Topics In Anomalies And Finiteness (fermion, Super Symmetry).

Abstract

This work consists of two parts. Part One studies the bounds on fermion masses when the mass is gained through spontaneous symmetry breaking. As a large mass is obtained by means of a large Yukawa coupling, mass renormalizations must be calculated with a non-perturbative technique. A chiral Yukawa model is solved to leading order in the large-N expansion. It is shown there is no limit on the physical mass due to any saturation mechanism, but there is a limit by virtue of the inherent cutoff in the theory. The doubt this casts on one method of quantization of anomalous gauge theories is discussed. Part Two investigates a class of theories obtained from finite N = 4 supersymmetric Yang-Mills theories by the addition of arbitrary soft-breaking terms. The one-loop renormalization group coefficients are obtained and the renormalization group equations solved for the running dimensionful couplings. It is shown that the conditions among the parameters that would guarantee finiteness at one loop are an infrared fixed hypersurface in parameter space. It is argued that imposition of these conditions constitutes an unnatural fine-tuning of parameters. Spontaneous symmetry breaking in the proposed models is studied. It is shown, through renormalization group analysis of the effective potential, that a minimum exists consistent with the requirement of stability. A relation among masses is obtained at the minimum. Because breaking the gauge symmetry gives masses to the non-abelian gauge bosons, while retaining a stable potential, the infrared problem of the ultravioletly finite theories is eliminated; for this reason, and by reason of naturalness, these models that retain finiteness in the gauge sector but allow running of the soft-breaking terms are recommended as the natural extentions of the supersymmetric theory.