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We develop the derivative expansion of the one-loop ${\cal N}=4$ SYMeffective action depending both on ${\cal N}=2$ vector multiplet and onhypermultiplet background fields. We get a new derivation of the complete${\cal N}=4$ supersymmetric low-energy effective action obtained inhep-th/0111062 and find subleading corrections to it. A problem of ${\cal N}=4$supersymmetry of the results is discussed. Using the formalism of ${\cal N}=2$harmonic superspace and exploring on-shell hidden ${\cal N}=2$ supersymmetry of${\cal N}=4$ SYM theory we construct the appropriate hypermultiplet-dependingcontributions. The hidden ${\cal N}=2$ supersymmetry requirements allow to geta leading, in hypermultiplet derivatives, part of the correct ${\cal N}=4$supersymmetric functional containing $F^{8}$ among the component fields.Comment: LaTeX, 26 pages; corrected typos; references adde

physical review. d. particles, fields, gravitation, and cosmology/physical review. d. particles and fields

One-loop effective action for<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi mathvariant="script">N</mml:mi><mml:mo>=</mml:mo><mml:mn>4</mml:mn><mml:mn /></mml:math>SYM theory in the hypermultiplet sector: Leading low-energy approximation and beyond

One-loop effective action forN=4SYM theory in the hypermultiplet sector: Leading low-energy approximation and beyond