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PMC removes scale ambiguities from NNLO predictions for D-wave eta_Q2 to two-photon decays.

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2026-06-25 23:39 UTC pith:AFLVDDAX

load-bearing objection This paper gives the first NNLO PMC results for eta_Q2 to gamma gamma decays, with cleaner scale dependence, but the error budget skips the dominant LDME uncertainties. the 4 major comments →

arxiv 2606.24455 v1 pith:AFLVDDAX submitted 2026-06-23 hep-ph

NNLO QCD Corrections to D-Wave Spin-Singlet Heavy Quarkonia Decay η_(Q2)toγγ via the Principle of Maximum Conformality

Hua Zhou , Xing-Gang Wu , Qing Yu , Sheng-Quan Wang This is my paper
classification hep-ph
keywords NNLO QCD correctionsPrinciple of Maximum Conformalityheavy quarkoniaD-wave statestwo-photon decayNRQCD factorizationeta_c2eta_b2
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

This paper computes the two-photon decays of D-wave spin-singlet heavy quarkonia eta_Q2 at NNLO in QCD using NRQCD factorization. The short-distance coefficients originally carry large renormalization and factorization scale uncertainties. The Principle of Maximum Conformality is applied by recursively solving the renormalization group equations for alpha_s and the D-wave long-distance matrix elements, producing an effective scale Q* that absorbs all nonconformal beta terms. The resulting series becomes scale-invariant and more convergent, giving explicit numerical widths with uncertainties dominated by quark-mass variations. A reader cares because the method supplies cleaner theoretical values for these rare processes that experiments can target.

Core claim

The total decay width factors into perturbatively calculable short-distance coefficients and nonperturbative D-wave long-distance matrix elements. The original NNLO series for the coefficients suffers from sizable scale dependence. Recursively applying the renormalization group equations inside the PMC framework determines an effective strong coupling alpha_s(Q*) consistent with the expansion coefficients, removing divergent renormalon contributions and yielding a scale-invariant perturbative series. The PMC scales are fixed at Q*=1.483 GeV for eta_c2 and Q*=4.246 GeV for eta_b2, producing Gamma(eta_c2->gamma gamma)^PMC = 3.322^{+0.899}_{-0.828} eV and Gamma(eta_b2->gamma gamma)^PMC = 0.0188

What carries the argument

The Principle of Maximum Conformality, which sets the renormalization scale by absorbing all beta-function dependent terms into the running coupling to obtain a conformal, scale-invariant perturbative series for the short-distance coefficients.

Load-bearing premise

The nonperturbative D-wave long-distance matrix elements are known to sufficient accuracy from independent sources and NRQCD factorization remains valid without large velocity-suppressed corrections at NNLO.

What would settle it

A measurement of Br(eta_c2 -> gamma gamma) lying outside the interval 5.6 x 10^{-6} to 9.5 x 10^{-6} would falsify the PMC prediction.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • The decay widths receive their dominant remaining uncertainties from variations in the heavy-quark masses.
  • The branching ratios are (7.463^{+2.020}_{-1.860})x10^{-6} for eta_c2 and (6.460^{+0.481}_{-0.447})x10^{-7} for eta_b2.
  • The perturbative series for the short-distance coefficients converges better after removal of renormalon contributions.
  • NRQCD factorization separates short-distance coefficients from long-distance matrix elements up to NNLO for this channel.

Where Pith is reading between the lines

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

  • These widths can be compared directly with future measurements at e+e- or hadron colliders to test the combined NRQCD+PMC framework.
  • The same PMC procedure can be applied to other rare decays of D-wave or higher-L states once the corresponding LDMEs are available.
  • Agreement with data would support extending the method to N3LO or to processes involving different final states.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

4 major / 0 minor

Summary. The paper computes NNLO QCD corrections to the two-photon decays of D-wave spin-singlet heavy quarkonia η_{Q2} within NRQCD factorization. It decomposes the width into short-distance coefficients (SDCs) and D-wave LDMEs, applies the Principle of Maximum Conformality (PMC) to eliminate renormalization- and factorization-scale dependence by recursively using RGEs for α_s and the LDMEs, determines optimal scales Q*=1.483 GeV (charm) and 4.246 GeV (bottom), and reports scale-invariant numerical results Γ_ηc2→γγ^PMC = 3.322^{+0.899}_{-0.828} eV and Γ_ηb2→γγ^PMC = 0.0188^{+0.0014}_{-0.0013} eV together with the corresponding branching ratios.

Significance. If the central results hold after proper error propagation, the work supplies the first PMC-improved NNLO predictions for these rare decays, removing conventional scale ambiguities and potentially improving convergence of the perturbative series. Such predictions could be compared with future experimental searches at e+e- or hadron colliders and help test the applicability of NRQCD factorization at NNLO for D-wave states.

major comments (4)
  1. [abstract] Abstract: the quoted uncertainties for Γ_ηc2→γγ^PMC = 3.322^{+0.899}_{-0.828} eV (and the analogous bottomonium result) are stated to arise only from Δm_Q and uncalculated higher orders. However, the width factorizes as Γ ~ |SDC_NNLO(Q*)|^2 × LDME; the D-wave LDMEs are external nonperturbative inputs whose typical 15–40% relative uncertainties are omitted from the error budget. This omission is load-bearing for the claimed precision.
  2. [abstract] Abstract (and numerical-results section): no explicit numerical values, sources, or references are supplied for the D-wave LDMEs (e.g. ⟨O^η_Q2(1D0)⟩ or equivalent) that enter the final widths. Without these inputs the numerical claims cannot be independently verified or reproduced.
  3. [abstract] Abstract: the claim that PMC “naturally improves the convergence of the perturbative series for SDCs” is asserted but not demonstrated by any explicit comparison of the conventional versus PMC series (e.g., term-by-term ratios or partial sums) in the provided text.
  4. Factorization and PMC implementation: the recursive application of RGEs to the D-wave LDMEs presupposes a consistent factorization-scale choice whose matching to the nonperturbative definition of the LDMEs is not independently verified; any mismatch would introduce an additional systematic uncertainty not quantified in the error budget.

Simulated Author's Rebuttal

4 responses · 0 unresolved

We thank the referee for the thorough review and constructive comments on our manuscript. We address each major comment point by point below, agreeing where revisions are needed to enhance clarity and completeness while defending our approach on substantive grounds.

read point-by-point responses

  1. Referee: [abstract] Abstract: the quoted uncertainties for Γ_ηc2→γγ^PMC = 3.322^{+0.899}_{-0.828} eV (and the analogous bottomonium result) are stated to arise only from Δm_Q and uncalculated higher orders. However, the width factorizes as Γ ~ |SDC_NNLO(Q*)|^2 × LDME; the D-wave LDMEs are external nonperturbative inputs whose typical 15–40% relative uncertainties are omitted from the error budget. This omission is load-bearing for the claimed precision.

    Authors: We agree that a complete error budget should incorporate the uncertainties from the input D-wave LDMEs. The abstract emphasized the perturbative improvements from PMC, but the full manuscript uses LDME values from the literature. We will revise the abstract and numerical-results section to propagate and quote the LDME uncertainties (typically 15-40%) alongside the existing ones, providing a more comprehensive total uncertainty. revision: yes

  2. Referee: [abstract] Abstract (and numerical-results section): no explicit numerical values, sources, or references are supplied for the D-wave LDMEs (e.g. ⟨O^η_Q2(1D0)⟩ or equivalent) that enter the final widths. Without these inputs the numerical claims cannot be independently verified or reproduced.

    Authors: The full manuscript provides the LDME values, sources, and references in the numerical-results section. To improve accessibility, we will add the explicit LDME numerical inputs and their references directly into the abstract (or a footnote) and ensure the numerical-results section cross-references them clearly for reproducibility. revision: yes

  3. Referee: [abstract] Abstract: the claim that PMC “naturally improves the convergence of the perturbative series for SDCs” is asserted but not demonstrated by any explicit comparison of the conventional versus PMC series (e.g., term-by-term ratios or partial sums) in the provided text.

    Authors: The manuscript text discusses the removal of renormalon contributions via PMC, but we acknowledge an explicit side-by-side comparison would strengthen the claim. We will add a dedicated paragraph or table in the revised manuscript showing term-by-term ratios and partial sums for both conventional and PMC series to demonstrate the improved convergence. revision: yes

  4. Referee: [—] Factorization and PMC implementation: the recursive application of RGEs to the D-wave LDMEs presupposes a consistent factorization-scale choice whose matching to the nonperturbative definition of the LDMEs is not independently verified; any mismatch would introduce an additional systematic uncertainty not quantified in the error budget.

    Authors: The PMC scale choice is determined self-consistently within the NRQCD factorization framework, with RGEs applied to both α_s and LDMEs to ensure scale invariance. This follows standard practice in PMC applications to effective theories. We will expand the discussion in the revised manuscript to clarify the matching procedure and note that any residual mismatch is absorbed into the LDME definition, but we do not introduce an additional ad-hoc systematic beyond the quoted higher-order errors. revision: partial

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The paper computes NNLO short-distance coefficients in NRQCD factorization, applies the PMC procedure (which absorbs non-conformal beta terms into the effective scale Q* via RGE recursion on alpha_s and LDMEs) to obtain a scale-invariant series, and multiplies by D-wave LDMEs stated to come from independent external sources. The quoted widths and branching ratios follow directly from these steps with uncertainties only from m_Q variations and higher-order estimates; no equation reduces the final numerical result to a fit or self-citation by construction. PMC is an external method applied here rather than a self-defined output, and no load-bearing uniqueness theorem or ansatz is smuggled via overlapping-author citation. The derivation chain therefore remains independent of its own inputs.

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 0 invented entities

The central claim rests on NRQCD factorization and the PMC scale-setting procedure, with LDMEs treated as external nonperturbative inputs whose concrete values are not supplied.

free parameters (2)
  • D-wave LDMEs
    Nonperturbative parameters required to obtain numerical widths from the SDCs; values and sources omitted from abstract.
  • Quark masses m_c and m_b
    Varied by ±0.07 GeV and ±0.06 GeV to estimate uncertainty; treated as external inputs.
axioms (2)
  • domain assumption NRQCD factorization theorem holds at NNLO for this decay
    Invoked to decompose the width into SDCs and LDMEs (abstract, factorization formalism paragraph).
  • domain assumption Recursive application of RGEs within PMC yields a unique optimal scale Q*
    Used to obtain the quoted Q* values and scale-invariant series.

pith-pipeline@v0.9.1-grok · 5991 in / 1538 out tokens · 28073 ms · 2026-06-25T23:39:38.551119+00:00 · methodology

0 comments
read the original abstract

In this paper, we perform a comprehensive study of the decay process $\eta_{Q2}\to\gamma\gamma$ for $D$-wave spin-singlet heavy quarkonia up to next-to-next-to-leading-order (NNLO) QCD corrections within the nonrelativistic QCD effective theory. Following its factorization formalism, the total decay width is decomposed into perturbatively calculable short-distance coefficients (SDCs) and nonperturbative $D$-wave long-distance matrix elements (LDMEs). The original NNLO series of SDCs suffers from sizable renormalization and factorization scale uncertainties. To eliminate such inherent scale ambiguities, we adopt the Principle of Maximum Conformality (PMC). We show that recursively applying the renormalization group equations for the running of $\alpha_s$ and $D$-wave LDMEs within the PMC framework yields an effective strong coupling $\alpha_s(Q_\ast)$ consistent with the expansion coefficients, resulting in a scale-invariant perturbative series. The determined PMC scales are $Q_\ast=1.483$ GeV for $\eta_{c2}$ and $Q_\ast=4.246$ GeV for $\eta_{b2}$. By removing divergent renormalon contributions, the PMC naturally improves the convergence of the perturbative series for SDCs. Our PMC predictions for the total decay widths are $\Gamma_{\eta_{c2}\to\gamma\gamma}^{\rm PMC} = 3.322^{+0.899}_{-0.828}\ \text{eV}$ and $\Gamma_{\eta_{b2}\to\gamma\gamma}^{\rm PMC} = 0.0188^{+0.0014}_{-0.0013}\ \text{eV}$. The uncertainties arise from variations of the charm and bottom quark masses $\Delta m_c=\pm 0.07$ GeV, $\Delta m_b=\pm 0.06$ GeV, as well as systematic errors from uncalculated higher-order corrections. The corresponding branching ratios are $\text{Br}(\eta_{c2}\to\gamma\gamma) = \big(7.463^{+2.020}_{-1.860}\big)\times 10^{-6}$ and $\text{Br}(\eta_{b2}\to\gamma\gamma) = \big(6.460^{+0.481}_{-0.447}\big)\times 10^{-7}$.

Figures

Figures reproduced from arXiv: 2606.24455 by Hua Zhou, Qing Yu, Sheng-Quan Wang, Xing-Gang Wu.

Figure 1
Figure 1. Figure 1: shows the factorization-scale dependence of the ηc2 decay width after incorporating the evolution effects of the relevant LDME h0|χ †K1D2 ψ(µf )|ηc2i. Compared with previous results in the literature, we demonstrate that the factorization-scale dependence of the ηc2 decay width can be completely eliminated once the scale evolu￾tion of the D-wave LDME is properly included. Such can￾cellation of the µf depen… view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. The total decay width Γ [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Variation of the total decay width Γ [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Total decay width of [PITH_FULL_IMAGE:figures/full_fig_p006_5.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Factorization scale [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗

discussion (0)

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Reference graph

Works this paper leans on

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