### Abstract

We present the complete four-loop four-point amplitude in N=4 super-Yang-Mills theory, for a general gauge group and general D-dimensional covariant kinematics, and including all nonplanar contributions. We use the method of maximal cuts-an efficient application of the unitarity method-to construct the result in terms of 50 four-loop integrals. We give graphical rules, valid in D dimensions, for obtaining various nonplanar contributions from previously-determined terms. We examine the ultraviolet behavior of the amplitude near D=11/2. The nonplanar terms are as well-behaved in the ultraviolet as the planar terms. However, in the color decomposition of the three- and four-loop amplitude for an SU(N_{c}) gauge group, the coefficients of the double-trace terms are better behaved in the ultraviolet than are the single-trace terms. The results from this paper were an important step toward obtaining the corresponding amplitude in N=8 supergravity, which confirmed the existence of cancellations beyond those needed for ultraviolet finiteness at four loops in four dimensions. Evaluation of the loop integrals near D=4 would permit tests of recent conjectures and results concerning the infrared behavior of four-dimensional massless gauge theory.

Original language | English (US) |
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Article number | 125040 |

Journal | Physical Review D - Particles, Fields, Gravitation and Cosmology |

Volume | 82 |

Issue number | 12 |

DOIs | |

State | Published - Dec 1 2010 |

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### All Science Journal Classification (ASJC) codes

- Nuclear and High Energy Physics
- Physics and Astronomy (miscellaneous)

### Cite this

*Physical Review D - Particles, Fields, Gravitation and Cosmology*,

*82*(12), [125040]. https://doi.org/10.1103/PhysRevD.82.125040