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V. Gogokhia

Publications and source records attributed to V. Gogokhia.

At least 19 recordsLinked to original sources
arXiv2026

The Mass Gap Approach to QCD. II. The non-perturbative renormalization theory for the massive gluon fields

The previously formulated the mass gap approach to QCD, based on a new insights into its ground state(vacuum), makes it possible to exactly define the dynamically generated gluon pole mass. We developed a non-perturbative multiplicative renormalization program for the corresponding massive full gluon propagator. Its asymptotic properties have been analyzed, including the perturbation theory limit to the free gluon state, i.e., no asymptotic freedom effect. The peculiarities of the mass-shell structure of the massive full gluon propagator has been discussed. The inconsistency of the canonical gauge in QCD is fixed. Our approach does not allow the massive gluons to be mass-shell objects. This prevents them to appear in the physical spectrum (confinement of the massive gluon states). The massive gluons may exist in the vacuum or inside hadrons only. Expression for massive full gluon propagator in Euclidean metric for the lattice simulations is present. It is also shown that the massive solution has a correct limit to the massless full gluon propagator, when the gluon pole mass is to be formally put zero.

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arXiv2026

The Mass Gap Approach to QCD.I. The true gauge and dynamical structures of its ground state

Assuming that a non-trivial quantum Yang-Mills theory exists, we have proved that it should have a mass gap $Δ^2 > 0$, indeed. The proof is based on the derivation of the novel constraint on any solution to QCD. It has been exactly and uniquely derived in the framework of the Slavnov-Taylor identities for the gauge particles Green's functions (propagators), involving the equation of motion for the full gluon propagator as well. The novel constraint has the two different solutions, coinciding only at high energies. The dynamical source of this difference has to be identified with the constant tadpole term, contributing to the full gluon self-energy. Just its renormalized version is conventionally called a mass gap. We prove that it cannot be disregarded from the theory and its ground state by any means. The perturbative renormalizability of QCD will not be affected by a new solution for the gluon equation of motion. We also provide the formulation of the self-consistency condition for the gauge choice in QCD. Finally, we discuss the interrelation of our advance results with the Jaffe-Witten's theorem.

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arXiv2018

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}

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arXiv2015

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 $ε_{LH}= 1.41$. This is a firm evidence of the first-order phase transition in $SU(3)$ pure gluon plasma. The heat capacity has a $δ$-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.

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arXiv2014

The Non-Perturbative Analytical Equation of State for SU(3) Gauge Theory

The effective potential approach for composite operators is generalized to non-zero temperature in order to derive the non-perturbative analytical equation of state for pure SU(3) Yang-Mills fields valid in the whole temperature range. Adjusting our parametrization of the gluon plasma pressure to the lattice pressure at high temperature for SU(3) Yang-Mills case, we have reproduced well our analytical curves and numbers not only for the pressure but for all other independent thermodynamic quantities as well in the whole temperature range $[0, \infty)$. We explicitly show that the pressure is a continuous function of the temperature across a phase transition at $T_c = 266.5 \MeV$. The entropy and energy densities have finite jump discontinuities at $T_c$ with latent heat $ε_{LH}= 1.414$. This is a firm evidence of the first-order phase transition in SU(3) pure gluon plasma. The heat capacity has a $δ$-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 limit at high temperatures rather rapidly and demonstrate the non-trivial dependence on the temperature below $T_c$. We predict the existence of the three massive and the two massless excitations, all of non-perturbative dynamical origin. One of the massive excitations has an effective mass $1.17 \GeV$ and the two others have the same effective mass $0.585 \GeV$, but are propagating in different ways.

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arXiv2012

Analytical derivation and numerical calculation of the $α_s$-order correction to the non-perturbative equation of state for SU(3) gluon matter

In our previous works the effective potential approach for composite operators has been generalized to non-zero temperature in order to derive the analytical equation of state for pure SU(3) Yang-Mills fields without quark degrees of freedom. In the absence of external sources this is nothing but the vacuum energy density. The key element of this derivation is the introduction of a temperature dependence into the expression for the bag constant. The non-perturbative part of the analytical equation of state does not depend on the coupling constant, but instead introduces a dependence on the mass gap. This is responsible for the large-scale dynamical structure of the QCD ground state. The perturbative part of the analytical equation of state does depend on the QCD fine-structure coupling constant $α_s$. Here we develop the analytical formalism, incorporating the perturbative part in a self-consistent way. It makes it possible to calculate the PT contributions to the equation of state in terms of the convergent series in integer powers of a small $α_s$. We also explicitly derive and numerically calculate the first perturbative contribution of the $α_s$-order to the non-perturbative part of the equation of state derived and calculated previously. The analytical equation of state or, equivalently, the gluon pressure is exponentially suppressed at low temperatures, while at temperature $T=T_c = 266.5 \ \MeV$ it has a maximum, if divided by $T^4/3$. It demonstrates a highly non-trivial dependence on the mass gap and the temperature near to $T_c$ and up to approximately $(3-4)T_c$. At very high temperatures its polynomial character is confirmed, containing the terms proportional to $T^2$ and $T$ with a non-analytical dependence on the mass gap.

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arXiv2010

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.

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arXiv2010

Vacuum Energy Density in the Quantum Yang - Mills Theory

Using the effective potential approach for composite operators, we have formulated a general method of calculation of the truly non-perturbative Yang-Mills vacuum energy density (this is, by definition, the Bag constant apart from the sign). It is the main dynamical characteristic of the QCD ground state. Our method allows one to make it free of the perturbative contributions ('contaminations'), by construction. We also perform an actual numerical calculation of the Bag constant for the confining effective charge. Its choice uniquely defines the Bag constant, which becomes free of all the types of the perturbative contributions now, as well as possessing many other desirable properties as colorless, gauge independence, etc. Using further the trace anomaly relation, we develop a general formalism which makes it possible to relate the Bag constant to the gluon condensate not using the weak coupling solution for the corresponding $β$ function. Our numerical result for the Bag constant shows a good agreement with other phenomenological estimates of the gluon condensate.

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arXiv2009

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 not 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.

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arXiv2009

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.

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arXiv2009

The color gauge invariance and a possible origin of a mass gap 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".

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arXiv2008

Comment to: "A note on failure of the ladder approximation to QCD" [Phys. Lett. B 640 (2006) 196 ]

In the paper [Hong-Shi Zong, Wei-Min Sun, Phys. Lett. B 640 (2006) 196], the authors claim that our proof of the inconsistency of the ladder approximation to QCD [Phys. Lett. B 611 (2005) 129] was incorrect. However, their claim is based on a derivation which contains a rough mathematical mistake, namely the unjustified change of variables in the divergent (though regularized) integrals. In this comment I will show this explicitly, so our conclusion that the ladder approximation to QCD is inconsistent remains, of course, correct.

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arXiv2008

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.

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arXiv2007

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.

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arXiv2007

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

We have investigated a closed system of equations for the quark propagator, obtained earlier within our general approach to QCD at low energies. It implies quark confinement (the quark propagator has no pole, indeed), as well as the dynamical breakdown of chiral symmetry (a chiral symmetry preserving solution is forbidded). This system can be solved exactly in the chiral limit. We have established the space of the smooth test functions (consisting of the Green's functions for the quark propagator and the corresponding quark-gluon vertex) in which our generalized function (the confining gluon propagator) becomes a continuous linear functional. It is a linear topological space $K(c)$ of the infinitely differentiable functions (with respect to the dimensionless momentum variable $x$), having compact support in the region $x \leq c$. We develop an analytical formalism, the so-called chiral perturbtion theory at the fundamental quark level, which allows one to find explicit solution for the quark propagator in powers of the light quark masses. We also develop an analytical formalism, which allows one to find the solution for the quark propagator in the inverse powers of the heavy quark masses. It justifies the use for the heavy quark propagator its free counterpart up to terms of the order $1/m_Q^3$, where $m_Q$ is the heavy quark mass. So this solution automatically possesses the heavy quark spin-flavor symmetry.

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arXiv2007

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.

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arXiv2007

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.

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