A two-loop test for the factorised S -matrix of planar N=4N=4

We check the recently proposed higher loop Bethe-ansatz for the sl(2)sl(2) sector of N=4N=4 at two loops by a direct perturbative calculation using N=2N=2 superfields in supersymmetric dimensional reduction. Our method can in principle address operators built from many elementary fields and of arbitrary twist. We work out in detail the spin three, twist three operator mixing problem at leading order in N and find agreement with the prediction based on integrability.