Complete N = 4 Structure of Low - Energy Effective Action in N = 4 Super Yang - Mills Theories

Using the N = 2 superfield approach, we construct full N = 4 supersymmetric low-energy effective actions for N = 4 SYM models, with both N = 2 gauge superfield strengths and hypermultiplet superfields included. The basic idea is to complete the known non-holomorphic effective potentials which depend only on N = 2 superfield strengths W and ¯ W to the full on-shell N = 4 invariants by adding the appropriate superfield hypermultiplet terms. We prove that the effective potentials of the form ln W ln ¯ W can be N = 4 completed in this way and present the precise structure of the corresponding completions. However, the effective potentials of the non-logarithmic form suggested in hep-th/9811017 and hep-th/9909020 do not admit the N = 4 completion. Therefore, such potentials cannot come out as (perturbative or non-perturbative) quantum corrections in N = 4 SYM models.