Iteration of planar amplitudes in maximally supersymmetric Yang-Mills theory at three loops and beyond

We compute the leading-color (planar) three-loop four-point amplitude of N=4supersymmetric Yang-Mills theory in 4 - 2 epsilon dimensions, as a Laurentexpansion about epsilon = 0 including the finite terms. The amplitude wasconstructed previously via the unitarity method, in terms of two Feynman loopintegrals, one of which has been evaluated already. Here we use theMellin-Barnes integration technique to evaluate the Laurent expansion of thesecond integral. Strikingly, the amplitude is expressible, through the finiteterms, in terms of the corresponding one- and two-loop amplitudes, whichprovides strong evidence for a previous conjecture that higher-loop planar N =4 amplitudes have an iterative structure. The infrared singularities of theamplitude agree with the predictions of Sterman and Tejeda-Yeomans based onresummation. Based on the four-point result and the exponentiation of infraredsingularities, we give an exponentiated ansatz for the maximallyhelicity-violating n-point amplitudes to all loop orders. The 1/epsilon^2 polein the four-point amplitude determines the soft, or cusp, anomalous dimensionat three loops in N = 4 supersymmetric Yang-Mills theory. The result confirms aprediction by Kotikov, Lipatov, Onishchenko and Velizhanin, which utilizes theleading-twist anomalous dimensions in QCD computed by Moch, Vermaseren andVogt. Following similar logic, we are able to predict a term in the three-loopquark and gluon form factors in QCD.Comment: 54 pages, 7 figures. v2: Added references, a few additional words about large spin limit of anomalous dimensions. v3: Expanded Sect. IV.A on multiloop ansatz; remark that form-factor prediction is now confirmed by other work; minor typos correcte