On powercounting in perturbative quantum gravity theories through color-kinematic duality

A bstract The standard argument why gravity is not renormalisable relies on direct powercounting of Feynman graphs to estimate the degree of UV divergence. In several (highly) supersymmetric examples the actual divergences have been shown to be considerably better. In these examples the improvement follows from a conjectured duality between color and kinematics. In this paper we initiate the systematic study of quite general powercounting under the assumption that color-kinematic duality exists. The main technical tool is a reformulation of the duality in terms of linear maps, modulo subtleties at loop level mostly inherent to the duality. This tool may have wider applications in both gauge and gravity theories, up to resolution of the subtleties. Here it is first applied to the large Britto-Cachazo-Feng-Witten (BCFW) shift behavior of gravity integrands constructed through the duality. Assuming color-kinematic duality and reasonable technical requirements hold these shifts are shown to be independent of loop order. This is a new quantitative measure for massive cancellations with respect to the Feynman graph expression. More speculatively, the same approach is then applied to provide estimates of the overall degree of UV divergence in quite general gravity theories, assuming the duality exists. The manifest cancellations obtained in these estimates depends on the exact implementation of the duality at loop level, especially on graph topology. The developed arguments apply to all multiplicity. Finally, some evidence for the duality to all loop orders is provided from an analysis of BCFW shifts of gauge theory integrands through Feynman graphs.