We investigate the continuum limit of the gauge-ball spectrum in the four-dimensional pure U(1) lattice gauge theory. In the confinement phase we identify various states scaling with the correlation length exponent ν ≃ 0.35. The square root of the string tension also scales with this exponent, which agrees with the non-Gaussian fixed point exponent recently found in the finite size studies of this theory. Possible scenarios for constructing a non-Gaussian continuum theory with the observed gaugeball spectrum are discussed. The 0 ++ state, however, scales with a Gaussian value ν ≃ 0.5. This suggests the existence of a second, Gaussian continuum limit in the confinement phase and also the presence of a light or possibly massless scalar in the non-Gaussian continuum theory. In the Coulomb phase we find evidence for a few gauge-balls, being resonances in multi-photon channels; they seem to approach the continuum limit with as yet unknown critical exponents. The maximal value of the renormalized coupling in this phase is determined and its universality confirmed.

The chiral phase transition induced by a charged scalar field is investigated numerically in a lattice fermion-gauge-scalar model with U(1) gauge symmetry, proposed recently as a model for dynamical fermion mass generation. For very strong gauge coupling the transition is of second order and its scaling properties are very similar to those of the Nambu-Jona-Lasinio model. However, in the vicinity of the tricritical point at somewhat weaker coupling, where the transition changes the order, the scaling behavior is different. Therefore it is worthwhile to investigate the continuum limit of the model at this point.

We investigate the critical behaviour of a three-dimensional lattice χU φ 3 model in the chiral limit. The model consists of a staggered fermion field, a U (1) gauge field (with coupling parameter β) and a complex scalar field (with hopping parameter κ). Two different methods are used: 1) fits of the chiral condensate and the mass of the neutral unconfined composite fermion to an equation of state and 2) finite size scaling investigations of the Lee-Yang zeros of the partition function in the complex fermion mass plane. For strong gauge coupling (β < 1) the critical exponents for the chiral phase transition are determined. We find strong indications that the chiral phase transition is in one universality class in this β interval: that of the three-dimensional Gross-Neveu model with two fermions. Thus the continuum limit of the χU φ 3 model defines here a nonperturbatively renormalizable gauge theory with dynamical mass generation. At weak gauge coupling and small κ, we explore a region in which the mass in the neutral fermion channel is large but the chiral condensate on finite lattices very small. If it does not vanish in the infinite volume limit, then a continuum limit with massive unconfined fermion might be possible in this region, too.

We investigate on the lattice the Yukawa models in 2 dimensions with Z(2) and U(1) symmetries. These models reduce to the usual and chiral Gross-Neveu models, respectively, when the kinetic and the selfcoupling terms of the scalar field are turned off. The numerical data and mean field arguments suggest that, at least for some range of the scalar field hopping parameter, fermion mass is dynamically generated for arbitrarily weak Yukawa coupling. The models are asymptotically free in this coupling, like the Gross-Neveu models, even when the scalar quartic selfcoupling is strong.