Abstract
It has recently been argued that there may be a nontrivial four-dimensional limit of the higher-dimensional Gauss--Bonnet and Lovelock interactions and that this might provide a loophole allowing for new four-dimensional gravitational theories, possibly without a standard Lagrangian. We investigate this claim by studying tree-level graviton scattering amplitudes, allowing us to draw conclusions independently of the Lagrangian. By taking four-dimensional limits of higher-dimensional scattering amplitudes of the Gauss--Bonnet theory, we find four-dimensional amplitudes that are different from general relativity; however, these amplitudes are not new since they all come from certain scalar-tensor theories. The nontrivial limit that does not lead to infinite strong coupling around flat space leads to $(\partial\phi)^4$ theory. We argue that there cannot be any six-derivative purely gravitational four-point amplitudes in any dimension other than those coming from Lovelock theory by directly constructing the on-shell amplitudes. In particular, there can be no new such amplitudes in four dimensions beyond those of general relativity. We also present some new results on the spin-averaged cross section for graviton-graviton scattering in general relativity and Gauss--Bonnet theory in arbitrary dimensions.
Highlights
According to Lovelock’s theorem [1,2], the most general theory of a single interacting massless graviton in any dimension leading to second-order equations of motion is given by a sum of the Lovelock-Lanczos terms, LðnÞ 1⁄4 n! 2n=2 δ1⁄2νμ11Á δ R νμnn νμ11νμ22 Rνμ33νμ44 Rνμnn−−11νμnn ;n 1⁄4 0; 2; 4; ...: ð1:1ÞIn D dimensions, the terms with n ≥ D are total derivatives that do not contribute to the equations of motion or on-shell amplitudes
We consider the cross section for unpolarized graviton-graviton scattering in general relativity (GR) and GaussBonnet theory in arbitrary dimensions, showing that finite results from Gauss-Bonnet are possible for D 1⁄4 4 only if the coupling is scaled in yet another way
The question of whether there exists a “novel 4D GaussBonnet theory” has been revived in Ref. [3]. Observables in such a theory are supposed to be obtained by taking a D → 4 limit of Gauss-Bonnet observables in general D after rescaling the Gauss-Bonnet coupling by α → α =ðD − 4Þ, leaving a finite contribution which differs from general relativity
Summary
According to Lovelock’s theorem [1,2], the most general theory of a single interacting massless graviton in any dimension leading to second-order equations of motion is given by a sum of the Lovelock-Lanczos terms, LðnÞ. We will take the view that there need not be a Lagrangian or equations of motion defined directly in four dimensions, but we instead use the above prescription for computing observables. We will argue that there can be no new purely gravitational four-point amplitudes beyond those given by the Lovelock theory (1.1) by directly constructing on-shell four-point graviton amplitudes with up to six derivatives in arbitrary dimensions, showing that there is no purely gravitational “novel 4D Gauss-Bonnet theory” from this point of view. We consider the cross section for unpolarized graviton-graviton scattering in GR and GaussBonnet theory in arbitrary dimensions, showing that finite results from Gauss-Bonnet are possible for D 1⁄4 4 only if the coupling is scaled in yet another way. The result is the following nonvanishing helicity amplitudes: 024029-2
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