Confinement Criterion Research Articles

Gauge orbits and the Coulomb potential

If the color Coulomb potential is confining, then the Coulomb field energy of an isolated color charge is infinite on an infinite lattice, even if the usual UV divergence is lattice regulated. A simple criterion for Coulomb confinement is that the expectation value of timelike link variables vanishes in the Coulomb gauge, but it is unclear how this criterion is related to the spectrum of the corresponding Faddeev-Popov operator, which can be used to formulate a quite different criterion for Coulomb confinement. The purpose of this article is to connect the two seemingly different Coulomb confinement criteria, and explain the geometrical basis of the connection.

Gribov no-pole condition, Zwanziger horizon function, Kugo-Ojima confinement criterion, boundary conditions, BRST breaking and all that

We aim to offer a kind of unifying view on two popular topics in the studies of nonperturbative aspects of Yang-Mills theories in the Landau gauge: the so-called Gribov-Zwanziger approach and the Kugo-Ojima confinement criterion. Borrowing results from statistical thermodynamics, we show that imposing the Kugo-Ojima confinement criterion as a boundary condition leads to a modified yet renormalizable partition function. We verify that the resulting partition function is equivalent with the one obtained by Gribov and Zwanziger, which restricts the domain of integration in the path integral within the first Gribov horizon. The construction of an action implementing a boundary condition allows one to discuss the symmetries of the system in the presence of the boundary. In particular, the conventional Becchi-Rouet-Stora-Tyutin symmetry is softly broken.

Infrared behavior of the Faddeev-Popov operator in Coulomb gauge QCD

We calculate the eigenvalue distribution of the Faddeev-Popov operator in Coulomb gauge QCD using quenched SU(3) lattice simulation. In the confinement phase, the density of the low-lying eigenvalues increases with lattice volume, and the confinement criterion is satisfied. Moreover, even in the deconfinement phase, the behavior of the FP eigenvalue density is qualitatively the same as in the confinement phase. This is consistent with the fact that the color-Coulomb potential is not screened in the deconfined phase.

Integrability conditions for nonautonomous quad-graph equations

This paper presents a systematic investigation of the integrability conditions for nonautonomous quad-graph maps, using the Lax pair approach, the ultra-local singularity confinement criterion and direct construction of conservation laws. We show that the integrability conditions derived from each of the methods are the one and the same, suggesting that there exists a deep connection between these techniques for partial difference equations.

The infrared behavior of lattice QCD Green's functions

We investigate different aspects of lattice QCD in Landau gauge using Monte Carlo simulations. In particular, we focus on the low momentum behavior of gluon and ghost propagators. The gauge group is SU(3). Different systematic effects on the gluon and ghost propagators are studied, e.g. the dependence on the choice of Gribov copies or the influence of dynamical Wilson fermions. We compare our data with results from studies of Dyson-Schwinger equations for the gluon and ghost propagators. We demonstrate that the infrared behavior of both propagators, as found in this thesis, is consistent with different criteria for confinement. However, the running coupling constant, given as a renormalization-group-invariant combination of the gluon and ghost dressing functions, does not expose a finite infrared fixed point. We also report on a first nonperturbative computation of the SU(3) ghost-gluon-vertex renormalization constant and on an investigation of the spectral properties of the Faddeev-Popov operator.

Unquenched Kogut-Susskind quark propagator in lattice Landau gauge QCD

Quark propagators of the unquenched Kogut-Susskind(KS) fermion obtained from the gauge configurations of the MILC collaboration are measured after Landau gauge fixing and using the Staple+Naik action. Presence of the $\bar q q$ condensates and $A^2$ condensates in the dynamical mass $M(q)$ and the quark wave function renormalization $Z_\psi(q^2)$ are investigated. We obtain the correlation of the renormalization factor of the running coupling taken at $\mu\sim 6$GeV and that of the quark wave function renormalization $Z_\psi(q^2)$ of the Staple+Naik action. The mass function $\displaystyle M(q)$ is finite at $q=0$ and its chiral limit is $\sim 0.38$GeV. We compared the results corrected by the scale of the vertex renormalization and the tadpole renormalization with the corresponding values obtained by the Asqtad action without renormalization and observed good agreement.Implication of infrared finite $Z_2(q)=1/Z_\psi(q^2)$ to the Kugo-Ojima confinement criterion is discussed.

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.

Structure of the confining solutions for SU(3)-Yang–Mills equations and confinement mechanism

Structure of exact solutions modelling confinement is discussed for SU(3)-Yang–Mills equations and uniqueness of such confining solutions is proved in a certain sense. Relationship of the obtained results to QCD and the confinement mechanism is considered. Incidentally the Wilson confinement criterion for the found solutions is verified and also numerical estimates for strength of magnetic colour field responsible for linear confinement are adduced in the ground state of charmonium.

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.

Infrared exponents and the running coupling of Landau gauge QCD and their relation to confinement

The infrared behaviour of the gluon and ghost propagators in Landau gauge QCD is reviewed. The Kugo-Ojima confinement criterion and the Gribov-Zwanziger horizon condition result 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 with corresponding lattice data obtained recently. The resulting running-coupling approaches a fix point in the infrared, α(0) = 8.92/N c . Two different fits for the scale dependence of the running coupling are given and discussed.

Complete integrability and singularity confinement of nonautonomous modified Korteweg–de Vries and sine Gordon mappings

A systematic investigation on the complete integrability of the nonautonomous discrete–discrete modified Korteweg–de Vries (ΔΔmKdV) and sine Gordon (ΔΔsG) mappings is presented using Lax pair technique and singularity confinement criteria. We derive conditions on the coefficients of Lax matrices such that the ΔΔmKdV and ΔΔsG admit a Lax representation separately. We then demonstrate that the obtained conditions for each of the nonautonomous equations are indeed precisely the necessary conditions to satisfy an ultra-local version of singularity confinement criteria. A similarity reduction of both the ΔΔmKdV and ΔΔsG is obtained and shown to lead a new discrete Painleve type of equations of higher order possessing Lax representation.

Exact localized solutions of quintic discrete nonlinear Schrödinger equation

We study a new quintic discrete nonlinear Schrödinger (QDNLS) equation which reduces naturally to an interesting symmetric difference equation of the form φ n+1 + φ n−1 = F( φ n ). Integrability of the symmetric mapping is checked by singularity confinement criteria and growth properties. Some new exact localized solutions for integrable cases are presented for certain sets of parameters. Although these exact localized solutions represent only a small subset of the large variety of possible solutions admitted by the QDNLS equation, those solutions presented here are the first example of exact localized solutions of the QDNLS equation. We also find chaotic behavior for certain parameters of nonintegrable case.

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, α(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.

Infrared and ultraviolet asymptotic solutions to gluon and ghost propagators in Yang–Mills theory

We examine the possibility that there may exist a logarithmic correction to the infrared asymptotic solution with power behavior which has recently been found for the gluon and Faddeev–Popov ghost propagators in the Landau gauge. We propose a new ansatz to find a pair of solutions for the gluon and ghost form factors by solving the coupled Schwinger–Dyson equation under a simple truncation. This ansatz enables us to derive the infrared and ultraviolet asymptotic solutions simultaneously and to understand why the power solution and the logarithmic solution is possible only in the infrared and ultraviolet limit, respectively. Even in the presence of the logarithmic correction, the gluon propagator vanishes and the ghost propagator is enhanced in the infrared limit, and the gluon–ghost–antighost coupling constant has an infrared fixed point (but with a different β function). This situation is consistent with Gribov–Zwanziger confinement scenario and color confinement criterion of Kugo and Ojima.

Monopole clusters, center vortices, and confinement in aZ2gauge-Higgs system

We propose to use the different kinds of vacua of the gauge theories coupled to matter as a laboratory to test confinement ideas of pure Yang-Mills theories. In particular, the very poor overlap of the Wilson loop with the broken string states supports the 't Hooft and Mandelstam confinement criteria. However in the Z(2) gauge-Higgs model we use as a guide we find that the condensation of monopoles and center vortices is a necessary, but not sufficient condition for confinement.

Confinement and amplification of acoustic waves in cubic heterostructures

We present the theory of acoustic phonon confinement in elastically anisotropic (cubic) quantum-well (QW) heterostructures grown in a direction of high symmetry. A general criterion for phonon confinement is derived. For ${\mathrm{S}\mathrm{i}/\mathrm{S}\mathrm{i}}_{0.5}{\mathrm{Ge}}_{0.5}/\mathrm{Si},$ $\mathrm{S}\mathrm{i}/\mathrm{G}\mathrm{e}/\mathrm{S}\mathrm{i}$ and $\mathrm{A}\mathrm{l}\mathrm{A}\mathrm{s}/\mathrm{G}\mathrm{a}\mathrm{A}\mathrm{s}/\mathrm{A}\mathrm{l}\mathrm{A}\mathrm{s}$ QW heterostructures, dispersion curves are obtained, and displacement fields corresponding to the confined phonons are studied in detail. It is found that the confinement of acoustic phonons in these QW layers is strong in the subterahertz and terahertz frequency ranges. The resulting description of phonon confinement is applied to analyze the amplification of confined modes by the drift of the two-dimensional carriers as a function of the phonon frequency, the temperature, and the parameters of heterostructure. The calculation shows that the amplification coefficient of the confined phonons can exceed ${10}^{3} {\mathrm{cm}}^{\ensuremath{-}1}$ for Si/Ge-based structures and ${10}^{2} {\mathrm{cm}}^{\ensuremath{-}1}$ for AlAs/GaAs-based structures.

Confinement and amplification of terahertz acoustic phonons in cubic heterostructures

A general criterion for phonon confinement is derived in the model of elastically anisotropic ( cubic) media. The results are applied to the calculation of the dispersion curves of the confined phonons in Si/Si 1− x Ge x /Si and AlAs/GaAs/AlAs heterostructures. For these structures, we show that the lowest-order phonon branches behave differently from those in the model of isotropic media. We have found that confinement is strong in the terahertz frequency range. For p-Si/SiGe/Si and n-AlAs/GaAs/AlAs quantum well heterostructures, we have studied the effect of amplification of confined high-frequency phonons by the drift of low-dimensional carriers. Two electron–phonon interaction mechanisms were taken into account: interaction via the deformation potential (p-SiGe and n-AlGaAs) and the piezoelectric interaction (n-AlGaAs). It was found that an amplification coefficient of the order of 10 2 cm −1 for the AlGaAs heterostructures and 10 3 cm −1 for the SiGe heterostructures can be obtained in spectrally-separated narrow amplification bands.

Similarity reduction, generalized symmetries and integrability of Belov–Chaltikian and Blaszak–Marciniak lattice equations

The Lie point symmetries of Belov–Chaltikian (BC) and Blaszak–Marciniak (BM) lattice equations is derived. Using the symmetries similarity reduction for both BC and BM lattice equations is obtained and show that each of the reduced equation possesses Lax representation and satisfies the singularity confinement criteria (a discrete version of the Painlevé property) indicating their integrability. Two particular solutions of both the reduced equations are given explicitly. A systematic investigation for nonclassical symmetries of BC and BM lattice equations is carried out. Further, a sequence of conserved densities and generalized symmetries of both BC and BM lattice equation are derived explicitly. The existence of a sequence of such symmetries is a predictor for integrability.

Verifying the Kugo-Ojima confinement criterion in Landau gauge Yang-Mills theory.

Expanding the Landau gauge gluon and ghost two-point functions in a power series we investigate their infrared behavior. The corresponding powers are constrained through the ghost Dyson-Schwinger equation by exploiting multiplicative renormalizability. Without recourse to any specific truncation we demonstrate that the infrared powers of the gluon and ghost propagators are uniquely related to each other. Constraints for these powers are derived, and the resulting infrared enhancement of the ghost propagator signals that the Kugo-Ojima confinement criterion is fulfilled in Landau gauge Yang-Mills theory.

Discrete Dynamical Systems¶Associated with Root Systems of Indefinite Type

A geometric charactrization of the equation found by Hietarinta and Viallet, which satisfies the singularity confinement criterion but which exhibits chaotic behavior, is presented. It is shown that this equation can be lifted to an automorphism of a certain rational surface and can therefore be considered to be a realization of a Cremona isometry on the Picard group of the surface. It is also shown that the group of Cremona isometries is isomorphic to an extended Weyl group of indefinite type. A method to construct the mappings associated with some root systems of indefinite type is also presented.