arXiv subjects

V. Gogohia

Publications and source records attributed to V. Gogohia.

At least 19 recordsLinked to original sources

arXiv2006

The existence of a mass gap in quantum Yang-Mills theory

The skeleton loop integrals which contribute into the gluon self-energy have been iterated (skeleton loops expansion) within the Schwinger-Dyson equation for the full gluon propagator. No any truncations/approximations as well as no special gauge choice have been made. It is explicitly shown that such obtained general iteration solution for the full gluon propagator can be exactly and uniquely decomposed as a sum of the two pricipally different terms. The first term is the Laurent expansion in integer powers of severe (i.e., more singular than $1/q^2$) infrared singularities accompanied by the corresponding powers of the mass gap and multiplied by the corresponding residues. The second (perturbative) term is always as much singular as $1/q^2$ and otherwise remaining undetermined. We have explicitly demonstrated that the mass gap is hidden in the above-mentioned skeleton loop integrals due to the nonlinear interaction of massless gluon modes. It shows explicitly up when the gluon momentum goes to zero. The appropriate regularization scheme has been applied in order to make a gauge-invariant existence of the mass gap perfectly clear. Moreover, it survives an infinite series summation of the relevant skeleton loop contributions into the gluon self-energy. The physical meaning of the mass gap is to be responsible for the large scale structure of the true QCD vacuum.

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arXiv2006

The mass gap and gluon confinement

In our previous publication [1-3] it has been proven that the general iteration solution of the Shwinger-Dyson equation for the full gluon propagator (i.e., when the skeleton loop integrals, contributing into the gluon self-energy, have to be iterated, which means that no any truncations/approximations have been made) can be algebraically (i.e., exactly) decomposed as the sum of the two principally different terms. The first term is the Laurent expansion in integer powers of severe (i.e., more singular than $1/q^2$) infrared singularities accompanied by the corresponding powers of the mass gap and multiplied by the corresponding residues. The standard second term is always as much singular as $1/q^2$ and otherwise remaining undetermined. Here it is explicitly shown that the infrared renormalization of the mass gap only is needed to render theory free of all severe infrared singularities in the gluon sector. Moreover, this leads to the gluon confinement criterion in a gauge-invariant way. As a result of the infrared renormalization of the mass gap in the initial Laurent expansion, that is dimensionally regularized, the simplest severe infrared singularity $(q^2)^{-2}$ survives only. It is multiplied by the mass gap squared, which is the scale responsible for the large scale structure of the true QCD vacuum. The $δ$-type regularization of the simplest severe infrared singularity (and its generalization for the multi-loop skeleton integrals) is provided by the dimensional regularization method correctly implemented into the theory of distributions. This makes it possible to formulate exactly and explicitly the full gluon propagator (up to its unimportant perturbative part).

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arXiv2005

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 survivies 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 $1/(q^2)^2$. Each severe (i.e., more singular than $1/q^2$) power-type IR singularity is accompanied by the corresponding powers of the mass gap. The standard second term is always as much singular as $1/q^2$, 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.

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arXiv2005

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.

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arXiv2004

Intrinsically nonperturbative QCD I. A pure dynamical theory of gluon confinement

We establish 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 $1/q^2$) infrared singularities. It reflects the zero momentum modes enhancement effect in the true QCD vacuum. Its existence exhibits a characteristic mass (the so-called mass gap), which is responsible for the scale of nonperturbative dynamics in the 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 $(q^2)^{-2}$. So, the smooth in the infrared limit the full gluon propagator is to be ruled out. Collective motion of all the purely transverse $virtual$ gluon field configurations with low-frequency components/large scale amplitudes is solely responsible for the color confinement phenomenon within our approach. At the microscopic, dynamical level these field configurations are saturated by the nonlinear fundamental four-gluon interaction. It just makes the full gluon propagator inevitably so singular in the infrared. The amplitudes of all the purely transversre severely singular $actual$ gluon field configurations are totally suppressed, leading thus to the confinement of gluons. We formulate exactly the gluon confinement criterion in a manifestly gauge-invariant way, taking into account the distribution nature of severe infrared singularities.

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arXiv2004

Severely infrared singular gluon propagator in QCD

We have explicitly shown that the infrared structure of the full gluon propagator in QCD is an infite sum over all severe (i.e., more singular than $1/q^2$) infrared singularities. It reflects the zero momentum modes enhancement effect in the true QCD vacuum. Its existence exhibits a characteristic mass (the so-called mass gap), which is responsible for the scale of nonperturbative dynamics in the QCD ground state. By an infrared renormalization of a mass gap only, the deep infrared structure of the full gluon propagator is saturated by the simplest severe infrared singularity, the famous $(q^2)^{-2}$. So, there is no smooth in the infrared limit the full gluon propagator. The main dynamical source of severe infrared singularities in the gluon propagator is the two-loop skeleton term of the corresponding equation of motion, which contains the four-gluon vertices only. Taking into account the distribution nature of severe infrared singularities, the gluon confinement criterion is formulated in a manifestly gauge-invariant way.

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arXiv2004

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 $1/q^2$) 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. It 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 the severe infrared singularities under the 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 $(q^2)^{-2}$. 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.

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arXiv2003

A general solution for the quark propagator in two-dimensional covariant gauge QCD

We have investigated a closed set of equations for the quark propagator, which has been obtained earlier within a new, nonperturbative approach to two-dimencional covariant gauge QCD. It is shown that this theory implies quark confinement (the quark propagator has no poles, indeed), as well as dynamical breakdown of chiral symmetry (a chiral symmetry preserving solution is forbidden). The above-mentioned set of equations can be exactly solved in the chiral limit. We develop an analytical formalism, the so-called chiral perturbation theory at the fundamental quark level, which allows one to find solution for the quark propagator in powers of the light quark masses. Each correction satisfies the differential equation, which can be formally solved. We develop also an analytical formalism which alows one to find solution for the quark propagator in the inverse powers of the heavy quark masses. IT coincides with free heavy quark propagator up to terms of order $1/m_Q^3$, where $m_Q$ is the heavy quark mass. So this solution automatically possesses the heavy quark flavor symmetry up to terms of order $1/m_Q$. At the same time, we have found a general solution for the heavy quark propagator, which by no means can be reduced to the free one.

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arXiv2002

Nonperturbative infrared multiplicative renormalizability of two-dimensional covariant gauge QCD

A nonperturbative approach to two-dimensional covariant gauge QCD is presented in the context of the Schwinger-Dyson equations and the corresponding Slavnov-Taylor identities. The distribution theory, complemented by the dimensional regularization method, is used in order to treat correctly the infrared singularities which inevitably appear in the theory. By working out the multiplicative renormalization program we remove them from the theory on a general ground and in a self-consistent way, proving thus the infrared multiplicative renormalizability of two-dimensional QCD within our approach. We also show explicitly how to formulate the bound-state problem and the Schwinger-Dyson equations for the gluon propagator and the triple gauge field proper vertex, all free from the infrared singularities.

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arXiv2002

A two-dimensional QCD in the covariant gauge

A nonperturbative approach to 2D covariant gauge QCD is presented in the context of the Schwinger-Dyson equations for quark and ghost propagators and the corresponding Slavnov-Taylor identities. The distribution theory, complemented by the dimensional regularization method, is used in order to correctly treat the severe infrared singularities which inevitably appear in the theory. By working out the multiplicative renormalization program we remove them from the theory on a general ground and in a self-consistent way, proving thus the infrared multiplicative renormalizability of 2D QCD within our approach. This makes it possible to sum up the infinite series of the corresponding planar skeleton diagrams in order to derive a closed set of equations for the infrared renormalized quark propagator. We have shown that complications due to ghost degrees of freedom can be considerable within our approach. It is shown exactly that 2D covariant gauge QCD implies quark confinement (the quark propagator has no poles, indeed) as well as dynamical breakdown of chiral symmety (a chiral symmetry preserving solution is forbidded). We also show explicitly how to formulate the bound-state problem and the Schwinger-Dyson equations for the gluon propagator and the triple gauge proper vertex, all free of the severe IR singularities.

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arXiv2001

How to explicitly introduce the Jaffe-Witten mass gap into the quantum Yang-Mills theory

We propose how to explicitly introduce the Jaffe-Witten mass gap into the quantum Yang-Mills theory. Through the full gluon propagator it is defined as such nonperturbative scale that when it formally goes to zero, then the perturbative phase survives in the theory only. The close link between mass gap and strong infrared singularities which are due to dominated in the QCD vacuum self-interaction of massless gluons is also discussed. This interaction leads thus to the zero momentum modes enhancement effect in the QCD nonperturbative vacuum. Using theory of distributions, we argue that strong nonperturbative infrared singularities can be put under control. A new, intrinsically nonperturbative phase in QCD is established.

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arXiv2001

A gauge invariant formulation of intrinsically nonperturbative QCD

Using a system of the corresponding Schwinger-Dyson equations of motion, a pure dynamical theory of quark confinement and spontaneous breakdown of chiral symmetry is formulated. It is based on dominated in the QCD vacuum self-interaction of massless gluons only, i.e., without involving some extra degrees of freedom. This interaction becomes strongly singular in the deep infrared domain leading thus to the enhancement of zero momentum modes in the nonperturbative QCD vacuum. Using theory of distributions, complemented by the dimensional regularization method, we have explicitly shown that strong infrared singularities can be put under control. In this way a new phase in QCD, intrinsically nonperturbative QCD which is manifestly gauge invariant, was discovered. As a result, a highly nontrivial dynamical and topological structure of the QCD vacuum has emerged within our approach. We have also explicitly shown how infrared multiplicative renormalization program should be done in order to self-consistently remove all the strong infrared singularities from the theory. The corresponding convergence conditions play a crucial role in this program. In this theory any physical observables are determined by such correlation functions from which all types of the perturbative contributions should be subtracted, by definition. Theory is not only infrared finite but it is free from the ultraviolet divergences as well. It has a mass gap $Δ> 0$, i.e., there are no physical states in the interval $(0, Δ)$. It explains confinement, spontaneous breakdown of chiral symmetry and other nonperturbative effects on a general ground and in self-consistent way.

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arXiv2001

Topological structure of chiral QCD vacuum

Using the trace anomaly relation, low-energy theorem and Witten-Veneziano formula, we have developed an analytical formalism which allows one to calculate the gluon condensate, the topological susceptibility and the mass of the $η'$ meson in the chiral limit as functions of the non-perturbative vacuum energy density. It is used for numerical evaluation of the chiral QCD topology within the QCD vacuum model consisting mainly of the quantum component given by the recently proposed zero modes enhancement (ZME) model and the classical component given by the the random instanton liquid model (RILM). We sum up both contributions into the total, nonperturbative vacuum energy density. A very good agreement with the phenomenological values of the topological susceptibility, the mass of the $η'$ meson in the chiral limit and the gluon condensate has been obtained. This puts the above mentioned QCD vacuum model on a firm phenomenological ground.

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arXiv2000

Mass of the $η'$ meson in the chiral limit in the zero momentum modes enhancement quantum model of the QCD nonperturbative vacuum

Using the trace anomaly and low energy relations, as well as the Witten-Veneziano formula for the mass of the $η'$ meson, we have developed a formalism which makes it possible to express the gluon condensate, the topological susceptibility and the mass of the $η'$ meson as a functions of the truly nonperturbative vacuum energy density which is one of the most important characteristics of the QCD true ground state. It was directly applied to the numerical evaluation of the chiral QCD vacuum topological structure within its quantum zero momentum modes enhancement model. A rather good agreement with phenomenological and experimental values of the above-mentioned quantities has been achieved. With the help of the Witten-Veneziano formula, we derived an absolute lower bounds for the pion decay constant and the mass of the $η'$ meson in the chiral limit. By introducing the most general parametrization of the gluon condensate, we also proposed how the correct $N_f$ (number of flavors) dependence of its phenomenologically extracted value could be restored.

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arXiv2000

Can instantons saturate the large mass of the $η'$ meson in the chiral limit?

Using the trace anomaly and low energy relations, as well as Witten-Veneziano formula for the mass of the $η'$ 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 $η'$ meson as a functions of the instanton number density in the chiral limit. We have explicitly shown that the topological susceptibility in this case is approximately one fours and one half of its phenomenological value at instanton number densities in the chiral limit, 0.5 fm^{-4} and 1.0 fm^{-4}, respectively. Thus the instanton contributions substantially underestimate the mass of the $η'$ meson which still remains large in the chiral limit. With the help of the above-mentioned Witten-Veneziano formula, we derived an absolute lower bounds for the pion decay constant and the mass of the $η'$ meson in the chiral limit.

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arXiv2000

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 integrating 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.

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arXiv2000

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.

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arXiv2000

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

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.

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