Next-to-leading-order correction to the effective action in N=2 gauge theories

The next-to-leading-order nonholomorphic contribution to the Wilsonian low-energy effective action in four-dimensional N=2 gauge theories with matter is discussed. In the semiclassical approximation, it is calculated by using the manifestly N=2 supersymmetric harmonic superspace approach. This perturbative one-loop correction is found to be in agreement with the N=1 superfield calculations of de Wit, Grisaru, and Rocek. In particular, the previously unknown numerical coefficient at the nonholomorphic contribution is determined for the SU(2) gauge group. Special attention is paid to the N=2 superconformal gauge theories, whose one-loop nonholomorphic contribution can be exact, even nonperturbatively. The coefficient in front of the next-to-leading-order term turns out to be zero in the case of the N=4 super-Yang-Mills theory, while it does not vanish in the case of finite N=2 supersymmetric gauge theories with four hypermultiplets in the fundamental representation of the gauge group.