Two-Loop Four-Gluon Amplitudes in N=4 Super-Yang-Mills

Using cutting techniques we obtain the two-loop N=4 super-Yang-Mills helicity amplitudes for four-gluon scattering in terms of scalar integral functions. The N=4 amplitudes are considerably simpler than corresponding QCD amplitudes and therefore provide a testing ground for exploring two-loop amplitudes. The amplitudes are constructed directly in terms of gauge invariant quantities and therefore remain relatively compact throughout the calculation. We also present a conjecture for the leading color four-gluon amplitudes to all orders in the perturbative expansion.