Abstract
We extract the long-range gravitational potential between two scalar particles with arbitrary masses from the two-to-two elastic scattering amplitude at 2nd Post-Minkowskian order in arbitrary dimensions. In contrast to the four-dimensional case, in higher dimensions the classical potential receives contributions from box topologies. Moreover, the kinematical relation between momentum and position on the classical trajectory contains a new term which is quadratic in the tree-level amplitude. A precise interplay between this new relation and the formula for the scattering angle ensures that the latter is still linear in the classical part of the scattering amplitude, to this order, matching an earlier calculation in the eikonal approach. We point out that both the eikonal exponentiation and the reality of the potential to 2nd post-Minkowskian order can be seen as a consequence of unitarity. We finally present closed-form expressions for the scattering angle given by leading-order gravitational potentials for dimensions ranging from four to ten.
Highlights
The perturbative series that naturally organizes the calculation of scattering amplitudes in quantum field theory offers a convenient tool to study the dynamics of such systems for weak gravitational fields without the need to consider the limit of small velocities, thanks to the Lorentz invariance of the amplitude
Starting from the elastic scattering amplitude of two scalar particles with arbitrary masses in Einstein gravity in an arbitrary number D of space-time dimensions, we isolated the terms that contribute in the classical limit by the method of regions
We extracted from them the long-range classical effective potential between the two scalar particles for arbitrary D by means of the Lippmann-Schwinger equation or, equivalently, by the technique of Effective Field Theory (EFT) matching
Summary
We derive the super-classical and classical parts of the one-loop amplitude. In appendix B we employ the method of expansion by regions to evaluate the classical limit of the one-loop integrals (2.13), (2.14) and (2.15) in arbitrary dimensions D and in a generic reference frame. It should be stressed that the above result for the triangle and box contributions (2.20), (2.21) is obtained from the expansion of the corresponding integrals in the soft region, as detailed in appendix B. Such integrals receive additional contributions from the hard region that are, proportional to positive integer powers of. We refer the reader to appendix B.3 for a detailed discussion of this comparison
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