  We compute the leading-color (planar) three-loop four-point amplitude of N=4
supersymmetric Yang-Mills theory in 4 - 2 epsilon dimensions, as a Laurent
expansion about epsilon = 0 including the finite terms. The amplitude was
constructed previously via the unitarity method, in terms of two Feynman loop
integrals, one of which has been evaluated already. Here we use the
Mellin-Barnes integration technique to evaluate the Laurent expansion of the
second integral. Strikingly, the amplitude is expressible, through the finite
terms, in terms of the corresponding one- and two-loop amplitudes, which
provides strong evidence for a previous conjecture that higher-loop planar N =
4 amplitudes have an iterative structure. The infrared singularities of the
amplitude agree with the predictions of Sterman and Tejeda-Yeomans based on
resummation. Based on the four-point result and the exponentiation of infrared
singularities, we give an exponentiated ansatz for the maximally
helicity-violating n-point amplitudes to all loop orders. The 1/epsilon^2 pole
in the four-point amplitude determines the soft, or cusp, anomalous dimension
at three loops in N = 4 supersymmetric Yang-Mills theory. The result confirms a
prediction by Kotikov, Lipatov, Onishchenko and Velizhanin, which utilizes the
leading-twist anomalous dimensions in QCD computed by Moch, Vermaseren and
Vogt. Following similar logic, we are able to predict a term in the three-loop
quark and gluon form factors in QCD.
