We use the duality between color and kinematics to simplify the construction of the complete four-loop four-point amplitude of $\mathcal{N}=4$ 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 $\mathcal{N}=8$ supergravity integrand, whose graph numerators are double copies (squares) of the $\mathcal{N}=4$ 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 $\mathcal{N}=4$ super-Yang-Mills amplitude. We determine the amplitude’s ultraviolet pole in the critical dimension of $D=11/2$, the same dimension as for $\mathcal{N}=4$ super-Yang-Mills theory. Strikingly, exactly the same combination of vacuum integrals (after simplification) describes the ultraviolet divergence of $\mathcal{N}=8$ supergravity as the subleading-in-$1/N_c^2$ single-trace divergence in $\mathcal{N}=4$ super-Yang-Mills theory.

• Received 6 February 2012

DOI:https://doi.org/10.1103/PhysRevD.85.105014