Ultraviolet Properties of N = 8 Supergravity at Five Loops

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.