Theories of quark confinement

It is conjectured that a non-Abelian gauge theory based on the color SU(3) group will confine quarks. Various techniques that have been applied to this question are reviewed. These include approximate methods based on strong coupling expansions of Hamiltonian and Euclidian lattice theories, instanton improvements on perturbation theory, and solutions of truncated Dyson-Schwinger equations for the gauge field propagator. Formal results based on electric-magnetic duality arguments and on the study of loop field theories are presented. Deconfinement at high temperatures, the inclusion of light quarks, and a possible reconciliation with a hypothetical discovery of free quarks are discussed.     

The theory of quark confinement

This is the second of the two last papers by V.N. Gribov concluding his 20 year long study of the problem of quark confinement in QCD. In this paper the analytic structure of quark and gluon Green functions is investigated in the framework of the theory of confinement based on the phenomenon of supercritical binding of light quarks. The problem of unitarity in a confining theory is discussed. The write-up remained unfinished and as such it is presented here. The author was planning to emphasise the link between the electroweak and strong interactions, and in particular the r?le of pions (Goldstone bosons) in confinement, to present an explicit solution for bound states, and to write down an analytic model for quark and gluon Green functions corresponding to confinement. Received: 15 February 1999 / Published online: 15 July 1999     

Infrared slavery and quark confinement

We pose the question of whether the so-called infrared slavery mechanism as, e.g., being manifest in non-Abelian gauge theories, necessarily confines quarks. Making a specific ansatz for the long-range forces, we solve the Schwinger-Dyson equation for the quark Green function. Besides having a confining solution, it appears that quarks may by-pass the long-range forces and be produced.     

Generation of gluons from quark confinement

We study a model of quark confinement defined by the vanishing of colour currents. The model is shown to be equivalent to quantum chromodynamics and this equivalence is interpreted as due to the compositeness of the colour gluons. The Green’s functions of the theory are found to contain nontrivial structure only for colour singlet composites which can be identified with hadrons.     

Topological symmetry breakdown and quark confinement

The manner by which topological symmetry breakdown leads to confinement in a non-Abelian gauge theory is discussed.     

From asymptotic freedom to quark confinement

Asymptotic freedom is the perturbative expression of paramagnetic screening by the QCD vacuum. When extended to large external fields the perturbative analysis leads to a catastrophe owing to the appearance of negative eigenvalues of the matrix describing the fluctuations of the gluonic field. This is met by a self-consistently determined magnetic-moment density. There results almost complete screening of the external field and the action is dominated by fluctuations of the magnetic moment. Such behaviour is more typical of very strong coupling e ⪢ 1 rather than that expected for e = O(1), the latter being the effective coupling at the QCD length scale. This kind of situation arises in the response to large Wilson loops, so that the derivation of the area law from strong-coupling lattice gauge theory can be rationalized.     

Locality,charges and quark confinement

General features of the Yang-Mills color theory suggested by Fritzsch, Gell-Mann and Leutwyler are discussed. It is shown that as a consequence of locality every observable is a color singlet and that the failure of the cluster property and consequently a q-$\text{q}$ potential not decreasing at infinity are possible, consistently with locality, just because of the non abelian gauge character. A generalized cluster property is proved for Yang-Mills local field theories.     

Classical quark confinement from general relativity

By assuming covariance of physical laws under (discrete) dilatations, it seems possible to describe strong and gravitational interactions in a unified way. An Einstein-type equation with “cosmological” term is for instance suggested for strong field inside hadrons, which yields - among other things - a classical quark confinement in a very natural way. Further consequences are briefly discussed.     

Gauss-law quark confinement and polar representation for scalar fields

The generators of gauge field and matter field gauge transformations for the SU(2) gauge group can be written as orbital and spin angular momenta respectively. The Gauss law requires that the sum of these momenta vanish. This is only possible if the state functionals are restricted to be single-valued or double-valued with respect to the gauge fields and the matter fields gauge variables at the same time. Such a restriction can be imposed on scalar matter fields by introducing a polar representation.     

A disorder parameter that test for confinement in gauge theories with quark fields

We propose to use a suitably defined vortex free energy as a disorder parameter in gauge field theories with matter fields. It is supposed to distinguish between the confinement phase, massless phase(s) and Higgs phase where they exist. The matter fields may transform according to an arbitrary representation of the gauge group. We compute the vortex free energy by series expansion for a Z₂ Higgs model and for SU(2) lattice models with quark or Higgs fields in the fundamental representation at strong coupling (confinement phase), and for the Z₂ Higgs model in the range of validity of low-temperature expansions (Higgs phase). The results are in agreement with the expected behavior.     

Temporary quark confinement in field theoretical bags

The problem of confining quarks within hadronic bound states is studied in detail in the context of local relativistic field theories exhibiting spontaneous breakdown of chiral symmetry. In particular, interacting systems of scalar and spinor fields in one-space-one time dimension and in three-space-one-time dimension with spherical symmetry are considered. For some such systems in the tree approximation bound state solutions are found. As we had conjectured earlier, the binding mechanism and the ensuing temporary confinement of quarks are seen to be intimately related to local vacuum excitations and the local restoration of manifest chiral symmetry (which is globally hidden). Properties of the solutions are discussed and various implications are drawn on aspects of the dynamics of reactions in which such bound states could participate as in-going or out-going particles.     

Classical solutions of a model of quark confinement

We find the classical solutions of a model of quark confinement defined by the vanishing of colour currents. Both plane-wave type of solutions extending all over space as well as string-type of solutions confined to restricted regions of space are found.     

Gauge-invariant formalism and quark confinement in QCD2

New gauge-invariant vector and spinor fields are introduced. Gauge-invariant quark propagator is defined in terms of these new fields. The equation for such a propagator, taken in 1/N approximation, does not require the introduction of an infrared regularization. As the regularization parameter in our approach there stands such a parameter which limit value corresponds to the gauge-invariant fields and translationally invariant quark propagator. It is shown that in this limit the pole of the gauge-invariant quark propagator shifts towards infinity what is usually treated as the confinement of a single quark.     

A dynamical mechanism for quark mixing and neutrino oscillations

We show that, if one assumes fermion generations to be given by a gauge symmetry, together with a certain Higgs mechanism for breaking it, then the known empirical features of quark and lepton mixing can be largely explained, including, in particular, the fact that the mixing (CKM) matrix element , responsible for the muon anomaly in atmospheric neutrinos, is near maximal, and much larger than its quark counterparts and , while the corner elements for both quarks () and leptons () are all very small. The mechanism also automatically gives a hierarchical fermion-mass spectrum which is intimately related to the mixing pattern. Received: 8 February 1999 / Published online: 15 July 1999     

On the Landau-Ginzburg type theory of quark confinement

We present a detailed investigation of the interaction of “magnetically” charged quarks in the vacuum of type II superconductivity (represented in a relativistic theory by a Higgs field). The analysis of confining forces is given in details. Spin dependence, relationship to other confining mechanisms and first quantization are discussed.     

Colored gluons,bjorken scaling and quark confinement

The logarithmic violations of scaling in the canonical gauge field theory of colored quarks and gluons are investigated. Upper bounds for the quark-gluon coupling constant are derived.     

Current algebra,baryons, and quark confinement

It is shown that ordinary baryons can be understood as solitons in current algebra effective lagrangians. The formation of color flux tubes can also be seen in current algebra, under certain conditions.