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 W̄ to the full on-shell N=4 invariants by adding the appropriate superfield hypermultiplet terms. We prove that the effective potentials of the form lnWlnW̄ 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.