N=4 Supersymmetric Gauge Theory in the Derivative Expansion

Maximally supersymmetric gauge theories have experienced renewed interest due to the AdS/CFT correspondence and its conjectured S-duality. These gauge theories possess a large amount of symmetry and have quasi-integrable properties. We derive the amplitudes in the derivative expansion of the spontaneously broken examples and perform all loop integrations. The S-matrix is found via an algebraic recursion and at each order is SL(2,Z) invariant.