Infrared features of gravitational scattering and radiation in the eikonal approach

Abstract

Following a semi-classical eikonal approach --- justified at transplanckian energies order by order in the deflection angle ${\Theta }_s\sim \frac{4G\sqrt{s}}{b}\equiv \frac{2R}{b}$ --- we investigate the infrared features of gravitational scattering and radiation in four space-time dimensions, and we illustrate the factorization and cancellation of the infinite Coulomb phase for scattering and the eikonal resummation for radiation. As a consequence, both the eikonal phase $2\delta (E,b)$ and the gravitational-wave (GW) spectrum $\frac{\mathrm{d}{E}^{GW}}{\mathrm{d}\omega }$ are free from infrared problems in a frequency region extending from zero to (and possibly beyond) $\omega =1/R$. The infrared-singular behavior of $4$-D gravity leaves a memory in the deep infrared region ($\omega R\ll \omega b<1$) of the spectrum. At $\mathcal{O}(\omega b)$ we confirm the presence of logarithmic enhancements of the form already pointed out by Sen and collaborators on the basis of non leading corrections to soft-graviton theorems. These, however, do not contribute to the unpolarized and/or azimuthally-averaged flux. At $\mathcal{O}({\omega }^2{b}^2)$ we find instead a positive logarithmically-enhanced correction to the total flux implying an unexpected maximum of its spectrum at $\omega b\sim 0.5$. At higher orders we find subleading enhanced contributions as well, which can be resummed, and have the interpretation of a finite rescattering Coulomb phase of emitted gravitons.

arXiv comment24 pages, 7 figures; in v2 added 1 figure and several comments

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