Gravitational amplitudes in the Regge limit: waveforms, shock waves and unitarity cuts

TL;DR

The paper delivers a unified Regge-theory framework for high-energy gravitational scattering, bridging quantum Regge/BFKL dynamics in the $t$-channel with classical $s$-channel multi-$H$ contributions, and translating these structures into a shock-wave formalism based on gravitational Wilson lines. By combining an exponential S-matrix approach with rapidity evolution, it yields a coherent description of both elastic and radiative amplitudes in MRK, including spin effects and Kerr-like spin multipoles. A key result is that the leading-log massive $2\to2$ amplitude at $5$PM with $2$SF matches the massless limit, and the formalism provides tree-level $2\to3$ waveforms for ultra-relativistic Kerr scattering, together with soft-theorem checks and Kerr spin extensions. The work clarifies the space-time realisation of high-energy gravity, exposes the interplay between $t$-channel BFKL dynamics and $s$-channel unitarity cuts, and sets the stage for systematic NLL analyses and waveform applications in gravitational phenomenology.

Abstract

Motivated by recent progress in the high-energy description of gravitational scattering, we develop a systematic Regge-theory framework for $2\to2+n$ amplitudes describing the scattering of two massive particles with $n$ graviton emissions, including spin effects. Working in the ultra-relativistic limit at leading logarithmic accuracy, the massive result smoothly reduces to its massless counterpart. We describe both quantum (Regge trajectory and BFKL $t$-channel evolution) and classical ($s$-channel multi-$H$ evolution) contributions using both an exponential representation of the S-matrix and a shock-wave formalism in light-cone quantisation. In the latter approach, gravitational Wilson lines evolve in rapidity space under a boost-invariant Hamiltonian, providing a space-time realisation of the high-energy dynamics and making contact with recent effective field theory descriptions in the forward limit. As an application, we compute the leading-logarithmic contribution to the massive spinless $2\to2$ amplitude at 5PM-2SF order, recovering the previously determined massless result, and derive the tree-level $2\to3$ amplitude and its associated scattering waveform for Kerr black holes in the ultra-relativistic limit.

Figures (10)

• Figure 1: Structure of the elastic amplitude $\mathcal{M}^{(0)}_{2\to2}$ in MRK, represented as two impact factors connected by a graviton line.
• Figure 2: Structure of the amplitude $\mathcal{M}^{(0)}_{2\to2+n}$ in MRK, represented as a ladder of impact factors $C_{i,j}$ connected by Lipatov currents $J^{\mu\nu}(q_r,k_r,q_{r+1})$ each emitting one graviton.
• Figure 3: Conventions for the tree-level five-point amplitude in MRK corresponding to the inelastic $2\rightarrow 3$ process involving two scalars and a graviton. All the rapidity gaps are large in this regime due to eq. \ref{['eq:MRK5pt2']}.
• Figure 4: The $H$ diagram. Vertical wavy lines denote Reggeized gravitons, and the horizontal blue line represents the on-shell soft graviton exchanged between the Lipatov currents $J$. After integrating over rapidities, the external lines contract to an effective vertex carrying the transverse momentum $q_{\perp}$, yielding the representation $\mathcal{H}_{\mathrm{GR}}(q_{\perp}^{2})$ in eq. (3.18).
• Figure 5: The BFKL ladder obtained by sewing two amplitudes $\mathcal{M}_{2\to2+n}^{(0)}$ in MRK. Vertical wavy lines denote Reggeized gravitons, while horizontal blue wavy lines represent soft graviton exchanges between Lipatov currents $J$.