$SU(4)$ harmonic superspace and supersymmetric gauge theory

We consider the harmonic superspace formalism in $N=4$ supersymmetry based on $SU(4)/SU(2)\times SU(2)\times U(1)$ harmonics, which was previously used in Abelian gauge theory. We propose a transformation of non-Abelian constraints in the standard $N{=}4$ superspace into a superfield equation for two basic analytic superfields: an independent strength $W$ of dimension one and a dimensionless harmonic four-prepotential $V$ of the $U(1)$ charge two. These constraint equations I explicitly depend on the Grassmann coordinates $\theta$, although they are covariant under nonstandard $N=4$ supersymmetry transformations. The component expansion of superfield equations I generates the known equations for physical fields of the $N=4$ supermultiplet, with the auxiliary fields vanishing or expressible in terms of physical fields on the mass shell. In the harmonic formalism of $N=4$ supergauge theory off the mass shell, we construct a gauge-invariant action $A(W,V)$ for two unconstrained non-Abelian analytic superfields $W$ and $V$; this action contains theta factors in each term and is invariant under the $SU(4)$ automorphism group and scaling transformations. At the level of component fields, this model acquires an interaction of two infinite-dimensional $N=4$ supermultiplets involving physical and auxiliary fields. The action $A(W,V)$ generates analytic equations of motion II, alternative to the superfield constraints I. Both sets of equations give equivalent equations for physical component fields of the $N=4$ gauge supermultiplet. We construct a nonlinear effective interaction for the Abelian harmonic superfield $W$.