Soft gravitational radiation from ultra-relativistic collisions at sub- and sub-sub-leading order

Using soft-graviton theorems a well-known zero-frequency limit (ZFL) for the gravitational radiation flux dE$^{GW}$/dω is re-derived and extended to order $ \mathcal{O}\left(\omega \right) $ and $ \mathcal{O}\left({\omega}^2\right) $ for arbitrary massless multi-particle collisions. The (angle-integrated, unpolarized) $ \mathcal{O}\left(\omega \right) $ correction to the flux turns out to be absent in the case of two-particle elastic collisions. The $ \mathcal{O}\left({\omega}^2\right) $ correction is instead non-vanishing and takes a simple general expression which is then applied to bremsstrahlung from two-particle elastic collisions. For a tree-level process the outcome is finite and consistent with expectations. Instead, if the tree-level form of the soft theorems is used at sub-sub-leading order even when the elastic amplitude needs an all-loop (eikonal) resummation, an unphysical infrared singularity occurs. Its origin can be traced to the infinite Coulomb phase of gravitational scattering in four dimensions. We briefly discuss how to get rid, in principle, of the unwanted divergences and indicate -without carrying out- a possible procedure to find the proper correction to the naive soft theorems. Nevertheless, if a simple recipe recently proposed for handling these divergences is adopted, we find surprisingly good agreement with results obtained independently via the eikonal approach to transplanckian-energy scattering at large (small) impact parameter (deflection angle), where such Coulomb divergences explicitly cancel out.