Abstract
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.
Highlights
Evanescent operators such as the Gauss-Bonnet term have vanishing perturbative matrix elements in exactly D 1⁄4 4 dimensions
Evanescent fields do not propagate in D 1⁄4 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
Summary
Φj is a scalar field and Hjμνρ and Hjμνρσ are the field strengths of the two- and three-form antisymmetric-tensor fields Ajμν and Ajμνρ. Standard gauge fixing for the two- and three-form actions, as well as for LEH, leads to a nontrivial ghost structure We avoid such complications by using the generalized unitarity method [11,12,13], which directly imposes appropriate D-dimensional physical-state projectors on the on-shell states crossing unitarity cuts. For a theory with n0 scalars, n2 two-forms and n3 threeforms coupled to gravity, the one-loop UV divergence takes the form of the GB term [1,2,7]. The matrix elements produced by Eq (5) vanish for four on-shell D 1⁄4 4 graviton polarization tensors This is because the GB combination is evanescent in D 1⁄4 4: It is a total derivative and vanishes when integrated over a topologically trivial space; pure Einstein gravity is finite at one loop [6]. For the case of antisymmetric tensors coupled to gravity, another relevant one-loop four-point divergence is that of two gravitons and two antisymmetric tensors, which is generated by the operator
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