The Five-Loop Four-Point Amplitude of N=4 super-Yang-Mills Theory

TL;DR

This work constructs the complete D-dimensional integrand of the five-loop four-point amplitude of N=4 super-Yang-Mills theory, including nonplanar contributions, and presents a compact explicit expression for the nonvanishing ultraviolet divergence in terms of three vacuum integrals.

Abstract

Using the method of maximal cuts, we construct the complete D-dimensional integrand of the five-loop four-point amplitude of N = 4 super-Yang-Mills theory, including nonplanar contributions. In the critical dimension where this amplitude becomes ultraviolet divergent, we present a compact explicit expression for the nonvanishing ultraviolet divergence in terms of three vacuum integrals. This construction provides a crucial step towards obtaining the corresponding amplitude of N = 8 supergravity useful for resolving the general ultraviolet behavior of supergravity theories.

Figures (4)

• Figure 1: Sample graphs for the five-loop four-point $\mathcal{N}=4$ sYM amplitude. The graph labels correspond to the ones used in the ancillary file AttachedFile.
• Figure 2: Sample N$^k$-maximal cuts for $k=0,1,2,3$. The exposed lines are all cut.
• Figure 3: Examples of simple cuts used to speed up the calculation. (a) is a two particle cut, (b) a box cut and (c) is a sample application of the new amplitude relations of ref. BCJ. The exposed lines are all cut.
• Figure 4: Some of the five-loop vacuum integrals that appear in intermediate steps. Only (a), (b) and (c) appear in the final UV divergence. Integral (j) has a nontrivial numerator factor, as indicated. The (blue) dots indicate that a propagator is squared.