The two-loop renormalization of the Yukawa sector for an arbitrary renormalizable field theory

The two-loop Yukawa coupling constant renormalization and the mass renormalization of the fermion for an arbitrary renormalizable field theory, including vector, scalar and of course fermion fields, has been calculated. This has been done in two different regularization schemes, i.e. dimensional regularization and dimensional reduction. Both calculations were done in the conventional way, and not with the use of the so-called background-field method. Only the reduction scheme is shown to preserve supersymmetry up to this level.Furthermore a simple example of a gauge theory is given to show that in general it is not possible to give an explicit transformation of the coupling constants in such a way that all β-functions become the same for the two regularization schemes, which proves that dimensional reduction, treated this way, is not a good regularization scheme.