Abstract
We consider Einstein gravity with the addition of $R^2$ and $R^{\mu \nu} R_{\mu \nu}$ interactions in the context of effective field theory, and the corresponding scattering amplitudes of gravitons and minimally-coupled heavy scalars. First, we recover the known fact that graviton amplitudes are the same as in Einstein gravity. Then we show that all amplitudes with two heavy scalars and an arbitrary number of gravitons are also not affected by these interactions. We prove this by direct computations, using field redefinitions known from earlier applications in string theory, and with a combination of factorisation and power-counting arguments. Combined with unitarity, these results imply that, in an effective field theory approach, the Newtonian potential receives neither classical nor quantum corrections from terms quadratic in the curvature.
Highlights
Much work has been devoted recently to studying the effects of possible modifications of Einstein-Hilbert (EH) gravity, see [1,2] for recent reviews
Quadratic and cubic corrections make an appearance in the effective gravitational action for closed strings in [3,4,5,6], and as counterterms at one loop in gravity coupled to matter [7] and at two loops in pure gravity [8,9]
The analysis of [3,4,5] showed that R2 or RμνRμν cannot be probed by looking at scattering amplitudes, since they can be removed by field redefinitions without influencing the S-matrix as a consequence of the S-matrix equivalence theorem, reviewed later, while the GB term, being topological in four dimensions, can be discarded
Summary
Much work has been devoted recently to studying the effects of possible modifications of Einstein-Hilbert (EH) gravity, see [1,2] for recent reviews. In the effective field theory approach the new Yukawa potentials induced by the quadratic terms are absent at tree level [30] This is best seen in momentum space, where the massive propagators are replaced by a polynomial in the momentum transfer squared, which in turn leads to local terms which give no contribution to long-range physics. Results obtained here are valid up to linear order in the small parameters a, and b, which is perfectly sufficient from an effective field theory point of view This argument is valid in D dimensions, it lends itself to an application of D-dimensional unitarity; 4See Fig. 1 of [38] for sample two-loop cut diagrams. We include an Appendix, containing the Feynman rules needed in the calculations
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