Kugo-Ojima confinement and QCD Green''s functions in covariant gauges

In Landau gauge QCD the Kugo-Ojima confinement criterion and its relations to the infrared behaviour of the gluon and ghost propagators are reviewed. It is demonstrated that the realization of this confinement criterion (which is closely related to the Gribov-Zwanziger 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. The results are in good agreement, also quantitatively, with corresponding lattice data obtained recently. The resulting running coupling approaches a fixed point in the infrared, (0) = 8.915/N_c. Solutions for the coupled system of Dyson-Schwinger equations for the quark, gluon and ghost propagators are presented. Dynamical generation of quark masses and thus spontaneous breaking of chiral symmetry takes place. In the quenched approximation the quark propagator functions agree well with those of corresponding lattice calculations. For a small number of light flavours the quark, gluon and ghost propagators deviate only slightly from the ones in quenched approximation. While the positivity violation of the gluon spectral function is manifest in the gluon propagator, there are no clear indications of analogous positivity violations for quarks so far.     

Dynamical chiral-symmetry breaking at T = 0 and T ≠ 0 in the Schwinger-Dyson equation with lattice QCD data

Dynamical chiral-symmetry breaking (DCSB) in QCD is investigated in the Schwinger-Dyson (SD) formalism based on lattice QCD data. From the quenched lattice data for the quark propagator in the Landau gauge, we extract the SD integral kernel function, the product of the quark-gluon vertex and the polarization factor in the gluon propagator, in an Ansatz-independent manner. We find that the SD kernel function exhibits the characteristic behavior of nonperturbative physics, such as infrared vanishing and strong enhancement at the intermediate-energy region around p 0.6GeV. The infrared and intermediate energy region (0.4GeV < p < 1.5GeV) is found to be most relevant for DCSB from analysis on the relation between the SD kernel and the quark mass function. We apply the lattice-QCD-based SD equation to thermal QCD, and calculate the quark mass function at the finite temperature. Spontaneously broken chiral symmetry is found to be restored at high temperature above 110 MeV.     

Chiral symmetry breaking in QCD

By applying bifurcation theory to a truncated Dyson-Schwinger equation for the quark propagator in massless QCD, we show that dynamical symmetry breaking occurs at a certain critical value of the coupling constant. Essential ingredients are (a) an effective dynamical mass for the gluon, and (b) a running coupling constant.     

Chiral symmetry breaking and pion properties at finite temperatures

Spontaneous and explicit chiral symmetry breaking is analyzed in Coulomb gauge QCD at finite temperatures, using an instantaneous approximation for the quark interaction and incorporating confinement through a running coupling constant. The thermodynamics of the quarks is treated approximatively by assuming that the momentum-dependent constituent quark mass sets the scale for thermodynamic fluctuations of colour singlet excitations. We investigate the class of a temperature independent and a temperature dependent interaction between quarks. In the chiral limit both temperature independent and a smooth temperature dependent interaction yields a second order chiral phase transition with critical exponents close to the values for a BCS super-conductor. For explicit chiral symmetry breaking we find a nearly constant pion mass below the transition temperature, but a strongly overdamped mode above. For a first order deconfining transition in the gluonic sector also the quark sector shows a first order chiral phase transition. The relevance of our results for relativistic heavy ion collisions is briefly discussed.     

The quark propagator from the Dyson-Schwinger equations: (I). The chiral solution

Within the framework of the Dyson-Schwinger equations in the axial gauge, and using a truncation procedure which respects the Ward-Takahashi identities, we study the effect that nonperturbative glue has on the quark propagator. We show that within this truncation scheme, the requirement of matching perturbative QCD at high momentum transfer leads to a multiplicatively renormalisable equation. Technically, the matching with perturbation theory is accomplished by the introduction of a transverse part to the quark-gluon vertex. In the case of an analytic gluon propagator, this truncation scheme can lead to chiral symmetry breaking only after the introduction of such a transverse vertex: massless solutions do not exist beyond a critical value of as. Using the gluon propagator that we previously obtained, we obtain small corrections to the quark propagator, which keeps a pole at the origin in the chiral phase.     

Verifying the Kugo-Ojima confinement criterion in Landau gauge Yang-Mills theory

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-Mills theory.     

Confining but chirally symmetric dense and cold matter

The possibility for existence of cold, dense chirally symmetric matter with confinement is reviewed. The answer to this question crucially depends on the mechanism of mass generation in QCD and interconnection of confinement and chiral symmetry breaking. This question can be clarified from spectroscopy of hadrons and their axial properties. Almost systematical parity doubling of highly excited hadrons suggests that their mass is not related to chiral symmetry breaking in the vacuum and is approximately chirally symmetric. Then there is a possibility for existence of confining but chirally symmetric matter. We clarify a possible mechanism underlying such a phase at low temperatures and large density. Namely, at large density the Pauli blocking prevents the gap equation to generate a solution with broken chiral symmetry. However, the chirally symmetric part of the quark Green function as well as all color non-singlet quantities are still infrared divergent, meaning that the system is with confinement. A possible phase transition to such a matter is most probably of the first order. This is because there are no chiral partners to the lowest lying hadrons.     

Confinement and dynamical chiral symmetry breaking in a non-perturbative renormalizable quark model

Inspired by the construction of the Gribov–Zwanziger action in the Landau gauge, we introduce a quark model exhibiting both confinement and chiral symmetry aspects. An important feature is the incorporation of spontaneous chiral symmetry breaking in a renormalizable fashion. The quark propagator in the condensed vacuum turns out to be of a confining type. Besides a real pole, it exhibits complex conjugate poles. The resulting spectral form is explicitly shown to violate positivity, indicative of its unphysical character. Moreover, the ensuing quark mass function fits well to existing lattice data. To further validate the physical nature of the model, we identify a massless pseudoscalar (i.e. a pion) in the chiral limit and present estimates for the ρ$\rho$ meson mass and decay constant.     

Diquark confinement in an extended NJL model

In a Nambu-Jona-Lasinio model supplemented with an infrared cutoff in addition to the ultraviolet cutoff we study the issue whether diquarks are confined when the model is extended beyond the rainbow-ladder approximation. The gap equation, obtained in a truncation scheme motivated via a non-trivial quark-gluon vertex function, is solved to determine the constituent quark mass if chiral symmetry is spontaneously broken. In a second step, the Bethe-Salpeter equations for mesons and diquarks beyond the ladder approximation are derived, taking care to preserve Goldstone's theorem in the pion channel. While the obtained masses of pseudo-scalar and vector mesons are only moderately shifted compared to the values in the ladder approximation, we observe that scalar diquarks disappear from the physical spectrum and therefore are confined. For axial-vector diquarks we observe indications that the same mechanism may also work, but the NJL model allows no conclusive answer in this channel.     

Dynamical Chiral Symmetry Breaking and Phase Transition of Cuprate Superconductors in QED3

With the coupled Dyson-Schwinger equations in the framework of the unified QED₃ theory, we study the phase transition between the antiferromagnet(AF) and the d-wave superconductor (dSC) of planar cuprates at T=0. By solving the coupled Dyson-Schwinger equations both analytically and numerically in rainbow approximation in Landau gauge and comparing the obtained results with that given in the 1/N expansion, we find that there exists a chiral symmetry breaking from dSC phase to AF phase when the quasi-fermion flavors N≤4 in half-filling and the AF phase can possibly coexist with the dSC phase in the underdoped region. By comparing the pressure between the coexistent AF-dSC phase and dSC phase, we find that AF-dSC coexisting phase is the stable phase, the AF phase can then coexist with the dSC phase.     

Real-Time Thermal Schwinger-Dyson Equation for Quark Self-energy in Landau Gauge

By means of a formal expression of Cornwall-Jackiw-Tomboulis effective potential for quark propagator at finite temperatures and finite quark chemical potentials, we derive the real-time thermal Schwinger-Dyson equation for quark propagator in Landau gauge. Denote the inverse quark propagator by A(p2)p - B(p2), we argue that, when temperature T is lower than the given infrared momentum cutoff pc, A(p2) = 1 is a feasible approximation and can be assumed in discussions of chiral symmetry phase transition problem in QCD.     

Cornwall-Jackiw-Tomboulis Effective Potential for Quark Propagator in Real-Time Thermal Field Theory and Landau Gauge

We complete the derivation of the Cornwall-Jackiw-Tomboulis effective potential for quark propagator at finite temperature and finite quark chemical potential in the real-time formalism of thermal field theory and in Landau gauge. In the approximation that the function A(p2) in inverse quark propagator is replaced by unity, by means of the running gauge coupling and the quark mass function invariant under the renormalization group in zero temperature Quantum Chromadynamics (QCD), we obtain a calculable expression for the thermal effective potential, which will be a useful means to research chiral phase transition in QCD in the real-time formalism.     

Cornwall-Jackiw-Tomboulis Effective Potential for Quark Propagator in Real-Time Thermal Field Theory and Landau Gauge

We complete the derivation of the Cornwall-Jackiw-Tomboulis effective potentiM for quark propagator at finite temperature and finite quark chemical potential in the real-time formalism of thermal field theory and in Landau gauge. In the approximation that the function A（p^2） in inverse quark propagator is replaced by unity, by means of the running gauge coupling and the quark mass function invariant under the renormalization group in zero temperature Quantum Chromadynamics （QCD）, we obtain a calculable expression for the thermal effective potential, which will be a useful means to research chiral phase transition in QCD in the real-time formalism.     

Infrared exponents and the running coupling of Landau gauge QCD and their relation to confinement

The infrared behaviour of the gluon and ghost propagators in Landau gauge QCD is reviewed. The Kugo-Ojima confinement criterion and the Gribov-Zwanziger horizon condition result 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. The results are in good agreement with corresponding lattice data obtained recently. The resulting running-coupling approaches a fix point in the infrared, . Two different fits for the scale dependence of the running coupling are given and discussed.Received: 30 September 2002, Published online: 22 October 2003PACS: 12.38.Aw General properties of QCD (dynamics, confinement, etc.) - 14.70.Dj Gluons - 12.38.Lg Other nonperturbative calculations - 11.15.Tk Other nonperturbative techniques - 02.30.Rz Integral equations     

Collective Perspective on Advances in Dyson-Schwinger Equation QCD

We survey contemporary studies of hadrons and strongly interacting quarks using QCD's Dyson-Schwinger equations, addressing the following aspects: confinement and dynamical chiral symmetry breaking; the hadron spectrum; hadron elastic and transition form factors, from small- to large-Q²; parton distribution functions; the physics of hadrons containing one or more heavy quarks; and properties of the quark gluon plasma.     

Collective Perspective on Advances in Dyson–Schwinger Equation QCD

We survey contemporary studies of hadrons and strongly interacting quarks using QCD's Dyson-Schwinger equations, addressing the following aspects: confinement and dynamical chiral symmetry breaking; the hadron spectrum; hadron elastic and transition form factors, from small-to large-Q2; parton distribution functions; the physics of hadrons containing one or more heavy quarks; and properties of the quark gluon plasma.     

Chiral symmetry breaking at finite temperature in Coulomb gauge QCD

Chiral symmetry breaking at finite temperature is analysed in Coulomb gauge QCD, using a suitably renormalised gap equation. In Coulomb gauge the gap equation is derived using the Ward identities and the Dyson equations for the vector and axial-vector vertices. Making the ladder approximation to the Bethe-Salpeter kernel relates the chiral symmetry breaking parameters to the static quark potential. It is thus possible to use a confining potential in the analysis of chiral symmetry breaking. We extend this to finite temperature. For a confining potential there is no chiral symmetry restoration at any finite temperature.     

Phase structure of two-flavor QCD at finite chemical potential

We study the phase diagram of two-flavor QCD at imaginary chemical potentials in the chiral limit. To this end we compute order parameters for chiral symmetry breaking and quark confinement. The interrelation of quark confinement and chiral symmetry breaking is analyzed with a new order parameter for the confinement phase transition. We show that it is directly related to both the quark density as well as the Polyakov loop expectation value. Our analytical and numerical results suggest a close relation between the chiral and the confinement phase transition.     

Aspects of the confinement mechanism in Coulomb-gauge QCD

Phenomenological consequences of the infrared singular, instantaneous part of the gluon propagator in the 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. In the limit of a vanishing infrared regulator the diquark masses diverge, while meson properties and diquark radii remain finite and well defined. These features are interpreted with respect to the resulting aspects of confinement for colored quark-quark correlations.