The mass gap and solution of the gluon confinement problem in QCD

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 propagator in the presence of a mass gap. It is mainly generated by the nonlinear interaction of massless gluon modes. The physical meaning of the mass gap is to be responsible for the large-scale (low-energy/momentum), i.e., nonperturbative structure of the true QCD vacuum. The direct nonlinear iteration solution of the transcendental equation for the full gluon propagator in the presence of a mass gap is present. We formulate a generl method how to restore the transversality of the full gluon propagator relevant for the nonperturbative QCD. It is explicitly shown that such a solution confines QCD. The exact and gauge-invariant criterion of gluon confinement is derived. The gauge-invariant quark confinement criterion is also formulated.Comment: 25 pages, no tables, no figures, rearangment of the materia, some additional clarification as well as a new reference adde

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.Comment: 16 pages, no figures, no tables, a few new references are adde

A few brief comments to arXiv:1005.3315 and arXiv:1005.3321

It is discussed the inconsistency of the so-called ladder-rainbow truncation scheme to QCD in the framework of the Dyson-Schwinger equation for the quark propagator and the Bethe-Salpeter equation for the bound state of mesons.Comment: 1 page, no tables and figure

I. The mass gap and solution of the quark confinement problem in QCD

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 explicitly on the gauge-fixing parameter. It is also free from all the types of the perturbative contributions ("contaminations"), which may appear at the fundamental quark-gluon level.Comment: no tables, no figures, 14 page

The color gauge invariance and a possible origin of the Jaffe-Witten mass gap in QCD

The physical meaning of a mass gap introduced by Jaffe and Witten is to be responsible for the large-scale (low-energy/momentum), i.e., the non-perturbative structure of the true QCD vacuum. In order to make the existence of a mass gap pefrectly clear it is defined as the difference between the regularized full gluon self-energy and its subtracted (also regularized) counterpart. The mass gap is mainly generated by the nonlinear interaction of massless gluon modes. A self-consistent violation of SU(3) color gauge invariance/symmetry is duscussed in order to realize a mass gap in QCD. For this purpose, we propose not to impose the transversality condition on the full gluon self-energy, while restoring the transversality of the full gluon propagtor relevant for the non-perturbative QCD at the final stage. At the same time, the Slavnov-Taylor identity for the full gluon propagator is always preserved. All this allows one to establish the general structure of the full gluon propagator in the presence of a mass gap. In this case, two independent types of formal solutions for the full gluon propagator have been established. 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.Comment: 19 pages, no tables, no figure

The non-perturbative equation of state for gluon matter

In order to derive equation of state for the pure SU(3) Yang-Mills fields from first principles, it is proposed to generalize the effective potential approach for composite operators to non-zero temperatures. It is essentially non-perturbative by construction, since it assumes the summation of an infinite number of the corresponding contributions. There is no dependence on the coupling constant, only a dependence on the mass gap, which is responsible for the large-scale structure of the QCD ground state. The equation of state generalizes the Bag constant at non-zero temperatures, while its nontrivial Yang-Mills part has been approximated by the generalization of the free gluon propagator to non-zero temperatures, as a first necessary step. Even in this case we were able to show explicitly that the pressure may almost continuously change its regime at $T^* = 266.5 MeV$.All the other thermodynamical quantities such as energy density, entropy, etc. are to be understood to have drastic changes in their regimes in the close vicinity of $T^*$. All this is in qualitative and quantitative agreement with thermal lattice QCD results for the pure Yang-Mills fields. We have firmly established the behavior of all the thermodynamical quantities in the region of low temperatures, where thermal lattice QCD calculations suffer from big uncertainties.Comment: 12 pages, 3 figure

The non-perturbative analytical equation of state for the gluon matter. I

The effective potential approach for composite operators has been generalized to non-zero temperatures in order to derive equation of state for the pure SU(3) Yang-Mills fields from first principles. In the absence of external sources it is nothing but the vacuum energy density. The key element of this derivation was an introduction of the temperature dependence into the expression for the Bag constant. Such obtained non-perturbative analytical equation of state for the gluon matter does not depend on the coupling constant, only the dependence on the mass gap, which is responsible for the large-scale structure of the QCD ground state, is present. The important thermodynamic quantities such as the pressure, energy and entropy densities, etc. have been calculated. We have shown explicitly that the pressure may continuously change its regime at $T_c=266.5 Mev$. All other thermodynamic quantities are to be understood to have drastic changes in their regimes at this point. The proposed analytical approach makes it possible to controll thermodynamics of the gluon matter at low temperatures below $T_c$ for the first time. We automatically reproduce the so-called "fuzy"-type bag models properties because of the mass gap explicit presence in our equation of state. We have also analytically calculated the NP gluon condensate as a function of temperature.Comment: 27 pages, 10 figures, one table, many new references are added as well as some old onces omitte

Analytic description of SU(3) lattice thermodynamics in the whole temperature range within the mass gap approach

A general approach how to analytically describe and understand $SU(3)$ lattice thermodynamics in the whole temperature range $[0, \infty)$ is formulated and used. It is based on the effective potential approach for composite operators properly extended to non-zero temperature and density. This makes it possible to introduce into this general formalism the mass gap, which is responsible for the large-scale dynamical structure of the QCD ground state. The mass gap dependent gluon plasma pressure adjusted by this approach to the corresponding lattice data is shown to be a continuously growing function of temperature being thus differentiable in every point of its domain. At the same time, the entropy and energy densities have finite jump discontinuities at some characteristic temperature T_c = 266.5 \ \MeV with latent heat $\epsilon_{LH}= 1.41$. This is a firm evidence of the first-order phase transition in $SU(3)$ pure gluon plasma. The heat capacity has a $\delta$-type singularity (an essential discontinuity) at $T_c$, so that the velocity of sound squared becomes zero at this point. All the independent thermodynamic quantities are exponentially suppressed below $T_c$ and rather slowly approach their respective Stefan-Boltzmann limits at high temperatures. Those thermodynamic quantities which are the ratios of their independent counterparts such as conformity, conformality and the velocity of sound squared approach their Stefan-Boltzmann limits rather rapidly and demonstrate a non-trivial dependence on the temperature below $T_c$. We also calculate the trace anomaly relation (the interaction measure) and closely related to it the gluon condensate, which are especially sensitive to the non-perturbative effects. An analytical description of the dynamical structure of $SU(3)$ gluon plasma is given.Comment: 38 pages, 15 figures, 1 table. The interpretation and numerical results are substantially change

The Hagedorn-type structure of the non-perturbative gluon pressure within the mass gap approach to QCD

We have shown in detail that the low-temperature expansion for the non-perturbative gluon pressure has the Hagedorn-type structure. Its exponential spectrum of all the effective gluonic excitations are expressed in terms of the mass gap. It is this which is responsible for the large-scale dynamical structure of the QCD ground state. The non-perturbative gluon pressure properly scaled has a maximum at some characteristic temperature T=T_c = 266.5 \ \MeV, separating the low- and high temperature regions. It is exponentially suppressed in the $T \rightarrow 0$ limit. In the $T \rightarrow T_c$ limit it demonstrates an exponential rise in the number of dynamical degrees of freedom. Its exponential increase behavior with temperature is valid only up to $T_c$. This makes it possible to identify $T_c$ with the Hagedorn-type transition temperature $T_h$, i.e., to put $T_h=T_c$ within the mass gap approach to QCD at finite temperature. The non-perturbative gluon pressure has a complicated dependence on the mass gap and temperature near $T_c$ and up to approximately $(4-5)T_c$. In the limit of very high temperatures $T \rightarrow \infty$ its polynomial character is confirmed, containing the terms proportional to $T^2$ and $T$, multiplied by the corresponding powers of the mass gap. \end{abstract}Comment: 14 pages, 1 figure, no tables, some unnecessary text has been removed as well as a few references have been cancelled. Short descriptions of the results of arXiv:1012.4157 and arXiv:1409.3375 are presented in sections II, III and I

Can instantons saturate the large mass of the η′\eta' meson in the chiral limit?

Using the trace anomaly relation, low energy theorem and Witten-Veneziano formula for the mass of the $\eta'$ meson, the chiral topology of the QCD nonperturbative instanton vacuum has been numerically evaluated. Our formalism makes it possible to express the topological susceptibility and the mass of the $\eta'$ meson in the chiral limit as a functions of the truly nonperturbative vacuum energy density due to instantons in various modifications. We have explicitly shown that the topological susceptibility due to instantons is one order of magnitude less than its phenomenological value. Also their contribution into the mass of the $\eta'$ meson in the chiral limit is about one third only. Thus the instanton contributions substantially underestimate the phenomenological value of the topological susceptibility and therefore cannot account for the large mass of the $\eta'$ meson alone. One has to look for completely $different from instantons$ types of the nonperturbative excitations of gluon field configurations in the QCD true vacuum in order to nicely saturate the above-mentioned important physical quantities