Infrared features of gravitational scattering and radiation in the eikonal approach

Following a semi-classical eikonal approach --- justified at transplanckian energies order by order in the deflection angle $\Theta_s\sim\frac{4G\sqrt{s}}{b} \equiv \frac{2 R}{b}$ --- 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.