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arxiv: 2409.05936 · v3 · pith:XC5NLJKHnew · submitted 2024-09-09 · ✦ hep-th

Partial Differential Equations for MHV Celestial Amplitudes in Liouville Theory

Igor Mol This is my paper

Pith reviewed 2026-05-23 21:11 UTC · model grok-4.3

classification ✦ hep-th
keywords celestial amplitudesLiouville theoryMHV amplitudescelestial OPEYang-Mills theoryloop correctionspartial differential equations
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0 comments X

The pith

The b² deformation of the celestial OPE in Liouville theory is isomorphic to the one-loop correction in pure Yang-Mills theory.

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

This paper derives partial differential equations that govern the n-point MHV celestial amplitudes for gluons and gravitons when parametrized by the Liouville coupling constant b. In the semiclassical limit where b approaches zero these amplitudes recover the tree-level MHV functions of Yang-Mills and gravity. The equations make it possible to compute the first corrections at order b squared, which turn out to be logarithmic. For gluons the resulting change in the celestial operator product expansion matches the one-loop correction already known from direct calculations in Yang-Mills theory. The authors therefore propose that an extension of celestial Liouville theory beyond the semiclassical regime captures the one-loop regime of pure Yang-Mills.

Core claim

The paper establishes partial differential equations for the b-parametrized MHV celestial amplitudes and shows that the O(b²) correction to the celestial OPE for gluons is isomorphic to the one-loop correction of the celestial OPE in pure Yang-Mills theory. It proposes that celestial Liouville theory extended beyond the semiclassical limit encodes the one-loop regime of pure Yang-Mills theory. A parallel computation for gravitons yields a deformation whose physical meaning is left open.

What carries the argument

The set of partial differential equations governing the n-point MHV celestial amplitudes parametrized by the Liouville coupling constant b.

If this is right

  • O(b²) corrections to the amplitudes are logarithmic for both gluons and gravitons.
  • The b²-deformed celestial OPE for gluons matches the one-loop OPE correction in Yang-Mills.
  • Celestial Liouville theory can be used to encode loop corrections in Yang-Mills.
  • The corresponding deformation for gravity has an unclear physical interpretation.

Where Pith is reading between the lines

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

  • Higher-order terms in b might correspond to higher-loop corrections in the four-dimensional theories.
  • The PDEs could provide a systematic way to resum or compute all-order celestial amplitudes.
  • Similar techniques might apply to other amplitudes beyond MHV.

Load-bearing premise

That the perturbative expansion in the Liouville coupling b directly corresponds to the perturbative loop expansion in four-dimensional Yang-Mills and gravity theories.

What would settle it

A direct calculation of the one-loop celestial OPE in Yang-Mills theory that differs from the b² deformation obtained from Liouville theory.

read the original abstract

In this note, we continue our study of Liouville theory and celestial amplitudes by deriving a set of partial differential equations governing the $n$-point MHV celestial amplitudes for gluons and gravitons, parametrised by the Liouville coupling constant $b$. These equations provide a systematic framework for computing the perturbative expansion in $b$ of the celestial amplitudes, which are known to reproduce the tree-level MHV $n$-point functions for pure Yang-Mills and Einstein gravity in the semiclassical $b\rightarrow0$ limit. We demonstrate that the $\mathcal{O}(b^{2})$ corrections are logarithmic for both gluons and gravitons. Furthermore, we compute the correction to the celestial operator product expansion (OPE) parametrised by $b^{2}$. In the case of gluons, the resulting deformation of the celestial OPE is shown to be isomorphic to the one-loop correction of the celestial OPE in pure Yang-Mills theory. We then propose that "celestial Liouville theory," extended beyond the semiclassical limit, encodes the one-loop regime of pure Yang-Mills theory. A formally analogous computation is performed for Einstein gravity to ascertain the deformation of the celestial OPE induced by a non-zero Liouville coupling constant. However, as we shall explain, the physical interpretation of this result remains an open problem due to the intricate nature of the loop-corrected holomorphic collinear limit in graviton scattering amplitudes.

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 derives a set of partial differential equations for the n-point MHV celestial amplitudes of gluons and gravitons in Liouville theory, parametrized by the Liouville coupling b. These PDEs are used to recover the tree-level MHV amplitudes in the semiclassical b→0 limit for pure Yang-Mills and Einstein gravity. The O(b²) corrections are shown to be logarithmic, and the b²-deformed celestial OPE is computed explicitly. For gluons, this deformation is isomorphic to the one-loop correction of the celestial OPE in pure Yang-Mills; the authors propose that celestial Liouville theory extended beyond b→0 encodes the one-loop regime of YM. An analogous computation is performed for gravity, but its physical interpretation is left open due to the structure of loop-corrected collinear limits in graviton amplitudes.

Significance. If the b-expansion in Liouville theory can be shown to correspond to the loop expansion in 4d YM, the PDE framework would offer a systematic method to generate higher-order corrections to celestial amplitudes. The explicit isomorphism between the b²-deformed gluon OPE and the known one-loop YM OPE correction is a concrete, verifiable result that strengthens the case at this order. The work builds on the established Liouville-celestial correspondence and provides a concrete computational tool (the PDEs) whose utility depends on the validity of the b-to-loop mapping.

major comments (2)
  1. [Abstract / proposal following OPE computation] The proposal that celestial Liouville theory encodes the one-loop regime of pure Yang-Mills (abstract and final paragraph) rests on the assumption that the Liouville coupling b supplies a parametrization in which successive powers of b map directly to successive loop orders. This mapping is used to interpret the b→0 limit as tree level and the O(b²) term as one-loop, but the manuscript does not derive the dictionary from first principles; the observed OPE isomorphism at O(b²) is consistent with the assumption yet does not independently establish the correspondence for general n-point amplitudes or higher orders in b.
  2. [Derivation of the PDEs (main text)] The derivation of the PDEs governing the b-parametrized MHV celestial amplitudes is presented as the central technical result, yet the available text provides no intermediate steps, explicit form of the PDEs, or verification that they indeed reproduce the known tree-level MHV amplitudes upon taking b→0. Without these details the claim that the PDEs furnish a systematic framework for the perturbative expansion in b cannot be assessed.
minor comments (2)
  1. [Abstract] The abstract states that O(b²) corrections are 'logarithmic' for both gluons and gravitons; a brief indication of the functional form (e.g., log(z) or log of cross-ratios) would clarify the nature of the correction already in the summary.
  2. [Gravity discussion] The gravity computation is described as 'formally analogous' but its interpretation is left open; a short statement of the precise obstruction (beyond the general remark on the holomorphic collinear limit) would help readers assess whether the same PDE framework can be applied or whether additional structures are required.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments. We address each major comment below, clarifying the scope of our proposal and committing to expand the technical details in a revised manuscript.

read point-by-point responses

  1. Referee: [Abstract / proposal following OPE computation] The proposal that celestial Liouville theory encodes the one-loop regime of pure Yang-Mills (abstract and final paragraph) rests on the assumption that the Liouville coupling b supplies a parametrization in which successive powers of b map directly to successive loop orders. This mapping is used to interpret the b→0 limit as tree level and the O(b²) term as one-loop, but the manuscript does not derive the dictionary from first principles; the observed OPE isomorphism at O(b²) is consistent with the assumption yet does not independently establish the correspondence for general n-point amplitudes or higher orders in b.

    Authors: We agree that the b-to-loop-order dictionary is not derived from first principles and remains conjectural. The proposal is motivated solely by the explicit isomorphism between the O(b²) deformation of the gluon celestial OPE and the known one-loop correction in pure Yang-Mills; this matching is presented as supporting evidence rather than a complete proof. The abstract and concluding paragraph frame the statement as a proposal. In revision we will qualify the language to make clear that the correspondence is conjectural, supported at this order by the OPE result, and that further verification for general n-point functions or higher orders in b would be required to establish it more broadly. revision: partial

  2. Referee: [Derivation of the PDEs (main text)] The derivation of the PDEs governing the b-parametrized MHV celestial amplitudes is presented as the central technical result, yet the available text provides no intermediate steps, explicit form of the PDEs, or verification that they indeed reproduce the known tree-level MHV amplitudes upon taking b→0. Without these details the claim that the PDEs furnish a systematic framework for the perturbative expansion in b cannot be assessed.

    Authors: The derivation of the PDEs is the central technical contribution. The note is concise, and we acknowledge that intermediate steps, the explicit PDEs, and the b→0 verification were not spelled out in sufficient detail. In the revised manuscript we will expand the relevant section to include: (i) the key steps deriving the PDEs from the Liouville correlators in the celestial basis, (ii) the explicit form of the PDEs for general n-point MHV amplitudes, and (iii) a direct check that the b→0 limit recovers the known tree-level MHV celestial amplitudes. This will make the systematic perturbative framework fully transparent. revision: yes

Circularity Check

1 steps flagged

Central proposal rests on unverified assumption that b-expansion in Liouville directly parametrizes loop expansion in 4d YM beyond the shown O(b²) OPE match

specific steps
  1. self citation load bearing [Abstract and Introduction (proposal paragraph)]
    "In this note, we continue our study of Liouville theory and celestial amplitudes by deriving a set of partial differential equations governing the n-point MHV celestial amplitudes for gluons and gravitons, parametrised by the Liouville coupling constant b. These equations provide a systematic framework for computing the perturbative expansion in b of the celestial amplitudes, which are known to reproduce the tree-level MHV n-point functions for pure Yang-Mills and Einstein gravity in the semiclassical b→0 limit. ... We then propose that 'celestial Liouville theory,' extended beyond the semicla"

    The b→0 limit reproducing tree-level MHV is stated as 'known' from prior self-cited work; the proposal that the same b-expansion encodes the one-loop regime (and thus that the O(b²) OPE isomorphism implies the full correspondence) reduces to that inherited parametrization assumption rather than an independent first-principles dictionary derived in this paper.

full rationale

The paper derives PDEs for MHV celestial amplitudes in Liouville theory parametrized by b, recovers tree-level in b→0, computes logarithmic O(b²) corrections, and shows the gluon OPE deformation at O(b²) is isomorphic to the one-loop YM correction. The proposal that celestial Liouville encodes the one-loop YM regime therefore depends on the assumption that powers of b map to loop orders. This mapping is inherited from the Liouville-celestial correspondence in the author's prior self-cited work rather than derived here; the O(b²) match is consistent with but does not establish the full dictionary. The gravity case is left open for the same reason. This is self-citation load-bearing for the central claim but the new PDEs and explicit O(b²) computation retain independent content, yielding a moderate circularity score.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The central claim rests on the Liouville parametrization of celestial amplitudes and the correspondence between the b-expansion and loop corrections; no new entities are postulated.

free parameters (1)
  • b
    Liouville coupling constant that parametrizes the celestial amplitudes and controls the perturbative expansion in the derived PDEs.
axioms (2)
  • domain assumption The semiclassical limit b→0 reproduces the tree-level MHV n-point functions for pure Yang-Mills and Einstein gravity.
    Invoked in the abstract as the base case against which the b² corrections are compared.
  • domain assumption Celestial amplitudes admit a parametrization by Liouville theory with coupling b that yields the stated PDEs.
    This is the foundational modeling choice that allows derivation of the governing equations.

pith-pipeline@v0.9.0 · 5780 in / 1743 out tokens · 39308 ms · 2026-05-23T21:11:23.291760+00:00 · methodology

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