ar X iv : h ep - t h / 99 03 02 4 v 1 2 M ar 1 99 9 Low - Energy Effective Action in N = 4 Super Yang - Mills Theory

We consider N = 4 supersymmetric Yang-Mills theory formulated in terms of N = 2 superfields in harmonic superspace. Using the background field method we define manifestly gauge invariant and N = 2 supersymmetric effective action depending on N = 2 strength superfields and develop a general procedure for its calculation in one-loop approximation. Explicit form for this effective action is found for the case of SU(2) gauge group broken down to U(1). Maximally extended N = 4 supersymmetric Yang-Mills (SYM) theory attracts much attention due to remarkable properties on quantum level and profound links with modern string/brane activity. It is natural to expect that an effective action describing many aspects of quantum theory, such as symmetry breaking, low-and high-energy behaviour, anomalies and so on, also must possess the many remarkable features in N = 4 SYM. Unfortunately, the effective action of N = 4 SYM was studied unsufficiently so far and ones know little both about its general structure and about its form in different approximations. One of the main obstacles to investigate the effective action in N = 4 SYM is absence of adequate quantum formulation such a theory. In our opinion, an efficient technique of carring out the calculations of effective action in N = 4 SYM preserving N = 4 supersymmetry manifestly is still undeveloped. The best we have at present is treatment of N = 4 SYM as N = 2 SYM coupled to a specific (hypermultiplet) matter and use the methods of N = 2 supersymmetric quantum field theory. Recently Dine and Seiberg [1] has shown that part of low-energy effective action of N = 4 SYM depending only on N = 2 superfield strengths W and ¯ W can be found exactly up to numerical factor on the base of restrictions imposed by N = 4 supersymmetry. To be more precise, it was shown that low-energy effective action Γ[W, ¯ W ] for N = 4 SU(2) theory is expressed in terms of non-holomorphic effective potential H(W, ¯ W) as follows