Infrared features of gravitational scattering and radiation in the eikonal approach

TL;DR

This work develops an infrared-finite eikonal framework for four-dimensional gravity in the transplanckian regime, showing that the infinite Coulomb phase cancels in both elastic scattering and gravitational radiation while the eikonal phase $2\delta(E,b)$ and the GW spectrum $\frac{dE^{GW}}{d\omega}$ remain well-behaved up to ${\omega\sim 1/R}$. Central to the approach is a unified single-graviton emission amplitude expressed in terms of shifted eikonal functions, enabling a consistent resummation into a coherent graviton state that preserves unitarity and yields a finite, physically meaningful radiation spectrum. The analysis identifies subleading infrared logarithms: a helicity-dependent ${\mathcal{O}}(\omega b)$ memory that cancels in unpolarized flux, and a positive ${\mathcal{O}}((\omega b)^2\log^2(\omega b))$ correction that drives a peak in the unpolarized spectrum around ${\omega b\approx 0.5}$, largely independent of the deflection angle ${\Theta_s}$ and the scale ${R}$. Numerical results corroborate the analytic predictions, showing robust agreement and revealing a peak structure and logarithmic corrections consistent with soft-graviton expectations, thereby providing a controlled view of memory effects and IR structure in gravitational bremsstrahlung within the eikonal framework.

Abstract

Following a semi-classical eikonal approach --- justified at transplanckian energies order by order in the deflection angle $Θ_s\sim\frac{4G\sqrt{s}}{b} \equiv \frac{2 R}{b}$ --- 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 $2δ(E,b)$ and the gravitational-wave (GW) spectrum $\frac{\mathrm{d}E^{GW}}{\mathrm{d}ω}$ are free from infrared problems in a frequency region extending from zero to (and possibly beyond) $ω=1/R$. The infrared-singular behavior of $4$-D gravity leaves a memory in the deep infrared region ($ωR \ll ωb < 1$) of the spectrum. At $\mathcal{O}(ωb)$ 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 $\mathcal{O}(ω^2 b^2)$ we find instead a positive logarithmically-enhanced correction to the total flux implying an unexpected maximum of its spectrum at $ωb \sim 0.5$. 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.

Figures (7)

• Figure 1: The scattering amplitude of two transplanckian particles (solid lines) in the eikonal approximation. Dashed lines represent (reggeized) graviton exchanges. The fast particles propagate on-shell throughout the whole eikonal chain. The angles ${\boldsymbol{\Theta}}_j \simeq \sum_{i=1}^{j-1} {\boldsymbol{\theta}}_i$ denote the direction of particle 1 w.r.t. the $z$-axis along the scattering process.
• Figure 2: Center-of-mass view of the collision at impact parameter ${\boldsymbol{b}}$ of particles 1 and 2 with associated emission of a graviton $q$. The polar angles $\Theta_s$ and $\theta$ are related to the 2D vectors ${\boldsymbol{\Theta}}_s$ and ${\boldsymbol{\theta}}$ as described in eq. \ref{['mompar']} and footnote \ref{['n:angles']}.
• Figure 3: Single-exchange emission diagram in ${\boldsymbol{Q}}$-space with deflection angles (a), and its transverse-space counterpart with final-state variables ${\boldsymbol{b}}$, ${\boldsymbol{x}}$ and the shifted impact parameter ${\boldsymbol{b}}-\frac{\omega}{E}{\boldsymbol{x}}$(b).
• Figure 4: The H diagram providing the first subleading correction to the eikonal phase.
• Figure 5: Picture of the polar and azimuthal angles in the transverse plane. ${\boldsymbol{\Theta}}_s$ and ${\boldsymbol{\theta}}$ correspond respectively to the projections of the unit-vectors $\hat{p}_1{}'$ and $\hat{q}$ on the $\langle x,y \rangle$-plane of fig. \ref{['f:3DimpactParam']}. In this configuration, all azimuthal angles $\phi_j$ and $\psi$ are positive.