Narrow Results

We develop a theory on a topologically nontrivial manifold which leads to different vacuum backgrounds at the field level. The different colors of the same quark flavor live in different backgrounds generated by the action of the torsion subgroup of H²(M, 2πℤ) on H²(M, 2πℤ) itself. This topological separation leads to a quark confinement mechanism which does not apply to the baryons as they turn out to live on the same vacuum state. The theory makes some topological assumptions on the spacetime manifold which are compared with the available data on the topology of the Universe.

The constituent quark model (CQM) has recently been widely used for the description of the internal structure of baryons. The baryon spectrum is usually well described, although various models are quite different. However, the study of baryon spectroscopy is not sufficient to distinguish among the various forms of quark dynamics. It would be interesting to consider the effect of an extra residual interaction among the quarks, which contains a dependence on isospin (or flavor). In the chiral constituent quark model the non-confining part of the potential is provided by the interaction with the Goldstone bosons, giving rise to a spin- and isospin-dependent part which is crucial for the description of the spectrum for energies lower than 1.7 GeV. In this work we have introduced an improved form of the hyperfine interaction and isospin dependent quark potential. The resulting description of the baryon spectrum was found to be satisfactory and we not only have included confinement potential at large separations, but also color charge and non-confining hyperfine interaction potentials. This combination of potentials yields spectra which are very close to the ones obtained in experiments.

We discuss a confining model for quark–antiquark system with a new color SU₃ gauge symmetry. New gauge transformations involve non-integrable phase factors and lead to the fourth-order gauge field equations and a linear potential. The massless gauge bosons have non-definite energies, which are not observable because they are permanently confined in quark systems by the linear potential. We use the empirical potentials of charmonium to determine the coupling strength of the color charge g_s and find . The rules for Feynman diagrams involve propagators with poles of order 2 associated with new gauge fields. The confining quark model may be renormalizable by power counting and compatible with perturbation theory.

At sufficiently large separation between a quark and an antiquark the quantum chromodynamics (QCD) string breaks into parts due to creation of light quark–antiquark pairs. We show that a background magnetic field affects the breaking of the QCD string oriented in the transverse directions with respect to the axis of the magnetic field. Using semiclassical approach we argue that above certain, the background magnetic field eB ≈0.3 GeV² breaking of the transverse string should become energetically unfavorable.

By making use of the background field method, we derive a novel reformulation of the Yang–Mills theory which was proposed recently by the author to derive quark confinement in QCD. This reformulation identifies the Yang–Mills theory with a deformation of a topological quantum field theory. The relevant background is given by the topologically nontrivial field configuration, especially, the topological soliton which can be identified with the magnetic monopole current in four dimensions. We argue that the gauge fixing term becomes dynamical and that the gluon mass generation takes place by a spontaneous breakdown of the hidden supersymmetry caused by the dimensional reduction. We also propose a numerical simulation to confirm the validity of the scheme we have proposed. Finally we point out that the gauge fixing part may have a geometric meaning from the viewpoint of global topology where the magnetic monopole solution represents the critical point of a Morse function in the space of field configurations.

We study quark confinement in a system of two parallel domain walls interpolating different color dielectric media. We use the phenomenological approach in which the confinement of quarks appears considering the QCD vacuum as a color dielectric medium. We explore this phenomenon in QCD₂, where the confinement of the color flux between the domain walls manifests, in a scenario where two 0-branes (representing external quark and antiquark) are connected by a QCD string. We obtain solutions of the equations of motion via first-order differential equations. We find a new color confining potential that increases monotonically with the distance between the domain walls.

A recently developed model for the QCD analytic invariant charge is compared with quenched lattice simulation data on the static quark–antiquark potential. By employing this strong running coupling one is able to obtain the confining quark–antiquark potential in the framework of the one-gluon exchange model. To achieve this objective a technique for evaluating the integrals of a required form is developed. Special attention is paid here to removing the divergences encountered the calculations. All this enables one to examine the asymptotic behavior of the potential at both small and large distances with high accuracy. An explicit expression for the quark–antiquark potential, which interpolates between these asymptotics, and satisfies the concavity condition, is proposed. The derived potential coincides with the perturbative results at small distances, and it is in a good agreement with the lattice data in the nonperturbative physically-relevant region. An estimation of the parameter Λ_QCD is obtained for the case of pure gluodynamics. It is found to be consistent with all the previous estimations of Λ_QCD in the framework of approach in hand.

We propose a novel type of color magnetic condensation originating from magnetic monopoles so that it provides the mass of off-diagonal gluons in the Yang-Mills theory. This dynamical mass generation enables us to explain the infrared Abelian dominance and monopole dominance by way of a non-Abelian Stokes theorem, which supports the dual superconductivity picture of quark confinement. Moreover, we show that the instability of Savvidy vacuum disappears by sufficiently large color magnetic condensation.

I give a brief review of advances in the strong interaction theory. This talk was delivered at the Conference in honor of Murray Gell-Mann's 80th birthday, 24–26 February 2010, Singapore.

There has been substantial progress in understanding confinement in a class of four-dimensional SU(N) gauge theories using semiclassical methods. These models have one or more compact directions, and much of the analysis is based on the physics of finite temperature gauge theories. The topology R³ × S¹ has been most often studied using a small compactification circumference L such that the running coupling g²(L) is small. The gauge action is modified by a double-trace Polyakov loop deformation term, or by the addition of periodic adjoint fermions. The additional terms act to preserve Z(N) symmetry and thus confinement. An area law for Wilson loops is induced by a monopole condensate. In the continuum, the string tension can be computed analytically from topological effects. Lattice models display similar behavior, but the theoretical analysis of topological effects is based on Abelian lattice duality rather than on semiclassical arguments. In both cases, the key step is reducing the low-energy symmetry group from SU(N) to the maximal Abelian subgroup U(1)^N-1 while maintaining Z(N) symmetry.

A new picture of quark confinement based on the instability of Coulomb phase at low energy was introduced by Volodya Gribov in the early nineties. In QCD the effective $\alpha$ coupling constant can reach very large values in the infrared regime what generates Coulomb phase instabilities. In the Gribov picture the instability leads to a vacuum decay into light quarks for coupling constants $\alpha$ larger than a critical value ${\alpha }_{\mathrm{\text{crit}}}=2N\pi (1-\sqrt{2/3})/(N^2-1)$, for SU(N) gauge theories. The instability of Coulomb phase can be derived from first principles in any non-Abelian gauge theory for $\alpha \ge \sqrt{2}$, a value which is larger than the Gribov critical value. In this paper we review the analytic derivation of the Gribov mechanism from first principles and analyze the effects of dynamical quarks in the instability of the Coulomb phase. The instabilities associated to light quarks turn out to appear at larger values of $\alpha$ than the ones induced from pure gluon dynamics, unlike it is expected in the standard Gribov scenario. The analytic results confirm the consistency of the picture where quark confinement is mainly driven by gluonic fluctuations.

In mean field approximation, the grand canonical potential of SU(3) Polyakov linear-$\sigma$ model (PLSM) is analyzed for chiral phase transition, ${\sigma }_l$ and ${\sigma }_s$ and for deconfinement order-parameters, $\varphi$ and ${\varphi }^{\ast }$ of light- and strange-quarks, respectively. Various PLSM parameters are determined from the assumption of global minimization of the real part of the potential. Then, we have calculated the subtracted condensates $({\mathrm{\Delta }}_{l,s})$. All these results are compared with recent lattice QCD simulations. Accordingly, essential PLSM parameters are determined. The modeling of the relaxation time is utilized in estimating the conductivity properties of the QCD matter in thermal medium, namely electric $[{\sigma }_{\mathrm{\text{el}}}(T)]$ and heat $[\kappa (T)]$ conductivities. We found that the PLSM results on the electric conductivity and on the specific heat agree well with the available lattice QCD calculations. Also, we have calculated bulk and shear viscosities normalized to the thermal entropy, $\xi /s$ and $\eta /s$, respectively, and compared them with recent lattice QCD. Predictions for $(\xi /s)/({\sigma }_{\mathrm{\text{el}}}/T)$ and $(\eta /s)/({\sigma }_{\mathrm{\text{el}}}/T)$ are introduced. We conclude that our results on various transport properties show some essential ingredients, that these properties likely come up with, in studying QCD matter in thermal and dense medium.

First, we give a gauge-independent definition of chromomagnetic monopoles in $SU(N)$ Yang–Mills theory which is derived through a non-Abelian Stokes theorem for the Wilson loop operator. Then we discuss how such magnetic monopoles can give a nontrivial contribution to the Wilson loop operator for understanding the area law of the Wilson loop average. Next, we discuss how the magnetic monopole condensation picture are compatible with the vortex condensation picture as another promising scenario for quark confinement. We analyze the profile function of the magnetic flux tube as the non-Abelian vortex solution of $U(N)$ gauge-Higgs model, which is to be compared with numerical simulations of the $SU(N)$ Yang–Mills theory on a lattice. This analysis gives an estimate of the string tension based on the vortex condensation picture, and possible interactions between two non-Abelian vortices.

In the preceeding works, we have given a non-Abelian dual superconductivity picture for quark confinement, and demonstrated the numerical evidences on the lattice. In this talk, we discuss the confinement and deconfinement phase transition at finite temperature in view of the dual superconductivity. We investigate chromomagnetic monopole currents induced by chromoelectric flux in both confinement and deconfinement phase by the numerical simulations on a lattice at finite temperature, and discuss the role of the chromomagnetic monopole in the confinement/deconfinement phase transition.

This work presents a model of center vortices, expressed by random lines in three-dimensional space–time. Monte Carlo methods were used to ensemble these closed random lines. The physical space wherein the vortex lines are defined is a cuboid including periodic boundary conditions. Reconnections are allowed aside from moving, growing, and shrinking of the vortex configuration. Therefore, the ensemble contains a variable number of closed vortex clusters, which are considered significant in realizing the deconfining phase transition. The average vortex density and length are studied as a function of vortex density $d$ by using this model. At different temperatures, potential $V(R)$ between quark and antiquark is studied as a function of distance $R$. This model is able to emphasize the confinement physics qualitative properties in $SU(2)$ Yang–Mills theory.

Quantum chromodynamics (QCD), the theory of the strong interactions, involves quarks interacting with non-Abelian gluon fields. This theory has many features that are difficult to impossible to see in conventional diagrammatic perturbation theory. This includes quark confinement, mass generation and chiral symmetry breaking. This paper is a colloquium level overview of the framework for understanding how these effects come about.

We investigate the contributions of the Cartan generators in the static potentials for various representations in the framework of the domain model of center vortices for SU(3) gauge theory. Using the center domains with the cores corresponding to only one Cartan generator $H_8$, already given as a particular proposal, leads to some concavities in the potentials for higher representations. Furthermore, the string tension of the fundamental representation is the same at Casimir scaling and $N$-ality regimes. We add the contribution of the other Cartan generator $H_3$ to the potentials and therefore these shortcomings can be eliminated. However, we discuss that at intermediate range of distances the potentials induced by only $H_8$ agree with the Casimir scaling better than those corresponding to both Cartan generators.

The confinement problem is studied using the thick center vortex model. It is shown that the $SU(3)$ Cartan subalgebra of the decomposed $G(2)$ gauge theory can play an important role in the confinement. The Casimir eigenvalues and ratios of the $G(2)$ representations are obtained using its decomposition to the $SU(3)$ subgroups. This leads to the conjecture that the $SU(3)$ subgroups also can explain the $G(2)$ properties of the confinement. The thick center vortex model for the $SU(3)$ subgroups of the $G(2)$ gauge theory is applied without the domain modification. Instead, the presence of two $SU(3)$ vortices with opposite fluxes due to the possibility of decomposition of the $G(2)$ Cartan subalgebra to the $SU(3)$ groups can explain the properties of the confinement of the $G(2)$ group both at intermediate and asymptotic distances which is studied here.

We discuss how the presence of a squeezed gluon vacuum might lead to quark confinement in the framework of the global color model of QCD. Using reduced phase space quantization of massive vector theory we construct a Lorentz-invariant and colorless squeezed gluon condensate and show that it induces a permanent, nonlocal quark interaction (delta-function in four-momentum space), which according to Munczek and Nemirovsky might lead to quark confinement. Our approach makes it possible to relate the strength of this effective confining quark interaction to the strength of the physical gluon condensate.

Inspired by Laughlin’s theory of the fractional quantum Hall effect, we explore the topological nature of the quark–gluon plasma (QGP) and the nucleons in the context of the Clifford algebra. In our model, each quark is transformed into a composite particle via the simultaneous attachment of a spin monopole and an isospin monopole. This is induced by a novel kind of meson endowed with both spin and isospin degrees of freedom. The interactions in the strongly coupled quark–gluon system are governed by the topological winding number of the monopoles, which is an odd integer to ensure that the overall wave function is antisymmetric. The states of the QGP and the nucleons are thus uniquely determined by the combination of the monopole winding number $m$ and the total quark number $N$. The radius squared of the QGP droplet is expected to be proportional to $mN$. We anticipate the observation of such proportionality in the heavy ion collision experiments.