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

Andrea Addazi, Massimo Bianchi, Gabriele Veneziano

TL;DR

The work connects soft-graviton theorems with the spectrum of gravitational radiation in ultra-relativistic collisions by extending the zero-frequency limit to ${ m O}()$ and ${ m O}()^2$, revealing that the ${ m O}()$ term vanishes for two-body elastic scattering while the ${ m O}()^2$ term yields a compact, covariant expression. It confirms the tree-level finite results and highlights IR subtleties in $D=4$ gravity, where naive soft-theorem extrapolations require modifications (e.g., Coulomb-phase regularization) to match eikonal analyses. The paper also verifies its predictions against ${ m N}=8$ supergravity amplitudes and discusses how IR-safe prescriptions can reconcile theory with known high-energy scattering results. These findings illuminate how soft-gravity theorems interface with classical radiation and transplanckian scattering, providing a framework for including subleading radiative corrections in gravitational bremsstrahlung. The results have implications for precision modeling of gravitational radiation in high-energy collisions and deepen the bridge between quantum soft theorems and classical gravitational wave physics.

Abstract

Using soft-graviton theorems a well-known zero-frequency limit (ZFL) for the gravitational radiation flux $dE^{GW}/d ω$ is re-derived and extended to order ${\cal O}(ω)$ and ${\cal O}(ω^2)$ for arbitrary massless multi-particle collisions. The (angle-integrated, unpolarized) ${\cal O}(ω)$ correction to the flux turns out to be absent in the case of two-particle elastic collisions. The ${\cal O}(ω^2)$ 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.

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

TL;DR

The work connects soft-graviton theorems with the spectrum of gravitational radiation in ultra-relativistic collisions by extending the zero-frequency limit to and , revealing that the term vanishes for two-body elastic scattering while the term yields a compact, covariant expression. It confirms the tree-level finite results and highlights IR subtleties in gravity, where naive soft-theorem extrapolations require modifications (e.g., Coulomb-phase regularization) to match eikonal analyses. The paper also verifies its predictions against supergravity amplitudes and discusses how IR-safe prescriptions can reconcile theory with known high-energy scattering results. These findings illuminate how soft-gravity theorems interface with classical radiation and transplanckian scattering, providing a framework for including subleading radiative corrections in gravitational bremsstrahlung. The results have implications for precision modeling of gravitational radiation in high-energy collisions and deepen the bridge between quantum soft theorems and classical gravitational wave physics.

Abstract

Using soft-graviton theorems a well-known zero-frequency limit (ZFL) for the gravitational radiation flux is re-derived and extended to order and for arbitrary massless multi-particle collisions. The (angle-integrated, unpolarized) correction to the flux turns out to be absent in the case of two-particle elastic collisions. The 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.

Paper Structure

This paper contains 16 sections, 190 equations, 1 figure.

Table of Contents

  1. Introduction
  2. The GW spectrum at leading order
  3. The ${{\cal O}}(\omega)$ correction to the GW spectrum
  4. Covariant calculation of $B_1$
  5. Breit-frame argument for $B_1 =0$ in two-body scattering
  6. Covariant argument for $B_1=0$ in two-body scattering
  7. The ${{\cal O}}(\omega^2)$ correction for a generic massless process
  8. Specializing to two-body collision processes
  9. General covariant result for $2 \rightarrow 2$ scattering
  10. Tree-level gravitational scattering
  11. Gravitational scattering in the leading eikonal approximation
  12. Elimination of (and finite terms from) infrared divergences
  13. Summary and outlook
  14. Computation of the "shift terms" arising in the ${{\cal O}}(\omega^{2})$ corrections
  15. Non covariant calculation of $P_{\mu} P_{\nu} \tilde{M}^{\mu \nu}$
  16. ...and 1 more sections

Figures (1)

  • Figure 1: The expression ${{f}}(x)$ appearing in the $B_{2}$ expression is displayed as a function of the cosine of the scattering angle $x=\cos \theta_{s}$.