Manifest ultraviolet behavior for the three-loop four-point amplitude ofN=8supergravity

Using the method of maximal cuts, we obtain a form of the three-loop four-point scattering amplitude of $\mathcal{N}=8$ supergravity in which all ultraviolet cancellations are made manifest. The Feynman loop integrals that appear have a graphical representation with only cubic vertices, and numerator factors that are quadratic in the loop momenta, rather than quartic as in the previous form. This quadratic behavior reflects cancellations beyond those required for finiteness, and matches the quadratic behavior of the three-loop four-point scattering amplitude in $\mathcal{N}=4$ super-Yang-Mills theory. By direct integration we confirm that no additional cancellations remain in the $\mathcal{N}=8$ supergravity amplitude, thus demonstrating that the critical dimension in which the first ultraviolet divergence occurs at three loops is $D_c=6$. We also give the values of the three-loop divergences in $D=7$, 9, 11. In addition, we present the explicitly color-dressed three-loop four-point amplitude of $\mathcal{N}=4$ super-Yang-Mills theory.