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arxiv: 2606.20433 · v1 · pith:LOO5PRLYnew · submitted 2026-06-18 · ✦ hep-th

Shadow Completion in Celestial OPEs

Reiko Liu , Zijian Liu , Wen-Jie Ma This is my paper

Pith reviewed 2026-06-26 15:52 UTC · model grok-4.3

classification ✦ hep-th
keywords celestial OPEshadow transformMellin basiscelestial holographyOPE consistencyshadow completiontree-level amplitudes
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The pith

Celestial OPEs do not close on Mellin-basis exchanges alone and must include shadow-basis representatives of the same bulk particle.

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

The paper shows that celestial operator product expansions, which encode how boundary operators multiply in celestial holography, fail to close consistently when restricted to standard Mellin-basis states. Consistency instead forces each exchanged bulk particle to contribute an additional term through its shadow transform, which supplies a distinct primary operator in the boundary theory. The resulting shadow coefficient is fixed by the ordinary coefficient through a universal shadow factor. This completion holds for scalars and extends to gluons and gravitons, with direct checks in explicit tree-level amplitudes.

Core claim

From OPE consistency, the ordinary celestial OPE does not close on Mellin-basis exchanges alone. Rather, the same exchanged bulk particle must also appear through its shadow-basis representative. This leads to a shadow-completed OPE, with the shadow OPE coefficient fixed by the ordinary collinear coefficient through the universal shadow factor. The shadow transform does not introduce new bulk degrees of freedom yet provides a distinct primary state in the boundary celestial theory.

What carries the argument

Shadow-completed OPE, in which the shadow transform of each Mellin-basis operator supplies a distinct primary whose coefficient is fixed by the universal shadow factor.

If this is right

  • The shadow OPE coefficient is determined by the ordinary collinear coefficient via the universal shadow factor.
  • The same completion applies to gluon and graviton exchanges.
  • The boundary Hilbert space must accommodate both Mellin and shadow representatives for each bulk particle.
  • The completion is verified explicitly in scalar 2 to n tree-level amplitudes and in a five-point example.

Where Pith is reading between the lines

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

  • Correlation functions built from the completed OPE would automatically incorporate shadow contributions at every order.
  • The structure may link to known relations between different conformal weights in two-dimensional CFTs.
  • Higher-loop or higher-point celestial amplitudes could provide further tests of the required shadow terms.

Load-bearing premise

That OPE consistency in the boundary theory requires the algebra to close on all primary states generated by the shadow transform without introducing new bulk degrees of freedom.

What would settle it

A concrete four-point celestial amplitude computed using only Mellin exchanges that nevertheless reproduces the full bulk result without any shadow contribution would contradict the claim.

Figures

Figures reproduced from arXiv: 2606.20433 by Reiko Liu, Wen-Jie Ma, Zijian Liu.

Figure 1
Figure 1. Figure 1: Tree-level 2 → n exchange contribution used to extract the 12-channel OPE from the regular celestial amplitude. The first incoming scalar is mass-regularized, the internal line denotes the s-channel propagator, and the shaded blob represents the remaining 1 → n subamplitude. 4.1 Regular celestial amplitude and split representation Regularized conformal basis. We briefly recall the regularized conformal bas… view at source ↗
Figure 2
Figure 2. Figure 2: Comb-channel tree diagram for the 2 → 3 scalar example. To check the shadow OPE analysis of Section 4.2, we now consider a concrete 2 → 3 example in massless ϕ 3 theory. The leading contribution to the amplitude is given by the tree-level exchange diagram in [PITH_FULL_IMAGE:figures/full_fig_p019_2.png] view at source ↗
read the original abstract

We argue that celestial OPEs must be supplemented by shadow-basis operators. Although the shadow transform does not introduce new bulk degrees of freedom, it provides a distinct primary state in the boundary celestial theory. From OPE consistency, we show that the ordinary celestial OPE does not close on Mellin-basis exchanges alone. Rather, the same exchanged bulk particle must also appear through its shadow-basis representative. This leads to a shadow-completed OPE, with the shadow OPE coefficient fixed by the ordinary collinear coefficient through the universal shadow factor. We discuss the corresponding boundary Hilbert-space interpretation, extend this argument to gluons and gravitons, and verify the shadow exchange directly in tree-level regular celestial amplitudes, including a scalar $2\rightarrow n$ analysis and an explicit five-point example.

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.

Referee Report

0 major / 3 minor

Summary. The manuscript argues that celestial OPEs must be supplemented by shadow-basis operators. Although the shadow transform introduces no new bulk degrees of freedom, it supplies a distinct primary in the boundary theory. OPE consistency on the celestial sphere implies that the ordinary celestial OPE does not close on Mellin-basis exchanges alone; the same bulk particle must also appear via its shadow-basis representative. The resulting shadow-completed OPE has its shadow coefficient fixed by the ordinary collinear coefficient through a universal shadow factor. The construction is verified directly in tree-level regular celestial amplitudes (scalar 2 o n processes and an explicit five-point example) and extended to gluons and gravitons, with a discussion of the corresponding boundary Hilbert-space interpretation.

Significance. If the central consistency argument holds, the result supplies a systematic completion of celestial OPEs that resolves closure without new bulk degrees of freedom. The explicit verifications in tree-level amplitudes and the parameter-free fixing of the shadow coefficient via a universal factor are strengths. The work could affect operator-algebra constructions in celestial holography.

minor comments (3)
  1. [§2] §2: the precise normalization convention for the shadow transform relative to the Mellin transform should be stated explicitly, including any overall constants that enter the universal factor.
  2. [§4.2] The five-point example in §4.2: while the presence of the shadow exchange is shown, the numerical coefficient matching to the universal factor is presented only graphically; an analytic comparison would improve clarity.
  3. [§5] The boundary Hilbert-space discussion in §5 would benefit from a short diagram or table contrasting the Mellin and shadow primaries for a single bulk exchange.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our work and the recommendation of minor revision. No specific major comments were raised in the report.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The derivation proceeds from requiring OPE consistency on the celestial sphere, showing that Mellin-basis exchanges alone do not close and that the same bulk particle must appear via its shadow representative, with the shadow coefficient fixed by the ordinary coefficient through a universal shadow factor. This is independently verified by explicit computation in tree-level regular celestial amplitudes (scalar 2→n and five-point cases) and extended to gluons/gravitons. No step reduces by construction to a fitted parameter, self-definition, or load-bearing self-citation; the shadow transform is treated as providing a distinct boundary primary without new bulk degrees of freedom, yielding an independent result rather than a renaming or ansatz smuggling.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Based on abstract only; the argument rests on domain assumptions about OPE closure and the distinctness of shadow primaries.

axioms (1)
  • domain assumption OPE consistency requires the algebra to close on all relevant boundary primary states including shadows
    Invoked to conclude that Mellin-basis alone is insufficient.

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Forward citations

Cited by 1 Pith paper

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