arxiv: 2604.27736 · v1 · submitted 2026-04-30 · ✦ hep-th

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A perturbative Liouville prescription for the celestial three-gluon amplitude

Grzegorz Biskowski , Franco Ferrari , Marcin R. Piatek , Artur R. Pietrykowski

Authors on Pith no claims yet

Pith reviewed 2026-05-07 06:09 UTC · model grok-4.3

classification ✦ hep-th

keywords celestial amplitudesLiouville theoryDOZZ three-point functionYang-Mills amplitudesperturbative expansionsoft limitsMellin transformdilaton background

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The pith

Resolving the Liouville-Mellin map ambiguity yields a controlled b-expansion of the celestial three-gluon amplitude whose leading term recovers tree-level Yang-Mills.

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

The paper derives the first two orders in a perturbative expansion in the Liouville coupling b for the celestial three-gluon amplitude in a dilaton background. The starting point is the exact Liouville DOZZ three-point function, but an ambiguity in how Liouville parameters map to Mellin variables must first be fixed by imposing global conformal covariance and consistency with the semiclassical limit. With the dictionary and operator normalizations uniquely determined, the leading term reproduces the known tree-level Yang-Mills amplitude when the total momentum is small. The order-b squared correction admits a closed-form expression in modified Bessel functions; its soft limit cleanly separates into geometric and logarithmic pieces. This construction supplies a systematic, computable route to finite-b corrections beyond the original STZ tree-level proposal.

Core claim

After fixing the Liouville-Mellin parameter dictionary by global conformal covariance and semiclassical compatibility, the b squared expansion of the full Liouville DOZZ three-point function produces a leading term that matches the tree-level Yang-Mills amplitude in the small total momentum limit, while the first correction term is expressible in closed form using modified Bessel functions whose soft-gluon limit separates into geometric and logarithmic contributions.

What carries the argument

The uniquely fixed map between Liouville parameters and Mellin variables, which converts the DOZZ three-point function into a controlled perturbative series in the Liouville coupling b.

If this is right

The same fixed dictionary supplies a systematic algorithm for computing higher orders in the b expansion of the same amplitude.
The closed-form one-loop term gives an explicit prediction for the first quantum correction to the celestial three-gluon vertex in the presence of a dilaton.
The separation of the soft limit into geometric and logarithmic pieces suggests that each contribution may be traceable to distinct sectors of the dual celestial theory.
The construction extends the STZ tree-level matching to a perturbative regime that remains computable order by order.

Where Pith is reading between the lines

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

The same resolution procedure could be applied to higher-point celestial amplitudes or to amplitudes involving different gauge-theory particles.
Numerical evaluation of the Bessel-function expression in the soft region could be compared with results from other celestial-holography techniques to test the map.
If the geometric-logarithmic split persists at higher orders, it may point to a natural factorization in the dual two-dimensional theory.

Load-bearing premise

That demanding global conformal covariance together with semiclassical compatibility uniquely determines both the Liouville-Mellin variable identification and the operator normalization.

What would settle it

An independent computation of the one-loop celestial three-gluon amplitude in the dilaton background, via standard perturbative methods, that fails to reproduce the modified-Bessel expression for the order-b squared correction.

read the original abstract

We study the celestial three-gluon amplitude in a dilaton background through the Mellin-Liouville formulation proposed by Stieberger, Taylor and Zhu (STZ). The original map contains an ambiguity in the identification of Liouville and Mellin variables; we resolve it by requiring global conformal covariance and compatibility with the semiclassical expansion of Liouville theory. This uniquely fixes the operator normalization and the parameter dictionary, and leads to a controlled expansion in the Liouville coupling $b$. Starting from the full Liouville DOZZ three-point function, we derive the leading and first subleading terms in the $b^2$ expansion. The leading term reproduces the tree-level Yang-Mills amplitude in the small total momentum limit, as anticipated in the STZ proposal. The one-loop correction can be written in closed form using modified Bessel functions, and its soft limit exhibits a clear separation into geometric and logarithmic contributions. The resulting framework extends the STZ proposal to finite-$b$ corrections in a consistent and computable way.

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.

Desk Editor's Note private letter to a colleague

They fix the map ambiguity in the STZ Liouville setup and extract an explicit closed-form b^2 correction for the three-gluon celestial amplitude.

read the letter

The paper resolves the identification between Liouville momenta and Mellin variables by requiring global conformal covariance plus consistency with the semiclassical small-b limit of Liouville theory. That choice fixes the operator normalizations and the parameter dictionary, after which they expand the exact DOZZ three-point function to order b squared. The leading term recovers the expected tree-level Yang-Mills celestial amplitude in the small-total-momentum regime. The O(b^2) piece comes out in closed form with modified Bessel functions, and its soft limit cleanly separates geometric and logarithmic contributions. This is the concrete new content: an explicit, computable first correction rather than just a formal proposal. The derivation stays controlled because it starts from the known DOZZ formula and applies only standard symmetry and semiclassical requirements; there is no data fitting or self-referential step. The calculation is therefore reproducible in principle and internally consistent. The main limitation is scope. Everything is done for three gluons only, so the map has not yet been stress-tested on higher-point functions or other amplitudes. The physical meaning of the dilaton background and the size of b in a realistic celestial setting also remain open for later work, but that is not a flaw in the present calculation. This is a narrow but solid technical extension for people already working on Liouville methods in celestial holography. A reader who wants explicit perturbative expressions in this framework will find the Bessel-function result and the soft-limit decomposition useful. It deserves peer review because the central steps are standard, the output is new and checkable, and no load-bearing assumption looks uncontrolled.

Referee Report

0 major / 3 minor

Summary. The manuscript develops a perturbative Liouville prescription for the celestial three-gluon amplitude in a dilaton background. The authors resolve the ambiguity in the Liouville-Mellin variable map by requiring global conformal covariance and compatibility with the semiclassical (small-b) expansion of Liouville theory. Starting from the exact DOZZ three-point function, they derive the leading and O(b²) terms in the expansion; the leading term reproduces the tree-level Yang-Mills amplitude in the small total momentum limit, while the subleading correction is expressed in closed form using modified Bessel functions whose soft limit separates geometric and logarithmic contributions.

Significance. If the derivation holds, the work supplies a controlled, parameter-free framework for finite-b corrections to celestial amplitudes, extending the STZ proposal. Notable strengths include the use of the exact DOZZ formula (no ad-hoc approximations) and the closed-form Bessel expression for the one-loop term, which enables analytic access to loop-level effects and a clean soft-limit decomposition. This could facilitate systematic higher-point calculations and strengthen the link between Liouville theory and celestial holography.

minor comments (3)

The explicit dictionary mapping Liouville momenta to Mellin variables (fixed by the covariance and semiclassical requirements) should be displayed in a table or boxed equation to aid reproducibility and verification of the leading-order match.
In the soft-limit analysis of the O(b²) term, a short comparison of the separated geometric and logarithmic pieces to the known soft theorems of Yang-Mills theory would clarify the physical interpretation.
A brief remark on the radius of convergence of the b-expansion or the regime where the small-total-momentum limit remains valid would help readers assess the practical range of the results.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of our manuscript, including the recognition of our resolution of the Liouville-Mellin map ambiguity via conformal covariance and semiclassical consistency, the controlled b-expansion derived from the exact DOZZ function, and the closed-form modified Bessel expression for the O(b²) term. The referee recommends minor revision but raises no specific major comments or criticisms. We will incorporate any minor editorial or presentational suggestions in the revised version.

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The paper begins with the standard, externally established DOZZ three-point function of Liouville theory and executes a perturbative series expansion in the Liouville parameter b. The sole additional input is the resolution of the Liouville-Mellin variable map via the independent requirements of global conformal covariance and consistency with the known semiclassical (small-b) limit of Liouville theory; these are standard symmetry and limiting-case constraints, not data fits or self-referential definitions. The leading-order term is verified to reproduce the tree-level Yang-Mills celestial amplitude in the small-total-momentum regime (a consistency check with the prior STZ proposal by unrelated authors), while the O(b²) correction is obtained in closed form. No step reduces by construction to a fitted parameter, a self-citation chain, or an ansatz smuggled from the authors' own prior work. The derivation therefore stands on external benchmarks and is not circular.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the DOZZ three-point function from Liouville theory (standard input) and the assumption that the ambiguity in the Liouville-Mellin map is uniquely resolved by global conformal covariance plus semiclassical compatibility; no new entities are postulated and no parameters are fitted to data.

axioms (2)

domain assumption Global conformal covariance must hold for the Liouville-Mellin map
Invoked to resolve the variable-identification ambiguity.
domain assumption The map must be compatible with the semiclassical expansion of Liouville theory
Used to uniquely fix the operator normalization and parameter dictionary.

pith-pipeline@v0.9.0 · 5490 in / 1665 out tokens · 132546 ms · 2026-05-07T06:09:24.086567+00:00 · methodology

discussion (0)

Reference graph

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