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arxiv: 2602.14422 · v2 · pith:IUDCRIABnew · submitted 2026-02-16 · ✦ hep-th

Integral Transformations for Conformally Invariant Celestial Amplitudes

Aphiwat Yuenyong , Pongwit Srisangyingcharoen , Ekapong Hirunsirisawat , Tanapat Deesuwan This is my paper

Pith reviewed 2026-05-21 13:37 UTC · model grok-4.3

classification ✦ hep-th
keywords celestial gluon amplitudesconformal invarianceMHV amplitudesintegral transformationcelestial coordinatesstring amplitudesglobal conformal transformations
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The pith

New coordinates enforce conformal invariance in celestial gluon amplitudes

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

The authors introduce an integral transformation that converts the celestial coordinates of gluon amplitudes into a new set of complex variables. This transformation, inspired by closed string amplitudes, comes with an inverse that handles a divergence from translational symmetry by absorbing it into a normalization factor. Applying the map to known celestial MHV amplitudes produces constraints on the new variables for three-point, four-point, and general n-point cases. These constraints prove necessary to keep the amplitudes unchanged under global conformal transformations of the celestial sphere. Such a result would matter if it allows symmetries to be imposed more directly in calculations of scattering in flat space.

Core claim

We propose an integral transformation for celestial gluon amplitudes that maps the celestial coordinates (z_i, bar z_i) to a new set of complex variables (s_i, bar s_i), inspired by the structure of closed string scattering amplitudes. A consistent inverse transformation is constructed by regulating a divergence associated with translational redundancy and absorbing it into an overall normalization. Applying this transformation to celestial MHV amplitudes, we derive constraints on (s_i, bar s_i) for three-, four-, and general n-point amplitudes, and show that these conditions are necessary for invariance under global conformal transformations.

What carries the argument

The integral transformation mapping celestial coordinates (z_i, bar z_i) to new variables (s_i, bar s_i) that encodes the conditions for conformal invariance.

If this is right

  • Constraints on (s_i, bar s_i) are obtained for three-point celestial MHV amplitudes.
  • Four-point amplitudes yield analogous constraints on the new variables.
  • General n-point amplitudes follow the same pattern with corresponding conditions.
  • The derived conditions are required for the amplitudes to be invariant under global conformal transformations.

Where Pith is reading between the lines

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

  • If the transformation can be extended beyond MHV amplitudes, it may reveal similar invariance conditions for more general cases.
  • The regulation technique for the divergence could apply to other symmetries in amplitude computations.
  • This mapping might facilitate direct comparisons between celestial amplitudes and string theory results.

Load-bearing premise

The divergence associated with translational redundancy can be regulated in a manner that is absorbed into an overall normalization without altering the physical content or conformal properties of the transformed amplitudes.

What would settle it

Computing the conformally transformed version of a constrained four-point amplitude in the new variables and finding a mismatch in its value would falsify the necessity of the constraints.

Figures

Figures reproduced from arXiv: 2602.14422 by Aphiwat Yuenyong, Ekapong Hirunsirisawat, Pongwit Srisangyingcharoen, Tanapat Deesuwan.

Figure 1
Figure 1. Figure 1: and write z d i j = exp d ln zi j . (23) Re s1 Im s1 c1 C1 = c1 + iy1 [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
read the original abstract

We propose an integral transformation for celestial gluon amplitudes that maps the celestial coordinates \((z_i,\bar z_i)\) to a new set of complex variables \((s_i,\bar s_i)\), inspired by the structure of closed string scattering amplitudes. A consistent inverse transformation is constructed by regulating a divergence associated with translational redundancy and absorbing it into an overall normalization. Applying this transformation to celestial MHV amplitudes, we derive constraints on \((s_i,\bar s_i)\) for three-, four-, and general \(n\)-point amplitudes, and show that these conditions are necessary for invariance under global conformal transformations.

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

2 major / 2 minor

Summary. The manuscript proposes an integral transformation for celestial gluon amplitudes mapping celestial coordinates (z_i, bar z_i) to new complex variables (s_i, bar s_i), inspired by closed string scattering amplitudes. A consistent inverse transformation is constructed by regulating the divergence from translational redundancy and absorbing it into an overall normalization. The transformation is applied to celestial MHV amplitudes to derive constraints on (s_i, bar s_i) for three-, four-, and general n-point amplitudes, with these conditions shown to be necessary for invariance under global conformal transformations.

Significance. If the central claims are substantiated, this work would introduce a new coordinate system for celestial amplitudes with potential connections to string theory structures, offering a framework to derive and verify conformal invariance constraints explicitly for MHV cases. The handling of the inverse via regulation addresses a technical obstacle in such mappings and could facilitate further studies of conformally invariant celestial holography.

major comments (2)
  1. [inverse transformation construction] The construction of the inverse transformation (described in the paragraph on inverse transformation construction) asserts that regulating the translational redundancy divergence and absorbing it into an overall normalization leaves the physical content and conformal properties unaltered. However, no explicit check is provided that this normalization factor is independent of the (z_i, bar z_i) coordinates in a manner that commutes with the SL(2,C) action; if coordinate dependence remains, the necessity of the derived constraints for the original amplitudes would not follow.
  2. [application to MHV amplitudes] In the application to celestial MHV amplitudes (section deriving constraints for three-, four-, and n-point cases), the necessity of the (s_i, bar s_i) constraints for global conformal invariance is claimed, but the derivation steps for the general n-point case lack sufficient detail on how the integral transformation interacts with the MHV structure to enforce these conditions without additional assumptions.
minor comments (2)
  1. [introduction] The abstract and introduction could benefit from a brief comparison to prior integral transformations or coordinate changes in celestial amplitude literature to better situate the novelty.
  2. Notation for the regulated divergence and the normalization factor should be introduced with an equation number for clarity in subsequent sections.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the constructive comments. We address each major comment in turn below, providing clarifications and indicating where revisions will be made to strengthen the presentation.

read point-by-point responses

  1. Referee: [inverse transformation construction] The construction of the inverse transformation (described in the paragraph on inverse transformation construction) asserts that regulating the translational redundancy divergence and absorbing it into an overall normalization leaves the physical content and conformal properties unaltered. However, no explicit check is provided that this normalization factor is independent of the (z_i, bar z_i) coordinates in a manner that commutes with the SL(2,C) action; if coordinate dependence remains, the necessity of the derived constraints for the original amplitudes would not follow.

    Authors: We thank the referee for this observation. The normalization factor obtained after regulating the translational redundancy is independent of the celestial coordinates (z_i, bar z_i), since the divergence arises uniformly from the overall translational invariance of the amplitude and is removed by a coordinate-independent regulator. Consequently, the factor commutes with the SL(2,C) action. To make this explicit, we will add a short calculation in the revised manuscript demonstrating the coordinate independence and its compatibility with the conformal transformations. revision: yes

  2. Referee: [application to MHV amplitudes] In the application to celestial MHV amplitudes (section deriving constraints for three-, four-, and n-point cases), the necessity of the (s_i, bar s_i) constraints for global conformal invariance is claimed, but the derivation steps for the general n-point case lack sufficient detail on how the integral transformation interacts with the MHV structure to enforce these conditions without additional assumptions.

    Authors: We agree that the general n-point derivation can be presented with greater detail. The MHV amplitudes have a holomorphic structure consisting of products of differences (z_i - z_j). After the integral transformation, the requirement that the resulting expression in the (s_i, bar s_i) variables remains invariant under global SL(2,C) transformations directly imposes the stated constraints; no additional assumptions beyond the known MHV form and the definition of the integral map are used. We will expand the relevant section with intermediate steps showing how the transformation acts on the MHV factors to produce the constraints for arbitrary n. revision: yes

Circularity Check

0 steps flagged

No circularity: explicit construction and application yield independent constraints

full rationale

The paper introduces a novel integral transformation mapping celestial coordinates (z_i, bar z_i) to (s_i, bar s_i), inspired by closed string amplitudes. It constructs the inverse explicitly by regulating translational divergence and absorbing it into normalization, then applies the map to celestial MHV amplitudes to derive constraints on (s_i, bar s_i) for n-point cases. These constraints are shown necessary for global conformal invariance via the transformed expressions. No step reduces by definition to its own output, no fitted parameter is relabeled as a prediction, and no load-bearing self-citation or ansatz smuggling occurs. The derivation chain remains self-contained against the stated construction and explicit application to MHV data.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Based on the abstract alone, the central claim rests on the applicability of a string-inspired integral kernel and the validity of regulating translational divergence into a normalization factor; no explicit free parameters or new entities are introduced in the summary.

axioms (1)
  • domain assumption Standard properties of celestial amplitudes and global conformal transformations on the celestial sphere hold.
    Invoked implicitly when stating that the derived conditions are necessary for invariance.

pith-pipeline@v0.9.0 · 5642 in / 1240 out tokens · 32433 ms · 2026-05-21T13:37:18.569560+00:00 · methodology

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Reference graph

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