Bi-harmonic superspace for N=4 d=4 super Yang-Mills

We develop N=4 d=4 bi-harmonic superspace and use it to derive a novel form for the low-energy effective action in N=4 super Yang-Mills theory. We solve the N=4 supergauge constraints in this superspace in terms of analytic superfields. Using these superfields, we construct a simple functional that respects N=4 supersymmetry and scale invariance. In components, it reproduces all on-shell terms in the four-derivative part of the N=4 SYM effective action; in particular, the F^4/X^4 and Wess-Zumino terms. The latter comes out in a novel SO(3) x SO(3)-invariant form.