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Conformal Partial Wave Expansion of Celestial Correlators

T0 review · 3 major / 5 minor · reviewed 2026-08-09 · deepseek-v4-flash

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Pith's one-line read Perturbative celestial correlation functions admit a conformal partial wave expansion with meromorphic spectral density, and therefore an expansion into conformal blocks.

desk verdict A real and useful CPWE result for celestial correlators, with an all-order claim that outruns the proof and leans on an imported cancellation. read the letter →

arxiv 2502.03087 v1 pith:ECD6M2SB submitted 2025-02-05 hep-th

Francesca Pacifico , Charlotte Sleight , Massimo Taronna This is my paper · ORCID
classification hep-th MSC 81T40
keywords celestialholographycorrelatorsconformalpartialwaveexpansionblocksAdSharmonicfunctionsFeynmanpropagatorbootstrapMellinamplitudes
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

This paper establishes that celestial correlation functions — holographic observables on the celestial sphere defined by extrapolating time-ordered Minkowski correlators — admit a conformal partial wave expansion with meromorphic spectral density at every order in perturbation theory. Because a meromorphic spectral density implies a discrete expansion into conformal blocks, the paper supplies a tool that lets flat-space celestial correlators be studied with the same bootstrap machinery used in AdS/CFT. The key move is to extend the harmonic-function, or spectral, decomposition of AdS bulk-to-bulk propagators to the Minkowski Feynman propagator, via the hyperbolic slicing of Minkowski space and analytic continuation from de Sitter to Euclidean AdS. Working out four-point contact and tree-level exchange diagrams for scalar fields, the paper extracts the direct-channel conformal block expansions; when all external and exchanged scalars are massless, the exchange collapses to a finite sum of conformal blocks, with special behaviour in d=2. The same framework is then used to propose a non-perturbative celestial bootstrap in which Lorentz unitarity appears as positivity of the spectral density.

What carries the argument

The central object is the AdS harmonic function decomposition of the Minkowski Feynman propagator (equation 3.26). It expresses the Feynman propagator, in each region of the hyperbolic slicing of Minkowski space, as an integral over ν of the Euclidean AdS harmonic function Ω_{ν,0} — an eigenfunction of the conformal Casimir — weighted by the radial spectral function $g^{{(m)}}$_{-d/2+iν}(R_1,R_2). This generalises the standard harmonic, or split-representation, decomposition of AdS bulk-to-bulk propagators to flat space. Its role is to factorise celestial correlators into co-dimension-one Euclidean AdS Witten diagrams times radial integrals that take the form of momentum-space Witten diagrams, making the meromorphicity of the spectral density in ν manifest.

What would settle it

Compute a one-loop, box-diagram celestial four-point correlator in a scalar theory using the same harmonic decomposition: if the spectral density in ν acquires branch cuts or poles incompatible with the asserted meromorphic structure, the central claim fails. A cheaper check is to evaluate the massless-exchange formula (1.11) in d=4 against a direct Mellin-Barnes integration; any mismatch would invalidate the contour reasoning of section 4.3.

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Extended reading notes

Core claim

The central claim is that perturbative celestial correlators of scalar fields in (d+2)-dimensional Minkowski space admit a conformal partial wave expansion with meromorphic spectral density, and hence a convergent conformal block expansion. The mechanism is the AdS harmonic function decomposition of the Feynman propagator (equation 3.26): in each contributing region of the hyperbolic slicing of Minkowski space, the Feynman propagator is rewritten, via analytic continuation, as an integral over a spectral parameter ν of Euclidean AdS harmonic functions Ω_{ν,0} multiplied by radial spectral functions that encode the mass and radial dependence. This reduces celestial correlators to the well-studied Euclidean AdS Witten-diagram form, so the spectral density is a product of a meromorphic reduced density from the radial integrals and the standard AdS spectral density; meromorphicity then follows. For four-point exchange diagrams with massless external scalars and a massive exchanged scalar, the conformal block expansion contains two infinite families of double-trace operators and two double-trace-like families that encode the exchanged massive state, in agreement with the corresponding celestial Mellin amplitude representation. When the exchanged scalar is also massless, the expansion truncates to a finite sum of conformal blocks for integer d, with derivatives of blocks contributing in d=2, and vanishes for even d>2 up to contact terms.

Load-bearing premise

The derivation assumes that, at every order in perturbation theory, the Minkowski Feynman propagator is exactly equivalent to a sum of Euclidean AdS bulk-to-bulk propagators after analytic continuation from the de Sitter region and cancellation of the other hyperbolic-slicing regions, a property established for the cases in reference [19].

Editorial extensions

If this is right

  • Four-point tree-level celestial exchange diagrams with massless external scalars admit conformal block expansions containing two infinite families of double-trace operators and two double-trace-like families with the scaling dimensions (1.9)–(1.10).
  • When the exchanged scalar is massless, the tree-level exchange diagram in integer dimensions d becomes a finite sum of conformal blocks; in d=2 it includes their derivatives as well (equations 1.11 and 4.33).
  • For even d>2 the all-massless exchange celestial correlator vanishes up to contact terms, a consequence of the zero-versus-pole mechanism in the bulk time integration.
  • The conformal partial wave expansion reproduces the celestial Mellin amplitude representation of the exchange diagram, providing a non-trivial consistency check of the analytic continuations used.
  • The same framework leads to a proposed non-perturbative celestial bootstrap, in which Lorentz unitarity manifests as positivity of the spectral density, ρ_J(ν) ≥ 0.

Reading between the lines

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

  • The harmonic decomposition of the Feynman propagator should extend beyond tree level, so the meromorphic-CPWE claim plausibly holds at all loop orders in scalar theories; a direct one-loop box diagram calculation would sharpen this prediction.
  • The appearance of derivatives of conformal blocks in the d=2 massless exchange hints that the corresponding celestial CFT may be logarithmic, with Jordan blocks in the OPE of massless primaries.
  • The same spectral argument should apply to exchanges of massless spinning fields, since the one-to-one map between massless fields in Minkowski space and massive fields on the hyperboloid has a known spinning analogue.
  • The vanishing of the massless exchange for even d>2 may be an artefact of contact-term conventions; with a different contact-term scheme, the finite conformal-block sum could acquire extra contributions.
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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 / 5 minor

Summary. The paper claims that celestial correlation functions, defined by Mellin extrapolation of Minkowski time-ordered correlators to the celestial sphere, admit in perturbation theory a conformal partial wave expansion with meromorphic spectral density, and hence also a conformal block expansion. The central technical step is a harmonic-function decomposition of the Minkowski Feynman propagator in the hyperbolic slicing, obtained by analytically continuing the EAdS propagator decomposition established in the authors' prior work. The paper works out the conformal partial wave and conformal block expansions for four-point contact and tree-level exchange diagrams, with massless external scalars, and finds that the massless-exchange case reduces to a finite sum of conformal blocks (plus derivatives for d=2). It closes with a speculative non-perturbative celestial bootstrap discussion.

Significance. If the central derivation is sound, the paper provides a useful bridge between celestial correlators and standard AdS/CFT spectral technology, with explicit checkable formulas. The tree-level exchange result in eq. (1.11), the d=2 derivative formula in eq. (4.33), and the contact-diagram delta-function constraints in eqs. (4.14)-(4.16) are concrete and falsifiable; the agreement with the Mellin representation of ref. [38] in Appendix D is a useful consistency check. The paper is clearly written and contains substantial technical detail. However, the all-order claim rests on the prior radial-reduction framework of ref. [19], and the derivation of the key decomposition (3.26) is not fully justified as it stands; these issues need to be addressed before the central claim can be regarded as established.

major comments (3)
  1. [§3.1, Eq. (3.26)] The derivation of the harmonic-function decomposition drops terms proportional to G^{AdS}_{d/2+iν} on the grounds that this propagator is holomorphic in the lower-half ν-plane. Holomorphicity alone does not imply that the corresponding contour integral vanishes; one must also control the behaviour of the integrand at infinity and the absence of poles crossing the contour. This is not a formality: Appendix D, Eq. (D.8), shows that the convergence of similar spectral integrals is nontrivial (the bound can be saturated for d=2), and the integrand of (3.23) does not decay uniformly on the real axis. Please provide an explicit contour-deformation argument, or state the precise contour and decay assumptions, that justifies the transition from (3.23) to (3.26).
  2. [Abstract and §5] The claim that 'in perturbation theory' celestial correlators admit a conformal partial wave expansion with meromorphic spectral density is not fully supported by the explicit computations in the paper, which cover only four-point contact diagrams (§4.1) and tree-level exchange diagrams (§3.2, §4.2-4.3). The all-order statement is imported from the radial reduction of ref. [19], specifically the cancellation of the A− and D− regions (eqs. (2.29)-(2.31)). The cross-check in Appendix D against the Mellin representation of [38] does not independently test these all-order steps, since [38] is formulated within the same framework. Either prove the all-order statement or explicitly qualify it as a conjecture modulo the assumptions of ref. [19].
  3. [§3.2 and §4.3] The conformal block expansion is extracted by deforming contours and collecting residues, but the legitimacy of the contour deformations is asserted rather than demonstrated. The paper notes that the reduced spectral density generally violates the symmetry property (3.18), and that for d≥4 subtractions may be needed ('keeping any required subtractions implicit', §3.2). In the massless-exchange case, eq. (4.27) forces poles to cross the real axis for every d, and the final result (4.29)/(4.33) is obtained from the asymmetric form (3.34). Since the central claim is that a well-defined conformal partial wave expansion with meromorphic spectral density exists, the contour prescriptions and any subtractions should be specified precisely rather than left implicit.
minor comments (5)
  1. [§4.3, Eq. (4.28)] The notation ρ^{(0,0|0|0,0)}_{...}(ν) appears without the bar used for the reduced spectral density in eq. (4.23); please clarify which object is meant.
  2. [Introduction, bullet list] The phrase 'As as result' should read 'As a result'.
  3. [§2, Eq. (2.29)] The arrow notation for analytic continuation is introduced but the direction of the arrow is not defined; a sentence explicitly defining '→' would help the reader.
  4. [§4.1, after Eq. (4.16)] The sentence 'the shadow scaling dimensions sum up to the dimension of the bulk Minkowski space' is slightly imprecise because the constraint is ∑(d−Δ_i)=d+2; consider rephrasing to avoid confusion.
  5. [§4.3, Eq. (4.33)] The derivative convention for f′(Δ_1+Δ_2) is not stated; since f is defined through N_{d/2+iν} G^{12,34}_{d/2+iν,0}, please specify how the derivative is taken and whether the conformal block is continued in its dimension.

Circularity Check

2 steps flagged · score 2.0 of 10

No definitional or fitted-input circularity in the tree-level derivation, but the all-order CPWE claim leans on same-author prior results [19,41,42] without re-derivation.

  1. self citation load bearing [Section 2, after eq. (2.14); used in eq. (B.2) and harmonic decomposition (3.26)]
    "Interestingly, in the perturbative computation of celestial correlators (2.1) the contributions from regions A− and D− in the hyperbolic slicing precisely cancel at all orders in perturbation theory [19]. We henceforth only consider contributions from internal points in regions A+ and D+."

    The all-order cancellation of the A− and D− hyperbolic regions is a theorem imported from ref. [19], authored by two of the present authors, and is not re-derived here. This cancellation is load-bearing: it restricts the Feynman-propagator decomposition to the A+ and D+ contributions, which is what allows the harmonic-function decomposition (3.26) and the resulting all-order CPWE claim. The explicit computations in this paper are tree-level only, and Appendix D checks consistency with the same-group Mellin representation [38], so the all-order statement is carried by the self-citation rather than by an independent derivation in this paper. This is reliance on prior work rather than an equivalence-by-construction, so it does not make the whole derivation circular.

  2. self citation load bearing [Section 5, first paragraph]
    "One can now use the same argument of [ 41, 42] to conclude that celestial correlators admit a meromorphic spectral decomposition, at least at the perturbative level."

    The central claim of the abstract — meromorphic spectral density, hence conformal-block expansion at all orders — is here transferred from refs. [41,42] by the same authors rather than proved anew. The paper explicitly evaluates only tree-level exchange and contact diagrams; the all-order meromorphicity conclusion is obtained by invoking the 'same argument' from [41,42]. That is a self-citational, load-bearing step. It is not a definitional reduction because the tree-level spectral densities and conformal-block sums are explicitly exhibited, but the all-order generality of the main result rests on this imported argument rather than on a calculation contained in this paper.

full rationale

The tree-level derivation in this paper is self-contained and non-circular: starting from the celestial correlator prescription (2.1) and the radial reduction reviewed in section 2, the authors construct the harmonic decomposition of the Feynman propagator (3.26), derive the spectral density for celestial exchange diagrams (3.30)–(3.32), and obtain the finite conformal-block sum for massless exchanges (4.29) by explicit residue calculus. No parameter is fitted and then renamed as a prediction, and no equation is asserted to follow from a definition that already contains the result. The consistency check in Appendix D reproduces the Mellin amplitude of [38], which is a useful internal cross-check but belongs to the same group, so it does not constitute an external benchmark. The only substantive circularity-related concern is the all-order part of the central claim: it depends on the A−/D− cancellation at all orders imported from [19] and on the meromorphic spectral-decomposition argument transferred from [41,42], both same-author prior results that are not re-proven here. Because the explicit tree-level analysis is genuine and the all-order extension is clearly labeled as relying on prior work, the appropriate verdict is minor self-citation load-bearing rather than constructional circularity.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The central claim rests on the prior radial-reduction results of [19] and standard AdS spectral theory, plus the explicit but unproven assumption of non-perturbative meromorphicity. No new particles, forces, or dimensions are introduced, and no parameters are fitted.

assumptions (4)
  • domain assumption The Feynman propagator in Minkowski space can be decomposed into EAdS bulk-to-bulk propagators via analytic continuation (eq 2.29), and contributions from regions A- and D- cancel at all orders (section 2, citing [19]).
    This radial reduction is the foundation for the harmonic function decomposition (3.26) and hence for the entire CPWE construction. It is taken from the authors' prior work [19].
  • standard math AdS harmonic functions satisfy the split representation and the identity (3.25) relating bulk-to-bulk propagators to harmonic functions.
    These are standard results in AdS/CFT spectral theory, cited to [33-35].
  • domain assumption The massless limit of the massive celestial correlators commutes with the Mellin transform and the CPWE.
    Used in section 4.3 to take m to 0 in the exchange spectral density; verified for the bulk-to-boundary propagator in appendix C.1 but assumed for the full diagram.
  • ad hoc to paper The conformal partial wave expansion with meromorphic spectral density continues to hold at the non-perturbative level (section 5).
    This is explicitly assumed to propose the celestial bootstrap; the paper states it is an open question.

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Cite this review

Pith. "Pith review of Conformal Partial Wave Expansion of Celestial Correlators." pith.science (2026). https://pith.science/paper/ECD6M2SB

@misc{pith2026250203087,
  author       = {Pith},
  title        = {Pith review of: Conformal Partial Wave Expansion of Celestial Correlators},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ECD6M2SB}},
  note         = {Machine review of arXiv:2502.03087}
}
abstract

A novel definition of holographic correlation functions on the celestial sphere of Minkowski space was recently introduced in arXiv:2301.01810 as the extrapolation of bulk time-ordered correlation functions to the celestial sphere. In this work, focusing on theories of scalar fields in $(d+2)$-dimensional Minkowski space, we show that in perturbation theory such celestial correlation functions admit a conformal partial wave expansion with meromorphic spectral density, and hence also an expansion into conformal blocks. This is achieved in the hyperbolic slicing of Minkowski space by extending the harmonic function (``spectral") decomposition of AdS bulk-to-bulk propagators to the Minkowski Feynman propagator. We study the conformal partial wave expansion of celestial correlators for four-point contact and tree-level exchange diagrams, and extract the contributions to their conformal block expansions in the direct channel. When all scalar fields are massless, the tree-level exchange diagram takes a remarkably simple form and is given by a finite sum of conformal blocks (and, for $d=2$, their derivatives as well). We also discuss the conformal partial wave expansion at the non-perturbative level, where Lorentz unitarity manifests as positivity of the spectral density.

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Forward citations

Cited by 1 Pith paper

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