Planar Amplitudes in Maximally Supersymmetric Yang-Mills Theory

The collinear factorization properties of two-loop scattering amplitudes in dimensionally regulated $N=4$ super-Yang-Mills theory suggest that, in the planar (’t Hooft) limit, higher-loop contributions can be expressed entirely in terms of one-loop amplitudes. We demonstrate this relation explicitly for the two-loop four-point amplitude and, based on the collinear limits, conjecture an analogous relation for $n$-point amplitudes. The simplicity of the relation is consistent with intuition based on the anti–de Sitter/conformal field theory correspondence that the form of the large-$N_c$ $L$-loop amplitudes should be simple enough to allow a resummation to all orders.

Figure 1

The collinear factorization of a two-loop amplitude. In each term in the sum, the left blob is a splitting amplitude, and the right blob an ($n-1$)-point scattering amplitude.

Figure 2

The scalar integral functions appearing in the (a) one- and (b) two-loop four-point $N=4$ amplitudes.