The ultraviolet behavior of N = 4 Yang-Mills and the power counting of extended superspace

Abstract

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. This leads one to expect that four-dimensional supergravity theories diverge at three loops. Some general issues about the meaning of finiteness in nonrenormalizable theories are discussed. In particular, we discuss the use of field redefinitions, the generalization of wave function renormalizations to nonrenormalizable theories, and whether counterterms should be used in calculations in finite theories.