Three-loop results in two-loop finite supersymmetric gauge theories

We investigate the three-loop infinity structure in the class of two-loop finite supersymmetric Yang-Mills theories. By means of explicit calculations of Feynman graphs in an N=1 superfield context, we show that the β-function of the gauge coupling does not receive any contribution at the three-loop level. By use of the Adler-Bardeen theorem it is argued that this is a general result, namely if the theory is finite at n loops then the gauge β-function will vanish at n+1 loops. However, we show that infinities do arise at three loops in the renormalizations of the matter fields in at least some of these theories. (orig.)