Full Gluon Propagator Research Articles

General exact solutions for the full gluon propagator in QCD with the mass gap

We have explicitly shown that Quantum Chromodynamics is a color gauge invariant theory with non-zero mass gap, which has been defined as the value of the regularized full gluon self-energy at a finite scale point. The mass gap itself is mainly generated by the nonlinear interaction of massless gluon modes. All this allows one to establish the structure of the full gluon propagator in the explicit presence of the mass gap. In this case, the two independent general types of formal solutions for the full gluon propagator as a function of the regularized mass gap have been found: (i) The nonlinear iteration solution at which the gluons remain massless is explicitly present. (ii) Existence of the solution with an effective gluon mass is also demonstrated.

Nonlinear Iteration Solution for the Full Gluon Propagator as a Function of the Mass Gap

We have explicitly shown that QCD is the color gauge invariant theory at non-zero mass gap as well. It has been defined as the value of the regularized full gluon self-energy at some finite point. The mass gap is mainly generated by the nonlinear interaction of massless gluon modes. All this allows one to establish the structure of the full gluon propagator in the explicit presence of the mass gap. In this case, the two independent general types of formal solutions for the full gluon propagator as a function of the regularized mass gap have been found. The nonlinear iteration solution at which the gluons remain massless is explicitly present. The existence of the solution with an effective gluon mass is also demonstrated.

Renormalization of the Mass Gap

The full gluon propagator relevant for the description of the truly non-perturbative QCD dynamics, the so-called intrinsically non-perturbative gluon propagator has been derived in our previous work. It explicitly depends on the regularized mass gap, which dominates its structure at small gluon momentum. It is automatically transversal in a gauge invariant way. It is characterized by the presence of severe infrared singularities at small gluon momentum, so the gluons remain massless, and this does no depend on the gauge choice. In this paper we have shown how precisely the renormalization program for the regularized mass gap should be performed. We have also shown how precisely severe infrared singularities should be correctly treated. This allowed to analytically formulate the exact and gauge-invariant criteria of gluon and quark confinement. After the renormalization program is completed, one can derive the gluon propagator applicable for the calculation of physical observables, processes, etc., in low-energy QCD from first principles.

The Color Gauge Invariance and a Possible Origin of Mass in QCD

The general scale parameter, having the dimensions of mass squared, is dynamically generated in the QCD gluon sector. It is introduced through the difference between the regularized full gluon self-energy and its value at some finite point. It violates transversality of the full gluon self-energy. The Slavnov-Taylor identity for the full gluon propagator, when it is given by the corresponding equation of motion, is also violated by it. So in order to maintain both transversality and the identity it should be disregarded from the very beginning, i.e., put formally zero everywhere. However, we have shown how to preserve the above-mentioned identity at non-zero mass squared parameter. This allows one to establish the structure of the full gluon propagator when it is explicitly present. Its contribution does not survive in the perturbation theory regime, when the gluon momentum goes to infinity. At the same time, its contribution dominates the structure of the full gluon propagator when the gluon momentum goes to zero. We have also proposed a method how to restore transversality of the relevant gluon propagator in a gauge invariant way, while keeping the mass squared parameter “alive”.

Dynamical generation of the mass gap in QCD

We have unambiguously established the dynamical source of the mass scale parameter (the mass gap) responsible for the large scale structure of the true QCD vacuum. At the microscopic, Lagrangian level it is the nonlinear fundamental four-gluon interaction. At the level of the corresponding equation of motion for the full gluon propagator, it is all the skeleton loop contributions into the gluon self-energy, which contain the four-gluon vertices. The key role of the four-gluon interaction is determined by the fact that this interaction survives when all the gluon momenta involved go to zero, while the three-gluon vertex vanishes in this limit. The mass gap and the corresponding infrared singularities are “hidden” in these terms, and they show up explicitly when the gluon momentum q goes to zero. The general iteration solution (i.e., when the relevant skeleton loop integrals have to be iterated) for the full gluon propagator unavoidably becomes the exact sum of the two terms. The first term is the Laurent expansion in the inverse powers of the gluon momentum squared, starting necessarily from the simplest one 1/(q2)2. Each severe (i.e., more singular than 1/q2) power-type IR singularity is accompanied by the corresponding powers of the mass gap. The standard second term is always as much singular as 1/q2, otherwise remaining undetermined. The inevitable existence of the first term makes just the principal difference between non-Abelian QCD and Abelian QED. Moreover, the infrared renormalization program of the theory leads to the gluon confinement criterion in the gauge-invariant way.

Failure of the ladder approximation to QCD

The proof of the failure of the ladder approximation to QCD is given in manifestly gauge-invariant way. This proof is valid for the full gluon propagator and for all types of quarks. The summation of the ladder diagrams within the Schwinger–Dyson integral equation for the quark–gluon vertex, on account of the corresponding Slavnov–Taylor identity, provides an additional constraint on the quark Schwinger–Dyson equation itself in the ladder approximation. It requires that there is neither running nor current quark masses in the ladder approximation. Thus, all the results based on the nontrivial (analytical or numerical) solutions to the quark Schwinger–Dyson equation in the ladder approximation should be reconsidered, and its use in the whole energy/momentum range should be abandoned.

Gluon confinement criterion in QCD

We fix 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/q2) infrared singularities. It reflects the zero momentum modes enhancement effect in the true QCD vacuum, which is due to the self-interaction of massless gluons. Its existence automatically exhibits a characteristic mass (the so-called mass gap). It is responsible for the scale of nonperturbative dynamics in the true QCD ground state. The theory of distributions, complemented by the dimensional regularization method, allows one to put severe infrared singularities under firm mathematical control. By an infrared renormalization of a mass gap only, the infrared structure of the full gluon propagator is exactly reduced to the simplest severe infrared singularity, the famous (q2)−2. Thus we have exactly established the interaction between quarks (concerning its pure gluon (i.e., nonlinear) contribution) up to its unimportant perturbative part. This also makes it possible for the first time to formulate the gluon confinement criterion and intrinsically nonperturbative phase in QCD in a manifestly gauge-invariant ways.

Erratum: Determination of the quantum part of the truly nonperturbative Yang-Mills vacuum energy density in covariant gauge QCD [Phys. Rev. D62, 076008 (2000)

Using the effective potential approach for composite operators, we have formulated a general method of calculation of the truly nonperturbative Yang-Mills vacuum energy density in the covariant gauge QCD ground state quantum models. It is defined as an integration of the truly nonperturbative part of the full gluon propagator over the deep infrared region (soft momentum region). A nontrivial minimization procedure makes it possible to determine the value of the soft cutoff in terms of the corresponding nonperturbative scale parameter, which is inevitably present in any nonperturbative model for the full gluon propagator. We have shown for specific models of the full gluon propagator explicitly that the use of the infrared enhanced and finite gluon propagators lead to the vacuum energy density which is finite, always negative and it has no imaginary part (stable vacuum), while the infrared vanishing propagators lead to unstable vacuum and therefore they are physically unacceptable.

Vacuum instability in the Abelian Higgs model with strings

Using the effective potential approach for composite operators, we have analytically evaluated the truly nonperturbative vacuum energy density in the Abelian Higgs model of dual QCD ground state. This quantity is defined as integrated out of the truly nonperturbative part of the full gluon propagator over the deep infrared region (soft momentum region). Defined in this way it is manifestly gauge invariant. We have explicitly shown that the corresponding effective potential always has an imaginary part. This means that the vacuum of this model with string contributions is unstable against quantum corrections.

Determination of the quantum part of the truly nonperturbative Yang-Mills vacuum energy density in covariant gauge QCD

Using the effective potential approach for composite operators, we formulate a general method of calculation of the truly nonperturbative Yang-Mills vacuum energy density in the covariant gauge QCD ground state quantum models. It is defined as an integration of the truly nonperturbative part of the full gluon propagator over the deep infrared region (soft momentum region). A nontrivial minimization procedure makes it possible to determine the value of the soft cutoff in terms of the corresponding nonperturbative scale parameter, which is inevitably present in any nonperturbative model for the full gluon propagator. We have shown for specific models of the full gluon propagator explicitly that the use of the infrared-enhanced and finite gluon propagators leads to the vacuum energy density which is finite, always negative, and has no imaginary part (stable vacuum), while the infrared vanishing propagators lead to an unstable vacuum and therefore they are physically unacceptable.

Structure of the Yang–Mills vacuum in the zero modes enhancement quantum model

We have formulated new quantum model of the QCD vacuum using the effective potential approach for composite operators. It is based on the existence and importance of such kind of the nonperturbative, topologically nontrivial excitations of gluon field configurations, which can be effectively correctly described by the q −4-type behavior of the full gluon propagator in the deep infrared domain. The ultraviolet part of the full gluon propagator was approximated by the asymptotic freedom to-leading order perturbative logarithm term of the running coupling constant. Despite the vacuum energy density remains badly divergent, we have formulated a method how to establish a finite (in the ultraviolet limit) relation between the two scale parameters of our model. We have expressed the asymptotic scale parameter as pure number times the nonperturbative scale, which is inevitably contained in any realistic Ansatz for the full gluon propagator.

How to calculate the quantum part of the truly nonperturbative Yang-Mills vacuum energy density in the axial gauge QCD

Using the effective potential approach for composite operators, we have formulated a general method how to calculate the truly nonperturbative vacuum energy density in the axial gauge QCD quantum models of its ground state. It is defined as integrated out the truly nonperturbative part of the full gluon propagator over the deep infrared region (soft momentum region). The nontrivial minimization procedure makes it possible to determine the value of the soft cutoff in terms of the corresponding nonperturbative scale parameter which is inevitably presented in any nonperturbative model for the full gluon propagator. If the chosen Ansatz for the full gluon propagator is a realistic one, then our method uniquely determines the truly vacuum energy density, which is always finite, automatically negative and it has no imaginary part (stable vacuum). We illustrate it by considering the Abelian Higgs model of dual QCD ground state. We have explicitly shown that the vacuum of this model without string contributions is unstable against quantum corrections.

NONPERTURBATIVE QUARK PROPAGATOR APPROACH TO QCD AT LARGE DISTANCES

A nonperturbative approach to QCD at large distances in the context of the Schwinger-Dyson equations and corresponding Slavnov-Taylor identity in the quark sector is presented. Making only one widely accepted assumption that the full gluon propagator becomes an infrared singular like (q2)−2 in the arbitrary covariant gauge, we find three and only three confinement-type solutions for the quark propagator (quark confinement theorem). Two of them vanish after the removal of the infrared regulation parameter. The third solution does not depend on this latter parameter, but it has no pole and it implies dynamical chiral symmetry breakdown (DCSB), which means a close connection between quark confinement and DCSB. We also show that multiplication solely by the quark infrared renormalization constant would make all the Green’s functions infrared finite (multiplicative renormalizability). The final forms of the renormalized (infrared finite) quark SD equations do not explicitly depend on a gauge-fixing parameter (“gauge invariance”). Our approach is free of ghost complications despite the fact that they play an essential role in nonperturbative dynamics. Our approach also implies the existence of a characteristic scale at which confinement, DCSB and other nonperturbative effects become essential. We solve explicitly the SD equation with corresponding ST identity for the above-mentioned IR finite quark propagator in the chiral limit and apply an effective potential in order to determine completely this solution.

Erratum: Gauge-invariant, nonperturbative approach to quark confinement and chiral-symmetry breaking in QCD

A gauge-invariant, nonperturbative approach to quark confinement and chiral-symmetry breaking in the context of the Schwinger-Dyson equations and corresponding Slavnov-Taylor identities is presented. Making only one widely accepted assumption that the full gluon propagator becomes an infrared singularity like {ital q}{sup {minus}4} at small momenta, we obtain a nonperturbative, gauge-invariant, infrared-finite quark propagator, which has no pole (confinement-type solution) and implies chiral-symmetry breaking (dynamical quark mass generation), so that we establish a close connection between these nonperturbative phenomena. We discover that the ghost degrees of freedom play an essential role in the dynamics of chiral-symmetry breaking, but nevertheless the quark propagator we got was free of ghost complications. In addition to the infrared-finite solution, we find also two infrared-vanishing (after the removal of the infrared regulation parameter) solutions for the quark propagator. For the dynamical (nonperturbative) quark mass we derive the expression which exhibits an essential singularity in the coupling constant in accordance with renormalization-group solutions in the infrared region.

Gauge-invariant, nonperturbative approach to quark confinement and chiral-symmetry breaking in QCD.

A gauge-invariant, nonperturbative approach to quark confinement and chiral-symmetry breaking in the context of the Schwinger-Dyson equations and corresponding Slavnov-Taylor identities is presented. Making only one widely accepted assumption that the full gluon propagator becomes an infrared singularity like ${q}^{\ensuremath{-}4}$ at small momenta, we obtain a nonperturbative, gauge-invariant, infrared-finite quark propagator, which has no pole (confinement-type solution) and implies chiral-symmetry breaking (dynamical quark mass generation), so that we establish a close connection between these nonperturbative phenomena. We discover that the ghost degrees of freedom play an essential role in the dynamics of chiral-symmetry breaking, but nevertheless the quark propagator we got was free of ghost complications. In addition to the infrared-finite solution, we find also two infrared-vanishing (after the removal of the infrared regulation parameter) solutions for the quark propagator. For the dynamical (nonperturbative) quark mass we derive the expression which exhibits an essential singularity in the coupling constant in accordance with renormalization-group solutions in the infrared region.

Quark confinement and dynamical breakdown of chiral symmetry in covariant gauge QCD

Abstract Making only one widely accepted assumption that the full gluon propagator becomes infrared singular like (q2)−2 at small momenta, we investigate in a gauge-invariant way the Schwinger-Dyson equation for the propagator, completed by the corresponding Slavnov-Taylor identity. We find three and only three confinement-type solutions for the quark propagator in the infrared region. We also establish a close connection between quark confinement and chiral symmetry breaking. We discover that the ghost degrees of freedom play an essential role in the dynamics of chiral symmetry beraking, but, nevertheless, the infrared-finite (it does not depend on the infrared regulation parameter ϵ in the ϵ→0+ limit), gauge-invariant and nonperturbative quark propagator we obtained was free of ghost complications.