Unified limiting form of graviton radiation at extreme energies

TL;DR

This work analyzes graviton radiation in four-dimensional transplanckian gravitational scattering at $E\gg M_P$ and small deflection, using a resummed eikonal $S$-matrix in impact-parameter space. The authors derive a unified limiting form for single-exchange graviton emission that smoothly connects soft and Regge limits via helicity-phase transformation laws, then extend to multi-graviton emission with a $b$-space factorization and coherent-state resummation. Including rescattering corrections yields a classical limit in agreement with recent results and reveals a Hawking-like, soft spectrum with characteristic energy $\hbar/R$, suppressed by coherence effects. The framework provides a coherent, quantum-consistent picture of radiation during near-collision transplanckian scattering and suggests a path toward understanding information transfer in the approach to gravitational collapse.

Abstract

We derive the limiting form of graviton radiation in gravitational scattering at transplanckian energies ($E\gg M_P$) and small deflection angles. We show that --- owing to the graviton's spin 2 --- such limiting form unifies the soft- and Regge- regimes of emission, by covering a broad angular range, from forward fragmentation to deeply central region. The single-exchange emission amplitudes have a nice expression in terms of the transformation phases of helicity amplitudes under rotations. As a result, the multiple-exchange emission amplitudes can be resummed via an impact parameter $b$-space factorization theorem that takes into account all coherence effects. We then see the emergence of an energy spectrum of the emitted radiation which, being tuned on $\hbar/R \sim M_P^2/E \ll M_P$, is reminiscent of Hawking's radiation. Such a spectrum is much softer than the one naïvely expected for increasing input energies and neatly solves a potential energy crisis. Furthermore, by including rescattering corrections in the (quantum) factorization formula, we are able to recover the classical limit and to find the corresponding quantum corrections. Perspectives for the extrapolation of such limiting radiation towards the classical collapse regime (where $b$ is of the order of the gravitational radius $R$) are also discussed.

Figures (15)

• 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: Picture and notation of generic exchange emission in (a) the soft and (b) the Regge limits.
• Figure 3: Schematics of dynamical regimes of graviton emission in transplanckian scattering with single-exchange (here $\hbar=1$). $\Delta_c$ is a parameter used in sec.\ref{['s:msarl']}.
• Figure 4: Double-exchange process. The angles describing the direction of the right-moving fast particle during the scattering process are shown on top of the upper line.
• Figure 5: Diagrammatic picture for the emission of a graviton from two fast-particle scattering in the Regge limit. The blob in the middle represents the Lipatov vertex, i.e., the effective reggeon-reggeon-graviton coupling.