Enhanced Ultraviolet Cancellations in N = 5 Supergravity at Four Loop

TL;DR

The paper identifies enhanced ultraviolet cancellations in N=4 and N=5 supergravity, showing that the four-loop four-point amplitudes of N=5 supergravity are ultraviolet finite in $D=4$ despite symmetry-based expectations. Using BCJ color–kinematics duality and the unitarity/double-copy framework, the authors construct the ${ m N}=5$ integrand via a direct product from ${ m N}=4$ sYM and an ${ m N}=1$ sector, and extract UV divergences with infrared regulation and FIRE5 reduction. Three-loop analyses reveal finiteness for both ${ m N}=4$ and ${ m N}=5$, driven by enhanced cancellations that cannot be displayed in covariant diagrams. At four loops, ${ m N}=4$ diverges in $D=4$ while ${ m N}=5$ remains finite, with cancellations that do not arise from standard symmetry arguments, suggesting a deeper color–kinematics structure underlying gravity amplitudes.

Abstract

We show that the four-loop four-point amplitudes of N = 5 supergravity are ultraviolet finite in four dimensions, contrary to expectations based on supersymmetry and duality-symmetry arguments. We explain why the diagrams of any covariant local formalism cannot manifestly exhibit the necessary cancellations for finiteness but instead require a new type of nontrivial ultraviolet cancellation that we call "enhanced cancellations". We also show that the three-loop four-point amplitudes in N = 4 and N = 5 supergravity theories display enhanced cancellations. To construct the loop integrand, we use the duality between color and kinematics. We apply standard methods for extracting ultraviolet divergences in conjunction with the FIRE5 integral reduction program to arrive at the four-loop results.

Figures (8)

• Figure 1: The basic loop-level Jacobi relation for either color or numerator factors given in Eq. (\ref{['BCJDuality']}). The basic identity can be embedded in a diagram at any loop order.
• Figure 2: Three-loop sample diagrams for maximal-cut power counting.
• Figure 3: Sample diagrams for power counting maximal cuts at five and seven loops.
• Figure 4: A four-loop diagram whose maximal-cut power count suggests that ${{\cal N}=5}$ supergravity should diverge in four dimensions, contrary to the behavior of the four-point amplitude as a whole.
• Figure 5: Contributing three-loop diagrams in ${{\cal N}=4}$ and ${{\cal N}=5}$ supergravity.