Analytic approximations to Hamiltonian lattice field theories (II). The weak-coupling behavior of U(1) theories

It is shown that at weak coupling physical quantities in hamiltonian U(1) lattive gauge (or global symmetric) theories of arbitrary dimension are provided as expectation values in a d ? 1 dimensional lagrangian Z(2) gauge (or spin) theory with calculable long-range interactions.Confinement and the existence of a magnetic mass gap are equivalent to the existence of infinite-range plaquette-plaquette (or link-link) correlations in the spin field. The existence of infinite range correlations is simply related to the dimension of the lattice and the transformation property of the order parameter. As expected, only the d = 2+1 U(1) gauge theory confines electric charges at all non-vanishing coupling.     

Progress in lattice gauge theory

Two topics of lattice gauge theory are reviewed. They include string tension and β-function calculations by strong coupling Hamiltonian methods for SU(3) gauge fields in 3 + 1 dimensions, and a 1/N-expansion for discrete gauge and spin systems in all dimensions. The SU(3) calculations give solid evidence for the coexistence of quark confinement and asymptotic freedom in the renormalized continuum limit of the lattice theory. The crossover between weak and strong coupling behavior in the theory is seen to be a weak coupling but non-perturbative effect. Quantitative relationships between perturbative and non-perturbative renormalization schemes are obtained for the O(N) nonlinear sigma models in 1 + 1 dimensions as well as the range theory in 3 + 1 dimensions. Analysis of the strong coupling expansion of the β-function for gauge fields suggests that it has cuts in the complex 1/g²-plane. A toy model of such a cut structure which naturally explains the abruptness of the theory's crossover from weak to strong coupling is presented. The relation of these cuts to other approaches to gauge field dynamics is discussed briefly.The dynamics underlying first order phase transitions in a wide class of lattice gauge theories is exposed by considering a class of models-P(N) gauge theories - which are soluble in the N → ∞ limit and have non-trivial phase diagrams. The first order character of the phase transitions in Potts spin systems for N #62; 4 in 1 + 1 dimensions is explained in simple terms which generalizes to P(N) gauge systems in higher dimensions. The phase diagram of Ising lattice gauge theory coupled to matter fields is obtained in a $\text{1}\text{N}$ expansion. A one-plaquette model (1 time-0 space dimensions) with a first-order phase transitions in the N → ∞ limit is discussed.     

A model regularizing gauge fields in the strong coupling region

We propose a model using an effective lagrangian for bilocal gauge extended fields, looking somehow as an average over all open strings with fixed end points. The locality of the action is retrieved as a range parameter, which acts as a regulator, shrinks, and the Yang-Mills theory is recovered in this limit. The system is invariant by rotation and translation, generalizes lattice gauge theories and allows regularized strong coupling expansions for Yang-Mills field theory.     

Non-perturbative to perturbative QCD via the FFBRST

Recently a new type of quadratic gauge was introduced in QCD in which the degrees of freedom are suggestive of a phase of abelian dominance. In its simplest form it is also free of Gribov ambiguity. However this gauge is not suitable for usual perturbation theory. The finite field dependent BRST (FFBRST) transformation is a method established to interrelate generating functionals for different effective versions of gauge fixed field theories. In this paper we propose a FFBRST transformation suitable for transforming the theory in the new quadratic gauge into the standard Lorenz gauge Faddeev–Popov version of the effective lagrangian. The task is made interesting by the fact that the effective lagrangian is invariant under two different BRST transformations which leads to suitable extension of the previous procedures to accomplish the required result. We are thus able to identify a field redefinition to go from a non-perturbative phase of QCD to perturbative QCD.     

A quenching scenario for lattice gauge theories

We study lattice gauge theories with complex, random and quenched couplings. Such theories are argued to have the same continuum limits as the annealed case. The first-order phase transitions are shown to be absent and the smoother cross-over behavior of the quenched theory leads to the universal scaling law.     

Local internal supersymmetry of the free Maxwell theory

The free Maxwell theory is shown to possess an extended gauge invariance consisting of local internal supersymmetry transformations in addition to the usual local phase transformations. The Maxwell lagrangian is derived as a particular gauge choice in the extended theory.     

On the nature of the phase transition in a class of lattice gauge theories

We investigate the connection between the phase transition recently found by Anthony in a variant of SU(2) lattice gauge theory and various mechanisms known to produce phase transitions in lattice gauge theories.     

Magnetically extended gauge theories

The gauge fields of two distinct O(3) gauge theories with spontaneous symmetry breaking are constrained to share an invariantly defined abelian-like “photon” sector. This magnetically extended lagrangian theory has O(3) × O(3) gauge invariance but only five gauge bosons: the “photon” and four further bosons that provide the electric and magnetic currents to which the “photon” couples. Generalizations to higher groups and comparison with a recent proposal of Montonen and Olive are made.     

On Gauge Invariant Theories of Gravitation

Theories of gravitation are called gauge invariant if the invariance of the gravitational field lagrangian with respect to gauge transformations of the gravitational field variables is independend of the invariance of this lagrangian with respect to the Einstein group of general coordinate transformations. They are bimetric theories because the coordinate covariance is ensured by constructing scalar densities relative to a globally flat background metric. Such a theory is represented by the PAUL-FIERZ equations for massless spin 2 particles. But this theory is inconsistent if nongravitational matter is enclosed as a source. All attempts to overcome this inconsistancy preserving gauge invariance lead to Einstein's GRT. We review this problem and compare the situation with a theory proposed by LOGUNOV showing that he overcomes the inconsistency of linear Einstein's equations by replacing the field variables by a gauge invariant combination of new ones, which turns out to be the first order form of v. FREUD'S superpotential.     

How to do mean field theory in Feynman gauge and doing it for U(1) with corrections to fourth order

It is demonstrated how mean field theory with corrections from fluctuations may be applied to lattice gauge theories in covariant gauges. By fixing the gauge at tree level the importance of fluctuations is decreased. This is understood as inclusion of terms of next-to-leading-order in d in the definition is the mean field tree approximation, d being the dimension of the lattice. The gauge group U(1) and Wilson's action are used as testing ground. Tree and one-loop results comparable to those previously obtained in axial gauge are obtained for d = 4. The next three correction terms to the free and plaquette energies are evaluated in Feynman gauge. The truncated asymptotic series thus obtained is compared to that of the ordinary weak coupling expansion. The mean field series gives, to those orders studied, a much better approximation. The location of phase transitions in 4d and 5d are predicted with 1% error bars.     

Effective lagrangian analysis of the chiral phase transition at finite density

Introducing a finite chemical potential μ for the quark number density ψ^°ψ, we study analytically the restoration of Π^° chiral symmetry as μ is varied. In the strong coupling limit, the effective lagrangian for SU(N) gauge theories coupled to fermion fields in d dimensions is derived for all N. In the case of SU(2) we predict a second order chiral symmetry restoration phase transition, whereas for all N?3 the transition is first order. Predictions are given for the critical values of the chemical potential μ.     

On the infrared behaviour of massless φ4 theory for small ϵ

In this paper we compare a formulation that gives the full infrared singular behaviour of massless φ⁴ theory for all continuous dimensions below four with the ?-expansion. These theories give very different result for small ?! We give then arguments that suggest that the ?-expansion when combined with the renormalization group may be based on too strong analyticity assumptions.     

Decay of the false vacuum at finite temperature

In this paper we study kinetics of the first-order phase transitions in gauge theories with spontaneous symmetry breaking. A general theory of the decay of the metastable phase (false vacuum) at a finite temperature is developed. A number of concrete examples are considered, which will make it possible to study kinetics of phase transitions in a wide class of theories without complicated computer calculations.     

A quantitative approach to low-energy quantum chromodynamics

A general method for solving the low-energy spectrum of an infrared unstable field theory is presented. The method involves a strong coupling expansion of the lattice approximation to the theory. Ultimately the results must be continued to zero-coupling constant in accord with the asymptotic freedom of such theories. The method is applied to the pure gauge field (glueball) part of quantum chromodynamics. The spectrum of lowest-lying states consists of a scalar and tensor which are almost degenerate and an axial vector with mass ≈1.6 times the scalar mass.The same procedure applied to the Abelian gauge theory yields unstable results which may indicate the presence of a phase transition.     

Properties of the anti-BRS symmetry in a general framework

The anti-BRS symmetry is investigated in a general and model-independent framework. A general expression for the anti-BRS charge, applicable to a large class of theories, is derived from the extended BRS algebra. It is given in the form of an ordinary BRS variation. At the quantum level this implies that the anti-BRS charge acts as a gauge generator and will have no influence upon unitarity, regardless of the lagrangian being anti-BRS invariant or not. It is further shown, that for arbitrary gauge choices, it is not possible to find a lagrangian of a standard form, which is simultaneously BRS and anti-BRS invariant. The necessary and sufficient conditions on the gauge fixing, to be given by a total BRS and anti-BRS variation, are also derived.     

Quantum phase transitions beyond the Landau-Ginzburg paradigm and supersymmetry

We make connections between studies in the condensed matter literature on quantum phase transitions in square lattice antiferromagnets, and results in the particle theory literature on abelian supersymmetric gauge theories in 2 + 1 dimensions. In particular, we point out that supersymmetric U(1) gauge theories (with particle content similar, but not identical, to those of theories of doped antiferromagnets) provide rigorous examples of quantum phase transitions which do not obey the Landau-Ginzburg-Wilson paradigm (often referred to as transitions realizing “deconfined criticality”). We also make connections between supersymmetric mirror symmetries and condensed matter particle-vortex dualities.     

M theory and singularities of exceptional holonomy manifolds

M theory compactifications on G₂ holonomy manifolds, whilst supersymmetric, require singularities in order to obtain non-Abelian gauge groups, chiral fermions and other properties necessary for a realistic model of particle physics. We review recent progress in understanding the physics of such singularities. Our main aim is to describe the techniques which have been used to develop our understanding of M theory physics near these singularities. In parallel, we also describe similar sorts of singularities in Spin(7) holonomy manifolds which correspond to the properties of three dimensional field theories. As an application, we review how various aspects of strongly coupled gauge theories, such as confinement, mass gap and non-perturbative phase transitions may be given a simple explanation in M theory.     

Symmetry behavior in gauge theories

A theory of phase transition with symmetry restoration in gauge theories at high temperature is investigated. The phase transition may be of the first or of the second order depending on relations between coupling constants. It is noted that the possible existense of a limiting temperature cannot prevent the high-temperature symmetry restoration. In the theories without neutral currents, symmetry also can be affected by a magnetic field. However in most of the models with neutral currents symmetry restoration takes place not due to a magnetic field but due to massive vector fields, created simultaneously by the magnetic field sources. It is pointed out that in most of the theories with neutral currents an increase of external currents lead to symmetry restoration, while an increase of density results in a further increase of symmetry breaking. In some cases critical values of temperature and external fields and currents appear to be extremely small. At certain relations between coupling constants radiative corrections lead to the absence of symmetry breaking in gauge theories even at zero temperature and in the absence of any other external factors. Strong constraints on masses and coupling constants for the symmetry in the Higgs model to be broken are obtained. It is shown that energy of substance is nonconserved due to energy “pumping” from the non-observable Bose-condensate in the processes under consideration.     

A lagrangian formulation of the Wess approach to supersymmetric Yang-Mills theories

We obtain a superspace lagrangian density which reproduces the Wess formulation of supersymmetric gauge theories. The present approach, which is a first-order formalism, gives rise to a lagrangian density polynomial in the Yang-Mills and auxiliary superfields and appears to be stable under quantization and renormalization.