Abstract
We apply the Hilbert series to extend the gravitational action for a scalar field to a complete, non-redundant basis of higher-dimensional operators that is quadratic in the scalars and the Weyl tensor. Such an extension of the action fully describes tidal effects arising from operators involving two powers of the curvature. As an application of this new action, we compute all spinless tidal effects at the leading post-Minkowskian order. This computation is greatly simplified by appealing to the heavy limit, where only a severely constrained set of operators can contribute classically at the one-loop level. Finally, we use this amplitude to derive the mathcal{O}left({G}^2right) tidal corrections to the Hamiltonian and the scattering angle.
Highlights
There has been significant progress made on the inclusion of spin effects
The layout of this paper is as follows: we begin in section 2 by presenting the full tidal actions for electromagnetism and gravity coupled to real scalars at quadratic order in the field strength or the Weyl tensor respectively
Through the Hilbert series we have been able to write down an action which includes all possible operators involving two real scalars and two Weyl tensors
Summary
This section is dedicated to the presentation of the tidal actions up to quadratic order in the field strengths or Weyl tensors respectively for QED or gravity coupled to a real scalar. We achieve the complete forms of these actions through application of the Hilbert series. We begin with a brief introduction to the Hilbert series before presenting the results of the series and corresponding tidal actions for QED and gravity. Technical details about the Hilbert series are postponed to appendix A
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