Evanescent Effects can Alter Ultraviolet Divergences in Quantum Gravity without Physical Consequences

Evanescent operators such as the Gauss-Bonnet term have vanishing perturbative matrix elements in exactly $D=4$ dimensions. Similarly, evanescent fields do not propagate in $D=4$; a three-form field is in this class, since it is dual to a cosmological-constant contribution. In this Letter, we show that evanescent operators and fields modify the leading ultraviolet divergence in pure gravity. To analyze the divergence, we compute the two-loop identical-helicity four-graviton amplitude and determine the coefficient of the associated (nonevanescent) $R^3$ counterterm studied long ago by Goroff and Sagnotti. We compare two pairs of theories that are dual in $D=4$: gravity coupled to nothing or to three-form matter, and gravity coupled to zero-form or to two-form matter. Duff and van Nieuwenhuizen showed that, curiously, the one-loop trace anomaly—the coefficient of the Gauss-Bonnet operator—changes under $p$-form duality transformations. We concur and also find that the leading $R^3$ divergence changes under duality transformations. Nevertheless, in both cases, the physical renormalized two-loop identical-helicity four-graviton amplitude can be chosen to respect duality. In particular, its renormalization-scale dependence is unaltered.

Figure 1

Representative diagrams of the (a) bare, (b) single-counterterm, and (c) double-counterterm insertions.