The Ultraviolet Critical Dimension of Half-Maximal Supergravity at Three Loops

TL;DR

This work determines the first dimension in which a three-loop four-point divergence appears for half-maximal ($Q=16$) supergravity, identifying $D_c=14/3$ and providing explicit divergent structures both for the pure theory and for cases with matter, parameterized by state-counting inputs $D_s$ and $N_f$ to access higher supersymmetry. The authors employ BCJ color–kinematics duality and the double-copy construction to build gravity integrands from gauge-theory data, and implement a divergence extraction pipeline that uses dimensional reduction, a uniform-mass IR regulator, and reduction to master integrals via FIREFIRE and Mellin–Barnes techniques. They present explicit three-loop divergences in $D=14/3$ for pure half-maximal gravity and for the $N_f$ piece, as well as divergences with various external multiplet configurations when matter is included; notably, the maximal theory with $D_s=10$ and $N_f=8$ cancels between the two contributions, aligning with known finiteness results. These results provide concrete data on enhanced cancellations and test the limits of the BCJ double-copy framework in higher-loop, higher-dimensional supergravity, informing expectations for UV behavior and guiding future investigations into the origin of these cancellations.

Abstract

We determine the minimum dimension at which divergences first occur in half-maximal 16-supercharge supergravity at three loops. For the four-point amplitudes, we find that the critical dimension is D=14/3 and give the explicit form of the divergence. We also give the divergence of an additional piece that gives us access to theories with higher degrees of supersymmetry, in particular 20- and 32-supercharge supergravities. We give the divergences of half-maximal supergravity when matter vector multiplets are included as well. Explicit forms of various divergences at one and two loops are also given.

Figures (3)

• Figure 1: The basic Jacobi relation between three diagrams for either color or numerator factors. The basic identity can be embedded into a diagram at any loop order.
• Figure 2: The contributing diagrams at (a) one loop and (b) two loops in the double-copy construction of the four-point amplitude in supergravity theories with 16 or more supercharges.
• Figure 3: The contributing diagrams to the three-loop four-point amplitudes in theories with 16 or more supercharges.