Light 0- Meson Solution of Bethe-Salpeter Equation Involving the Vacuum Condensates in QCD

We dkcuss the Bethe-Salpeter equation involving the vacuum condensates in QCD in the backgrouud fields. The bound state equation for the light 0^- meson (π and K) is solved in the ladder approximation of single gluon exchange, where the gluon propagator is calculated in tree level of a gluon condensate and a quark condensate. The kernel is divided into two parts: perturbative and non-pert,urbative. The non-perturbative part is determined by the vacuum condensate. Pseudo-scalar meson spectrum and decay constants are in good agreement with data.     

Correction to Goldstein Equation from the Gluon Condensate Diagram

An exact Goldstein solution of the Bethe-Salpeter equation including the gluon condensate in the background field QCD is discussed.We find that the gluon condensate diagram makes quark and anti-quark more close to form a bound state.     

Chiral and deconfinement transition from correlation functions: SU(2) vs. SU(3)

We study a gauge-invariant order parameter for deconfinement and the chiral condensate in SU(2) and SU(3) Yang–Mills theory in the vicinity of the deconfinement phase transition using the Landau gauge quark and gluon propagators. We determine the gluon propagator from lattice calculations and the quark propagator from its Dyson–Schwinger equation, using the gluon propagator as input. The critical temperature and a deconfinement order parameter are extracted from the gluon propagator and from the dependency of the quark propagator on the temporal boundary conditions. The chiral transition is determined using the quark condensate as order parameter. We investigate whether and how a difference in the chiral and deconfinement transition between SU(2) and SU(3) is manifest.     

Plane-wave,coordinate-space,and moment techniques in the operator-product expansion: Equivalence,improved methods,and the heavy quark expansion

We compare plane-wave, coordinate-space and moment methods for evaluating operator-product expansion (OPE) coefficients of the light-quark and gluon condensates. Equivalence of these methods for quark condensate contributions is proven to all orders in the quark mass parameterm. The three methods are also shown to yield equivalent gluon condensate contributions to two-current correlation functions, regardless of the gauge chosen for external gluon fields in the coordinate space approach. An improved method for evaluating quarkcondensate OPE coefficients is presented for several (two-current) correlation functions. Gauge-dependent Green functions are also discussed. It is shown that contradictory expressions for the gluon-condensate contribution to the quark propagator occurring from the plane-wave and coordinate-space approaches yield identical relations between the heavy-quark and gluon condensates, as anticipated from the gauge invariance of the heavy-quark expansion.     

Gluon condensates at finite baryon densities and temperature

We derive here the equation of state for quark matter with a nontrivial vacuum structure in QCD at finite temperature and baryon density. Using thermofield dynamics, the parameters of thermal vacuum and the gluon condensate function are determined through minimisation of the thermodynamic potential, along with a self-consistent determination of the effective gluon and quark masses. The scale parameter for the gluon condensates is related to the SVZ parameter in the context of QCD sum rules at zero temperature. With inclusion of quarks in the thermal vacuum the critical temperature at which the gluon condensate vanishes decreases as compared to that containing only gluons. At zero temperature, we similarly obtain the critical baryon density for the same to be about 0.36 fm^?3.     

Quark Gluon Condensate,Virtuality and Susceptibility of QCD Vacuum

We study vacuum of QCD in this work. The structure of non-local quark vacuum condensate, values of various local quark and gluon vacuum condensates, quark-gluon mixed vacuum condensate, quark and gluon virtuality in QCD vacuum state, quark dynamical mass and susceptibility of QCD vacuum state to external field are predicted by use of the solutions of Dyson-Schwinger equations in “rainbow” approximation with a modeling gluon propagator and three different sets of quark-quark interaction parameters. Our theoretical predictions are in good agreement with the correspondent empirical values used widely in literature, and many other theoretical calculations. The quark propagator and self-energy functions are also obtained from the numerical solutions of Dyson-Schwinger equations. This work is centrally important for studying non-perturbative QCD, and has many important applications both in particle and nuclear physics.     

Hadron resonance gas and nonperturbative QCD vacuum at finite temperature

Nonperturbative QCD vacuum with two light quarks at finite temperature was studied in a hadron resonance-gas model. Temperature dependences of the quark and gluon condensates in the confined phase were obtained. It is shown that the quark condensate and one-half (chromoelectric component) of the gluon condensate are evaporated at the same temperature corresponding to the quark-hadron phase transition. With allowance for the temperature shift of hadron masses, the critical temperature was found to be T _c ?190 MeV.     

Vacuum Condensates from the Modification of Global Color Symmetry Model

Based on the quark propagator in the instanton dilute liquid approximation, we have determined the quark condensate , the mixed quark gluon condensate g,μνσ^μνq) and the four-quark condensate at the mean-field level in the framework of global color symmetry model. The numerical calculation shows that our result is compatible with the range obtained within other nonperturbative approaches. In particular, we have found that even at the mean-field level the naive vacuum saturation approximation is not a good approximation when we consider nonlocal four-quark condensate.     

Quark condensate contributions to the gluon self-energy and the ϱ meson sum rule

The operator-product expansion will be employed to obtain the lowest-order, quark condensate component of both the gluon self-energy and the ? meson correlation function to all orders in the quark mass parameter. Field-theoretic aspects of the self-energy and correlation function will be considered, and physical effects of the quark condensate upon gluon mass generation will be examined.     

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.     

Power law running of the effective gluon mass

The dynamically generated effective gluon mass is known to depend non-trivially on the momentum, decreasing sufficiently fast in the deep ultraviolet, in order for the renormalizability of QCD to be preserved. General arguments based on the analogy with the constituent quark masses, as well as explicit calculations using the operator-product expansion, suggest that the gluon mass falls off as the inverse square of the momentum, relating it to the gauge-invariant gluon condensate of dimension four. In this article we demonstrate that the power law running of the effective gluon mass is indeed dynamically realized at the level of the non-perturbative Schwinger-Dyson equation. We study a gauge-invariant non-linear integral equation involving the gluon self-energy, and establish the conditions necessary for the existence of infrared finite solutions, described in terms of a momentum-dependent gluon mass. Assuming a simplified form for the gluon propagator, we derive a secondary integral equation that controls the running of the mass in the deep ultraviolet. Depending on the values chosen for certain parameters entering into the Ansatz for the fully dressed three-gluon vertex, this latter equation yields either logarithmic solutions, familiar from previous linear studies, or a new type of solutions, displaying power law running. In addition, it furnishes a non-trivial integral constraint, which restricts significantly (but does not determine fully) the running of the mass in the intermediate and infrared regimes. The numerical analysis presented is in complete agreement with the analytic results obtained, showing clearly the appearance of the two types of momentum dependence, well-separated in the relevant space of parameters. Several technical improvements, various open issues, and possible future directions, are briefly discussed.     

Low-energy relation for the trace of the energy–momentum tensor in QCD and the gluon condensate in a magnetic field

The nonperturbative QCD vacuum in a magnetic field has been studied. A low-energy relation for the trace of the energy–momentum tensor in the magnetic field has been obtained. It has been shown that the derivatives of the quark and gluon contributions to the trace of the energy–momentum tensor with respect to the magnetic field coincide with each other. The magnetic field dependence of the gluon condensate has been calculated in the limits of strong and weak fields.     

Chiral-symmetry breaking and gluon condensation

With the help of the Ward-Takahashi identities associated with a specific non linear chiral transformation of the quark fields we show, in the framework of massless QCD, that in addition to scale invariance breaking gluon condensation implies dynamical chiral symmetry breaking, that is to say the appearance of a non vanishing quark condensate. Moreover the dynamically generated quark mass is directly connected to the quark and gluon condensates.     

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.     

Zero mode solutions of quark Dirac equations in QCD as the sources of chirality violating condensates

It is demonstrated, that chirality violating condensates in massless QCD arise from zero mode solutions of Dirac equations in arbitrary gluon fields. Basing of this idea, the model is suggested, which allows one to calculate quark condensate magnetic susceptibilities in the external constant electromagnetic field.     

QCD中等时方程的赝标介子解

本文讨论在背景场量子色动力学框架中的等时方程赝标介子解. 方程的积分核由微扰和非微扰贡献两部分组成．微拔部分是通常的单胶子交换图的贡献；非微扰部分的贡献来自于最低阶的夸克和胶子凝聚效应. 当输入合理的夸克质量参数和耦合常数时，可以得到与实验相符合的赝标介子谱．     

Non-perturbative Quark Propagator and the Study of the Non-trivial Q2 Dependence in the Nucleon Structure Functions

In consideration of the lowest order non-perturbative effect due to the quark condensate <qq>and gluon condensate on the quark propagator,we calculate QCD non-pertubative quark propagator under the chain approximation.Using the obtained non-perturbative quark propagator,we analyse the non-perturbative effect in the nucleon structure functions and show the non-trivial Q²-dependence in the nucleon structure functions.     

GORDON DECOMPOSITIONS FOR γ-TYPE MATRICES AND SOME OF THEIR APPLICATIONS

In this paper, some decomposotion formulas of γ-type matrices are derived based on Gordon identities. As an illustration of the application of these formulas, the one gluon exchange potential between quark and antiquark is rederived which appears to be different from that obtained by Faessler, et al. As another illustration, the correct reduction of the γ-matrices appearing in the kernel function of Bethe-Salpeter equation for quark-antiquark system is achieved and yields an expression different from that derived by Mitre whose calculation was grounded on a wrong decomposition formula for one γ-type matrix.     

Chiral symmetry breaking in an instantaneous approximation to Coulomb gauge QCD

We analyze the interplay between explicit and spontaneous chiral-symmetry breaking in Coulomb gauge QCD. Quark and pseudoscalar meson properties are investigated, using an instantaneous approximation to gluon exchange, with momentum-dependent coupling constants and current quark masses in agreement with the full QCD renormalization group equations. We show how a finite momentum-dependent constituent quark mass can be defined even for a confining interaction between the quarks, and derive an integral equation for this constituent mass from the renormalized Dyson-Schwinger equations. This equation is shown to be equivalent to a gap equation derived in a Bogoliubov-Valatin variational method from the model's hamiltonian. Including momentum-dependent current masses also ensures a finite value for the quark condensate. We report numerical results for a purely confining and for a Richardson potential for the Coulombic part of the quark-antiquark potential. Transverse gluons are included in the Breit approximation, neglecting retardation. As a confining Breit interaction leads to an infrared inconsistency in the model, and since there is mounting evidence for a dynamical gluon mass, such a mass is included. Numerical results for the constituent quark mass for one flavour, for different values of the current mass, are reported, together with the corresponding energy densities, quark condensates, pseudoscalar meson masses and pseudoscalar meson decay constants. The results are encouraging from a phenomenological point of view.     

Shifman-Vainshtein-Zakharov moments and quark-antiquark potentials

From the SVZ moments, for heavy quark-antiquark systems, we extract an “equivalent” non-relativistic potential. It increases as the fourth power of the distance and linearly with the quark mass. Numerical integration of the “equivalent” Schrödinger equation leads to much higher estimates of the gluon condensate parameter ø₁ than the original inverse power moment method or even the Borel transformed exponential version.