Absence of Three-Loop Four-Point Divergences in N=4 Supergravity

Abstract

We compute the coefficient of the potential three-loop divergence in pure N=4 supergravity and show that it vanishes, contrary to expectations from symmetry arguments. The recently uncovered duality between color and kinematics is used to greatly streamline the calculation. We comment on all-loop cancellations hinting at further surprises awaiting discovery at higher loops.

Figures (3)

• Figure 1: A sample cut at three loops displaying cancellations in $\mathcal{N}=4$ supergravity special to four dimensions.
• Figure 2: The twelve graphs appearing in the three-loop $\mathcal{N}=4$ sYM amplitude BCJLoop and in gravity amplitudes obtained from these using the double-copy formula.
• Figure 3: The basis of vacuum integrals for one through three loops.