Evanescent operators such as the Gauss-Bonnet term have vanishing perturbative matrix elements in exactly D ¼ 4 dimensions. Similarly, evanescent fields do not propagate in D ¼ 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 ¼ 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. DOI: 10.1103/PhysRevLett.115.211301 PACS numbers: 04.65.+e, 11.15.Bt, 11.25.Db, 12.60.Jv Although theories of quantum gravity have been studied for many decades, basic questions about their ultraviolet (UV) structure persist. One subtlety is the trace anomaly [1] which, at one loop, provides the coefficient of the GaussBonnet (GB) term. The physical significance of this relationship has not been settled, however. In particular, Duff and van Nieuwenhuizen showed that the trace anomaly changes under duality transformations of p-form fields, suggesting that theories related through such transformations are quantum-mechanically inequivalent [2]. In response, Siegel argued that this effect is a gauge artifact and, therefore, not physical [3]; Fradkin, Tseytlin, Grisaru et al. have also argued that duality should hold at the quantum level [4]. Furthermore, for D ¼ 4 external states, one-loop divergences in gravity theories coupled to two-form antisymmetric tensors are unchanged under a duality transformation relating two-forms to zero-form scalars [5]. However, as we shall see, intuition based on one-loop analyses can be deceptive.As established in the seminal work of 't Hooft and Veltman [6], pure gravity is finite at one loop because the only available on-shell counterterm is the GB term, which integrates to zero in a topologically trivial background. While amplitudes with external matter fields diverge at one loop, amplitudes with only external gravitons remain finite. At two loops, however, pure gravity diverges, as demonstrated explicitly by Goroff and Sagnotti [7] and confirmed by van de Ven [8].In this Letter, we investigate the UV properties of the two-loop amplitude for the scattering of four identicalhelicity gravitons, including the effect of p-form duality transformations. We use ...

The coefficient of the dimensionally regularized two-loop R 3 divergence of (nonsupersymmetric) gravity theories has recently been shown to change when non-dynamical three forms are added to the theory, or when a pseudo-scalar is replaced by the anti-symmetric 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 in renormalized scattering processes. In particular, the dependence of the two-loop four-graviton scattering amplitude on the renormalization scale is simple. In this paper, 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.

No abstract