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

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