A two loop calculation in the N = 4 supersymmetric Yang Mills theory is performed in various dimensions. The theory is found to be two-loop finite in six dimensions or less, but infinite in seven and nine dimensions. The six-dimensional result can be explained by a formulation of the theory in terms of N = 2 superfields. The divergence in seven dimensions is naively compatible with both N=2 and N=4 superfield power counting rules, but is of a form that cannot be written as an on-shell N=4 superfield integral. The hypothesized N=4 extended superfield formalism therefore either does not exist, or at least has weaker consequences than would have been expected. By analogy, four-dimensional supergravity theories are expected to be infinite at three loops.
Some general issues about the meaning of finiteness in nonrenormalizable theories are discussed. In particular, the use of field redefinitions, the generalization of wavefunction renormalizations to nonrenormalizable theories, and whether counterterms should be used in calculations in "finite" theories are studied. It is shown that theories finite to n loops can have at most simple-pole divergences at n + 1 loops.
A method for simplifying the calculation of infinite parts of Feynman diagrams is developed. Based on the observation that counterterms are local functions, all integrals are reduced to logarithmically divergent ones with no dependence on masses or external momenta. The method is of general use, and is particularly effective for many-point Green functions at more than one loop.
