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

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

Problem and Motivation
The paper addresses the theoretical description of gravitational scattering in the ultra-relativistic regime, specifically where the center-of-mass energy $\sqrt{s}$ is much larger than the momentum transfer $\sqrt{|t|}$ and the particle masses ($s \gg |t|, m^2$). While Post-Minkowskian (PM) and Self-Force (SF) expansions provide complementary perturbative descriptions of two-body dynamics, they encounter limitations in the high-energy limit. Fixed-order expansions fail to capture the dynamics of highly relativistic encounters, particularly where large logarithms of the form $\log(s/|t|)$ dominate. These logarithms arise from specific diagrammatic structures, such as the "H-diagram" (involving soft graviton exchanges across eikonal ladders) and multi-H diagrams, which scale as $O(Gs^2(G^2 s \log(s/|t|))^N)$. The authors aim to develop a systematic framework to organize these high-energy expansions for $2 \to 2$ and $2 \to 2+n$ amplitudes, including spin effects, to recover known massless results and extend them to massive spinning particles (Kerr black holes).

Methodology
The authors employ a unified framework combining Regge theory and shock-wave formalism in light-cone quantization. The methodology proceeds through several key steps:

1. Multi-Regge Kinematics (MRK) and Factorization: The scattering process is analyzed in MRK, characterized by strong ordering of rapidities ($y_0 \gg y_1 \gg \dots \gg y_{n+1}$) and comparable transverse momenta. In this limit, amplitudes factorize into impact factors (coupling external particles to the $t$-channel) and Lipatov currents (central emission vertices). The paper establishes this factorization for massive scalars, Kerr black holes, and gravitons.
2. Regge Theory and the $\hat{N}$ Operator: The $S$-matrix is reformulated in an exponential representation, $\hat{S} = \exp(i\hat{N}/\hbar)$, where $\hat{N}$ contains only two-massive-particle-irreducible (2MPI) contributions. The authors construct generating functionals for $\hat{N}$ matrix elements in MRK. This allows for the systematic iteration of tree-level amplitudes via unitarity cuts:
   • $t$-channel iteration: Leads to the gravitational BFKL equation, describing the evolution of Reggeized gravitons. The authors note that the Regge trajectory $\alpha(q_\perp)$ introduces purely quantum corrections (scaling with $\hbar$) and does not contribute to classical observables in the limit $\hbar \to 0$.
   • $s$-channel iteration: Corresponds to the "multi-H" diagrams (iterated three-particle unitarity cuts). These diagrams generate the classical leading logarithmic contributions scaling as $O(Gs^2(G^2 s \log(s/|t|))^N)$.
3. Shock-Wave Formalism: The authors adapt the QCD shock-wave formalism to gravity. Highly boosted states are represented by gravitational Wilson lines. The evolution of these lines in rapidity space is governed by a boost-invariant Hamiltonian $\hat{H}$.
   • The Hamiltonian consists of a Regge trajectory term ($\hat{R}_1$) and a boost term ($\hat{R}_2$) involving the gravitational kernel $H_{GR}$.
   • The authors demonstrate that the perturbative expansion of the shock-wave amplitude reproduces the Regge-theory results, including the BFKL ladder and the multi-H diagrams.
   • Crucially, they show that the gravitational Wilson lines and the boost Hamiltonian break the $SL(2, \mathbb{C})$ conformal invariance present in QCD due to the dimensionful nature of the gravitational coupling $\kappa$.
4. Waveform Computation: Using the shock-wave formalism, the authors compute the tree-level $2 \to 3$ amplitude and extract the associated scattering waveform in the ultra-relativistic limit. They incorporate spin effects by utilizing the impact factors for Kerr black holes, which relate to the Newman-Janis shift of the impact parameter.

Key Contributions and Results

• Unified Framework: The paper provides a systematic Regge-theory framework for $2 \to 2+n$ amplitudes with $n$ graviton emissions, valid for both massive and massless particles and including spin effects.
• Massive to Massless Reduction: It is shown that at the 5PM order in the 2SF (second self-force) sector, the leading-logarithmic contribution to the massive $2 \to 2$ amplitude smoothly reduces to the previously determined massless result. This confirms the universality of leading logarithms in the Regge limit.
• Shock-Wave Equivalence: The authors explicitly derive the gravitational BFKL equation and the tower of classical multi-H diagrams using the shock-wave formalism. They identify the boost Hamiltonian as the generator of rapidity evolution, providing a space-time realization of high-energy dynamics.
• Spin-Dependent Waveforms: For the first time, the formalism is extended to include spin effects in the computation of scattering waveforms. The authors derive the tree-level waveform for Kerr black holes in the ultra-relativistic limit. The result is obtained by applying a Newman-Janis shift to the impact parameter of the spinless waveform.
• Soft Theorem Recovery: The computed waveforms are shown to reproduce the high-energy limits of the leading (Weinberg) and subleading (logarithmic) soft graviton theorems, providing a non-trivial consistency check.
• Conformal Symmetry Breaking: The paper explicitly demonstrates that, unlike in QCD, the gravitational Regge evolution is not conformally invariant in the transverse plane due to the dimensionful coupling constant.

Significance
The paper claims significance in establishing a bridge between different theoretical approaches to high-energy gravity. By unifying Regge theory, unitarity cuts, and shock-wave methods, it offers a coherent picture of how classical gravitational observables (like waveforms and impulses) emerge from the high-energy limit of quantum amplitudes. The ability to recover the massless limit from massive calculations and to incorporate spin effects into the ultra-relativistic waveform provides a robust tool for understanding the dynamics of compact binary systems in regimes where fixed-order perturbation theory breaks down. The work also clarifies the distinction between classical and quantum contributions in the high-energy expansion, specifically isolating the classical multi-H diagrams from the quantum BFKL evolution.