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Conformal blocks from celestial graviton amplitudes

T0 review · 3 major / 6 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read A single-valued celestial graviton correlator is built by shadow transform and completed to crossing symmetry, and its inverse shadow is the double copy of the gluon amplitude.

desk verdict A solid, mostly computational extension of the gluon shadow-correlator program to gravitons, with the single-valued completion as the one genuinely load-bearing ansatz; worth serious refereeing if the long identities and the J=-1 state get checked. read the letter →

arxiv 2501.05805 v3 pith:CP2HZBLH submitted 2025-01-10 hep-th

Iustin Surubaru , Bin Zhu This is my paper · ORCID
classification hep-th
keywords celestialholographyshadowtransformsingle-valuedcorrelatorsconformalblocksgravitonamplitudesOPEdoublecopyconformallysoftlimit
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 proposes an analytic, single-valued four-graviton correlator for celestial holography, replacing the distributional correlator that standard celestial amplitudes produce. Starting from the tree-level MHV four-graviton amplitude, the authors shadow-transform one of the graviton operators and, in the conformally soft limit, complete the resulting correlator to a single-valued, crossing-symmetric object. They decompose it into conformal blocks in all three channels and find that only integer-spin states are exchanged, with OPEs matching the known celestial graviton OPEs plus one new $J=-1$ operator. An integral representation lets them invert the shadow transform, yielding a simple single-valued celestial graviton amplitude proportional to $\bar{z}/(z(1-z))$, which they identify as the double copy of the corresponding single-valued gluon amplitude.

What carries the argument

The central object is the shadow transform, which maps a primary operator of dimension $\Delta$ and helicity $J$ to one of dimension $2-\Delta$ and helicity $-J$ by integrating against a kernel; applying it to one graviton in the tree-level MHV four-graviton celestial amplitude removes the distributional support $\delta(z-\bar{z})$ and leaves an analytic correlator. The single-valued completion (3.5) then adds a second term $S_2(x)\bar{I}_2(\bar{x})$ chosen so that the branch cut of the analytically continued holomorphic part near $x=1$ is cancelled by the matching antiholomorphic block, following the gluon construction of [56]. The Coulomb-gas-type integral representation (3.25), with holomorphic and antiholomorphic exponents differing by integers, is the mechanism that permits inverting the shadow transform; a change of variables recasts the integral as the shadow transform of the simple amplitude $\bar{z}/(z(1-z))$. The double-copy identification uses the KLT/BCJ relation, writing that amplitude as $z\bar{z}$ times the product of two color-ordered single-valued gluon amplitudes.

What would settle it

Compute the four-graviton celestial amplitude in a concrete bulk theory with broken translation invariance, such as gravity coupled to photons on a nontrivial background, and check whether its conformally soft shadow limit reproduces (3.5) and the $\bar{z}/(z(1-z))$ amplitude, including the $J=-1$ OPE term; if no bulk process produces that operator, the completion is not physical.

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Extended reading notes

Core claim

The paper's central claim is that the single-valued completion of the shadow four-graviton correlator in the conformally soft shadow limit, equation (3.5), is the desired analytic celestial correlator: it is single-valued on the whole complex plane, crossing symmetric under the three channel maps, and its conformal block decomposition in every channel contains only exchanged states with integer spin, in contrast to the continuous-spin states that appear if one simply analytically continues the uncompleted shadow correlator. From this correlator the authors extract leading OPEs that match the known celestial graviton OPEs, with one new spin $J=-1$ operator whose presence they interpret as a hint of a bulk background (possibly photons coupled to gravity). The Coulomb-gas-type integral representation of the single-valued correlator allows them to invert the shadow transform, obtaining a single-valued celestial graviton amplitude proportional to $\bar{z}/(z(1-z))$, which up to constants equals the double copy of the single-valued gluon amplitude of [56].

Load-bearing premise

The argument assumes that the particular single-valued completion chosen to cancel the branch cut—selected by following the gluon template—is the physically correct celestial correlator, not just one ad hoc completion.

Editorial extensions

If this is right

  • The single-valued correlator (3.5) is crossing symmetric and blocks-expands with integer spins only in all three channels, so continuous-spin states are an artifact of naively continuing the uncompleted shadow correlator.
  • The leading OPEs reproduce the standard celestial graviton OPEs, so the single-valued completion is consistent with the known celestial CFT operator algebra; the only new ingredient is the $J=-1$ operator in the opposite-helicity channel.
  • The inverse shadow yields a single-valued celestial graviton amplitude proportional to $\bar{z}/(z(1-z))$, a double copy of the gluon result, providing a graviton analogue of the single-valued gluon correlator.
  • The integral representation opens the way to applying Coulomb-gas/Dotsenko-Fateev techniques to graviton correlators and to constructing more general graviton correlators from gluon ones via the double copy.

Reading between the lines

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

  • If the $J=-1$ operator is real, the single-valued correlator likely describes gravity on a nontrivial background; a concrete test would be computing the tree-level celestial four-graviton amplitude in Einstein-Maxwell theory and checking whether the same operator appears in the OPE.
  • The double-copy structure at four points suggests that higher-point single-valued graviton correlators could be built as KLT products of single-valued gluon correlators, which would provide a bootstrap route to gravity correlators.
  • Generalizing the single-valued completion beyond the conformally soft shadow limit (general $\lambda_1$) would test whether the integer-spin spectrum and the $J=-1$ operator persist or are special to the soft point.
  • The analogy with minimal-model Coulomb-gas integrals raises the possibility that the single-valued graviton correlator satisfies a null-vector differential equation, which could link the construction to Liouville-type descriptions of celestial gravity.
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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 / 6 minor

Summary. The paper computes the shadow transform of the four-graviton MHV celestial amplitude with one conformally soft shadow operator, expands the result in two-dimensional conformal blocks in the compatible channel, then constructs a single-valued completion following the gluon construction of [56]. The single-valued correlator is block-decomposed in all three channels, shown to satisfy crossing symmetry, and rewritten in a Coulomb-gas-like integral representation. The integral representation is used to invert the shadow transform, yielding a 'single-valued celestial graviton amplitude' proportional to \bar z/(z(1-z)), from which a double-copy relation with the single-valued gluon amplitude is observed. The paper also extracts leading OPEs and finds the standard celestial graviton OPEs plus one new J=-1 operator.

Significance. If the construction is accepted, the paper provides a rare example of an analytic, crossing-symmetric celestial graviton correlator with only integer-spin exchanges and a simple double-copy structure. The computations are long but internally cross-checked: single-valuedness is verified near x=1 and x=infinity, crossing relations (3.7) are imposed and checked, and the leading OPEs reproduce known results except for the new J=-1 operator. However, the central novelty rests on an ansatz for the single-valued completion whose bulk origin is left open, so the significance is conditional on whether that completion is physically justified rather than a formal device.

major comments (3)
  1. [Section 3.1, Eq. (3.5)] The single-valued completion is introduced by adding S2(x) I2(\bar x) to S1(x) I1(\bar x), with S2 fixed by Eq. (3.6) to cancel the branch-cut of S1 I1 near x=1. The paper does not show that this is the unique single-valued completion, nor that it corresponds to a known bulk amplitude. Single-valuedness only fixes branch-cut discontinuities; in principle one can add single-valued terms with the same allowed singularities without spoiling the crossing relations (3.7). The paper itself states in Sec. 4.2 that the new J=-1 operator 'might correspond to photons coupled to gravity' and leaves the bulk connection for future work. Since the integer-spin spectra in the (14<->32)2 and (13<->24)2 channels, the OPE (3.21)/(4.20), and the double-copy relation (4.18) are all properties of this specific completion, the central claims are conditional on an unproven assumption. Please either derive the completion from a physical principle (for example, from the Banerjee-Ghosh differential equations mentioned in Sec. 5) or explicitly frame the results as properties of one possible completion.
  2. [Section 4, Eq. (4.9)] The 'single-valued celestial graviton amplitude' is obtained by inverting the shadow of the completed correlator using the change of variables (4.4), which is chosen so that the integral takes the shadow form. It is not demonstrated that this amplitude is the Mellin transform of any known tree-level amplitude, and the double-copy relation (4.18) is an observation about the simple form \bar z/(z(1-z)) rather than a derived equivalence between celestial amplitudes. Please clarify the status of (4.9) as a proposal, and check whether the amplitude satisfies the known celestial graviton Ward identities or soft theorems beyond the leading OPE comparison.
  3. [Section 2.2, Eqs. (2.32)-(2.33)] The general-λ1 conformal block decomposition is stated after a 'tedious computation' without intermediate steps. The coefficients contain many gamma functions and alternating signs, and the final expression (2.32) involves a triple sum. To make the paper self-contained and verifiable, please provide a derivation in an appendix or supplementary material, or at least include a computer-algebra verification of the decomposition. This matters because the specialized limit λ1=i in Sec. 3 relies on the same hypergeometric identities, and the reader currently cannot check the main technical result without redoing the computation.
minor comments (6)
  1. [Section 3.2, Eq. (3.18)] In the last sum of Eq. (3.18), the block is labeled K42_31 whereas all other blocks in that equation use K24_31; please fix the label.
  2. [Section 3.3] 'Coulumb gas formulation' should be 'Coulomb gas formulation'.
  3. [Section 2.1, Eq. (2.14)] The powers of x and \bar x in the prefactor of Eq. (2.14) do not appear to match the exponent that follows from Eq. (2.11) after setting λ1=i; please double-check the algebra so that Eq. (2.14) is consistent with Eq. (2.15).
  4. [Section 3.1, Eqs. (3.3)-(3.4)] The definition of \bar I2(\bar x) in Eq. (3.3) and its analytic continuation in Eq. (3.4) would benefit from an explicit statement of which prefactors are stripped and which are part of the conformal block normalization, since the second term of (3.4) has the same (1-\bar x)^{-1+iλ4} factor as \bar I1(\bar x).
  5. [Section 4.2] The OPEs (4.19) and (4.20) are identical to (3.20) and (3.21); it would help the reader if the paper stated explicitly that the single-valued shadow correlator and the inverted-shadow correlator yield the same leading OPEs.
  6. [General] There are several typographical inconsistencies in the subscripts and superscripts of the conformal blocks, e.g. K21_34 vs K12_34 and K24_31 vs K42_31; a careful proofreading pass is recommended.

Circularity Check

0 steps flagged · score 0.0 of 10

The paper is a self-contained computation from the known MHV graviton amplitude; the single-valued completion is an explicitly flagged ansatz whose consequences are derived, not assumed, so no circularity is found.

full rationale

The derivation chain starts from the known celestial MHV graviton amplitude (2.1) and computes its shadow transform (2.9) by an explicit integral; the conformal block decompositions in Section 2 are direct hypergeometric manipulations of that integral. The single-valued completion (3.5)-(3.6) is explicitly constructed: S2 is fixed by the requirement that the branch cut visible in the analytic continuation (3.1)-(3.4) cancels, and the paper transparently states that the new J=-1 operator 'is a consequence of the shadow correlator single-valuedness (3.5)' and leaves its bulk interpretation open. An ansatz chosen for convenience, with its consequences computed explicitly, is not circular: none of the downstream block decompositions, OPEs, or the inverse-shadow amplitude (4.7)-(4.9) is used as input to justify the completion. The integral representation (3.25) is an identity matching the explicit form (3.5), and the double copy (4.18) is read off from the resulting closed form (4.9), not imposed. The citation of [56] for the single-valued construction is self-citation, but [56] is an independent published gluon computation whose method is re-derived here, so it is not load-bearing in a circular sense. The unresolved physical interpretation of the J=-1 operator is a limitation, not a circularity.

Assumptions & free parameters 3 free parameters · 4 assumptions · 1 invented entities

The central computation begins from the known MHV graviton amplitude (an input from prior literature). The new output is the shadow transform and block decomposition, which are genuine derivations. The single-valued completion is an ansatz whose only justification is cancellation of monodromy and agreement with the gluon construction, which introduces a circularity burden because the subsequent OPEs are read off from that same constructed object. The only invented entity is the J=-1 operator, which is a completion artifact that is not independently evidenced.

free parameters (3)
  • soft-shadow tuning condition lambda1 = i (i.e., Delta1 = 1, shadow conformally soft) = lambda1 = i
    The main single-valued construction is done only in the soft shadow limit lambda1 = i; the general-lambda1 single-valued completion is not constructed. This restriction is chosen for tractability, and the paper explicitly says the soft case yields the series of interesting consequences.
  • single-valued completion coefficients (choice of S2/I2 combination) = coefficients in b_k and related terms
    The completion (3.5) is fixed by requiring cancellation of the (1-x)^(-1) branch-cut term, leaving no free numeric parameter in the final formula but the overall normalization (2+i*lambda2)B(i*lambda3,i*lambda4)/pi is chosen to match the leading OPE. This is a hand-fitted completion ansatz, not a parameter-free derivation.
  • normalization of the inverted single-valued amplitude (overall constant) = (2+i*lambda2) B(i*lambda3, i*lambda4)/pi
    The overall normalization of (4.8)/(4.9) is chosen so the leading OPE reproduces the standard celestial OPE coefficient B(Delta3-1, Delta4-1). It is not derived from an amplitude computation.
assumptions (4)
  • domain assumption Tree-level MHV celestial four-graviton amplitude (2.1) from [10,83] is the correct starting point.
    Used as the input to the shadow transform; the paper computes transforms of this known amplitude, not a new bulk amplitude.
  • domain assumption Shadow transform and shadow conformal basis are the right resolution of distributional support (method of [55]).
    Assumed at the outset in Sec. 2; the shadow transform kernel (2.5) is taken as the standard one.
  • ad hoc to paper The single-valued completion of a shadow correlator is obtained by adding the S2*I2 term, following the gluon construction of [56].
    The completion (3.5) is an ansatz; the paper checks that it is single-valued near 1 and infinity, but does not derive it from a physical principle or bulk computation.
  • standard math Conformal block decompositions use the standard SL(2,C) block formula (2.18) from [86] and the hypergeometric identities (2.21)-(2.23) and Appell identity (2.31).
    These are established mathematical tools; the specific applications are the paper's work.
invented entities (1)
  • J = -1 operator O^epsilon_{Delta3+Delta2-1,J=-1} appearing in the opposite-helicity graviton OPE
    purpose: Needed to make the single-valued completion crossing symmetric; it also suggests a possible coupling to photons/background fields.
    The operator appears in the OPE (3.21)/(4.20) as a consequence of single-valuedness. The paper does not provide an independent derivation of its existence from bulk physics; it is a new state in a constructed correlator, and its interpretation is left open.

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Pith. "Pith review of Conformal blocks from celestial graviton amplitudes." pith.science (2026). https://pith.science/paper/CP2HZBLH

@misc{pith2026250105805,
  author       = {Pith},
  title        = {Pith review of: Conformal blocks from celestial graviton amplitudes},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/CP2HZBLH}},
  note         = {Machine review of arXiv:2501.05805}
}
read the original abstract

Four-point gluon and graviton correlators in celestial holography are famously non-analytic, having distributional support. In this work, we propose an alternative graviton correlator that is analytic and displays several desirable properties. We compute the four-point correlator involving one graviton shadow operator and three graviton primary operators from the celestial four-point graviton amplitudes at tree-level. We perform the conformal block decomposition for the shadow correlator in the compatible channel. For the case when the shadow operator is conformally soft, we compute the single-valued completion of the shadow correlator and perform the conformal block decomposition of the single-valued shadow correlator in all channels. We find an integral representation of the single-valued shadow correlator, which allows us to invert the shadow transform to find the single-valued celestial graviton amplitude. We study various properties of the single-valued celestial graviton amplitude. Interestingly, it exhibits a double copy structure in relation to its counterpart gluon amplitude.

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Forward citations

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Reference graph

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