A nonperturbative study of three dimensional quantum electrodynamics with N flavours of fermion

Abstract

This work is concerned with the breaking of chiral symmetry in gauge theories and the associated generation of a dynamical mass scale. We investigate this phenomenon in the context of a simple model, three dimensional QED, where the complicating factor of infinite renormalisations is absent. This model possesses an intrinsic scale, set by the coupling [e(^2)] = M, and it is the relationship between this and the dynamically generated mass scale that is of interest. The chiral symmetry breaking mechanism is investigated using the Schwinger Dyson equations which are then truncated in a nonperturbative manner using the Ball-Chiu vertex ansatz. The complexity of the resulting coupled fermion-photon system means that the photon is initially replaced by its perturbative form. Numerical investigations of this simplified system then reveal the existence of an exponential relationship, in terms of the dimensionless parameter N, between the intrinsic and dynamical mass scales, m ~ e(^2) exp(-cN). Contrary to the assertions of Appelquist et al the wavefunction renormalisation was found to be nonperturbative and crucial in determining this behaviour. The sensitivity of this mechanism to the nonperturbative behaviour of the photon is investigated. A simple analysis shows it to be far stronger than previously expected. This is confirmed by a numerical analysis of the coupled photon-fermion system which suggest the relationship between the two scales in the theory is of the form m ~ e(^2) exp(-cN(^2)). This model therefore illustrates how a large hierarchy of scales may naturally occur in a gauge theory, for instance N=3 m/a ~ 10(^-5). Finally an investigation of the gauge dependence of this approach is initiated. The softening of the photon in the low momentum region is shown to amplify automatically any inadequacy of the vertex ansatz by factors of O(a/m) in all but the Landau gauge. It is therefore expected that any incomplete vertex form will result in the generation of a "critical gauge", ɛ(_e), below which chiral symmetry breaking solutions will not exist. A path of further investigation is suggested.