arxiv: 2603.26199 · v2 · submitted 2026-03-27 · ✦ hep-ph

Recognition: 2 theorem links

· Lean Theorem

Complete Next-to-Next-to-Leading-Order QCD Correction to J/psi to 3γ Decay

Chao Zeng , Bin Gong , Jian-Xiong Wang , Ruichang Niu , Xu-Dong Huang , Cong Li

Authors on Pith no claims yet

Pith reviewed 2026-05-14 23:03 UTC · model grok-4.3

classification ✦ hep-ph

keywords NNLO QCD correctionJ/psi to three photonsNRQCD factorizationdecay widthcharmoniumrenormalization scaleBESIII measurement

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The pith

Amplitude-level NRQCD factorization yields the first complete NNLO QCD correction to J/ψ → 3γ, producing a positive decay width of 0.96 eV that aligns with BESIII data.

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

The paper proposes amplitude-level NRQCD factorization as a way to obtain positive rates for exclusive decays where conventional perturbative QCD calculations turn negative. It carries out the first full next-to-next-to-leading-order QCD computation for the J/ψ decaying into three photons, adding the known relativistic corrections up to order alpha_s v squared. The resulting width sits at 0.96 eV with a large asymmetric uncertainty driven by the choice of renormalization scale. The same framework is applied to the upsilon decay into three photons. This approach directly addresses a long-standing sign problem in these processes and brings theory into closer agreement with precise experimental measurements.

Core claim

The central claim is that the complete NNLO QCD correction to the partial width Gamma(J/psi to 3 gamma) equals 0.96^{+4.32}_{-0.13} eV when combined with the O(alpha_s v^2) relativistic correction, obtained via amplitude-level NRQCD factorization; the same procedure gives Gamma(Upsilon to 3 gamma) = 0.0086^{+0.0028}_{-0.0006} eV, and the dominant remaining uncertainty arises from renormalization-scale variation.

What carries the argument

amplitude-level NRQCD factorization, which reorganizes the perturbative expansion so that the decay amplitude is factored before squaring and thereby eliminates unphysical negative rates at higher orders.

If this is right

The partial width for J/ψ → 3γ becomes positive and lies closer to the measured value than all lower-order results.
The same framework produces a definite prediction for the Υ → 3γ width.
Renormalization-scale dependence remains the leading theoretical uncertainty and points to the need for still higher orders.
The method supplies a consistent way to compute other exclusive charmonium and bottomonium decays that previously suffered from negative rates.

Where Pith is reading between the lines

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

The same factorization may be tested on additional exclusive processes such as two-photon decays or production channels where sign problems appear.
If scale variation shrinks at N3LO, it would support the claim that NNLO marks the onset of reliable convergence for these decays.
The approach could be combined with lattice inputs for the wave function at the origin to reduce parametric uncertainties further.

Load-bearing premise

The assumption that amplitude-level NRQCD factorization supplies a systematic prescription that removes negative rates and that the perturbative series has begun to converge once the NNLO term is included.

What would settle it

A next-to-next-to-next-to-leading-order calculation that shifts the central value of the width by an amount comparable to or larger than the present NNLO correction would indicate that the series has not yet stabilized.

Figures

Figures reproduced from arXiv: 2603.26199 by Bin Gong, Chao Zeng, Cong Li, Jian-Xiong Wang, Ruichang Niu, Xu-Dong Huang.

**Figure 2.** Figure 2: FIG. 2. Renormalization scale dependence of partial decay [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗

read the original abstract

We address the long-standing problem of negative decay and production rates in perturbative QCD for exclusive processes by proposing amplitude-level NRQCD factorization as a systematic prescription. Building on this, we present the first complete next-to-next-to-leading-order (NNLO) QCD correction to the decay $J/\psi \to 3\gamma$. The resulting partial width, $\Gamma(J/\psi \to 3\gamma) = 0.96^{+4.32}_{-0.13}$ eV, combines this NNLO contribution with the known up to $\mathcal{O}(\alpha_s v^2)$ relativistic correction and shows markedly improved agreement with the high-precision BESIII measurement. In the same way, $\Gamma(\Upsilon \to 3\gamma) = 0.0086^{+0.0028}_{-0.0006}$ eV is obtained. The dominant theoretical uncertainty originates from the renormalization scale variation, underscoring the challenge of perturbative convergence at this order and the necessity for future higher-order calculations.

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

The paper delivers the first complete NNLO QCD correction to J/ψ → 3γ via amplitude-level NRQCD factorization, but the huge asymmetric scale uncertainty shows the series has not stabilized yet.

read the letter

The main thing to know is that the authors have completed the first NNLO QCD calculation for the decay J/ψ to three photons and combined it with relativistic corrections to get a width closer to the BESIII data. The catch is that the renormalization scale uncertainty remains huge and asymmetric at this order. They introduce amplitude-level NRQCD factorization as a way to avoid the negative decay rates that appear in conventional pQCD treatments of exclusive processes. This seems to work at the level they have calculated, and they apply the same method to the Υ decay as well. The technical effort involved in evaluating the two-loop contributions and reducing the integrals is substantial, and the paper shows they have done that work. The result they quote is 0.96 eV with errors +4.32 and -0.13 from scale variation. The central value improves the agreement with experiment, but the error band is wider than the value itself. This points to slow perturbative convergence, which is a common issue in these heavy quarkonium decays where the strong coupling is not small. The soft spot is exactly that scale dependence. If the NNLO term was supposed to stabilize the prediction, the variation should have shrunk more than it did. The assumption that the new factorization prescription systematically resolves the negativity problem holds for the orders they computed, but the remaining scale sensitivity suggests uncalculated higher orders are still important. This paper is for specialists in quarkonium physics and higher-order perturbative calculations. A reader who needs an updated theoretical prediction for this specific channel will find it directly useful, though they will have to carry the large theory uncertainty. It deserves a serious referee. The calculation is new and the details of the amplitude factorization need expert checking, even if the final error bar is not as small as one would like. Recommendation: Send it to peer review rather than desk reject.

Referee Report

2 major / 1 minor

Summary. The manuscript claims to resolve negative decay rates in perturbative QCD for exclusive processes via amplitude-level NRQCD factorization. It presents the first complete NNLO QCD correction to J/ψ → 3γ, yielding Γ(J/ψ → 3γ) = 0.96^{+4.32}_{-0.13} eV that combines with O(α_s v^2) relativistic corrections to show markedly improved agreement with BESIII data; a parallel result is given for Υ → 3γ. The dominant uncertainty is stated to arise from renormalization-scale variation.

Significance. If the scale dependence were shown to stabilize and the factorization prescription were validated at this order, the result would constitute a notable technical advance in higher-order calculations for quarkonium radiative decays, potentially enabling more reliable predictions for similar exclusive processes.

major comments (2)

[Abstract] Abstract: The quoted result Γ(J/ψ → 3γ) = 0.96^{+4.32}_{-0.13} eV has an upper renormalization-scale uncertainty (+4.32) more than four times larger than the central value. This asymmetric band directly undermines the claim of 'markedly improved agreement' with the BESIII measurement, because the theoretical prediction spans a range too broad for a conclusive comparison.
[Abstract] Abstract and introduction: The assertion that amplitude-level NRQCD factorization provides a systematic prescription resolving negative rates is load-bearing for the central claim, yet the persistence of large μ_R sensitivity at NNLO indicates that leading scale logarithms are not fully canceled; an explicit demonstration that the factorization eliminates this dependence at the present order is required.

minor comments (1)

[Abstract] The precise interval over which the renormalization scale μ_R is varied to obtain the quoted asymmetric uncertainty should be stated explicitly.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive comments. We address each major comment below and have revised the manuscript to incorporate the suggestions where appropriate.

read point-by-point responses

Referee: [Abstract] Abstract: The quoted result Γ(J/ψ → 3γ) = 0.96^{+4.32}_{-0.13} eV has an upper renormalization-scale uncertainty (+4.32) more than four times larger than the central value. This asymmetric band directly undermines the claim of 'markedly improved agreement' with the BESIII measurement, because the theoretical prediction spans a range too broad for a conclusive comparison.

Authors: We acknowledge the validity of this observation. The large upper uncertainty reflects the substantial renormalization scale dependence remaining at NNLO, which is a known challenge in higher-order calculations for quarkonium decays. Although the central value shows better alignment with the BESIII data, the broad theoretical band does limit the strength of the comparison. In response, we will revise the abstract to use 'improved agreement' instead of 'markedly improved agreement' and add a clarifying sentence about the dominant uncertainty from scale variation. This is a partial revision focused on the presentation. revision: partial
Referee: [Abstract] Abstract and introduction: The assertion that amplitude-level NRQCD factorization provides a systematic prescription resolving negative rates is load-bearing for the central claim, yet the persistence of large μ_R sensitivity at NNLO indicates that leading scale logarithms are not fully canceled; an explicit demonstration that the factorization eliminates this dependence at the present order is required.

Authors: The amplitude-level NRQCD factorization offers a systematic way to factorize the decay amplitude, which by construction prevents the appearance of negative decay rates that can occur in direct perturbative calculations of exclusive processes. The large μ_R sensitivity at NNLO does indicate that not all scale logarithms are resummed at this order. We will include in the revised manuscript an explicit comparison of the scale dependence at LO, NLO, and NNLO orders, demonstrating the reduction achieved and clarifying that the factorization addresses the negativity issue while scale dependence is mitigated order-by-order. This addition will be made to the introduction and results sections. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is a direct perturbative calculation

full rationale

The paper computes the NNLO QCD correction to J/ψ → 3γ using standard Feynman diagram techniques within amplitude-level NRQCD factorization, building explicitly on known lower-order terms. The central result Γ(J/ψ → 3γ) = 0.96^{+4.32}_{-0.13} eV is obtained from this expansion plus O(α_s v^2) relativistic corrections, with the dominant uncertainty arising from renormalization-scale variation rather than any fitted parameter or self-referential definition. No step reduces the output to an input by construction, and the improved agreement with BESIII data is presented as a numerical outcome, not an imposed condition. The derivation chain remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The central result depends on the validity of NRQCD at NNLO and the choice of renormalization scale for the perturbative series.

free parameters (1)

renormalization scale
Varied over a range to estimate uncertainty, which is the dominant error source.

axioms (1)

domain assumption Amplitude-level NRQCD factorization is a valid systematic prescription for exclusive processes
Proposed in the paper to address negative rates.

pith-pipeline@v0.9.0 · 5494 in / 1163 out tokens · 55947 ms · 2026-05-14T23:03:26.788201+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

We present the first complete next-to-next-to-leading-order (NNLO) QCD correction to the decay J/ψ → 3γ. ... amplitude-level NRQCD factorization ... Γ(J/ψ → 3γ) = 0.96^{+4.32}_{-0.13} eV
IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

The dominant theoretical uncertainty originates from the renormalization scale variation

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

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