Complete high-frequency radiated-energy analysis

Complete the analysis of the radiated-energy spectrum for trans-Planckian massless-particle scattering by deriving the complementary \(\chi\)-collinear high-frequency limit and performing the angular integration over the \(\xi\)-collinear, \(\chi\)-collinear, and Multi-Regge emission regions.

Background

The paper proves convergence of the angular spectrum of radiated energy at leading Self-Force order in the high-frequency region collinear with the primary shockwave direction, corresponding to ξ⋅k→∞\xi\cdot k\to\infty. The proof relies on cancellations among the free and interacting waveform contributions after resumming the Post-Minkowskian expansion.

That result does not cover the opposite collinear direction, χ⋅k→∞\chi\cdot k\to\infty, nor the full angular integration. The remaining analysis must account for all relevant emission regions, including the Multi-Regge region, and is directly connected to the broader question of whether the total radiated energy remains finite in trans-Planckian massless scattering.

References

A similar derivation for \chi\cdot k \to \infty emission should, for completeness, also be carried out, as well as the angular integration which requires expansions in all the emission regions -- \xi-collinear, \chi-collinear and Multi-Regge. We leave the complete analysis for future work.

— Gravitational Radiation from Shockwave Scattering in the Self-Force EFT  (2609.09378 - Rosi, 8 Sep 2026) in Section 4, subsection “Trans-Planckian energy limit: convergence of the energy spectrum at high frequencies in the collinear region”; reiterated in Section 5, Conclusions