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REVIEW 3 major objections 4 minor 42 references

Quantum graviton scattering with definite helicities in the null surface formulation. III: Fourth-order recursion and ultraviolet finiteness

T0 review · 3 major / 4 minor · reviewed 2026-08-02 · deepseek-v4-flash

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Pith's one-line read Fourth-order recursion completes the graviton tree amplitude

desk verdict The paper has a genuinely new structural idea—the generalized Jordan–Pauli relation with advanced-cone corrections—but the tree-level amplitude is not actually derived: the corrections are left uncomputed and the final partial-fraction check is algebraically wrong. read the letter →

arxiv 2605.24512 v2 pith:B6UOBTGP submitted 2026-05-23 hep-th

C. N. Kozameh , G. O. Depaola This is my paper · ORCID
classification hep-th PACS 04.60.-m
keywords nullsurfaceformulationgravitonscatteringtree-levelamplitudeultravioletfinitenessasymptoticshearloopcountinghelicityamplitudesunitarity
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper completes a three-part program that computes quantum graviton scattering directly from null-cone data at past and future infinity, without off-shell propagators or loop-momentum integrals. It derives the fourth-order correction to the outgoing shear and shows that the tree-level 2→2 amplitude is exhausted by contributions from the second-, third-, and fourth-order fields; after imposing four-momentum conservation externally, those three pieces add to the standard result −κ²s³/(4tu). It also proves a loop-counting formula, L=(n1+n2−6)/2, which says no field beyond fourth order contributes at tree level, and a power-counting theorem, K(n)∼ω_ext/q^{n−2}, which makes the L-loop integrand scale as dq/q^{4L}. If these hold, the expansion is ultraviolet finite at every loop order without regularization. A sympathetic reader would care because the result points to a version of perturbative quantum gravity in which UV divergences never arise, and in which the S-matrix is recursively determined by a single free datum.

What carries the argument

The generalized retarded-advanced cone matching equation — the master equation — is the central object. It equates the advanced and retarded null-cone integrals with the linear cut term, after the free-field parts cancel between the two cones; the surviving terms are the advanced-cone corrections from previously determined scattering fields. The same equation yields the kernel-scaling relation K(n)∼ω_ext/q^{n−2} inductively, while the operator-counting identity L=(n1+n2−6)/2 classifies loop order. The inverse of the angular derivative operator on the sphere converts cone sources into shear coefficients and produces the denominators that become s, t, u on shell.

What would settle it

Evaluate, at order n=4, the sum of scattering-correction cut terms Σ_{j=2}^{3} Z+_{4,cut}|_{δσ+_j} in the master equation; if it is not a boundary term that vanishes after integration over the sphere, then the master equation misses contributions to σ+_4 and the tree-level identification fails. Alternatively, compute the one-loop (δaout_4)^2 integrand in the uniform radial UV region and check whether it scales as dq/q^4 rather than dq/q^{4L}; a mismatch would falsify the all-orders UV-finiteness theorem.

Watch

Extended reading notes

Core claim

The central claim is that the perturbative null-surface formulation determines every coefficient of the outgoing shear from the free incoming datum, and at fourth order the computation closes the tree-level 2→2 graviton amplitude. Concretely, σ+_4 is obtained by inverting the second-order angular derivative operator on the sphere, using three retarded cone pairs and, for the first time, advanced-cone corrections built from the nonzero second- and third-order scattering fields; the full kernel is twice the retarded contributions plus those advanced corrections. The resulting matrix element M(24), together with M(22) and M(33), equals −κ²s³/(4tu) after the kinematic substitution that maps the

Load-bearing premise

The load-bearing premise is that all scattering-correction cut terms, built from the earlier nontrivial fields δσ+_j, cancel as boundary or total-derivative terms in the master equation; if that cancellation is not exact, σ+_4 picks up additional contributions and the computed kernels, and hence the tree-level identification, are incomplete.

Editorial extensions

If this is right

  • Tree-level 2→2 graviton scattering is fully accounted for by the second-, third-, and fourth-order shear fields; no higher-order field can contribute at zero loop order.
  • Imposing four-momentum conservation externally on the null-cone denominators reproduces the standard amplitude −κ²s³/(4tu), with the three pole channels s, t, u individually identified.
  • Every L-loop contribution to 2→2 scattering scales as ∫ dq/q^{4L}, so the ultraviolet part vanishes as the regulator is removed and convergence improves with loop order; no renormalization is needed.
  • A complete one-loop amplitude requires the fifth- and sixth-order shear fields in addition to the fourth; this paper identifies the required operator sectors but leaves those fields for future work.
  • The S-matrix is unitary order by order, because each order's generator is Hermitian as a consequence of the real kernels and the exponential expansion of the unitary operator.

Reading between the lines

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

  • If the cancellation of scattering-correction cut terms holds at every order, the entire perturbative S-matrix is fixed by the single free incoming datum at past null infinity; no additional asymptotic input would be needed at any loop order.
  • The null-denominator structure that produces UV convergence is also what generates soft limits; checking whether the recursion reproduces the universal soft-graviton factor order by order would be a direct test of the framework beyond the tree amplitude.
  • Because all internal gravitons are on-shell and fixed by frequency deltas, the expansion reorganizes the amplitude as a sum over causal histories; evaluating the one-loop (δaout_4)^2 integrand explicitly would test the claimed dq/q^4 scaling before the missing fifth- and sixth-order fields are computed.
  • The paper uses flat null cones; if the cones are replaced by the self-corrected ones determined by the recursion, the inductive denominator growth could change, which would alter the UV power counting — a direction the paper itself names as future work.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 4 minor

Summary. This third paper in a trilogy claims to complete the null-surface formulation (NSF) computation of the tree-level 2→2 graviton scattering amplitude by determining the fourth-order Bondi shear σ_4^+. The central assertions are: (i) the tree amplitude is M_tree = M^(22)+M^(33)+M^(24) = −κ² s³/(4tu), with M^(24) derived from the new fourth-order computation; (ii) the NSF expansion is UV-finite at all loop orders, with the L-loop integrand scaling as dq/q^{4L} (Theorems V.2 and V.4); and (iii) a generalized Jordan–Pauli relation determines σ_n^+ recursively from the free incoming data, with new advanced-cone corrections first appearing at n=4. The paper provides a detailed channel classification, explicit retarded-cone kernel assembly in Appendices B–G, and consistency identities (101) and (107) that the form factors must satisfy.

Significance. If the advertised equality with the Weinberg–DeWitt amplitude and the all-order UV finiteness were established, this would be a significant result: an on-shell, manifestly UV-finite formulation of graviton scattering without off-shell propagators would be a major alternative to covariant perturbative quantum gravity. The paper contains useful structural ideas — the recursive factor formula, the generalized Jordan–Pauli decomposition, the loop-counting classification, and the retarded-cone kernel assembly. However, the central result is conditional: the advanced-cone corrections that are claimed to complete σ_4^+ are never explicitly computed, and the paper itself repeatedly states that this computation remains. In addition, Eq. (114) contains an algebraic inconsistency in the DeWitt-amplitude check. The significance is therefore prospective rather than demonstrated.

major comments (3)
  1. [Secs. IIE–IIF, Eq. (29), Eq. (87), Eq. (102), Sec. V.E] The advanced-cone corrections δK^{(4),adv}_{(1,3)} and δK^{(4),adv}_{(2,2)} are introduced in Eq. (29) and claimed to be part of the complete fourth-order kernel, but they are never explicitly evaluated. After Eq. (87) the text says the correction 'needs to be added to complete M^{(24)}'; after Eq. (101) it says the advanced correction 'must be verified against the DeWitt amplitude separately'; and Sec. V.E states that 'only the explicit computation of the advanced cone corrections δM_{t,u} remains to fully determine the numerical values of N_{t,u}+M_{t,u}.' Since M^{(24)} is one of the three tree-level pieces in Eq. (39), the advertised equality M_tree = −κ² s³/(4tu) is conditional on a computation that is absent from the manuscript. This is a load-bearing gap, not a presentation issue.
  2. [Appendix A, Step 2b, Eq. (A15)/Eq. (11)] The derivation of the master equation depends on the assertion that all scattering-correction cut terms, Σ_{j=2}^{n−1} Z^+_{n,cut}|_{δσ^+_j}, cancel as boundary/total-u-derivative terms. The text provides only a verbal explanation ('they appear as ˙σ^+·Z^+_j, which is a total u-derivative') and concludes that they cancel. No explicit computation of these terms at n=4 is shown. If this cancellation is not exact, σ_4^+ receives additional contributions not present in Eq. (18), and the computed kernels and the tree-level identification are incomplete. This is a second unproven step in the central derivation.
  3. [Eq. (114), Sec. V.H] The partial-fraction identity used to confirm the DeWitt amplitude is algebraically false as written. With the numerators of Eq. (113), the sum P_s/s + (N_t+M_t)/t + (N_u+M_u)/u equals −κ²/4 [s³/(tu)+s³/t²+s³/u²], not −κ²/4 s³/(tu). The passage to the final expression requires the extra terms s³/t²+s³/u² to vanish, which they do not for generic t,u. Moreover, the t-channel residue quoted in Eq. (90) is of order s²/t (after using u≈−s), whereas Eq. (113) assigns N_t+M_t ~ s³/t², which is more singular. The DeWitt-amplitude verification in Sec. V.H therefore fails as written.
minor comments (4)
  1. [Sec. V.D] The text refers to 'Case 2 of Sec. 15.4' and 'Sec. 15.4' which appears to be a nonexistent section number; the intended reference is likely Sec. V.D.
  2. [Eq. (29)] The phrase 'absent in the original computation of this paper' is confusing. It should clarify that the reference is to the retarded-cone computation alone, not to an earlier version of the same paper.
  3. [Theorem V.2, Eq. (51)] The loop-counting formula L=(n1+n2−6)/2 gives L=−1 for the (δa_2^out)^2 term, which is then treated as a special tree-level case. The theorem is stated with 'if and only if n1+n2 ∈ {4,6}', but the formula itself does not cover the case n1+n2=4. The presentation should state the exceptional case explicitly rather than presenting Eq. (51) as universal.
  4. [Sec. II.A and Eq. (4)] The notation σ_j^+[σ^-] = −σ̄_j^-(ζ̂) + δσ_j^+[σ^-] is introduced, but the retarded mirror term σ̄_j^-(ζ̂) is not defined until later. A short definition or reference would improve readability.

Circularity Check

3 steps flagged · score 6.0 of 10

Tree-level amplitude is imposed via consistency with the known DeWitt amplitude, guaranteed by the authors' own unitarity citation, while the advanced-cone corrections needed for M(24) are left uncomputed.

  1. self citation load bearing [Sec. V.A Eq. (39) and Sec. V.F Remark after Eq. (101); Theorem V.6]
    "Their sum reproduces the full DeWitt amplitude [7]: Mtree = M(22) + M(33) + M(24) = − κ2/4 s3/tu, with consistency guaranteed by the unitarity of the S-matrix (Theorem V.6). ... The retarded part of the conditions is guaranteed by the Hermiticity of δT(3) (established in Ref. [5]); the advanced corrections provide additional contributions that must be verified separately."

    Theorem V.6 is stated as an 'if and only if' conditional on Eqs. (101)–(107) and on the uncomputed advanced-cone corrections δM_t,u. The retarded half of that condition is then waved through by citing Ref. [5], a companion paper by the same authors, rather than by a computation in this text. The central equality Mtree = −κ²s³/(4tu) is therefore not independently derived here; it is asserted on the authority of the authors' own prior unitarity claim, which is not machine-checked or externally benchmarked in the present paper.

  2. fitted input called prediction [Sec. V.F after Eq. (101); Sec. V.H after Eq. (114); Sec. V.F 'Effect of the advanced cone correction']
    "Comparing with the tree amplitude (90), consistency requires: [Eq. (101)]. ... δMt ... modifies the left-hand side of (101) and must be verified against the DeWitt amplitude separately. ... The identification of Ps, Nt, Mt, Nu, Mu with the explicit NSF form factors then gives the on-shell values of Hbar_sum, Hdiff, and Hbar^(II,IV)_3."

    The direction of the verification is inverted: instead of evaluating σ+4 and then comparing with the DeWitt amplitude, the NSF combination is required to equal the known partial-fraction numerator, and the on-shell form-factor values are inferred from that equality. The paper explicitly says the δMt piece, which belongs to the complete kernel, must be verified against the DeWitt amplitude separately. Thus the announced 'reproduction' of the tree amplitude is a consistency condition imposed on the target, not a prediction derived from computed inputs.

1 more flagged steps
  1. other [Sec. II Remark after Eq. (36); Sec. II.F Eq. (28); Sec. V.D Eq. (87); Sec. V.E after Eq. (89)]
    "The operators δa_out_{4,±} are therefore complete once these corrections are computed explicitly from δσ+2=σ+2 (Ref. [5]) and δσ+3=σ+3 (Ref. [6]). ... This form is correct; only the explicit computation of the advanced cone corrections δMt,u remains to fully determine the numerical values of Nt,u+Mt,u."

    The complete kernel (29), the operator δa_out4 (34), and the M(24) matrix element (86) all contain the advanced-cone terms δK^(4),adv, but those terms are never evaluated: Eq. (28) leaves F^(δ) unspecified, and Eqs. (87)/(102) express the needed kernels through unevaluated δK^(4),adv. Since M(24) is one of the three tree-level pieces in Theorem V.2, the central claimed equality is conditional on a computation absent from the paper. This is an omitted computation rather than a definitional identity, but it is the explicit limitation that makes the central 'prediction' a target to be fitted rather than an independent result.

full rationale

The paper contains genuine non-circular structural content: Theorem V.2's loop-counting formula L=(n1+n2−6)/2 is a purely combinatorial statement about operator sectors, and Theorem V.4's UV power counting is an independent scaling argument (modulo the same uncomputed kernels). However, the central physical result — that the three NSF matrix elements sum to −κ²s³/(4tu) — is not actually completed in the text. The advanced-cone corrections, which the paper itself identifies as necessary for M(24), are declared 'complete' only after they are 'computed explicitly,' and that computation never appears. The consistency checks in Eqs. (101) and (107) are written as conditions the NSF form factors 'must' satisfy, with the retarded part guaranteed by the authors' own Ref. [5] and the advanced part deferred to a separate verification against the DeWitt amplitude. In addition, Appendix A's Step 2b asserts that the cut terms Σ_{j=2}^{n−1} Z+_{n,cut}|_{δσ+j} cancel as boundary terms, but the paper provides no explicit verification for that cancellation at n=4; this is a technical assumption, not a circular step by itself. Taken together, the claimed tree-level amplitude is imposed via a consistency identity and a self-citation chain, rather than derived from fully computed kernels, warranting a partial-circularity score of 6.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

No numeric constants are fitted to data, but the central claim rests on the NSF framework of prior self-cited papers, on an asserted cancellation of cut terms, and on a gauge choice that kills BMS supertranslations.

assumptions (5)
  • domain assumption The null-surface formulation of Refs. [3,5,6] correctly describes graviton scattering; σ+_2 and σ+_3 are accepted as inputs.
    The paper's fourth-order computation takes the second- and third-order out-fields from the authors' own companion papers [5,6] as known, without independent derivation.
  • ad hoc to paper All scattering-correction cut terms Σ_{j=2}^{n−1} Z+_{n,cut}|_{δσ+_j} cancel as boundary terms.
    Appendix A Step 2b asserts this cancellation ('they appear as ∂_u derivative acting on Z+_j and therefore do not contribute') without a complete proof; the master equation (A15) depends on it.
  • domain assumption The unitarity of the S-matrix established in Ref. [5] is valid and sufficient to guarantee identities (101)–(107).
    This is a self-cited unpublished companion result; the identities are not independently verified here.
  • domain assumption In the UV limit the null denominators ℓ±·P never vanish for generic kinematics and scale as q.
    Used in Theorem V.4's power counting; collinear and special kinematic configurations are not analyzed.
  • domain assumption Integration constants F_n and f_n in the cone-cut equations can be set to zero by flat-space boundary conditions.
    Appendix A.2; this gauge choice is needed for uniqueness of the matching solution, and BMS supertranslations are otherwise a degeneracy.

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Pith. "Pith review of Quantum graviton scattering with definite helicities in the null surface formulation. III: Fourth-order recursion and ultraviolet finiteness." pith.science (2026). https://pith.science/paper/B6UOBTGP

@misc{pith2026260524512,
  author       = {Pith},
  title        = {Pith review of: Quantum graviton scattering with definite helicities in the null surface formulation. III: Fourth-order recursion and ultraviolet finiteness},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/B6UOBTGP}},
  note         = {Machine review of arXiv:2605.24512}
}
abstract

We extend the perturbative null-surface formulation (NSF) scattering map to fourth order and derive an all-order recursion for the quantum cut. After the antipodal matching, both cone sources are evaluated on the same retarded solution determined by the free incoming radiative data. The perturbative NSF equations therefore determine every coefficient $Z_n$ from that data, without introducing new independent asymptotic information. The partial cut $Z_{[N]}=\sum_{j=1}^{N}\ep^j Z_j$ defines the cumulative operator $U_{\omega,[N]}=\exp[-\ii\omega Z_{[N]}]$. An exact factor recursion for this operator gives a generating formula for $\delta a_{n,\lambda}^{\mathrm{out}}$ in terms of the order-$n$ cone source and lower-order operators. The scalar flux term $\Sigma$, which begins quadratically, is included on the cut side of the matching equation; its free quadratic part cancels between future and past infinity and it never introduces a new order-$n$ radiative operator. For smooth smeared radiative data, every finite-order cut is well defined and self-adjoint, so $U_{\omega,[N]}$ and its recursive factors are unitary and bounded. The frequency powers in the perturbative coefficients are thus part of the expansion of a bounded unitary operator, rather than separate ultraviolet enhancements. At fourth order we formally determine $\delta a_{4,\lambda}^{\mathrm{out}}$ and identify the mixed one-loop sector $\mathcal M_{24}=\mathcal M^{(24)}+\mathcal M^{(42)}$. A general radial power-counting proposition proves ultraviolet finiteness at arbitrary perturbative order for the flat-cone two-vertex sectors. In particular, $\mathcal M_{24}$ and the previously obtained $\mathcal M_{33}$ both scale as $\int^\infty \dd K/K^4$ in the uniform radial ultraviolet region.

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Works this paper leans on

42 extracted references · 5 linked inside Pith

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    applying all canonical contractions[ain λ (⃗k), a†in λ′ (⃗k′)] =δ λλ′ 2ω δ(3)(⃗k− ⃗k′), which fix the momenta of internal lines on-shell; and

  2. [2]

    Ret. sources

    The general structure of the type-δcorrection from the cross terms is: δK (4),adv (2,2),(n) = ηabK (12) a k4b ℓ+c(ˆk4) (K(12) +k 4)c F (i)(⃗k1, ⃗k2, ζ)¯F (δ)(⃗k3, ⃗k4, ζ) + c.c.,(28) where ¯F (δ) is the kernel ofδσ+ 2 =σ + 2 , i.e.(ð ¯ðS Ω +S A), and the denominator carriesℓ+ as the signature of the advanced cone. The corrections contribute to all six cha...

  3. [3]

    vertices

    applying all frequency deltasδ(ω′ − | ±⃗ki ± ⃗kj ± · · · |)from the kernels ofδaout n , which constrain the on-shell combinations. A contribution hasL= 0(tree) ifallinternal momenta are fixed. It hasL= 1if exactly one three-momentum integration remains free after all fixings, and so on: L= (internal contractions)−(independent constraints from frequency de...

  4. [4]

    Graviton 1 emits a spacelike virtual graviton (gravitational vertex, orderε)

  5. [5]

    The virtual graviton propagates to finite distance (not toI+)

  6. [6]

    It is absorbed by graviton 2, which emits secondary null/timelike radiation (vertex, orderε); 36

  7. [7]

    Steps 1 and 3 together account for the extra factorε2 inM t,u relative toM s

    This secondary radiation propagates causally toI+ and is recorded asσ+. Steps 1 and 3 together account for the extra factorε2 inM t,u relative toM s. a. Limitation.This argument relies on the flat-space notion of timelike vs. spacelike propagation, which is globally well-defined in Minkowski space. In an asymptotically flat spacetime, themetricisdynamical...

  8. [8]

    The advanced solution(Z +,Ω +)uses free dataσ + atI +; the retarded solution(Z −,Ω −)uses free dataσ − atI −

    The metric matching condition The NSF constructs the spacetime metric from the null cone cutZand the conformal factorΩvia: gab(x) = Ω2 hab[Z],(A1) whereh ab[Z]is the conformal metric determined by the null surfacesZ= const. The advanced solution(Z +,Ω +)uses free dataσ + atI +; the retarded solution(Z −,Ω −)uses free dataσ − atI −. Imposingg + ab(x) =g − ...

Show all 42 references
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    (11) of Ref

    Integration constants and gauge fixing The NSF field equation forZn (Eq. (11) of Ref. [3]) involves∂2/∂s2 in the affine parameter sof Minkowski space. Integrating twice insintroduces two arbitrary functions of integration at each ordern:

  2. [10]

    ABondi supertranslationf n(ζ, ¯ζ): an arbitrary function on the sphere, reflecting the BMS gauge freedom at null infinity. 42

  3. [11]

    Both are fixed simultaneously by requiring that in the absence of radiation (σ± = 0) the spacetime is flat

    A functionF n(xaℓ+ a , ζ,¯ζ) =F n(u, ζ,¯ζ): an arbitrary function of the Bondi retarded timeuand the angular coordinates, arising from the first integration ins. Both are fixed simultaneously by requiring that in the absence of radiation (σ± = 0) the spacetime is flat. For fla...

  4. [12]

    (A15) explicitly

    Explicit derivation of the master equation We now derive Eq. (A15) explicitly. The NSF equation forZ± n (Eq. (5) of Ref. [3]) is: ¯ð2ð2Z ± n =ð 2σ±(Z ± n−1, ζ) +¯ð2¯σ±(Z ± n−1, ζ) + Z ±∞ 0 2ð¯ðδΩ ± n + n−1X j=1 ηab∂aΛ± j ∂b ¯Λ± n−j ds.(A7) where, using the parametric form of t...

  5. [13]

    The free parts of the cut cancel:Z+ n,cut|σ+ 1 + ˆZ − n,cut|σ− = 0

  6. [14]

    The scattering corrections Pn−1 j=2 Z + n,cut|δσ+ j are boundary terms that cancel (Step 2b below)

  7. [15]

    , δσ+ n−1

    What remains is precisely the master equation (A15), which determinesδσ+ n from the cone integrals built withδσ+ 2 , . . . , δσ+ n−1. The recursive structure is therefore manifest: at each ordernone solves (A15) forδσ+ n using δσ + 2 , . . . , δσ+ n−1 already determined at pre...

  8. [16]

    Decomposition ofΛ − 1 Using the quantum operator identificationσ −(⃗k)→ p 4πG/ω ain +(⃗k)and¯σ −(⃗k)→ p 4πG/ω ain −(⃗k)(established in Sec. I): Λ− 1 (x, ζ) = s 4πG ωζ ωζ 2 ain +(⃗kζ)e ixakζa + Z d3k 2ω G2,+2(ζ, ˆk) r 4πG ω ain −(⃗k)e −ixaka.(B4) The first term istype-δ(localiz...

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    Pair(1,3):η ab∂aΛ− 1 ∂b ¯Λ− 3 ¯Λ− 3 has three pieces: ¯Λ− 3 = ¯Λ− 3,kn + ¯Λ− 3,¯σ− 3 + ¯Λ− 3,cone,(B5) each of which is cubic inain ±. a. Type-δcontribution ofΛ − 1.Thea in + part ofΛ − 1 localizes at ˆk1 =ζ ′. Contracting with∂ a ¯Λ− 3: ηab∂aΛ− 1,δ ∂b ¯Λ− 3 =i Z d3k1 2ω1 d3k2...

  10. [18]

    The nine cross products of the pair(2,2)

    Pair(2,2):η ab∂aΛ− 2 ∂b ¯Λ− 2 SinceΛ − 2 = Λ− 2,kn + Λ− 2,σ + Λ− 2,cone, this pair generates3×3 = 9cross products: ¯Λ− 2,kn ¯Λ− 2,σ ¯Λ− 2,cone Λ− 2,kn (1) kn×kn (2) kn×σ(3) kn×cone Λ− 2,σ (4)σ×kn (5)σ×σ(6)σ×cone Λ− 2,cone (7) cone×kn (8) cone×σ(9) cone×cone TABLE VIII. The nin...

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    All kernels areK (4),(1,3) with momenta relabeled

    Pair(3,1):η ab∂aΛ− 3 ∂b ¯Λ− 1 This is the Hermitian conjugate of pair(1,3)undera in + ↔a †in − ,a in − ↔a †in + ,G 2,+2 ↔ G−2,−2. All kernels areK (4),(1,3) with momenta relabeled. 48 Channel Operatorsω ′ F (σ) (I)a in +ain −ain +ain − |k1 −k 2 +k 3 −k 4|diff×diff (II)a in +ai...

  12. [20]

    Direct pair(1,3)

    Group 1: direct terms∂ 2 r Λ− k ∂2 r ¯Λ− j a. Direct pair(1,3). δ ˜Ω− 4,(n),(1,3) = (ℓ−ck1c)2 (ℓ−cK (234) c )2 [ℓ−cK (n) c ]2 ain +(⃗k1)· ¯H (n) 3 (ˆk1; ⃗k2, ⃗k3, ⃗k4).(F2) b. Direct pair(2,2). δ ˜Ω− 4,(n),(2,2) = (ℓ−cK (12) c )2 (ℓ−cK (34) c )2 [ℓ−cK (n) c ]2 X i,j F (i) (n)(...

  13. [21]

    Group 2: coupling withδΩ − 3 δ ˜Ω− 4,(n),Ω3 = Z 4Y i=1 d3ki 2ωi h (ℓ−ck1c)2(ℓ−ck2c)2 + (ℓ−ck1c)(ℓ−ck2c)·(ℓ −ck1c)(ℓ−ck2c) i ×a in +(⃗k1)ain −(⃗k2)· ˜Ω− 3,(n′)(⃗k3, ⃗k4)δ (n′⊗diff/sum),(n).(F5)

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    CompleteK (4),Ω (n)

    Group 3: coupling withδΩ − 2 δ ˜Ω− 4,(n),Ω2 = (ℓ−cK (12) c )2 (ℓ−ck3c)2 [ℓ−cK (n) c ]2 ˜Ω− 2 (⃗k1, ⃗k2)· G(n)(⃗k3, ⃗k4, ζ′) + (ℓ−ck1c)2 (ℓ−cK (34) c )2 [ℓ−cK (n) c ]2 G(n)(⃗k1, ⃗k2, ζ′)· ˜Ω− 2 (⃗k3, ⃗k4).(F6) a. CompleteK (4),Ω (n) . K (4),Ω (n) = I d2ζ ′ G2,0(ζ, ζ′) ℓ−c(ζ ′)K...

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    Ultraviolet-finite perturbative expansion of quantum gravity at null infinity,

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

    2026

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    Quantum graviton scattering with definite helicities in the null surface formulation,

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

    2026

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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)

    2023

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Reviewed August 2, 2026 · model on record in the stance chip above.