Color-Kinematics Duality and Sudakov Form Factor at Five Loops for $\mathcal{N}=4$ Supersymmetric Yang-Mills Theory

Using color-kinematics duality, we construct for the first time the full integrand of the five-loop Sudakov form factor in $\mathcal{N}=4$ super-Yang-Mills theory, including nonplanar contributions. This result also provides a first manifestation of the color-kinematics duality at five loops. The integrand is explicitly ultraviolet finite when $D<26/5$, coincident with the known finiteness bound for amplitudes. If the double-copy method could be applied to the form factor, this would indicate an interesting ultraviolet finiteness bound for $\mathcal{N}=8$ supergravity at five loops. The result is also expected to provide an essential input for computing the five-loop nonplanar cusp anomalous dimension.

Figure 1

Cubic graphs for four-point tree amplitudes.

Figure 2

Color-kinematics related graphs at loop level.

Figure 3

Four master graphs for the five-loop Sudakov form factor, labeled by (a), (b), (c), and (d).

Figure 4

Nontrivial unitarity cuts. Each blob denotes a tree amplitude or form factor. (a)–(e) are heptuple cuts. (f) is the most involving hextuple cut.

Figure 5

Unitarity check of subtle 1PR graphs. (a) is a 1PR graph. (b) and (c) are two other graphs that contribute to the corresponding cut.

Figure 6

Sample graphs of five-loop Sudakov form factor. (a) is an example of fish graphs. (b), (c), and (d) are three sample graphs whose numerators are given in Eqs. (9) and (10).