Construction of one-loop N=4 SYM effective action on the mixed branch in the harmonic superspace approach

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We develop a systematic approach to construct the one-loop N=4 SYM effective action depending on both N=2 vector multiplet and hypermultiplet background fields. Beginning with the formulation of N=4 SYM theory in terms of N=2 harmonic superfields, we construct the one-loop effective action using the covariant N=2 harmonic supergraphs and calculate it in N=2 harmonic superfield form for constant Abelian strength Fmn and corresponding constant hypermultiplet fields. The hypermultiplet-dependent effective action is derived and given by integral over the analytic subspace of harmonic superspace. We show that each term in the Schwinger-De Witt expansion of the low-energy effective action is written as integral over full N=2 superspace.