Ultraviolet Properties of N = 8 Supergravity at Five Loops

TL;DR

The paper determines the five-loop ultraviolet behavior of N=8 supergravity, showing that the first divergence occurs in D_c = 24/5 and corresponds to a D^8 R^4 counterterm, with no evidence of enhanced cancellations at this order. It achieves this by constructing an improved five-loop integrand through a generalized double-copy, performing a vacuum-expansion of the integrand, and executing a full IBP reduction organized by an SL(5) symmetry to extract the UV pole. A striking outcome is that the leading UV contributions can be expressed entirely in terms of a small set of vacuum master integrals with coefficients dictated by diagram automorphisms, and that cross-loop consistency relations tie higher-loop divergences to lower-loop vacuum structures. These patterns not only validate the five-loop result but also suggest powerful avenues for simplifying higher-loop calculations and understanding the ultraviolet structure of maximal supergravity in four dimensions.

Abstract

We use the recently developed generalized double-copy construction to obtain an improved representation of the five-loop four-point integrand of $N = 8$ supergravity whose leading ultraviolet behavior we analyze using state of the art loop-integral expansion and reduction methods. We find that the five-loop critical dimension where ultraviolet divergences first occur is $D_c=24/5$, corresponding to a $D^8 R^4$ counterterm. This ultraviolet behavior stands in contrast to the cases of four-dimensional $N = 4$ supergravity at three loops and $N = 5$ supergravity at four loops whose improved ultraviolet behavior demonstrates enhanced cancellations beyond implications from standard-symmetry considerations. We express this $D_c=24/5$ divergence in terms of two relatively simple positive-definite integrals reminiscent of vacuum integrals, excluding any additional ultraviolet cancellations at this loop-order. We note nontrivial relations between the integrals describing this leading ultraviolet behavior and integrals describing lower-loop behavior. This observation suggests not only a path towards greatly simplifying future calculations at higher loops, but may even allow us to directly investigate ultraviolet behavior in terms of simplified integrals, avoiding the construction of complete integrands.

Figures (21)

• Figure 1: The three four-point diagrams participating in either color or numerator Jacobi identities.
• Figure 2: Sample maximal and next-to-maximal cuts. The exposed lines connecting the blobs are taken to be on shell delta-functions.
• Figure 3: Sample N$^{k}$MCs used in the construction of five-loop four-point amplitudes. The exposed lines connecting the blobs are taken to be on-shell delta-functions.
• Figure 4: New contribution found via the method of maximal cuts can be assigned to contact terms. The labels (X: Y) correspond to the labeling of Ref. FiveLoopN8Integrand and refer to the level and contact diagram number.
• Figure 5: An example illustrating the notation in Eq. (\ref{['DoubleCopyCut']}). Expanding each of the two four-point blob gives a total of nine diagrams. The label N$^{2}$MC 867 refer to 867th diagram of the 2nd level cuts, and the $n_{i,j}$ correspond to labels used in the cut. The shaded thick (blue and red) lines are the propagators around which BCJ discrepancy functions are defined.