Two-loop renormalization of quantum gravity simplified

The coefficient of the dimensionally regularized two-loop $R^3$ divergence of (nonsupersymmetric) gravity theories has recently been shown to change when nondynamical three-forms are added to the theory, or when a pseudoscalar is replaced by the antisymmetric two-form field to which it is dual. This phenomenon involves evanescent operators, whose matrix elements vanish in four dimensions, including the Gauss-Bonnet operator which is also connected to the trace anomaly. On the other hand, these effects appear to have no physical consequences for renormalized scattering processes. In particular, the dependence of the two-loop four-graviton scattering amplitude on the renormalization scale is simple. We explain this result for any minimally-coupled massless gravity theory with renormalizable matter interactions by using unitarity cuts in four dimensions and never invoking evanescent operators.

Figure 1

Representative four-point diagrams for (a) the bare contribution, and (b) the single-GB-counterterm, (c) double-GB-counterterm, and (d) $R^3$-counterterm insertions needed to remove all divergences.

Figure 2

Renormalization of on-shell Yang-Mills amplitudes at one loop requires (a) the bare amplitude and (b) an $F^2$ counterterm, for which representative contributions are shown.

Figure 3

The $s$-channel two-particle cuts (a) and (b) from which we can extract the logarithmic parts of the two-loop four-point identical-helicity four-graviton amplitude. The exposed lines are placed on shell and are in four dimensions.

Figure 4

Representative contributions to the three-particle cut. This cut generates no new $\ln {\mu }^2$ contributions to the $R^3$ operator for the identical-helicity four-graviton amplitude.