⚛️ high-energy theory

The classical two-loop amplitude at leading logarithmic accuracy for graviton emission in multi-Regge kinematics

Using the gravitational shockwave formalism, this paper computes the classical part of the $\mathcal{O}(G^{7/2})$ five-point amplitude for graviton emission in the collision of two massless scalars within multi-Regge kinematics, decomposing the result into two- and three-particle reducible terms and verifying its consistency with the ultrarelativistic limit of massive amplitudes via the optical theorem.

Original authors: Federico Buccioni, Vittorio Del Duca, Emanuele Rosi

Published 2026-10-23

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CC BY 4.0

Original authors: Federico Buccioni, Vittorio Del Duca, Emanuele Rosi

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Gravity is the force that holds the universe together, yet it remains the most elusive of nature's fundamental interactions. While we can easily observe its effects on falling apples or orbiting planets, understanding how gravity behaves when two massive objects crash into each other at nearly the speed of light has long been a theoretical nightmare. This extreme scenario, where objects move so fast that time and space themselves seem to warp in complex ways, is crucial for interpreting the ripples in spacetime known as gravitational waves. These waves, detected by observatories on Earth, carry the fingerprints of such violent collisions. To decode them, scientists need precise mathematical descriptions of what happens when two objects scatter off one another, a task that requires peeling back layers of complexity that standard physics equations struggle to handle.

In this new work, a team of researchers has taken a significant step forward by calculating a specific, high-level interaction between two massless particles that emit a graviton, the theoretical particle that carries the force of gravity. They focused on a particular high-speed regime where the particles move in a way that simplifies the chaos of their collision, allowing them to isolate the most dominant effects. By using a sophisticated mathematical framework that treats the particles as lines moving through a distorted space, the team was able to compute the classical part of this interaction at a level of detail previously unattainable. Their result reveals that the radiation emitted during such a collision is not a single, uniform event but is composed of two distinct physical structures, one of which acts as a radiative version of a famous diagram used to describe gravitational deflection.

The researchers began by setting up a scenario involving two massless particles, which can be thought of as idealized, ultra-fast objects, colliding and emitting a single graviton. In the real world, these particles might represent the extreme conditions found just before two black holes merge, or the behavior of light-speed particles in the early universe. The challenge lies in the sheer number of variables involved; as the particles interact, they exchange energy and momentum in ways that create an infinite number of possible outcomes. To make sense of this, the team employed a technique called the shockwave formalism. This approach views the fast-moving particles as creating a shockwave in the fabric of spacetime, similar to a sonic boom created by a supersonic jet, but in the gravitational field. This shockwave allows the complex interaction to be broken down into simpler, manageable pieces that can be calculated step by step.

Using this method, the team calculated the probability of this specific event occurring at a very high level of precision, known as the two-loop level. In the language of physics, this means they accounted for the effects of virtual particles popping in and out of existence during the collision, a necessary step to get an accurate picture of the classical behavior of gravity. They found that the total result splits into two distinct parts. The first part is a "two-particle reducible" term, which essentially describes a quantum correction layered on top of a classical exchange. This part, while mathematically necessary, does not contribute to the actual classical data observed in scattering experiments. The second part is a "three-particle reducible" term, which is the true star of the show. This component represents the radiative counterpart to a well-known diagram called the H-diagram, which has been studied for decades as a way to understand how objects deflect each other under gravity.

The significance of this finding lies in the nature of that second term. It carries the classical radiative information, meaning it describes the theoretical gravitational radiation emitted during the collision. However, the researchers point out a crucial nuance: when applying the KMOC formalism to translate these amplitudes into observable waveforms, both the two-particle and three-particle reducible contributions to this specific leading-logarithmic amplitude actually yield a null contribution to the final emitted waveform. They demonstrated that this result can be derived in two different ways: either by using the shockwave formalism directly or by looking at the collision of massive particles and taking the limit where they become massless. This dual confirmation strengthens the idea that the laws governing these high-speed interactions are universal, regardless of whether the colliding objects have mass or not, at this specific level of approximation.

However, the authors are careful to note the limits of their discovery. Their calculation is strictly accurate only for the leading logarithmic terms, which are the most dominant effects in this high-energy regime. They point out that the next level of detail, which would include smaller corrections, is much harder to calculate and might not follow the same universal rules. In fact, they suggest that the elegant universality they found might break down when looking at these finer details, particularly when trying to predict the exact shape of the gravitational waves emitted by massive objects like black holes. This implies that while their work provides a solid foundation for understanding the broad strokes of high-speed gravitational scattering, the full picture of gravitational wave emission will require even more complex tools and methods to capture the subtle nuances that lie beyond the leading effects.

Ultimately, this paper provides a clearer map of the gravitational landscape at extreme speeds. By isolating the classical part of the interaction and showing how it relates to established diagrams like the H-diagram, the researchers have offered a new tool for interpreting the data from gravitational wave detectors. Their work confirms that even in the most violent and relativistic collisions, the underlying structure of gravity retains a surprising order, governed by specific mathematical patterns that can be unraveled with the right theoretical framework. This clarity is essential for the next generation of gravitational wave astronomy, where distinguishing the subtle signals of merging black holes from the background noise of the universe will depend on having the most precise predictions possible.

Technical Summary: The Classical Two-Loop Amplitude at Leading Logarithmic Accuracy for Graviton Emission in Multi-Regge Kinematics

Problem Statement
This work addresses the calculation of the classical part of the five-point amplitude describing the emission of a graviton in the collision of two massless scalar particles within Einstein-Hilbert gravity. The study focuses on the Multi-Regge Kinematics (MRK) limit, where emissions are strongly ordered in rapidity while transverse momenta remain comparable. Specifically, the authors compute the amplitude at order $O(G^{7/2})$ (corresponding to two loops in the perturbative expansion) at leading logarithmic (LL) accuracy. This calculation is motivated by the need to understand the ultrarelativistic limit of the two-body problem, where the Post-Minkowskian (PM) expansion requires resummation due to the non-commutativity of the high-energy ( $\gamma \to \infty$ ) and weak-field ( $G \to 0$ ) limits. While the conservative scattering angle is known up to 4PM, and the radiative waveform is known at one-loop, the classical two-loop radiative amplitude in the high-energy regime had not been explicitly derived in this formalism.

Methodology
The authors employ the gravitational shockwave formalism, a framework recently transposed from QCD to gravity. The methodology relies on the following key components:

Shockwave and Wilson Lines: The two colliding massless particles are modeled as Wilson lines (coherent states of gravitons) moving along straight-line trajectories. The interaction is described by the time-ordered correlator of these Wilson lines.
Rapidity Evolution: To handle the rapidity divergences inherent in the high-energy limit, the formalism introduces a rapidity evolution equation governed by a boost Hamiltonian $\hat{H}$ . This Hamiltonian evolves the Wilson lines through the rapidity gap $L \sim \log(s/t)$ .
Reggeon Fields: The Wilson lines are expanded in terms of "reggeon" fields ( $\tilde{W}$ ), which represent the instantaneous interactions on the shockwave front. The evolution of these fields is controlled by operators $\hat{R}^{(m)}_{i,j}$ , which describe the interaction of $j$ reggeons evolving into $i$ reggeons at $m$ loops.
Operator Product Expansion (OPE): The emission of the real graviton is encoded via the OPE between the graviton creation operator $\hat{a}_\lambda(k)$ and the reggeon fields. At leading order, this insertion corresponds to the gravitational Lipatov Central Emission Vertex (CEV).
Kinematic Parametrization: The calculation utilizes a specific parametrization of the five-point kinematics in the MRK limit, characterized by a small parameter $x \to 0$ . The transverse dynamics are reduced to a two-dimensional Euclidean space ( $d=2-2\epsilon$ ) and parametrized by complex variables $z$ and $\bar{z}$ .
Integral Reduction: The resulting amplitude involves two-loop three-point integrals in the transverse plane. These are reduced to a set of 11 Master Integrals (MIs) using Integration-by-Parts (IBP) identities and further reduced to 5 independent MIs using the permutation symmetry group $S_3$ . The solutions are expressed in terms of single-valued polylogarithms (SVPLs).

Key Contributions and Results
The paper derives the classical two-loop five-point amplitude, $M^{(2,2),\lambda}_{2\to3}$ , at leading logarithmic accuracy. The result is decomposed into two distinct physical structures:

Two-Particle Reducible (2PR) Term: This term arises from the evolution of a single reggeon via the one-loop Regge trajectory operator $\hat{R}^{(1)}_{1,1}$ . It represents the convolution of a quantum correction (the Regge trajectory) with a classical exchange. The authors note that this term does not contribute to classical scattering observables directly.
Three-Particle Reducible (3PR) Term: This term arises from the connected evolution of the two-reggeon state via the gravitational BFKL kernel operator $\hat{R}^{(1)}_{2,2}$ . This structure is identified as the radiative counterpart of the famous "H-diagram" (the first correction to massless scattering at $O(G^3)$ ), with the graviton emission constrained to the MRK region.

Explicit Results:

The amplitude is expressed in terms of the rapidity gaps $L_1$ and $L_2$ and the transverse variables $z, \bar{z}$ .
The authors provide explicit expressions for the coefficients $A^\oplus_{2PR}$ and $A^\oplus_{3PR}$ (Equations 4.15 and 4.16), which involve poles in $\epsilon$ (dimensional regularization) and single-valued functions including the Bloch-Wigner dilogarithm $D_2(z, \bar{z})$ .
The result satisfies the Weinberg soft factorization theorem. In the limit where the emitted graviton becomes soft, the amplitude correctly factorizes into the product of the classical two-loop four-point conservative amplitude and the Weinberg soft current.
The authors demonstrate that the same result can be obtained by taking the ultrarelativistic limit of the massive classical amplitude using the optical theorem, supporting the conjectured universality of leading logarithmic amplitudes in classical gravity.

Significance and Claims
The paper claims to provide the first derivation of the classical two-loop radiative amplitude in the high-energy limit using the shockwave formalism. Its significance lies in:

Extending the Shockwave Formalism: It successfully applies the formalism to radiative processes (graviton emission) at two loops, moving beyond the tree-level and one-loop results previously established.
Universality Check: It reinforces the connection between massless and massive scattering in the ultrarelativistic limit, showing that the leading logarithmic classical amplitude for massive scalars converges to the massless result derived here.
Structural Insight: It clarifies the separation between 2PR and 3PR contributions, identifying the 3PR term as the radiative generalization of the H-diagram.

Limitations and Modest Claims
The authors are explicit about the scope and limitations of their work:

Leading Logarithmic Accuracy: The result is strictly valid at leading logarithmic accuracy in the MRK parameter $x$ . The authors state that reconstructing subleading logarithms via ansatz is impractical due to the complexity of the five-point parameter space.
Classical vs. Quantum: At one-loop, the formalism yields only superclassical or quantum terms; the classical contribution only appears at two loops.
Waveform Contribution: Citing the KMOC-Cristofoli-Gonzo formalism, the authors conclude that both the 2PR and 3PR contributions to the leading logarithmic amplitude yield a null contribution to the emitted gravitational waveform. Consequently, a next-to-leading treatment is necessary to capture physical waveform effects.
Universality Breakdown: They expect the universality between massless and massive amplitudes to break down beyond the leading logarithmic approximation, implying that different methods may be required for massive-massive scattering waveforms at high velocities.

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