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

Abstract

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 ...

Two-loop gravity amplitudes from four dimensional unitarity

“…We obtain the same coefficient of ln(µ 2 ) for the four-point amplitude as in ref. [13]. We also obtain the coefficient of ln(µ 2 ) for the five-point amplitude and show that this matches to the same R…”

“…Additionally when coupled to scalars and vectors a 2 was simply proportional to the the difference between the number of bosonic and fermionic degrees of freedom. As was argued in ref [13], the coefficient a 2 has physical content since after renormalisation the amplitude depends upon a 2 but not a 1 .…”

“…A feature of the computation was the appearance of sub-divergences which were cancelled diagram by diagram. In [13] this computation was revisited using evanescent operators which arise at one-loop to remove the sub-divergences in the four-point two-loop amplitude. For gravity coupled to various matter multiplets they found UV terms…”

Manifesting enhanced cancellations in supergravity: integrands versus integrals

“…However, this is of secondary concern because usually we are interested in studying the very first potential divergence of a supergravity theory. (There are some subtleties with evanescent effects feeding into divergences which require some care [2,3].) The most interesting cases, such as N = 8 supergravity at five loops in D = 24/5, automatically have no subdivergences because of a lack of lower-loop divergences.…”

Non-renormalization for non-supersymmetric black holes

“…Indeed, the non-trivial Euler number (2.7) makes it more interesting. For example, there is a one loop contribution even in pure gravity (some recent discussions include [22,23]). …”