Quantum graviton scattering with definite helicities in the null surface formulation, Part II: Third-order scattering and the exchange channels \author{C.~N.~Kozameh \and G.~O.~Depaola}

Carlos Kozameh; Gerardo Depaola

arxiv: 2605.19764 · v1 · pith:7VUDQVMFnew · submitted 2026-05-19 · ✦ hep-th

Quantum graviton scattering with definite helicities in the null surface formulation, Part II: Third-order scattering and the exchange channels author{C.~N.~Kozameh and G.~O.~Depaola}

Carlos Kozameh , Gerardo Depaola This is my paper

Pith reviewed 2026-05-20 04:22 UTC · model grok-4.3

classification ✦ hep-th

keywords graviton scatteringnull surface formulationhelicity amplitudestree-level amplitudeWick contractionsasymptotic modesexchange channelsBondi shear

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The pith

Null surface formulation derives the tree-level graviton amplitude 16πG s³/(tu) with both t- and u-channel poles from a single Wick contraction.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper computes the third-order quantum operator for the Bondi shear in the null surface formulation of general relativity, keeping definite graviton helicities. For the specific scattering process with incoming and outgoing helicities (+,-) it shows that the expectation value of the product of these operators reproduces the full tree-level amplitude including the exchange poles. Both the t-channel and u-channel poles emerge together from one Wick contraction of the NSF kernels over the sphere, without any separate propagator or diagram. The s³ dependence and the overall factor 16πG follow directly from the null-cone geometry and the Ashtekar normalization of asymptotic modes. This construction also identifies the operators as terms in the Baker-Campbell-Hausdorff expansion of the S-matrix, thereby verifying unitarity at this order.

Core claim

In the null surface formulation the third-order operators daout{3,±} are constructed recursively from the scattering equation in four helicity channels. Their contribution to the two-graviton process h⁺(K₁) + h⁻(K₂) → h⁺(K′₃) + h⁻(K′₄) yields, after a single explicit Wick contraction, the amplitude M^(33)|(+,−→+,−) = 16πG s³/(tu). The angular integral over S² remains finite by construction, no propagator is introduced, and both exchange poles appear simultaneously as a direct consequence of the null-cone geometry; the coefficient 16πG is fixed by the Ashtekar normalization of the asymptotic modes.

What carries the argument

The third-order operators daout{3,±} obtained from the recursive NSF equations, whose Wick contractions generate the helicity-specific amplitude without separate Feynman diagrams.

If this is right

Both t- and u-channel poles arise from one Wick contraction rather than two separate diagrams.
The s³ dependence and 1/(tu) pole structure are exact geometric consequences of the null-cone formulation.
Unitarity at order ε² follows structurally from the identification of the operators with the BCH expansion.
The remaining piece of the tree-level amplitude is supplied by the second-fourth order cross term M^(24) treated in Part III.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The simultaneous emergence of both exchange poles suggests that the null-surface geometry encodes crossing symmetry at the level of the kernels themselves.
Extending the same Wick-contraction procedure to higher orders could yield loop corrections whose pole structure is likewise fixed by the sphere integrals.
The approach may supply a geometric route to the factorization properties observed in celestial amplitudes for gravitons.

Load-bearing premise

The recursive NSF equations allow a consistent definition of the third-order operators that can be identified with terms in the Baker-Campbell-Hausdorff expansion of the S-matrix, and the Ashtekar normalization supplies the exact overall coefficient without further adjustment.

What would settle it

An independent helicity-projected computation of the same two-graviton tree amplitude in standard covariant perturbation theory that fails to reproduce exactly 16πG s³/(tu) with the same normalization.

read the original abstract

We compute the third-order Bondi shear $\sigma^+_3$ in the null surface formulation (NSF) of general relativity with definite graviton helicities. The quantum operator $\daout{3,\pm}$ is derived explicitly in terms of the four helicity channels (I)--(IV) of the scattering equation, and compared with the helicity-summed result of Ref.~\cite{PRL2026}. Applied to two-graviton scattering, the contribution $\langle\daout{3,+}\,\daout{3,-}\rangle$ for the process $h^+(K_1)+h^-(K_2)\to h^+(K'_3)+h^-(K'_4)$ generates simultaneously the $t$- and $u$-channel poles of the tree-level graviton amplitude. An explicit Wick-contraction calculation (Appendix~\ref{app:Wick}) shows that the NSF kernels yield \begin{equation*} \mathcal{M}^{(33)}\big|_{(+,-\to+,-)} = 16\pi G\,\frac{s^3}{tu}, \end{equation*} from first principles, with the angular integration over $S^2$ manifestly finite and no propagator introduced. The pole structure $1/(tu)$ and the $s^3$ dependence are exact consequences of the null-cone geometry; the coefficient $16\pi G$ follows from the Ashtekar normalization of the asymptotic modes, in analogy with Ref.~\cite{PRD2026partI}. Both $t$- and $u$-channel poles arise simultaneously from a single Wick contraction; in covariant perturbation theory they arise from two separate Feynman diagrams. The completion of the tree-level amplitude via $\mathcal{M}^{(24)}=\langle\daout{2}\cdot\daout{4}\rangle$ is carried out in Part~III~\cite{PRD2026partIII}. Unitarity at order $\varepsilon^2$ is verified; the operators $\daout{n,\pm}$ are identified as the terms of the Baker-Campbell-Hausdorff expansion of $S^\dagger\ain{\pm}S$, establishing unitarity as a structural consequence of the recursive NSF equations~\cite{SmatrixNSF}.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This paper pushes the null surface formulation to explicit third-order definite-helicity graviton operators and recovers the tree-level exchange amplitude from a single Wick contraction.

read the letter

This paper takes the null surface formulation to third order for gravitons with definite helicities. The key result is that the third-order operators produce the tree-level amplitude 16πG s³/(tu) for the (+,- → +,-) process through a single Wick contraction, with both exchange poles appearing together. They derive the explicit form of daout{3,±} from the four channels in the scattering equation and compare it to the summed version from their prior work. The calculation avoids standard propagators and keeps the S² integrals finite by construction from the null geometry. The overall factor follows from the usual Ashtekar normalization of the modes. This gives a different route to the amplitude that might help with higher orders or boundary conditions at null infinity. The unitarity check at order ε² is a plus, as it ties back to the recursive structure. The potential weak point is whether the recursively defined operators exactly match the third-order terms in the BCH expansion of S† a S without missing or extra pieces. The paper states the identification and builds on the earlier NSF setup, but doesn't perform an independent expansion here to cross-verify at this order. If that link holds, the result is solid; otherwise the amplitude match could be off. This is for specialists in asymptotic gravity and alternative approaches to quantum scattering. A reader looking for geometric derivations of known amplitudes will find it relevant. It deserves peer review because the computation is specific and the outcome aligns with established physics. Recommendation: Send it to referees with a request to check the operator identification against the S-matrix expansion.

Referee Report

1 major / 1 minor

Summary. The manuscript computes the third-order Bondi shear operator daout{3,±} in the null surface formulation of GR with definite graviton helicities, expressing it via the four helicity channels (I)–(IV) of the scattering equation. For the process h+(K1) + h−(K2) → h+(K′3) + h−(K′4), the vacuum expectation value ⟨daout{3,+} daout{3,−}⟩ is evaluated by explicit Wick contraction in the appendix, yielding the tree-level amplitude M^(33)|(+,−→+,−) = 16πG s³/(tu). Both t- and u-channel poles arise simultaneously from a single contraction; the s³ dependence and 1/(tu) pole structure follow from the null-cone geometry. Unitarity is verified at O(ε²), and the operators are identified with terms in the Baker-Campbell-Hausdorff expansion of S† ain S.

Significance. If the operator identification is confirmed, the result is significant because it derives the known graviton scattering amplitude from the NSF recursion without propagators or separate Feynman diagrams. Strengths include the explicit Wick-contraction calculation, the manifest finiteness of the S² angular integration, and the geometric origin of the pole structure and s³ dependence. The Ashtekar normalization supplies the coefficient 16πG without additional fitting, and unitarity emerges structurally from the recursive equations. This advances the NSF program by providing a concrete, helicity-resolved check at third order.

major comments (1)

[Main text (operator identification and unitarity section)] § on operator identification and unitarity verification (near the end of the main text, following the Wick contraction result): The manuscript states that daout{n,±} are the terms of the Baker-Campbell-Hausdorff expansion of S† ain S and that this identification follows from the recursive NSF equations. While unitarity is verified at O(ε²), no explicit third-order expansion of the S-matrix operator is performed to cross-check that the coefficients in channels (I)–(IV) exactly reproduce the BCH terms without extra contact terms or normalization adjustments. This identification is load-bearing for the claim that the computed M^(33) is the physical tree-level amplitude.

minor comments (1)

[Abstract and main text near Eq. (the amplitude result)] The appendix reference to the Wick contraction (Appendix A) is mentioned but the main text could briefly summarize the key steps of the contraction that produce the simultaneous t- and u-poles, to improve readability for readers not consulting the appendix immediately.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and for identifying the operator identification as a load-bearing claim. We address this point directly below and agree that additional clarification will strengthen the presentation.

read point-by-point responses

Referee: The manuscript states that daout{n,±} are the terms of the Baker-Campbell-Hausdorff expansion of S† ain S and that this identification follows from the recursive NSF equations. While unitarity is verified at O(ε²), no explicit third-order expansion of the S-matrix operator is performed to cross-check that the coefficients in channels (I)–(IV) exactly reproduce the BCH terms without extra contact terms or normalization adjustments. This identification is load-bearing for the claim that the computed M^(33) is the physical tree-level amplitude.

Authors: The identification follows by construction from the recursive NSF equations, which were shown in the cited reference [SmatrixNSF] to generate the unitary S-matrix operator via the BCH expansion order by order. The four helicity channels (I)–(IV) are defined by the scattering equation whose kernels are fixed by the null-surface geometry and the Ashtekar normalization; no free coefficients remain that could introduce extra contact terms. The explicit Wick contraction in Appendix A yielding exactly M^(33) = 16πG s³/(tu) with the correct pole structure therefore constitutes a non-trivial cross-check that the third-order coefficients match the BCH requirements. We acknowledge that an explicit third-order expansion of S† ain S itself is not written out in the present text. To address the referee’s concern we will add a short paragraph in the operator-identification section that recalls the recursive derivation, notes the absence of adjustable normalizations, and states that the amplitude match confirms consistency at this order. revision: yes

Circularity Check

2 steps flagged

Operator identification relies on self-citation to prior NSF recursion; coefficient from analogous prior normalization

specific steps

self citation load bearing [Abstract]
"Un itarity at order ε² is verified; the operators daout{n,±} are identified as the terms of the Baker-Campbell-Hausdorff expansion of S† ain{±}S, establishing unitarity as a structural consequence of the recursive NSF equations~[SmatrixNSF]."

The identification that allows <daout{3,+} daout{3,-}> to be read as the graviton amplitude is asserted by reference to the recursive NSF equations in the cited prior framework rather than by performing an independent third-order expansion of the S-matrix operator in this manuscript.
self citation load bearing [Abstract]
"the coefficient 16πG follows from the Ashtekar normalization of the asymptotic modes, in analogy with Ref.~[PRD2026partI]."

The overall normalization factor that converts the geometric kernel result into the standard amplitude is imported by analogy from the authors' prior Part I rather than re-derived or independently fixed within the present calculation.

full rationale

The explicit derivation of daout{3,±} from the four helicity channels of the scattering equation, followed by the Wick contraction yielding M^(33) = 16πG s³/(tu) with poles from null-cone geometry, is self-contained and produces the claimed pole structure without fitting. However, the interpretation of this quantity as the physical tree-level amplitude depends on identifying the recursively defined operators with BCH terms in S† ain S (justified only by citation to the NSF framework) and on the overall 16πG factor taken from Ashtekar normalization in analogy to prior work. This is a moderate self-citation dependence on a load-bearing interpretive step, but the central algebraic result retains independent geometric content.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the recursive structure of the null surface formulation equations and the standard Ashtekar normalization of asymptotic modes. No new free parameters or invented entities are introduced; the result follows from the geometry of null surfaces and the quantization framework established in earlier references.

axioms (2)

domain assumption The recursive NSF equations define the operators daout{n,±} consistently with the S-matrix.
Invoked when identifying the operators with the Baker-Campbell-Hausdorff expansion.
domain assumption Ashtekar normalization of asymptotic modes fixes the overall coefficient 16πG.
Used to obtain the prefactor in analogy with Part I.

pith-pipeline@v0.9.0 · 6031 in / 1551 out tokens · 64928 ms · 2026-05-20T04:22:53.349041+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

The NSF scattering equations provide, at each order n, an operator δa_out^{n,±} satisfying (δa_out^{n,±})† = δa_out^{n,±}†. ... unitarity of S to all orders
IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

angular integration over S² manifestly finite ... null-cone geometry

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

12 extracted references · 12 canonical work pages

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B. C. Hall,Lie Groups, Lie Algebras, and Representations: An Elementary Introduc- tion, 2nd ed., Graduate Texts in Mathematics222(Springer, New York, 2015)

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A. Ashtekar,Asymptotic Quantization(Bibliopolis, Naples, 1987)

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work page 2023
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C. N. Kozameh and G. O. Depaola, “Quantum graviton scattering with definite helic- ities in the null surface formulation,” Phys. Rev. D (submitted, 2026)

work page 2026
[6]

Ivancovich, C

J. Ivancovich, C. Kozameh, and E. Newman, J. Math. Phys.30, 45 (1989)

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Ultraviolet-finite perturbative expansion of quan- tum gravity at null infinity,

C. N. Kozameh and G. O. Depaola, “Ultraviolet-finite perturbative expansion of quan- tum gravity at null infinity,” Phys. Rev. Lett. (submitted, 2026)

work page 2026
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Quantum theory of asymptotically flat space- times,

C. N. Kozameh and L. Zapata-Altuna, “Quantum theory of asymptotically flat space- times,” Phys. Rev. D112, 026021 (2025)

work page 2025
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C. N. Kozameh and G. O. Depaola, “Quantum graviton scattering with definite helici- ties in the null surface formulation. Part III: Fourth-order scattering and the complete tree-level graviton-graviton amplitude,” Phys. Rev. D (submitted, 2026)

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B. C. Hall,Lie Groups, Lie Algebras, and Representations: An Elementary Introduc- tion, 2nd ed., Graduate Texts in Mathematics222(Springer, New York, 2015)

work page 2015

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M. E. Peskin and D. V. Schroeder,An Introduction to Quantum Field Theory (Addison-Wesley, Reading, MA, 1995)

work page 1995

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Ashtekar,Asymptotic Quantization(Bibliopolis, Naples, 1987)

A. Ashtekar,Asymptotic Quantization(Bibliopolis, Naples, 1987)

work page 1987

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Nonlinear scattering of classical gravitons,

M. Bordcoch, C. N. Kozameh, and T. A. Rojas, “Nonlinear scattering of classical gravitons,” Phys. Rev. D107, 104026 (2023)

work page 2023

[5] [5]

Quantum graviton scattering with definite helic- ities in the null surface formulation,

C. N. Kozameh and G. O. Depaola, “Quantum graviton scattering with definite helic- ities in the null surface formulation,” Phys. Rev. D (submitted, 2026)

work page 2026

[6] [6]

Ivancovich, C

J. Ivancovich, C. Kozameh, and E. Newman, J. Math. Phys.30, 45 (1989)

work page 1989

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Ultraviolet-finite perturbative expansion of quan- tum gravity at null infinity,

C. N. Kozameh and G. O. Depaola, “Ultraviolet-finite perturbative expansion of quan- tum gravity at null infinity,” Phys. Rev. Lett. (submitted, 2026)

work page 2026

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Quantum theory of asymptotically flat space- times,

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work page 2025

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Quantum graviton scattering with definite helici- ties in the null surface formulation. Part III: Fourth-order scattering and the complete tree-level graviton-graviton amplitude,

C. N. Kozameh and G. O. Depaola, “Quantum graviton scattering with definite helici- ties in the null surface formulation. Part III: Fourth-order scattering and the complete tree-level graviton-graviton amplitude,” Phys. Rev. D (submitted, 2026)

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S. Weinberg, inGeneral Relativity: An Einstein Centenary Survey, edited by S. W. Hawking and W. Israel (Cambridge University Press, 1979)

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BCH expansion for the gravitational S-matrix in the null surface formulation,

C. N. Kozameh and G. O. Depaola, “BCH expansion for the gravitational S-matrix in the null surface formulation,” Working notes (2026)

work page 2026

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Quantum theory of gravity. III. Applications of the covariant theory,

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