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Evanescent operators such as the Gauss-Bonnet term have vanishingperturbative matrix elements in exactly D=4 dimensions. Similarly, evanescentfields do not propagate in D=4; a three-form field is in this class, since itis dual to a cosmological-constant contribution. In this Letter, we show thatevanescent operators and fields modify the leading ultraviolet divergence inpure gravity. To analyze the divergence, we compute the two-loopidentical-helicity four-graviton amplitude and determine the coefficient of theassociated (non-evanescent) R^3 counterterm studied long ago by Goroff andSagnotti. We compare two pairs of theories that are dual in D=4: gravitycoupled to nothing or to three-form matter, and gravity coupled to zero-form orto two-form matter. Duff and van Nieuwenhuizen showed that, curiously, theone-loop conformal anomaly --- the coefficient of the Gauss-Bonnet operator ---changes under p-form duality transformations. We concur, and also find that theleading R^3 divergence changes under duality transformations. Nevertheless, inboth cases the physical renormalized two-loop identical-helicity four-gravitonamplitude can be chosen to respect duality. In particular, itsrenormalization-scale dependence is unaltered.

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