### Abstract

We use the duality between color and kinematics to simplify the construction of the complete four-loop four-point amplitude of N=4 super-Yang-Mills theory, including the nonplanar contributions. The duality completely determines the amplitude's integrand in terms of just two planar graphs. The existence of a manifestly dual gauge-theory amplitude trivializes the construction of the corresponding N=8 supergravity integrand, whose graph numerators are double copies (squares) of the N=4 super-Yang-Mills numerators. The success of this procedure provides further nontrivial evidence that the duality and double-copy properties hold at loop level. The new form of the four-loop four-point supergravity amplitude makes manifest the same ultraviolet power counting as the corresponding N=4 super-Yang-Mills amplitude. We determine the amplitude's ultraviolet pole in the critical dimension of D=11/2, the same dimension as for N=4 super-Yang-Mills theory. Strikingly, exactly the same combination of vacuum integrals (after simplification) describes the ultraviolet divergence of N=8 supergravity as the subleading-in-1/Nc2 single-trace divergence in N=4 super-Yang-Mills theory.

Original language | English (US) |
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Article number | 105014 |

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

Volume | 85 |

Issue number | 10 |

DOIs | |

State | Published - May 17 2012 |

### All Science Journal Classification (ASJC) codes

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

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## Cite this

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

*85*(10), [105014]. https://doi.org/10.1103/PhysRevD.85.105014