Non-perturbative studies of gauge theories : their renormalisation and hierarchies of scales
Two aspects of gauge theories are studied in the non-perturbative regime; firstly, using a set of pre-determined, approximate renormalised Feynman rules, the divergent parts of the O(α(_8)) virtual graphs of the process e(^+)e" → qq are determined to explicitly test whether multiplicative renormalisation is preserved by these rules. The calculation is performed using dimensional regularisation in 2(2 - ɛ) dimensional Euclidean space, where the divergences appear as 1/ɛ(^n) poles as ɛ → 0 Though the corrections to both the fermion-photon vertex and to the final state self energy are shown to have 1/ɛ singularities, the coefficients of these are quite different. This mis-match in singular behaviour signals the breakdown of multiplicative renormalisation, which, in turn, implies that the physical process is not guaranteed to be finite and the rules used are in admissable as a set of consistent Feynman rules. The second investigation is to solve numerically the Schwinger-Dyson equation for the fermion propagator in QED in three (Euclidean) dimensions. The aim being to study the scale of dynamical mass generation. To control infrared divergences the 1/N (flavour) expansion is used and to close the equation vertex and gauge propagator are approximated by their lowest order forms in 1/N. Numerical solutions for the fermion self energy and wavefunction renormalisation are determined. The latter is found not to be suppressed by O(1/N), contrary to the expectation of Appelquist et al, and the coupled equation for these functions has to be solved. It is then found that a mass scale is dynamically generated and that a scale hierarchy between it and the dimensionful coupling, α, of many orders of magnitude exists (typically m/a ~ 10(^-7) for N(_F)=5). Thus showing, albeit in a simplified 'toy' model, how large scale hierarchies can 'naturally' occur in gauge theories with spontaneously broken symmetries.