Perturbative expansion of N<8 Supergravity
David C. Dunbar, James H. Ettle, Warren B. Perkins
TL;DR
The paper investigates the UV structure of four-dimensional gravity theories with $\mathcal{N}=6$ and $\mathcal{N}=4$ supersymmetry by computing one-loop $n$-point amplitudes using IR-consistent integral bases and unitarity methods. It finds that $\mathcal{N}=6$ amplitudes have a reduced loop-momentum power count $d_{\text{eff}}=n-3$ and can be expressed purely in terms of truncated box functions (with no bubbles or three-mass triangles in MHV), while $\mathcal{N}=4$ amplitudes exhibit $d_{\text{eff}}=n$ and necessarily contain a nonzero rational term $R_n$ (even at four points, $R_4\neq0$). These results imply improved UV behavior for $\mathcal{N}=6$ relative to naive expectations and reveal a qualitatively different structure for $\mathcal{N}=4$ due to rational components. Extending the analysis to three loops via power-count estimates suggests possible finiteness in $D=4$ for these theories, though explicit multi-loop calculations are required to confirm this. Overall, the work sharpens the understanding of loop-level cancellations in $N<8$ supergravity and informs finiteness conjectures for less supersymmetric gravity theories.
Abstract
We characterise the one-loop amplitudes for N=6 and N=4 supergravity in four dimensions. For N=6 we find that the one-loop n-point amplitudes can be expanded in terms of scalar box and triangle functions only. This simplification is consistent with a loop momentum power count of n-3, which we would interpret as being n+4 for gravity with a further -7 from the N=6 superalgebra. For N=4 we find that the amplitude is consistent with a loop momentum power count of n, which we would interpret as being n+4 for gravity with a further -4 from the N=4 superalgebra. Specifically the N=4 amplitudes contain non-cut-constructible rational terms.