Explorations in Scalar Fermion Theories: $\beta$-functions, Supersymmetry and Fixed Points

Results for $\beta$-functions and anomalous dimensions in general scalarfermion theories are presented to three loops. Various constraints on theindividual coefficients for each diagram following from supersymmetry areanalysed. The results are used to discuss potential fixed points in the$\varepsilon$-expansion for scalar fermion theories, with arbitrary numbers ofscalar fields, and where there are just two scalar couplings and one Yukawacoupling. For different examples the fixed points follow a similar pattern asthe numbers of fermions is varied. For diagrams with subdivergences there areextensive consistency constraints arising from the existence of a perturbative$a$-function and these are analysed in detail. Further arbitrary schemevariations which preserve the form of $\beta$ functions and anomalousdimensions in terms of 1PI diagrams are also discussed. The existence of linearand quadratic scheme invariants is demonstrated and the consistency conditionare shown to be expressible in terms of these invariants.