magnetic moments are computed. The masses and magnetic moments of the single heavy baryons are found t…
We study Dick quark-antiquark potential (up to a color factor) $V_D(r)={-α_s \over r} + gf \sqrt{N_c \over {2(N_c-1)}} \ln[exp(2mr)-1]$ in the heavy meson sector. This potential emerges from an effective dilaton-gluon coupling inspired from string theory and proves to be linearly rising at large distances which ensures quark confinement. The semi-relativistic wave equation which appears in the theory of relativistic quark-antiquark bound states is used. This equation is cast into a constituent second order Schrödinger-like equation with the inclusion of relativistic corrections up to order $(v…
An introduction is given into current lattice investigations of the non-perturbative gluon and ghost propagators, in the light of the Gribov-Zwanziger and Kugo-Ojima scenarios of confinement, in the context of results obtained from the non-perturbative Dyson-Schwinger approach in the continuum and in connection with the vortex mechanism of confinement.
We derive a gauge-invariant low-energy effective model of the SU(2) Yang-Mills theory. We find that the effective gluon propagator belongs to the Gribov-Stingl type, irrespective of the gauge choice. In the maximally Abelian gauge, especially, we demonstrate that the model exhibits both quark confinement and gluon confinement: the Wilson loop average has area law and the Schwinger function violates reflection positivity. Moreover, we give a formula for the string tension calculable from the gluon propagator of the gauge-invariant field strength and gives a good estimate for the string tension.…
We review some of the most recent work on confinement in 4d gauge theories with a massive scalar field (dilaton). Emphasis is put on the derivation of confining analytical solutions to the Coulomb problem versus dilaton effective couplings to gauge terms. It is shown that these effective theories can be relevant to model quark confinement and may shed some light on confinement mechanism. Moreover, the study in the heavy meson sector of an interquark potential derived from Dick Model, proves that phenomenological investigation of this mechanism is more than justified and deserves more efforts.
We have investigated a closed system of equations for the quark propagator, obtained earlier within our general approach to QCD at low energies. It implies quark confinement (the quark propagator has no pole, indeed), as well as the dynamical breakdown of chiral symmetry (a chiral symmetry preserving solution is forbidded). This system can be solved exactly in the chiral limit. We have established the space of the smooth test functions (consisting of the Green's functions for the quark propagator and the corresponding quark-gluon vertex) in which our generalized function (the confining gluon p…
We consider the effects of the Faddeev-Popov determinant in the Coulomb gauge on the confinement properties of the QCD vacuum. We show that the the determinant is needed to regularize the otherwise divergent functional integrals near the Gribov horizon but still enables large field configurations to generate IR enhanced running coupling. The physical gluon propagator is found to be strongly suppressed in the IR consistent with expectations from lattice gauge calculations.
We calculate the lattice quark propagator in Coulomb gauge both from dynamical and quenched configurations. We show that in the continuum limit both the static and full quark propagator are multiplicatively renormalizable. From the propagator we extract the quark renormalization function Z(|p|) and the running mass M(|p|) and extrapolate the latter to the chiral limit. We find that M(|p|) practically coincides with the corresponding Landau gauge function for small momenta. The computation of M(|p|) can be however made more efficient in Coulomb gauge; this can lead to a better determination of …
The gluon propagator is investigated in Landau and in maximal center gauge for the gauge group SU(2) by means of lattice gauge simulations. We find that Gribov ambiguities arising from the implementation of Landau gauge have a small influence on the propagator. In agreement with previous findings, we obtain that the small momentum behavior is dominated by a mass, which is of order $(1.48 \pm 0.05) \sqrtσ$, where $σ$ is the string tension. By removing the confining vortices from the full Yang-Mills ensemble, we convert full YM-theory into a theory which does not confine quarks. We find that in …
The propagator of confined quarks is calculated for timelike momenta by transforming Minkowski Greens functions to the Temporal Euclidean space. Based on the framework of the Schwinger-Dyson equations the QCD quark propagator is obtained in two approximations which differ by assuming behaviour of gluon propagator. In both studied cases we get universal result for the light quarks: The quark mass function becomes complex bellow expected perturbative threshold, the obtained absolute value of the infrared mass is $M\simeq Λ_{QCD} $ with the infrared phase $\simeq {π\over 2}$. Permanent confinemen…
Using the previously derived confining gluon propagator, the corresponding system of equations determining the quark propagator is derived. The system of equations consists of the Schwinger-Dyson equation for the quark propagator itself, which includes the zero momentum transfer quark-gluon vertex. It is complemented by the Slavnov-Taylor identity for this vertex. The quark equation depends explicitly on the mass gap, determining the scale of the truly nonperturbative dynamics in the QCD ground state. The obtained system of equations is manifestly gauge-invariant, i.e., does not depend explici…
We investigate the behavior of the one body propagator in SQED. The self energy is calculated using three different methods: i) the simple bubble summation, ii) the Dyson-Schwinger equation, and iii) the Feynman-Schwinger represantation. The Feynman-Schwinger representation allows an {\em exact} analytical result. It is shown that, while the exact result produces a real mass pole for all couplings, the bubble sum and the Dyson-Schwinger approach in rainbow approximation leads to complex mass poles beyond a certain critical coupling. The model exhibits confinement, yet the exact solution still …
We propose to realize a mass gap in QCD not imposing the transversality condition on the full gluon self-energy, while preserving the color gauge invariance condition for the full gluon propagator. Since due to color confinement the gluon is not a physical state, none of physical observables/processes in low-energy QCD will be directly affected by such a temporary violation of color gauge invriance/symmetry. In order to make the existence of a mass gap perfectly clear the corresponding subtraction procedure is introduced. All this llows one to etablish the general structure of the full gluon p…
Analytical properties of the photon propagator in QED3 in the N^1-approximation are considered. It is noticed that in given approximation if the dynamical fermions mass is nonzero this propagator has a single pole at the point k^2=0 and a branch point at k^2=-4m^2 (m is the fermion mass). When m=0, the propagator gets the pole at the spacelike domain of the variable k^2. It witnesses the unstability of the state conforming to the photon and leads to the screening of the source field at r speeds to infinity. The approximation by the rational function of the photon propagator containing the vacu…
The properties of elementary particles are encoded in their respective propagators and interaction vertices. For a SU(2) gauge theory coupled to a doublet of fundamental complex scalars these propagators are determined in both the Higgs phase and the confinement phase and compared to the Yang-Mills case, using lattice gauge theory. Since the propagators are gauge-dependent, this is done in the Landau limit of 't Hooft gauge, permitting to also determine the ghost propagator. It is found that neither the gauge boson nor the scalar differ qualitatively in the different cases. In particular, the …
We study the lattice gauge boson propagator of 3D compact QED in Landau gauge at zero and non-zero temperature. Non-perturbative effects are reflected by the generation of a mass $m$, by an anomalous dimension $α$ and by the photon wave function renormalisation $Z$. These effects can be attributed to monopoles: they are absent in the propagator of the regular part of the gauge field. The rôle of Gribov copies is carefully investigated.
We study the gauge boson propagator in the dual Abelian Higgs theory which confines electric charges. The confinement is due to dual Abrikosov-Nielsen-Olesen strings. We show that the infrared double pole 1/(p^2)^2 in the propagator is absent due to the confinement phenomenon. Instead, specific angular singularities signal the eventual set in of the confinement. These angular singularities are a manifestation of the confining strings.
We study the Abelian projection of quark confinement in SU(3) quenched lattice QCD, in terms of the dual superconductor picture. In the maximal Abelian gauge, we perform the Cartan decomposition of the non-Abelian gauge field on a $32^4$ lattice with spacing $a \simeq0.058, 0.10$ fm (i.e., $β=6.4, 6.0$), and investigate the interquark potential $V(r)$, the Abelian part $V_{\mathrm{Abel}}(r)$, and the off-diagonal part $V_{\mathrm{off}}(r)$. For the potential analysis, we use both on-axis data and several types of off-axis data, with larger numbers of gauge configurations. Remarkably, we find a…
Recent investigations of the Dyson-Schwinger equations and Monte-Carlo lattice calculations resulted in a coherent description of the fully dressed gluon, ghost and quark propagators in Landau gauge QCD. In the Dyson-Schwinger approach the infrared behaviour of these propagators is determined analytically. For finite spacelike momenta the gluon, ghost and quark propagators are compared to available corresponding results of lattice Monte-Carlo calculations. For all three propagators an almost quantitative agreement is found. These results for the non-perturbative propagators allow an analytical…
Analytical functions for the propagators of QCD, including a set of chiral quarks, are derived by a one-loop massive expansion in the Landau gauge, and are studied in Minkowski space, yielding a direct proof of positivity violation and confinement from first principles. Complex conjugated poles are found for the gluon propagator.
We establish exactly and uniquely the infrared structure of the full gluon propagator in QCD, not solving explicitly the corresponding dynamical equation of motion. By construction, this structure is an infinite sum over all possible severe (i.e., more singular than $1/q^2$) infrared singularities. It reflects the zero momentum modes enhancement effect in the true QCD vacuum. Its existence exhibits a characteristic mass (the so-called mass gap), which is responsible for the scale of nonperturbative dynamics in the QCD ground state. The theory of distributions, complemented by the dimensional r…
Local formulations of quantum field theory provide a powerful framework in which non-perturbative aspects of QCD can be analysed. Here we report on how this approach can be used to elucidate the general analytic features of QCD propagators, and why this is relevant for understanding confinement.
The chirally symmetric overlap quark propagator is explored in Coulomb gauge for the first time. This gauge is especially well suited for studying the interrelation between confinement and chiral symmetry breaking, since confinement can be attributed to the infrared divergence of the vector dressing function of the inverse quark propagator. Using quenched gauge field configurations on a $20^4$ lattice, the dressing functions of the quark propagator as well as the dynamical quark mass function are evaluated, also in the chiral limit. Chiral symmetry is artificially restored by removing the low …
General properties of the light and heavy quark propagators have been investigated in the context of string tension interaction. Confinement, chiral symmetry breaking, spectral properties of the propagator are analytically studied and numerically validated. We show that the propagator is analytic in the infrared region even for massless quarks with a non zero radius of convergence. Emergence of more than one mass scale is exemplified. Massless limit of the quark propagator does exhibit critical behaviour.
The Kugo-Ojima confinement criterion and its relation to the infrared behaviour of the gluon and ghost propagators in Landau gauge QCD are reviewed. The realization of this confinement criterion (which in Landau gauge relates to Zwanziger's horizon condition) results from quite general properties of the ghost Dyson-Schwinger equation. The numerical solutions for the gluon and ghost propagators obtained from a truncated set of Dyson-Schwinger equations provide an explicit example for the anticipated infrared behaviour. These results are in good agreement, also quantitatively, with corresponding…
Analyticity of gluon and Faddeev--Popov ghost propagators and their form factors on the complex momentum-squared plane is exploited to continue analytically the ultraviolet asymptotic form calculable by perturbation theory into the infrared non-perturbative solution. We require the non-perturbative multiplicative renormalizability to write down the renormalization group equation. These requirements enable one to settle the value of the exponent characterizing the infrared asymptotic solution with power behavior which was originally predicted by Gribov and has recently been found as approximate…
The structure of the quark propagator of $QCD$ in a confining background is not known. We make an Ansatz for it, as hinted by a particular mechanism for confinement, and analyze its implications in the meson and baryon correlators. We connect the various terms in the Källen-Lehmann representation of the quark propagator with appropriate combinations of hadron correlators, which may ultimately be calculated in lattice $QCD$. Furthermore, using the positivity of the path integral measure for vector like theories, we reanalyze some mass inequalities in our formalism. A curiosity of the analysis i…
In this article, we calculate the dressed quark propagator with the flat bottom potential in the framework of the rain-bow Schwinger-Dyson equation. Then based on the nonperturbative dressed quark propagator, we calculate the $π$ decay constant and the quark condensate. The $π$ decay constant is an important parameter in describing the interplay between dynamical symmetry breaking and confinement, while the quark condensate is an order parameter for dynamical chiral symmetry breaking. To implement confinement, we prove that the dressed quark propagator has no poles on the real timelike $p^2$ a…
Phenomenological consequences of the infrared singular, instantaneous part of the gluon propagator in Coulomb gauge are investigated. The corresponding quark Dyson-Schwinger equation is solved, neglecting retardation and transverse gluons and regulating the resulting infrared singularities. While the quark propagator vanishes as the infrared regulator goes to zero, the frequency integral over the quark propagator stays finite and well-defined. Solutions of the homogeneous Bethe-Salpeter equation for the pseudoscalar and vector mesons as well as for scalar and axial-vector diquarks are obtained…
Novel order parameters for the confinement-deconfinement phase transition of quenched QCD and fundamentally charged scalar QCD are presented. Similar to the well-known dual condensate, they are defined via generalized matter propagators with $U(1)$-valued boundary conditions. The order parameters are easily accessible with functional methods. Their validity and accessibility is explicitly demonstrated by numerical studies of the Dyson-Schwinger equations for the matter propagators. Even in the case of heavy scalar matter, where the propagator does not show a signature of the phase transition, …
We study a model Dyson-Schwinger equation for the quark propagator closed using an {\it Ansatz} for the gluon propagator of the form \mbox{$D(q) \sim q^2/[(q^2)^2 + b^4]$} and two {\it Ansätze} for the quark-gluon vertex: the minimal Ball-Chiu and the modified form suggested by Curtis and Pennington. Using the quark condensate as an order parameter, we find that there is a critical value of $b=b_c$ such that the model does not support dynamical chiral symmetry breaking for $b>b_c$. We discuss and apply a confinement test which suggests that, for all values of $b$, the quark propagator in the m…
On the lattice we study the gauge boson propagator of three dimensional compact QED in Landau gauge at zero and non-zero temperature. The non-perturbative effects are taken into account by the generation of a mass, by an anomalous dimension and by the photon wave function renormalization. All these effects can be attributed to the monopoles: they are absent in the propagator of the singularity-free part of the gauge field. We assess carefully the Gribov copy problem for the propagator and the parameters emerging from the fits.
We derive a gauge-invariant low-energy effective model of the Yang-Mills theory. We find that the effective gluon propagator belongs to the Gribov-Stingl type and agrees with it when a mass term which breaks nilpotency of the BRST symmetry is included. We show that the effective model with gluon propagator of the Gribov-Stingl type exhibits both quark and gluon confinement: the Wilson loop average has the area law and the Schwinger function violates reflection positivity. However, we argue that both quark and gluon confinement can be obtained even in the absence of such a mass term.
In QCD both the quark and ghost propagators are important for governing the non-perturbative dynamics of the theory. It turns out that the dynamical properties of the quark and ghost fields impose non-perturbative constraints on the analytic structure of these propagators. In this work we explicitly derive these constraints. In doing so we establish that the corresponding spectral densities include components which are multiples of discrete mass terms, and that the propagators are permitted to contain singular contributions involving derivatives of $δ(p)$, both of which are particularly releva…
The Gluon and Ghost propagators in Landau gauge QCD are investigated using the Schwinger-Dyson equation approach. Working in Euclidean spacetime, we solve for these propagators using a selection of vertex inputs, initially for the ghost equation alone and then for both propagators simultaneously. The results are shown to be highly sensitive to the choices of vertices. We favor the infrared finite ghost solution from studying the ghost equation alone where we argue for a specific unique solution. In order to solve this simultaneously with the gluon using a dressed-one-loop truncation, we find t…
After a derivation of low-energy limit of QCD, being this a non-local Nambu-Jona-Lasinio model, we are able to show that confinement emerges as a two-loop correction to the gluon propagator. One-gluon exchange is not enough as recently shown in literature about studies on the gluon propagator in the Landau gauge.
Using the Schwinger-Dyson equations the possible infrared behaviour of the gluon propagator is studied. Previous work performed in axial gauges is reviewed, the approximations needed detailed and the difficulties of their justification discussed. We then turn to the Landau gauge and investigate the possibility of a gluon propagator less singular than $1/p^{2}$ when $p^{2} \rightarrow 0$. We find that this infrared softened behaviour of the gluon propagator is inconsistent; only an infrared enhanced gluon, as singular as $1/p^{4}$ when $p^{2} \rightarrow 0$ is consistent with the truncated Schw…
QCD perturbation theory ignores the compact nature of $SU(3)$ gauge group that gives rise to the periodic $θ$-vacuum of the theory. We propose to modify the gluon propagator to reconcile perturbation theory with the anomalous Ward identities for the topological current in the $θ$-vacuum. As a result, the gluon couples to the Veneziano ghost describing the tunneling transitions between different Chern-Simons sectors of the vacuum; we call the emerging gluon dressed by ghost loops a "glost". We evaluate the glost propagator and find that it has the form $G(p) = (p^2 + χ_{top}/p^2)^{-1}$ where $χ…
We study the correlation between confinement and dynamical chiral-symmetry breaking (D$χ$SB) in the dual Ginzburg-Landau (DGL) theory using the effective potential formalism. The DGL theory is an infrared effective theory based on the dual Higgs mechanism, and provides the nonperturbative gluon propagator, which leads to the linear quark potential. The screening effect for the confining potential can be obtained by introducing the infrared cutoff corresponding to the hadron size. We formulate the effective potential for D$χ$SB in the DGL theory, and find the vacuum instability against quark co…
High accuracy numerical results for the SU(2) gluonic form factor are previewed for the case of Landau gauge. I focus on the information of quark confinement encoded in the gluon propagator.
Some years ago Gribov developed the so-called supercritical light quark confinement scenario. Based on physical arguments he conjectured a drastic change in the analytical properties of the quark propagator when the back-reaction of Goldstone bosons (pions) is considered. We investigate this scenario and provide numerical solutions for the quark propagator in the complex plane with and without the pion back-reaction. We find no evidence for the scenario Gribov advocated. As an aside we present a novel method to solve the quark Dyson-Schwinger equation in the complex plane and discuss new chara…
We summarise an analysis of the infrared regime of Landau gauge QCD by means of a flow equation approach. The infrared behaviour of gluon and ghost propagators is evaluated. The results provide further evidence for the Kugo-Ojima confinement scenario. We also discuss their relation to results obtained with other functional methods as well as the lattice.
Some recent work on the implications of a dilaton in 4d gauge theories are revisited. In part I of this paper we see how an effective dilaton coupling to gauge kinetic term provides a simple attractive mechanism to generate confinement. In particular, we put emphasis on the derivation of confining analytical solutions and look into the problem how dilaton degrees of freedom modify Coulom potential and when a confining phase occurs. In part II, we solve the semi-relativistic wave equation, for Dick interquark potential using the Shifted l-expansion technique (SLET) in the heavy quarkonium secto…
We study Landau-gauge Yang-Mills theory by means of a nonperturbative vertex expansion of the quantum effective action. Using an exact renormalization group equation, we compute the fully dressed gluon and ghost propagators to lowest nontrivial order in the vertex expansion. In the mid-momentum regime, $p^2\sim\mathcal{O}(1)\text{GeV}^2$, we probe the propagator flow with various {\em ansätze} for the three- and four-point correlations. We analyze the potential of these truncation schemes to generate a nonperturbative scale. We find universal infrared behavior of the propagators, if the gluon …
We rewrite the Zwanziger horizon condition in terms of the Kugo-Ojima parameter for color confinement. This enables one to explain which value of the Kugo-Ojima parameter is allowed if the horizon condition is imposed. Although all the calculations are performed in the limit of vanishing Gribov parameter for simplicity, the obtained value is consistent with the result of numerical simulations. Consequently, the ghost propagator behaves like free and the gluon propagator is non-vanishing at low momenta, in harmony with recent lattice results and decoupling solution of the Schwinger-Dyson equati…
We examine the possibility that there may exist a logarithmic correction to the infrared asymptotic solution with power behavior which has recently been found for the gluon and Faddeev-Popov ghost propagators in the Landau gauge. We propose a new Ansatz to find a pair of solutions for the gluon and ghost form factors by solving the coupled Schwinger-Dyson equation under a simple truncation. This Ansatz enables us to derive the infrared and ultraviolet asymptotic solutions simultaneously and to understand why the power solution and the logarithmic solution is possible only in the infrared and u…
Expanding the Landau gauge gluon and ghost two-point functions in a power series we investigate their infrared behavior. The corresponding powers are constrained through the ghost Dyson-Schwinger equation by exploiting multiplicative renormalizability. Without recourse to any specific truncation we demonstrate that the infrared powers of the gluon and ghost propagators are uniquely related to each other. Constraints for these powers are derived, and the resulting infrared enhancement of the ghost propagator signals that the Kugo-Ojima confinement criterion is fulfilled in Landau gauge Yang-Mil…
We study the analytical structure of the fermion propagator in planar quantum electrodynamics coupled to a Chern-Simons term within a four-component spinor formalism. The dynamical generation of parity-preserving and parity-violating fermion mass terms is considered, through the solution of the corresponding Schwinger-Dyson equation for the fermion propagator at leading order of the 1/N approximation in Landau gauge. The theory undergoes a first order phase transition toward chiral symmetry restoration when the Chern-Simons coefficient $θ$ reaches a critical value which depends upon the number…
