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We develop a novel bi-harmonic $\mathcal{N}=4$ superspace formulation of the$\mathcal{N}=4$ supersymmetric Yang-Mills theory (SYM) in four dimensions. Inthis approach, the $\mathcal{N}=4$ SYM superfield constraints are solved interms of on-shell $\mathcal {N}=2$ harmonic superfields. Such an approachprovides a convenient tool of constructing the manifestly $\mathcal{N}=4$supersymmetric invariants and further rewriting them in $\mathcal{N}= 2$harmonic superspace. In particular, we present $\mathcal{N}=4$ superfield formof the leading term in the $\mathcal{N}=4$ SYM effective action which was knownpreviously in $\mathcal{N}=2$ superspace formulation.

New bi-harmonic superspace formulation of $4D, \mathcal{N}=4$ SYM theory