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We present an expression for the leading-color (planar) four-loop four-pointamplitude of N=4 supersymmetric Yang-Mills theory in 4-2 e dimensions, in termsof eight separate integrals. The expression is based on consistency ofunitarity cuts and infrared divergences. We expand the integrals around e=0,and obtain analytic expressions for the poles from 1/e^8 through 1/e^4. We givenumerical results for the coefficients of the 1/e^3 and 1/e^2 poles. Theseresults all match the known exponentiated structure of the infrareddivergences, at four separate kinematic points. The value of the 1/e^2coefficient allows us to test a conjecture of Eden and Staudacher for thefour-loop cusp (soft) anomalous dimension. We find that the conjecture isincorrect, although our numerical results suggest that a simple modification ofthe expression, flipping the sign of the term containing zeta_3^2, may yieldthe correct answer. Our numerical value can be used, in a scheme proposed byKotikov, Lipatov and Velizhanin, to estimate the two constants in thestrong-coupling expansion of the cusp anomalous dimension that are known fromstring theory. The estimate works to 2.6% and 5% accuracy, providingnon-trivial evidence in support of the AdS/CFT correspondence. We also use theknown constants in the strong-coupling expansion as additional input to provideapproximations to the cusp anomalous dimension which should be accurate tounder one percent for all values of the coupling. When the evaluations of theintegrals are completed through the finite terms, it will be possible to testthe iterative, exponentiated structure of the finite terms in the four-loopfour-point amplitude, which was uncovered earlier at two and three loops.Comment: 72 pages, 15 figures, v2 minor correction

Four-loop planar amplitude and cusp anomalous dimension in maximally supersymmetric Yang-Mills theory