Abstract
We explicitly compute the tree-level on-shell four-graviton amplitudes in four, five and six dimensions for local and weakly nonlocal gravitational theories that are quadratic in both, the Ricci and scalar curvature with form factors of the d'Alembertian operator inserted between. More specifically we are interested in renormalizable, super-renormalizable or finite theories. The scattering amplitudes for these theories turn out to be the same as the ones of Einstein gravity regardless of the explicit form of the form factors. As a special case the four-graviton scattering amplitudes in Weyl conformal gravity are identically zero. Using a field redefinition, we prove that the outcome is correct for any number of external gravitons (on-shell $n-$point functions) and in any dimension for a large class of theories. However, when an operator quadratic in the Riemann tensor is added in any dimension (with the exception of the Gauss-Bonnet term in four dimensions) the result is completely altered, and the scattering amplitudes depend on all the form factors introduced in the action.
Highlights
On the other hand, since it is quite difficult to propose good and unambiguous quantities which could play the role of observables in pure quantum gravity, we concentrate our attention on hypothetical experiments of graviton scattering described in the perturbative framework of quantum field theory around flat Minkowski spacetime
When an operator quadratic in the Riemann tensor is added in any dimension the result is completely altered, and the scattering amplitudes depend on all the form factors introduced in the action
Since it is quite difficult to propose good and unambiguous quantities which could play the role of observables in pure quantum gravity, we concentrate our attention on hypothetical experiments of graviton scattering described in the perturbative framework of quantum field theory around flat Minkowski spacetime
Summary
Where operators in the third set are called killers, because they are crucial in making the theory finite in any dimension They are used to kill the beta functions. The minimal choice for a finite and unitary theory of quantum gravity in four dimensions may consist of terms with γ = 3 (and a = 1) in the kinetic part This alone leads to one-loop super-renormalizable quantum nonlocal gravity. We recently proposed, following [29], another class of weakly nonlocal possibly finite theories, which are constructed entirely from kinetic terms (only weakly nonlocal operators quadratic in curvature appear), without local or nonlocal gravitational potential V cubic in curvature or higher.
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