Abstract
Using mathcal{N} = 8 supergravity as a theoretical laboratory, we extract the 3PM gravitational eikonal for two colliding massive scalars from the classical limit of the corresponding elastic two-loop amplitude. We employ the eikonal phase to obtain the physical deflection angle and to show how its non-relativistic (NR) and ultra-relativistic (UR) regimes are smoothly connected. Such a smooth interpolation rests on keeping contributions to the loop integrals originating from the full soft region, rather than restricting it to its potential sub-region. This task is efficiently carried out by using the method of differential equations with complete near-static boundary conditions. In contrast to the potential-region result, the physical deflection angle includes radiation-reaction contributions that are essential for recovering the finite and universal UR limit implied by general analyticity and crossing arguments. We finally discuss the real emission of massless states, which accounts for the imaginary part of the 3PM eikonal and for the dissipation of energy-momentum. Adopting a direct approach based on unitarity and on the classical limit of the inelastic tree-level amplitude, we are able to treat mathcal{N} = 8 and General Relativity on the same footing, and to complete the conservative 3PM eikonal in Einstein’s gravity by the addition of the radiation-reaction contribution. We also show how this approach can be used to compute waveforms, as well as the differential and integrated spectra, for the different radiated massless fields.
Highlights
Struction is concerned, especially in the presence of supersymmetry, while systematic methods based on differential equations and integration-by-parts (IBP) identities automatise to a great extent the evaluation of the Feynman integrals
In contrast to the potential-region result, the physical deflection angle includes radiation-reaction contributions that are essential for recovering the finite and universal UR limit implied by general analyticity and crossing arguments
Following [13], but performing the calculation of the various loop diagrams taking into account the full soft region — rather its potential part — we computed the 2 → 2 scattering amplitude and the associated eikonal Re 2δ2 in massive N = 8 supergravity at 3PM level and we found that the resulting deflection angle is free of log(s)-divergences, restoring agreement with [32] and the universality in the ultrarelativistic limit
Summary
The first inequality ensures that quantum effects be negligible, i.e. suppressed by the ratio between the Compton wavelength and the size of the objects under consideration, while the last inequality allows us to focus on a regime of weak gravitational interactions, i.e. of near-forward scattering In this near-forward limit, namely for small momentum transfer q2 → 0, each of the above contributions to the amplitude in eq (2.19) can be schematically expanded as follows. The most natural interpretation of the singular, super-classical terms is that such contributions arise from the formal small-G expansion of an exponential of the type e2iδ that oscillates infinitely rapidly in the classical limit, where its phase Re 2δ becomes large To see that this is the case, one needs first to rewrite the amplitude in terms of the (D − 2)-dimensional impact-parameter bμ according to. This is the basic relation that we are going to employ in the following in order to retrieve the scattering angle χ from the eikonal
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