Tadeusz Bałaban

The mass gap for Higgs models on a unit lattice

Abstract An isomorphism is established between eertain compact and noncompact formulations of Abelian gauge theory on a lattice. For weak coupling, the mass gap predicted by the Higgs mechanism is then established.

Regularity and decay of lattice Green's functions

In the paper we study a class of lattice, covariant Laplace operators with external gauge fields. We prove that these operators are positive and that their Greens functions decay exponentially. They also have regularity properties similar to continuous space Greens functions. All the bounds are uniform in the lattice spacing.

Renormalization group approach to lattice gauge field theories. I. Generation of effective actions in a small field approximation and a coupling constant renormalization in four dimensions

We study four-dimensional pure gauge field theories by the renormalization group approach. The analysis is restricted to small field approximation. In this region we construct a sequence of localized effective actions by cluster expansions in one step renormalization transformations. We construct also β-functions and we define a coupling constant renormalization by a recursive system of renormalization group equations.

(Higgs)2, 3 quantum fields in a finite volume

We consider a Euclidean model of interacting scalar and vector fields in two and three dimensions, and prove a lower bound for vacuum energy in a lattice approximation. The bound is independent of a lattice spacing; it is proved with the help of renormalization transformations in Wilson-Kadanoff form. It extends in principal also to generating functional for Schwinger functions.

Propagators and renormalization transformations for lattice gauge theories. II

Lattice gauge theories may be looked at as perturbations of the theory of a vector field with a Gaussian action. We study this theory here and in following papers obtaining crucial results for understanding the renormalization group method in more complicated non-Abelian gauge field theories.

Renormalization of the Higgs Model: Minimizers, Propagators and the Stability of Mean Field Theory*

We study the effective actionsS(k) obtained byk iterations of a renormalization transformation of the U(1) Higgs model ind=2 or 3 spacetime dimensions. We identify a quadratic approximationSQ(k) toS(k) which we call mean field theory, and which will serve as the starting point for a convergent expansion of the Greens functions, uniformly in the lattice spacing. Here we show how the approximationsSQ(k) arise and how to handle gauge fixing, necessary for the analysis of the continuum limit. We also establish stability bounds onSQ(k), uniformly ink. This is an essential step toward proving the existence of a gap in the mass spectrum and exponential decay of gauge invariant correlations.

Averaging operations for lattice gauge theories

Usually renormalization group transformations are defined by some averaging operations. In this paper we study such operations for lattice gauge fields and for gauge transformations. We are interested especially in characterizing some classes of field configurations on which the averaging operations are regular (e.g., analytic). These results will be used in subsequent papers on the renormalization group method in lattice gauge theories.

Propagators for lattice gauge theories in a background field

We prove regularity and decay properties for propagators connected with the renormalization group method in lattice gauge theories. These propagators depend on an external gauge field configuration, called a background field.

Ultraviolet stability of three-dimensional lattice pure gauge field theories

We prove the ultraviolet stability for three-dimensional lattice gauge field theories. We consider only the Wilson lattice approximation for pure Yang-Mills field theories. The proof is based on results of the previous papers on renormalization group method for lattice gauge theories.

Spaces of regular gauge field configurations on a lattice and gauge fixing conditions

We consider spaces of lattice gauge field configurations satisfying gauge invariant regularity conditions, and intersections of these spaces with a surface given by gauge fixing conditions. We prove that if these conditions are chosen properly then configurations belonging to the intersection are small and regular.