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

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Federico Buccioni, Vittorio Del Duca, Emanuele Rosi

High Energy Physics - Theory · Submitted 22 Oct 2026 · 130 references

Pith's verdict: A genuinely new two-loop MRK radiative amplitude from the shockwave formalism, with a real but addressable completeness gap at the O(kappa) OPE truncation; referee it rather than desk reject.

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arXiv ID2610.31378 (version 1)
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CitationEmanuele Rosi, Federico Buccioni, Vittorio Del Duca. "The classical two-loop amplitude at leading logarithmic accuracy for graviton emission in multi-Regge kinematics." arXiv:2610.31378 (2026). https://pith.science/paper/CJOF5LAE
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The paper

The paper claims

At two loops, the classical amplitude for graviton emission in multi-Regge kinematics splits into a two-particle reducible piece and a radiative H-diagram piece, and its soft limit matches Weinberg's current times the two-body amplitude.

The reading

Two massless particles passing each other at ultrarelativistic speed radiate gravitons, and this paper computes the classical amplitude for that emission at two loops, order $G^{7/2}$, in multi-Regge kinematics, where final-state particles are strongly ordered in rapidity. The central result, eq. (4.1) evaluated in eqs. (4.15)–(4.16), is that the leading-logarithmic part of the amplitude splits into two pieces of different physical origin: a two-particle reducible term, in which a one-loop quantum correction dresses a classical graviton exchange, and a three-particle reducible term, the radiative counterpart of the $H$-diagram. The same answer is obtained as the ultrarelativistic limit of the massive five-point amplitude by applying the optical theorem, in line with the conjectured universality of leading-logarithmic classical amplitudes. The soft limit of the result reproduces the Weinberg current times the known two-loop two-body amplitude, and when the amplitude is converted into a waveform observable neither term contributes at this order, so the leading logarithms drop out of radiative observables.

What carries the argument

The load-bearing mechanism is the gravitational shockwave formalism. Each colliding scalar is a Wilson line at a fixed transverse impact parameter; the lines are expanded in reggeon fields, coherent single-graviton states, and connected by the rapidity evolution operator $e^{-\hat H L}$, whose boost Hamiltonian $\hat H = \hat R^{(1)}_{1,1} + \hat R^{(1)}_{2,2} + O(\kappa^4)$ contains the one-loop graviton Regge trajectory insertion $\hat R^{(1)}_{1,1}$ and the connected two-reggeon interaction $\hat R^{(1)}_{2,2}$ given by the gravitational BFKL kernel. The emitted graviton enters through the operator product expansion (3.13) between its creation operator $\hat a_\lambda$ and a reggeon field, whose leading term is Lipatov's central emission vertex; the amplitude is then the time-ordered correlator (3.12). The power counting of the ansatz (2.22) — one factor of $x^{-1}$ per reggeon, $x^{-4}$ from the two-body Fourier transform, and one rapidity logarithm per Hamiltonian insertion — singles out eq. (4.1) as the first classical contribution and reduces the transverse dynamics to two-loop massless three-point integrals in $d=2-2\varepsilon$ dimensions, whose eleven master integrals collapse to five and evaluate to single-valued polylogarithms in $z$.

Extended reading notes

Core claim

The paper's central claim is that equation (4.1), $$$M^{{(2,2),\lambda}}$_{2\to 3} = -L_1\big\langle \$Phi_2^{{(2)}}$\,\hat a_\$\lambda$\big[\hat $R^{{(1)}}$_{1,1}+\hat $R^{{(1)}}$_{2,2}\big]\$Phi_1^{{(2)}}$\big\rangle - L_2\big\langle \$Phi_2^{{(2)}}$\,\big[\hat $R^{{(1)}}$_{1,1}+\hat $R^{{(1)}}$_{2,2}\big]\,\hat a_\$\lambda$\,\$Phi_1^{{(2)}}$\big\rangle ,$$ with rapidity gaps $L_1\simeq\log(s_{45}/Q^2)$ and $L_2\simeq\log(s_{34}/Q^2)$, is the leading-logarithmic contribution to the classical part of the two-loop radiative five-point amplitude in the ultrarelativistic limit. Evaluated in eqs. (4.15) and (4.16), it yields explicit expressions in the complex transverse variable $z$ built from logarithms and the Bloch-Wigner dilogarithm $D_2(z,\bar z)$, with spurious collinear poles cancelling from the final answer. The amplitude splits into a two-particle reducible term, in which the one-loop graviton Regge trajectory dresses a classical exchange, and a three-particle reducible term, the radiative counterpart of the $H$-diagram built on the gravitational BFKL kernel. The authors show the same expression is obtained as the ultrarelativistic limit of the imaginary part of the massive five-point amplitude via the optical theorem, and that the soft limit factorises into the Weinberg current times the known two-loop leading-logarithmic four-point amplitude (eqs. (4.21)–(4.22)). They further state that neither the 2PR nor the 3PR term contributes to the emitted waveform in the KMOC-Cristofoli-Gonzo formalism, so the leading logarithms cancel from radiative observables.

Load-bearing premise

The load-bearing premise is that the operator product expansion for graviton emission, eq. (3.16), is complete at leading order in the gravitational coupling $\kappa$: every $O(\kappa^2)$ correction to the emission operator is deferred, explicitly, to future work, and the claim that eq. (4.1) captures the full leading-logarithmic two-loop amplitude holds only if none of those omitted terms generates additional leading logarithms at this order.

Editorial extensions

If this is right

  • The $O(G^{7/2})$ graviton-bremsstrahlung amplitude in multi-Regge kinematics is now known in closed form at leading-logarithmic accuracy: two rapidity logarithms multiplying weight-two single-valued polylogarithms in the transverse variable $z$.
  • The optical-theorem equivalence (4.3) carries the conjectured universality of leading-logarithmic classical amplitudes from elastic two-body scattering to radiative five-point processes.
  • Because neither the 2PR nor the 3PR term contributes to the emitted waveform in the KMOC-Cristofoli-Gonzo formalism, the first radiative information at this order must come from subleading logarithms.
  • The soft limit factorises into the Weinberg current times the two-loop leading-logarithmic four-point amplitude, tying the radiative result to the $H$-diagram and Regge-trajectory pieces of the elastic amplitude.
  • Each insertion of $\hat R^{(1)}_{1,1}$ or $\hat R^{(1)}_{2,2}$ produces one rapidity logarithm and one factor of $\kappa^2$, so the construction extends to higher-loop leading-logarithmic radiative amplitudes by adding Hamiltonian insertions between the emission operator and the two Wilson lines.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Because the leading logarithms cancel from the waveform, a usable $O(G^{7/2})$ radiation prediction requires the next-to-leading-logarithmic terms, and the paper's expectation of a breakdown in the massless/massive universality at that order marks it as the natural place to look for genuinely new radiative dynamics.
  • The recurrence of the same five master integrals at every loop order suggests the transverse-space evaluation can be pushed to higher accuracy with existing integral technology, with higher-weight single-valued polylogarithms in $z$ replacing the Bloch-Wigner dilogarithm.
  • If the optical-theorem equivalence is a general principle rather than an accident of this order, leading-logarithmic radiative amplitudes are fully determined by elastic Regge data — the trajectory and the BFKL kernel — plus the emission vertex, meaning bremsstrahlung adds no independent information beyond the two-body amplitude.
  • A concrete test would be to check, at the next order, whether the expected breakdown of universality shows up first in the 2PR or the 3PR sector; the answer would identify whether the Regge trajectory or the $H$-diagram dynamics is the limiting ingredient.
Original abstract

Using the gravitational shockwave formalism, we compute the classical part of the $\mathcal{O}(G^{7/2})$ five-point amplitude at leading logarithmic accuracy describing the emission of a graviton in the collision of two massless scalars in multi-Regge kinematics (MRK). The result splits into a two-particle reducible term and a three-particle reducible one, the latter being the radiative counterpart of the $H$-diagram with the emission taken in MRK limit. The same amplitude can be equivalently calculated as the ultrarelativistic limit of the corresponding massive amplitude by applying the optical theorem according to the conjectured universality of the leading logarithmic amplitudes in classical gravity.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-10-23: 1 request, 3,154 input tokens.

What the abstract itself states, as probabilities: claims first 0.04 · states limitation 0.12 · human participants 0.01 · controlled comparison 0.02 · randomized assignment 0.01 · states quantitative result 0.10 · compares with prior results 0.78 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; .

Using the gravitational shockwave formalism, we compute the classical part of the $\mathcal{O}(G^{7/2})$ five-point amplitude at leading logarithmic accuracy describing the emission of a graviton in the collision of two massless scalars in multi-Regge kinematics (MRK). The result splits into a two-particle reducible term and a three-particle reducible one, the latter being the radiative counterpart of the $H$-diagram with the emission taken in MRK limit. The same amplitude can be equivalently calculated as the ultrarelativistic limit of the corresponding massive amplitude by applying the optical theorem according to the conjectured universality of the leading logarithmic amplitudes in classical gravity.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 Using the gravitational shockwave formalism, we compute the classical part of the $\mathcal{O}(G^{7/2})$ five-point amplitude at leading logarithmic accuracy describing the emission of a graviton in the collision of two massless scalars in multi-Regge kinematics (MRK). contribution 1.00 0.96 0.56 asserted 1.00
s1 The result splits into a two-particle reducible term and a three-particle reducible one, the latter being the radiative counterpart of the $H$-diagram with the emission taken in MRK limit. result 0.92 0.86 0.16 asserted 1.00
s2 The same amplitude can be equivalently calculated as the ultrarelativistic limit of the corresponding massive amplitude by applying the optical theorem according to the conjectured universality of the leading logarithmic amplitudes in classical gravity. method 0.73 0.36 0.18 asserted 0.67

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 0.97.

What was examined, and what was not

sources read
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not examined
  • full text
  • figures and tables
  • references and citing sentences
  • relations to other papers
levels run
  • P abstract
not run
  • S sections
  • T body sentences
  • C citations
  • R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

Figures
Figure 1. Kinematics of the massless five-point amplitude with all outgoing convention. The average momenta of the massless and spinless particles are named pi , while qi their difference. We use the double wavy line to describe a graviton in the multi-Regge region. 2.2 Multi-Regge limit Multi-Regge kinematics isolates the high-energy regime in which emissions are strongly or￾dered in rapidity, while their transverse momenta remain comparable. For the process (2.1), it corresponds to the ordering P + 5 ≫ P + 4 ≫ P + 3 , P − 5 ≪ P − 4 ≪ P − 3 , |P3⊥| ∼ |P4⊥| ∼ |P5⊥| , (2.8) or, equi
Figure 2. The five-point amplitude Mλ 2→3 obtained from the shockwave formalism (3.12). At fixed order in perturbation theory, the blue Wilson line interacts with the gravitational field through N1 reggeon fields, which evolve via Hˆ ⊃ RˆN1,N2 into N2 reggeons; afterwards, one selects the emission aˆ λ ·WN2 → WN3 via the operator product expansion; the produced reggeons interact again via the Hamiltonian RˆN3,N4 to regularise the rapidity divergences, and finally the remaining N4 reggeons are absorbed by the red Wilson line. The operator Rˆ (1) 2,2 is more involved, and reads Rˆ (1) 2,2 = − κ…
Figure 3. The figure above is a graphical representation of the amplitudes in equation (4.1), specifically those in the first line, in which the operators Rˆ (1) i,j act by evolving the upper Wilson line in rapidity. These diagrams are proportional to L1. Diagrams such as (A) are obtained by evolving a single reggeon at a time via Rˆ (1) 1,1 , whereas diagram (B) evolves, irreducibly, the pair of reggeons emitted by the upper particle. Similarly, the second line of equation (4.1) is obtained by applying TP, that is, by flipping these diagrams upside down. In particular, we see that diagram (

Figures reproduced from arXiv:2610.31378 by the authors.

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