Abstract
We employ the curvature expansion of the quantum effective action for gravity-matter systems to construct graviton-mediated scattering amplitudes for non-minimally coupled scalar fields in a Minkowski background. By design, the formalism parameterises all quantum corrections to these processes and is manifestly gauge-invariant. The conditions resulting from UV-finiteness, unitarity, and causality are analysed in detail and it is shown by explicit construction that the quantum effective action provides sufficient room to meet these structural requirements without introducing non-localities or higher-spin degrees of freedom. Our framework provides a bottom-up approach to all quantum gravity programs seeking for the quantisation of gravity within the framework of quantum field theory. Its scope is illustrated by specific examples, including effective field theory, Stelle gravity, infinite derivative gravity, and Asymptotic Safety.
Highlights
At this stage, it is an open debate under which conditions gravity may be formulated as a quantum field theory
We will focus on scattering in the effective field theory framework, classical Stelle gravity, infinite derivative gravity, and renormalisation group (RG) improvement from Asymptotic Safety
We have used the form factor parameterisation of the quantum effective action Γ to construct the most general amplitude for a two-scalar-to-two-scalar process mediated by gravitons in a Minkowski background
Summary
We start our investigation with a brief introduction to form factors in curved spacetime. A detailed discussion can be found in [44]. Form factors arise naturally in the computation of loop corrections. A textbook example is the electron self-energy in QED which introduces a non-trivial momentum dependence in the electron propagator. Form factors are functions of momentum invariants entering the propagators and interaction vertices. The Fourier transform allows to switch from the momentum-space representation of the form factors to a position-space representation where the corresponding functions depend on partial derivatives. The correspondence principle provides a natural generalisation to curved space, replacing the partial derivatives by covariant derivatives
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