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.     

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.     

Deconfinement in dense two-color QCD

We study SU(2) lattice gauge theory with two flavors of Wilson fermion at non-zero chemical potential μ and low temperature on a 8³×16 system. We identify three régimes along the μ-axis. For μ≲m_π/2 the system remains in the vacuum phase and all physical observables considered remain essentially unchanged. The intermediate régime is characterised by a non-zero diquark condensate and an associated increase in the baryon density, consistent with what is expected for Bose–Einstein condensation of tightly bound diquarks. We also observe screening of the static quark potential here. In the high-density deconfined régime we find a non-zero Polyakov loop and a strong modification of the gluon propagator, including significant screening in the magnetic sector in the static limit, which must have a non-perturbative origin. The behaviour of thermodynamic observables and the superfluid order parameter are consistent with a Fermi surface disrupted by a BCS diquark condensate. The energy per baryon as a function of μ exhibits a minimum in the deconfined régime, implying that macroscopic objects such as stars formed in this theory are largely composed of quark matter.     

Bound diquarks and their Bose–Einstein condensation in strongly coupled quark matter

We explore the formation of diquark bound states and their Bose–Einstein condensation (BEC) in the phase diagram of three-flavor quark matter at nonzero temperature, T, and quark chemical potential, μ  . Using a quark model with a four-fermion interaction, we identify diquark excitations as poles of the microscopically computed diquark propagator. The quark masses are obtained by solving a dynamical equation for the chiral condensate and are found to determine the stability of the diquark excitations. The stability of diquark excitations is investigated in the T–μ$T\text{–}\mu$ plane for different values of the diquark coupling strength. We find that diquark bound states appear at small quark chemical potentials and at intermediate coupling strengths. Bose–Einstein condensation of non-strange diquark states occurs when the attractive interaction between quarks is sufficiently strong.     

Critical Number of Fermion Flavors at Finite Chemical Potential in QED3

We propose a new method for calculating the dressed fermion propagator at finite chemical potential in QED3 under the rainbow approximation of Dyson-Schwinger equation. In the above approximation, we show that the dressed fermion propagator at finite chemical potential # has the form S（p） = iγ.p^-A（p^-2） ＋ B（ p^-2） with p^-μ= （p^-1p3 ＋ iμ）. Using this form of fermion propagator at nonzero chemical potential, we investigate the Dyson-Schwinger equation for the dressed fermion propagator at finite chemical potential and study the effects of the chemical potential on the critical number of the fermion flavors.     

Dyson-Schwinger approach to color superconductivity at finite temperature and density

We investigate the phases of dense QCD matter at finite temperature with Dyson-Schwinger equations for the quark propagator for N _f = 2 + 1 flavors. For the gluon propagator we take a fit to quenched lattice data and add quark-loop effects perturbatively in a hard-thermal-loop-hard-dense-loop approximation. We consider 2SC and CFL-like pairing with chiral up and down quarks and massive strange quarks and present results for the condensates and the phase diagram. We find a dominant CFL phase at chemical potentials larger than 500-600MeV. At lower values of the chemical potential we find a 2SC phase, which also exists in a small band at higher temperatures for larger chemical potentials. With values of 20–30 MeV, the critical temperatures to the normal phase turn out to be quite small.     

A Modified Approach for Calculating Dressed Quark Propagator at Finite Chemical Potential

Based on the rainbow approximation of Dyson-Schwinger equation and the assumption that the full inverse quark propagator at finite chemical potential is analytic in the neighborhood of μ = 1, it is proved that the dressed quark propagator at finite chemical potential μ can be written as G0^-1 [μ] =iγ·p↑-A（p↑-^2） ＋B（p↑-^2） with p↑-μ= （p↑-p4 ＋iμ）. From the dressed quark propagator at finite chemical potential in Munczek model the bag constant of a baryon and the scalar quark condensate are evaluated. A comparison with previous results is given.     

Behavior of Vacuum Polarization of Gauge-Boson and Wavefunction Renormalization Factor of Fermion in Different Phases of QED3

We investigate the behavior of the vacuum polarization of the gauge-boson Π and the wave-function renormalization factor of the fermion A in QED₃, using the coupled Dyson-Schwinger equations for the gauge-boson and fermion propagator. Using several different ansätze for the fermion-gauge-boson vertex, we find that the wave-function renormalization factor A and especially the vacuum polarization Π have different behaviors in the dynamical chiral symmetry breaking phase and in the chiral symmetric phase and hence in the phenomenological applications of QED₃ one should choose different forms of gauge-boson propagator for these two phases. We also find that when adopting a specific ansätze of the fermion-gauge-boson vertex (ansätze (3)) the vacuum polarization function equals its one-loop perturbative result in the chiral symmetric phase. This fact suggests that in QED₃ the Wigner vacuum corresponds to the perturbative vacuum.     

Numerically Solving Quark-Loop Effects on Dressed Gluon Propagator in Chiral Limit

We do a numerical calculation on the quark-loop effects on the dressed gluon propagator in the chiral limit. It is found that the quark-loop effects on the dressed gluon propagator are significant in solving the quark propagator in the rainbow approximation of the Dyson-Schwinger equation. The approach we used here is quite general and can also be used to calculate both the chemical potential and current quark mass dependence of the dressed gluon propagator.     

Critical behavior in the dense planar Nambu--Jona-Lasinio model

We present results of a Monte Carlo simulation of a (2+1)-dimensional Nambu-Jona-Lasinio model. In the vacuum phase, the diquark condensate vanishes linearly as a function of diquark source j as expected, but simulations in a region with nonzero baryon density suggest a power-law scaling infinity j(alpha) and hence a critical system for all mu > mu(c). There is no signal for superfluidity. Comparisons are drawn with the pseudogap phase in cuprate superconductors. We also measure the dispersion relation E(k) for fermionic excitations, and find results consistent with a sharp Fermi surface. Any gap Delta is constrained to be much less than the constituent quark mass scale Sigma(0).     

Calculation of Partition Function of QCD at Finite Chemical Potential

In equilibrium statistical field theory, the partition function has fundamental importance. In this paper we propose a direct and general method for calculating the partition function and equation of state of QCD at finite chemical potential. It is found that the partition function is totally determined by the dressed quark propagator at finite chemical potential up to a multiplicative constant. From this a criterion for the phase transition between the Nambu and the Wigner phases is obtained. This general method is applied to two specific cases： the free quark theory and QCD with a model dressed quark propagator having confinement features. In the first case, the standard Fermi distribution at T = 0 is reproduced. In the second case, we apply the conclusion in previous works to obtain the dressed quark propagator at finite chemical potential and find the unphysical result that the baryon number density vanishes for all values of chemical potential. The reason for this result is discussed.     

Effect of Fermion Velocity on Phase Structure of QED3

Dynamical chiral symmetry breaking (DCSB) in thermal QED₃ with fermion velocity is studied in the framework of Dyson-Schwinger equations. By adopting instantaneous approximation and neglecting the transverse component of gauge boson propagator at finite temperature, we numerically solve the fermion self-energy equation in the rainbow approximation. It is found that both DCSB and fermion chiral condensate are suppressed by fermion velocity. Moreover, the critical temperature decreases as fermion velocity increases.     

Influence of Finite Chemical Potential on Critical Boson Mass in QED3

Using the coupled Dyson-Schwinger equation for the fermion propagator at finite chemical potential μ, we investigate the fermion chiral condensate when the gauge boson mass is nonzero in QED3. We show that the chiral symmetry restores when the boson mass is large enough, and the critical boson mass depends little on μ.     

Low-energy dynamics in ultradegenerate QCD matter

We study the low-energy behavior of QCD Green functions in the limit that the baryon chemical potential is much larger than the QCD scale parameter LambdaQCD. We show that there is a systematic low-energy expansion in powers of (omega/m)(1/3), where omega is the energy and m is the screening scale. This expansion is valid even if the effective quark-gluon coupling g is not small. The expansion is purely perturbative in the magnetic regime |k| > k0. If the external momenta and energies satisfy |k| approximately k0, planar, Abelian ladder diagrams involving the full quark propagator have to be resummed but the corresponding Dyson-Schwinger equations are closed.     

Chiral and deconfinement phase transitions of two-flavour QCD at finite temperature and chemical potential

We present results for the chiral and deconfinement transition of two flavor QCD at finite temperature and chemical potential. To this end we study the quark condensate and its dual, the dressed Polyakov loop, with functional methods using a set of Dyson-Schwinger equations. The quark propagator is determined self-consistently within a truncation scheme including temperature and in-medium effects of the gluon propagator. For the chiral transition we find a crossover turning into a first order transition at a critical endpoint at large quark chemical potential, μEP/TEP≈3. For the deconfinement transition we find a pseudo-critical temperature above the chiral transition in the crossover region but coinciding transition temperatures close to the critical endpoint.     

The quark-gluon vertex in Landau gauge QCD: Its role in dynamical chiral symmetry breaking and quark confinement

The infrared behavior of the quark-gluon vertex of quenched Landau gauge QCD is studied by analyzing its Dyson-Schwinger equation. Building on previously obtained results for Green functions in the Yang-Mills sector, we analytically derive the existence of power-law infrared singularities for this vertex. We establish that dynamical chiral symmetry breaking leads to the self-consistent generation of components of the quark-gluon vertex forbidden when chiral symmetry is forced to stay in the Wigner-Weyl mode. In the latter case the running strong coupling assumes an infrared fixed point. If chiral symmetry is broken, either dynamically or explicitly, the running coupling is infrared divergent. Based on a truncation for the quark-gluon vertex Dyson-Schwinger equation which respects the analytically determined infrared behavior, numerical results for the coupled system of the quark propagator and vertex Dyson-Schwinger equation are presented. The resulting quark mass function as well as the vertex function show only a very weak dependence on the current quark mass in the deep infrared. From this we infer by an analysis of the quark-quark scattering kernel a linearly rising quark potential with an almost mass independent string tension in the case of broken chiral symmetry. Enforcing chiral symmetry does lead to a Coulomb type potential. Therefore, we conclude that chiral symmetry breaking and confinement are closely related. Furthermore, we discuss aspects of confinement as the absence of long-range van der Waals forces and Casimir scaling. An examination of experimental data for quarkonia provides further evidence for the viability of the presented mechanism for quark confinement in the Landau gauge.     

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.     

Thermodynamics of chiral symmetry at low densities

The phase diagram of two-color QCD as a function of temperature and baryon chemical potential is considered. Using a low-energy chiral Lagrangian based on the symmetries of the microscopic theory, we determine, at the one-loop level, the temperature dependence of the critical chemical potential for diquark condensation and the temperature dependence of the diquark condensate and baryon density. The prediction for the temperature dependence of the critical chemical potential is consistent with the one obtained for a dilute Bose gas. The associated phase transition is shown to be of second order for low temperatures and first order at higher temperatures. The tricritical point at which the second order phase transition ends is determined. The results are carried over to QCD with quarks in the adjoint representation and to ordinary QCD at a non-zero chemical potential for isospin.     

Diquark condensation in dense adjoint matter

We study SU(2) lattice gauge theory at non-zero chemical potential with one staggered quark flavor in the adjoint representation. In this model the fermion determinant, although real, can be both positive and negative. We have performed numerical simulations using both hybrid Monte Carlo and two-step multibosonic algorithms, the latter being capable of exploring sectors with either determinant sign. We find that the positive determinant sector behaves like a two-flavor theory, with the chiral condensate rotating into a two-flavor diquark condensate for , implying a superfluid ground state. Good agreement is found with analytical predictions made using chiral perturbation theory. In the ‘full’ model there is no sign of either onset of baryon density or diquark condensation for the range of chemical potentials we have considered. The impact of the sign problem has prevented us from exploring the true onset transition and the mode of diquark condensation, if any, for this model. Received: 28 September 2001 / Published online: 23 November 2001     

Validity of Parametrized Quark Propagator

Based on an extensively study of the Dyson-Schwinger equations for a fully dressed quark propagator in the “rainbow”approximation, a parametrized fully dressed quark propagator is proposed in this paper. The parametrized propagator describes a confining quark propagator in hadron since it is analytic everywhere in complex p2-plane and has no Lemmann representation. The validity of the new propagator is discussed by comparing its predictions on selfenergy functions A/(p2), Bl(p2) and effective mass M$(p2) of quark with flavor f to their corresponding theoretical results produced by Dyson-Schwinger equations. Our comparison shows that the parametrized quark propagator is a good approximation to the fully dressed quark propagator given by the solutions of Dyson-Schwinger equations in the rainbow approximation and is convenient to use in any theoretical calculations.