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REVIEW 3 major objections 4 minor 2 cited by

The paper claims that Regge-pole residues of flat-space scattering amplitudes fix the OPE data of the celestial CFT through an explicit dictionary.

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

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2026-08-03 04:54 UTC pith:CPAJ4KIJ

load-bearing objection Well-packaged and honest proposal for a celestial Regge dictionary, but the contour integral at its center is divergent under the stated assumptions and the phase in (5.6) doesn't match the derivation — plausible physics, unfinished mathematics. the 3 major comments →

arxiv 2602.03573 v2 pith:CPAJ4KIJ submitted 2026-02-03 hep-th

Celestial Regge theory

Eduardo Casali , Riccardo Giordana Pozzi This is my paper
classification hep-th
keywords celestial holographyRegge theoryMellin transformscattering amplitudescelestial CFTOPE coefficientsFroissart-Gribov expansionconformal blocks
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The paper aims to show that the Regge limit of flat-space scattering amplitudes and the Regge limit of celestial CFT correlation functions are two faces of the same data. It introduces a contour reformulation of the Mellin transform that evaluates the celestial correlator from the analytic structure of the bulk amplitude, leading to a celestial dispersion relation and a celestial Froissart-Gribov expansion. In the Regge limit the reduced correlator takes the same functional form as the standard CFT Regge limit, and matching the two yields an explicit relation between the bulk Regge-pole residues and the boundary OPE coefficients. If the relation holds, the celestial CFT's OPE data in this kinematic regime are determined by the S-matrix rather than being independently adjustable.

Core claim

The paper's central claim is that the Regge limit of flat-space scattering amplitudes has a precise counterpart on the celestial sphere, and that the two are linked by an explicit formula. Using a contour representation of the Mellin transform, the authors derive a dispersion relation for the celestial correlator and then show that, in the limit of small cross-ratio, the celestial Regge contribution takes the same form as the standard CFT Regge limit. Matching the two expansions gives the dictionary sigma±(nu) = i pi e^(2 pi i j(nu)) 2^(2 j(nu) - beta - 5) Delta(nu)^(beta/2 - 1) / sin(pi beta/2) e^(pi i (Delta12 - Delta34)/2) K_{Delta(nu),j(nu)} rho±(nu), where rho±(nu) are the residues of t

What carries the argument

The engine of the argument is the contour-trick Mellin transform. For a function f with no singularities on the integration domain, the paper rewrites the standard Mellin transform as M[f](s) = pi/sin(pi s) times the contour integral of (-r)^(s-1) f(r) around the positive real axis. This turns the energy integral in the celestial correlator into a contour integral in a complex Mandelstam variable, so that poles and discontinuities of the bulk amplitude can be read off directly. The same device converts the Froissart-Gribov partial-wave expansion into a celestial expansion, and after a further deformation of the contour to the principal series mu = 1 + i nu, it produces the Regge-limit integr

Load-bearing premise

The load-bearing premise is the working hypothesis that the Regge residues rho±(mu) are holomorphic and exponentially bounded and that the trajectory is exactly linear with j1 > 0 and 1 - j0 - j1 < 0, so the contour can be deformed from the negative real axis to the principal series; if any part fails, the leading celestial Regge term is not the CFT-like integral and the dictionary collapses.

What would settle it

Evaluate the dictionary in a concrete theory with a known Regge pole: compute the bulk partial-wave residue rho±(nu) from the Froissart-Gribov projection and the celestial OPE residue sigma±(nu) from the conformal partial-wave projection, and check the equality at the pole. A theory whose Regge trajectory is not exactly linear, or whose residues are not exponentially bounded, would make the contour deformation to mu = 1 + i nu invalid, and the leading Regge term would fail, falsifying the dictionary.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • Celestial OPE data in the Regge limit are fixed by bulk Regge residues through the explicit dictionary, so the boundary theory is not free in this sector.
  • The contour-trick Mellin transform yields a celestial dispersion relation, so celestial correlators can be reconstructed from poles and discontinuities of the bulk amplitude without performing the energy integral explicitly.
  • At the level of the Regge pole, the dictionary extends to the partial amplitudes, meaning the complex-spin data of the S-matrix and the conformal data of the celestial CFT become the same object in this limit.
  • Because the Regge residues carrying the discontinuity contributions are loop-generated in crossing-symmetric theories, the dictionary implies that celestial OPE data encode genuinely quantum, beyond-tree-level dynamics.

Where Pith is reading between the lines

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

  • An extension the paper leaves implicit is that the same contour trick could be used to define and compute celestial correlators in other kinematic regimes, effectively using contour deformations to implement energy cutoffs.
  • If the dictionary survives contact with explicit examples, string-like amplitudes with exactly linear Regge trajectories are a natural laboratory: they avoid the working hypothesis and could yield closed-form celestial OPE coefficients.
  • The paper notes that gravity's dominant eikonal limit lies outside its scope; a parallel celestial eikonal dictionary would be the natural next step, and the contour-trick Mellin transform is a natural tool for it.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 4 minor

Summary. This paper proposes a dictionary between the Regge limit of flat-space scattering amplitudes and the Regge limit of the putative celestial CFT. The authors introduce an alternative 'contour trick' representation of the Mellin transform that defines the celestial correlator through a contour integral in a Mandelstam variable, leading to a celestial dispersion relation. Applying this to the Regge amplitude, they obtain a principal-series integral (4.10) whose integrand is governed by the bulk Regge residue ρ±(μ). After reviewing the conformal Regge limit, they compare (4.10) with the CFT expression (5.3) and read off the central relation (5.6), expressing the celestial OPE coefficient σ±(ν) in terms of the bulk Regge residue ρ±(ν). The derivation is explicit and carefully signposted, with the main analyticity assumptions labelled as a working hypothesis.

Significance. If valid, Eq. (5.6) is a concrete and useful entry in the flat-space holography dictionary: it determines boundary conformal data from bulk Regge residues and gives a celestial analogue of conformal Regge theory. The contour-trick representation is elegant and may be of independent use. The paper is honest about its working assumptions and does not overclaim beyond the Regge-pole approximation. However, the central derivation is conditional on a convergence property that is not established (see major comments), so the result is currently a plausible dictionary rather than a proven theorem.

major comments (3)
  1. [§4.3, Eq. (4.10)] The convergence of the vertical-line integral is not established. With j(μ)=j0+j1μ and μ=1+iν, the factor e^{iπj(ν)}α±(ν) grows like e^{π j1|ν|} for ν→−∞ after the sin/cos denominators are taken into account; the factor ζ^{1−j(ν)} has unit modulus up to a fixed power. Consequently (4.10) converges only if |ρ±(1+iν)| decays at least as e^{−π j1|ν|}. 'Exponentially bounded' does not imply this, and no such decay is derived from S-matrix properties. The same asymptotic controls the horizontal arcs in the deformation from C(R−+iε) to Re μ=1, so the issue also affects (4.7). If (4.10) is not a convergent integral, the comparison with (5.3) and the dictionary (5.6) is not valid. Please prove the additional decay from known analyticity/boundedness assumptions or weaken the claim accordingly.
  2. [§5.2 / Appendix E] The matching of the CFT and CCFT Regge limits requires an analytic continuation of the distribution δ(z−z̄). The paper defines this via the restriction homomorphism D(C)→D(R+) in Appendix E, so that monodromies act only on the function h(r) on the support. This is a prescription, not a consequence of the bulk amplitude; the phase factors in (5.6) (e.g. e^{2πij(ν)} and e^{πi(Δ12−Δ34)/2}) depend on it. The text acknowledges that different prescriptions may change overall factors, but (5.6) is presented as an equality. Please either show that the relation is independent of the distribution prescription, or state clearly which prescription defines the dictionary and what ambiguity remains.
  3. [§4.3, working hypothesis] The 'working hypothesis' on ρ±(μ) and the linear trajectory j(μ)=j0+j1μ is imported without physical justification. The derivation of the celestial Regge term (4.10) depends on these conditions, and no example or argument is given that they hold for amplitudes with Regge poles beyond the general motivation in Appendix B.4. If these hypotheses fail, the central claim is conditional. I would ask the authors to either justify them from S-matrix analyticity or provide a class of amplitudes where they hold.
minor comments (4)
  1. [§2] Typo: 'weather' should be 'whether' in the sentence 'on weather the momenta are incoming or outgoing.'
  2. [Introduction, Eq. for α−] The displayed expression for α−(ν) in the Introduction uses ρ−(ν), while later in §4.3 the notation is ρ−(µ). Make the argument of ρ± consistent.
  3. [§5.1, Eqs. (5.4)–(5.5)] The notation 'iπ2' is ambiguous (iπ² vs iπ·2). In the subsequent derivation it appears to mean iπ², but this should be written as iπ^2 for clarity.
  4. [§5.2] When setting μ=1+iν in the principal series, the branch of (−iε+μ)^{β/2−1} must be specified. The phase in (5.6) may depend on this choice, so please state the chosen branch and check consistency with the contour orientation.

Circularity Check

0 steps flagged

No significant circularity found: (5.6) is a genuine dictionary between bulk Regge residues and celestial OPE data, not a redefinition.

full rationale

The paper's central relation (5.6) connects two objects defined by different expansions: the bulk Regge residues ρ±(ν), arising from the Froissart–Gribov/Sommerfeld–Watson decomposition of T(s,t), and the conformal OPE data σ±(ν), defined as residues of c(Δ,J) in the conformal partial wave expansion of the celestial correlator. These are not the same quantity by definition; the relation follows by explicitly matching the bulk Mellin transform (4.10) with the CFT Regge expression (5.3), producing nontrivial prefactors (powers of 2, K_{Δ,j}, phases) rather than a tautology. There are no author self-citations used as load-bearing support; the cited Regge/CFT results [7,8,9,15,18,19] are external literature. The 'working hypothesis' on holomorphy and exponential boundedness of ρ± in §4.3 is a convergence/validity assumption for the contour deformation, not a circular redefinition. The paper's own caveat in §5.2 that the matching prescription is 'motivated purely by an educated guess' concerns non-uniqueness of the dictionary, not circularity. The skeptical concern about e^{iπj(ν)} growth making the integral in (4.10) divergent is a technical correctness question; even if fatal, it would invalidate the derivation rather than make it circular. No step in the derivation reduces to its own inputs.

Axiom & Free-Parameter Ledger

2 free parameters · 5 axioms · 0 invented entities

The central dictionary (5.6) rests on: standard S-matrix analyticity assumptions (poles + branch cuts), the cut-off/residue structure of the Froissart-Gribov projection; a working hypothesis that Regge residues are holomorphic and exponentially bounded; a linear Regge trajectory with specific signs; the existence of an SL(2,C) conformal partial wave expansion of celestial correlators; and a distributional analytic-continuation prescription for the delta function. No new entities are introduced. The two free parameters j0 and j1 are not fitted but are constrained by inequalities.

free parameters (2)
  • j0 (Regge intercept)
    Intercept of the assumed linear Regge trajectory j(μ)=j0+j1μ; not fixed by the paper; enters the celestial Regge limit and the dictionary (5.6). Conditions j1>0 and 1-j0-j1<0 are imposed in §4.3.
  • j1 (Regge slope)
    Slope of the assumed linear Regge trajectory; same status as j0, constrained only by inequalities in §4.3.
axioms (5)
  • domain assumption Scattering amplitude is analytic except for poles and branch cuts; arcs at infinity can be neglected in contour manipulations
    Used to derive (3.7) and the celestial dispersion relation (4.3); violated by Regge-growing amplitudes, as noted in §4.3.
  • ad hoc to paper Regge residues ρ±(μ) are holomorphic and exponentially bounded
    Stated in §4.3 as a working hypothesis; required for the contour deformation from C(R-) to the principal series (4.10).
  • ad hoc to paper Regge trajectory is linear: j(μ)=j0+j1μ with j1>0 and 1-j0-j1<0
    Assumed in §4.3 to make the ζ^{1-j(ν)} contribution leading; selects (4.10) as the celestial Regge limit.
  • domain assumption The celestial correlator admits an SL(2,C) conformal partial wave expansion over the principal series with coefficients σ±(ν)
    Used in §5.1; based on [18,19].
  • ad hoc to paper A δ(z-z̄)-supported distribution can be analytically continued by restricting to its support
    Needed to define monodromies of the celestial correlator in §5.2; the prescription is constructed in Appendix E.

pith-pipeline@v1.3.0-alltime-deepseek · 30751 in / 18055 out tokens · 171782 ms · 2026-08-03T04:54:29.459004+00:00 · methodology

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read the original abstract

Exploiting the analytic properties of scattering amplitudes, we provide an alternative but equivalent definition of the standard Mellin transform used to obtain celestial correlation functions. From this representation, we identify a celestial dispersion relation that relates the reduced correlation function to the poles and discontinuities of the bulk amplitude, and we present a novel expansion for the celestial correlator from an integral transform of the Froissart-Gribov expansion on the bulk. By drawing an analogy with the standard CFT case, we define the celestial Regge limit and identify the relevant celestial CFT data in terms of the partial amplitudes governing the bulk Regge limit.

Figures

Figures reproduced from arXiv: 2602.03573 by Eduardo Casali, Riccardo Giordana Pozzi.

Figure 1
Figure 1. Figure 1: Contour modification from C(s) to C in the complex s-plane. Setup with no massive interactions and branch cuts are in red. where we have considered branch cuts that are slightly tilted by a small parameter ϵ0 in order to avoid the overlap with the on-shell poles. By setting ϵ0 = 0 we obtain the standard expression for the discontinuity on the real axis, figure 1. For crossing symmetric amplitudes, one can … view at source ↗
Figure 2
Figure 2. Figure 2: Contour modification for the modified Mellin transform. We move from the contour C −1 (R + + iε) to the one around the pole at µ = m2 − iϵ Evaluating the ω-integral one finds Z ∞ 0 dω ωβ−1 T (s = ω 2 , z) = Z C(R++iε) dµ 2πi T (z, µ) −π(−µ + iε) β/2−1 2 sin(πβ/2)  . The reduced correlator is then written as g(β, z) = 2 −3−β π sin(πβ/2)z 2 Z C−1(R++iε) dµ 2πi (−µ + iε) β/2−1 T (z, µ), (4.2) where we absor… view at source ↗
Figure 3
Figure 3. Figure 3: Contour modification from the discontinuity to the principal series. In blue the poles in α ±(µ). Assuming ζ > 0, the delta is evaluated for θ = {0, π, 2π, ..}. We can restrict it to the fundamental domain θ ∈ [0, 2π). Moreover, note that for θ = π, z = −ζ =⇒ z < 0, which is not possible considered that we take Re(z) ∈ (0, 1). It follows that δ(z − z¯) = 1 2ζ δ(θ). Putting all together the Regge limit of t… view at source ↗
Figure 4
Figure 4. Figure 4: Initial configuration where all operators are spacelike separated and 1 and 2 are inside the causal diamond. z¯ = 4¯ρ (1 + ¯ρ) 2 , 1 − z¯ = (1 − ρ¯) 2 (1 + ¯ρ) 2 , ρ¯ = 1 − √ 1 − z¯ 1 + √ 1 − z¯ , and similarly for z and ρ. In terms of these coordinates, the OPE converges for ρ and ρ¯ within the unit disc. From these expression, it will be clearer, later, how monodromies on z or z¯, relates different confi… view at source ↗
Figure 5
Figure 5. Figure 5: Lorentzian configuration where 1 and 2 are sent to the lightcones of 4 and 3 respectively. In red, the effect of taking the monodromy (1−z¯) → e 2πi(1−z¯) or, equivalently, ρ¯ → 1/ρ¯ from the line integral is subleading, [7, 8, 15], we obtain f ± ∆i (z, z¯) ≈ − Z ∞ −∞ dν 2π π σ±(ν) sin(πj(ν))  G1+iν,j(ν)(z, z¯) ± (−1)j(ν)G1+iν,j(ν)(z, z¯)  , where the overall minus comes from the clock-wise contour aroun… view at source ↗
Figure 6
Figure 6. Figure 6: Causal configuration specified by 0 < ρ < 1 < ρ <¯ ∞. In black the Regge limit associated to small cross ratios z, z¯. where, in order to keep a streamlined expression, we keep some contributions in terms of ∆(ν) = 1 + iν. The coefficients γ ±(ν) encodes the conformal data and are defined by γ +(ν) = e −iπj(ν)/2 σ +(ν) 4 sin(πj(ν)/2) e πi ∆12−∆34 2 iπ2 K∆(ν),j(ν) , (5.4) γ −(ν) = e −iπj(ν)/2 σ −(ν) 4 i cos… view at source ↗
Figure 7
Figure 7. Figure 7: Effect of mapping back to spacelike configuration, ρ → 1/ρ, after the Regge limit continuing around z = 1, which, in terms of the radial coordinates ρ, ρ¯, corresponds to the transformation ρ → 1/ρ. Together with the small-cross-ratio limit, this maps the Regge configuration, where the two pairs of insertions are causally related, to a configuration in which all operators lie on the real line and are mutua… view at source ↗

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