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REVIEW 4 major objections 4 minor 21 references

Comment on "Improved measurement of $\eta/\eta'$ mixing in $B_{(s)}^0\to J/\psi \eta^{(\prime)}$ decays"

T0 review · 4 major / 4 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read A quark-annihilation contribution, not a glueball in the eta-prime, explains the new LHCb branching-fraction ratios.

desk verdict A compact, honest comment that fits an existing W-exchange model to new LHCb data and questions the glueball interpretation, but the load-bearing amplitude coefficients are imported and unexamined. read the letter →

arxiv 2507.19872 v1 pith:R52GO5TR submitted 2025-07-26 hep-ph hep-exhep-lat

Ying Chen , Geng Li This is my paper · ORCID
classification hep-phhep-exhep-lat
keywords eta-eta'mixingglueballU_A(1)anomalyBmesondecaysW-exchangeannihilationbranchingfractionratiosLHCbQCD
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This comment argues that the new LHCb measurements of the ratios of $B_{(s)}^0\to J/\psi\eta^{(\prime)}$ branching fractions do not require the $\eta'$ to contain a light pseudoscalar glueball. The authors add a W-exchange (quark-annihilation) contribution to the usual colour-suppressed tree amplitude and fit its relative size to a single real number, $\alpha=A_{\mathrm{EA}}/A_{\mathrm{CT}}=-0.039(10)$, which reproduces all four measured ratios. With this one parameter the $\eta$–$\eta'$ mixing angle $\phi_P$ extracted from the different ratios becomes consistent, within about $39.5^\circ$ to $44.2^\circ$. The physical motivation is the QCD $U_A(1)$ anomaly, which can make gluonic production of the light pseudoscalars significant; if the interpretation holds, the large $\eta'$–glueball mixing angle inferred by LHCb is an artifact of neglecting this annihilation effect.

What carries the argument

The central object is the EA diagram, the W-exchange (light-quark annihilation) topology in $B_{(s)}^0\to J/\psi\eta^{(\prime)}$, whose coupling to the flavor-singlet pseudoscalars is enhanced by the QCD $U_A(1)$ anomaly. The machinery is the set of amplitude relations, taken from Ref. [3], that give the EA contribution coefficients $(1,2,4)$ across the four channels relative to the colour-suppressed tree amplitude $A_{\mathrm{CT}}$: $A(B_s^0\to J/\psi\eta)\propto -\sin\phi_P(A_{\mathrm{CT}}-A_{\mathrm{EA}})$, $A(B_s^0\to J/\psi\eta')\propto \cos\phi_P(A_{\mathrm{CT}}+2A_{\mathrm{EA}})$, $A(B^0\to J/\psi\eta)\propto \cos\phi_P(A_{\mathrm{CT}}+A_{\mathrm{EA}})$, and $A(B^0\to J/\psi\eta')\propto \sin\phi_P(A_{\mathrm{CT}}+4A_{\mathrm{EA}})$. Assuming $\alpha=A_{\mathrm{EA}}/A_{\mathrm{CT}}$ is real, these relations turn each branching-fraction ratio into a function of $\phi_P$ and $\alpha$; the product $R_d R_s$ fixes $\alpha$ alone, and the individual ratios then give $\phi_P$. The negative sign of $\alpha$ is traced to the extra quark loop in the EA diagram and to a lattice study of $D_s\to\eta'$ form factors that exhibits a similarly signed disconnected contribution.

What would settle it

A lattice QCD calculation of the W-exchange (quark-line disconnected) contributions to $B_{(s)}^0\to J/\psi\eta^{(\prime)}$ amplitudes, or an experimental measurement of the relative strong phase between the two amplitudes via interference effects in related decay distributions, would settle the claim: if $\alpha$ were found to have a modulus or phase far from the fitted real value, the claimed consistency of $\phi_P$ would disappear.

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Extended reading notes

Core claim

The paper's central claim is that the four LHCb ratios of $B_{(s)}^0\to J/\psi\eta^{(\prime)}$ branching fractions can be understood without $\eta'$–glueball mixing once the W-exchange (light quark annihilation) amplitude $A_{\mathrm{EA}}$ is included. With the amplitude relations $A(B_s^0\to J/\psi\eta)\propto -\sin\phi_P(A_{\mathrm{CT}}-A_{\mathrm{EA}})$ and $A(B_s^0\to J/\psi\eta')\propto \cos\phi_P(A_{\mathrm{CT}}+2A_{\mathrm{EA}})$, together with the analogous relations for the $B^0$ modes, and assuming $\alpha=A_{\mathrm{EA}}/A_{\mathrm{CT}}$ is real, the fit gives $\alpha=-0.039(10)$. The $\phi_P$ values extracted from $R_d,R_s,R_\eta,R_{\eta'}$ then come out as $41.6(1.2)^\circ$, $44.2(1.4)^\circ$, and $39.5(2.2)^\circ$, mutually consistent within one $\sigma$, whereas the glueball-mixing interpretation forces a large mixing angle $\phi_G\approx 28^\circ$ that conflicts with lattice QCD studies. The paper therefore offers a parameter-economical alternative: the correction is small but systematically lowers the $\eta'$ rates by 16–32 percent, recovering the standard two-state $\eta$–$\eta'$ mixing picture.

Load-bearing premise

The argument's load-bearing assumption is that the W-exchange amplitude contributes to the four decay channels with fixed relative weights (1, 2, 4) and exactly the same phase as the dominant tree amplitude; if the weights or the no-phase assumption are wrong, the fitted $\alpha=-0.039$ and the convergence of $\phi_P$ values do not follow.

Editorial extensions

If this is right

  • The four LHCb ratios are reproduced without putting any glueball component into $\eta'$, so the large glueball mixing angle $\phi_G\approx 28^\circ$ is unnecessary.
  • The $\eta$–$\eta'$ mixing angle extracted from the $B^0$ and $B_s^0$ channels becomes consistent, at $39.5(2.2)^\circ$ to $44.2(1.4)^\circ$, instead of spanning $36.6(7)^\circ$ to $46.9(6)^\circ$.
  • The annihilation effect is sizable: it lowers the $B^0\to J/\psi\eta'$ rate by about 32 percent, lowers $B_s^0\to J/\psi\eta'$ by about 16 percent, and shifts the two $\eta$ rates by about 8 percent, so future fits of these decays should include it.
  • The negative sign of $\alpha$ indicates the two diagrams have opposite signs, consistent with the extra quark loop in the annihilation diagram; this can be checked against lattice determinations of disconnected contributions.
  • If the interpretation holds, the $\eta'$ remains close to the standard flavor-octet/singlet mixing picture, with no need for a light pseudoscalar glueball near the $\eta'$ mass.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The same $U_A(1)$-enhanced annihilation mechanism could affect other $B$ decays with light pseudoscalars, such as $B\to J/\psi\,\pi^0$ or charm decays with $\eta/\eta'$; the paper does not work those out, but the pattern of coefficients would be a natural next test.
  • A dedicated lattice calculation of the quark-line disconnected matrix elements for these $B$-decay amplitudes would give an independent, parameter-free estimate of $\alpha$; agreement with $-0.039(10)$ would validate the mechanism, while a complex or much larger result would demand a modified version.
  • Future improved measurements of the four ratios could distinguish the two interpretations more sharply, because the glueball-mixing ansatz and the annihilation ansatz predict different correlations among $R_d,R_s,R_\eta,R_{\eta'}$ as precision improves.
  • The argument suggests the size of the annihilation correction should depend on the mass of the produced pseudoscalar, so lighter states such as $\pi^0$ would probe the same operator in a different kinematic regime.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 4 minor

Summary. This manuscript is a comment on the recent LHCb measurement of ratios of branching fractions for B_(s)^0 -> J/psi eta^(prime) decays. The authors propose that the data can be described without the eta'-glueball mixing ansatz by adding a W-exchange (light quark annihilation, EA) amplitude to the dominant colour-suppressed tree amplitude. Using amplitude relations imported from Ref. [3], they define alpha = A_EA/A_CT and fit it from the product R_d R_s, obtaining alpha = -0.039(10). They then extract the eta-eta' mixing angle phi_P from each of the four measured ratios, reporting values that are mutually consistent within one sigma, and argue that this consistency is better than the phi_P values obtained in the LHCb paper without the EA contribution. The central claim is that the U_A(1) anomaly-enhanced annihilation effect can replace the eta'-glueball mixing interpretation.

Significance. If the central claim is correct, the paper offers an economical alternative to the large eta'-glueball mixing angle phi_G ~ 28 deg preferred by the LHCb analysis, and it is consistent with lattice QCD predictions of a heavy pseudoscalar glueball and with lattice results showing small eta-glueball mixing. The analysis is transparent, the algebra is simple, and the input data are public. The paper also makes a falsifiable prediction: the pattern of rate modifications (roughly 8%, 16%, and 32%) for the four channels, which can be tested with additional experimental input or with more detailed QCD calculations of the annihilation amplitude. However, the significance is conditional on the validity of the amplitude coefficients in Eq. (7) and on the reality of alpha, both of which are assumed rather than derived or quantified in the manuscript.

major comments (4)
  1. [Eq. (7)] The relative EA amplitude coefficients (1, 2, 4) for B_s -> J/psi eta, B_s -> J/psi eta', B^0 -> J/psi eta, B^0 -> J/psi eta' are imported from Ref. [3] without derivation. These coefficients are load-bearing: the factor (1+4 alpha)^2/(1+2 alpha)^2 in R_eta' is what shifts phi_P(R_eta') from 36.6 deg to 39.5 deg in Eq. (9). The physical motivation given in the text, that the EA couples through gluonic interactions enhanced by the U_A(1) anomaly, suggests a flavour-singlet production amplitude whose strength for eta and eta' is governed by singlet wave-function overlaps, which would give the same EA coefficient for a given meson in B^0 and B_s decays. The pattern in Eq. (7) does not follow from the stated picture alone. The authors should either derive the coefficients from the flavour wave functions and the CT/EA topologies, or state explicitly that they are assumed from Ref. [3] and discuss the sensitivity of the phi_P convergence to the coefficient of the B^0 -> J/psi eta' amplitude relative to the other channels.
  2. [Eq. (8) and following text] The assumption that alpha = A_EA/A_CT is real is stated in one sentence after Eq. (8) and is not quantified. If alpha carries a strong phase, the product R_d R_s fixes a complex combination and the ratios in Eq. (8) change; the extracted phi_P values and the claimed convergence would be modified. The authors should provide an estimate of the phase uncertainty, for example by using the lattice results of Ref. [21] to bound the strong phase, or by performing a two-parameter (Re alpha, Im alpha) fit to the four ratios and showing that the resulting phi_P values remain consistent. Without this, the numerical results in Eq. (9) are only valid under an unquantified assumption.
  3. [Paragraph after Eq. (9)] The claimed improvement from including the EA contribution is presented qualitatively: the three phi_P values in Eq. (9) are said to be 'consistent within one-sigma' and the convergence is judged by eye. Since the phi_P values are derived from overlapping data and shared fitted parameters, a quantitative comparison is required. The authors should perform a joint fit to the four ratios with the model in Eq. (8), with alpha and phi_P as free parameters (or with alpha fixed from R_d R_s), and compute a chi-square or likelihood ratio relative to the no-EA model of Eq. (4). This would also allow the uncertainties on the percentage rate changes to be propagated properly.
  4. [Rate-change paragraph] The statements that the EA contribution increases/decreases the decay rates by roughly 8%, 16%, and 32% are quoted without uncertainties. The fitted alpha = -0.039(10) has a 25% relative error, so the central rate changes carry sizable uncertainties. For example, the 32% decrease for B^0 -> J/psi eta' would range approximately from 20% to 45% at 1 sigma. Propagating the alpha uncertainty through the expressions in Eq. (8) would make the quantitative claim precise and is necessary for the result to be used by other authors.
minor comments (4)
  1. [Abstract and text] There are several typographical errors: 'contrain' in the abstract should be 'constrain', 'syammetry' after Eq. (3) should be 'symmetry', 'machanism' in Section 2 should be 'mechanism', 'phenomnon' should be 'phenomenon', 'pesudoscalar' should be 'pseudoscalar', 'pervious' should be 'previous', 'through' in 'is through gluonic interaction' and 'though' in the figure caption appear to be misspelled, and 'In constrast' should be 'In contrast'. These should be corrected.
  2. [Reference list] Reference [9] is missing its publication year: 'JHEP 10, 170' should include the year (2012). Please verify all bibliographic entries against the published versions.
  3. [Eq. (4)] The typesetting of Eq. (4) is ambiguous: 'cot2ϕP 2' and 'tan2ϕP 2' are difficult to read. Please format the expressions as cot^2(phi_P)/2 and tan^2(phi_P)/2, or as the intended ratios, so that the factors of 1/2 (which enter the comparison with Eq. (9)) are clear.
  4. [Fig. 1 caption] The caption states the EA diagram couples 'though gluonic interaction'; this should read 'through gluonic interaction'. Also, the sentence describing the two panels could clarify that the EA diagram is the W-exchange contribution.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: α is fitted to R_d R_s and then tested against independent R_η, R_η′ combinations; self-citations are motivational, not load-bearing.

full rationale

The paper's derivation is self-contained. It takes the amplitude decomposition Eq. (7) from external Ref. [3] (not the authors' prior work), derives the modified ratios Eq. (8), and fits the single new parameter α=A_EA/A_CT to the product R_d R_s, which is independent of ϕ_P. The subsequent central check—extracting ϕ_P from R_η and R_η′ and showing consistency with ϕ_P(R_d/R_s)—uses measured ratios that were not used to fix α, so the agreement is an out-of-sample consistency test rather than a relation forced by construction. The quoted rate modifications after Eq. (8) are restatements of the fitted α, but the paper does not advertise them as independent predictions. The cited lattice articles coauthored by the present authors (Refs. [7,10,11,12,17–20]) are used only to motivate the UA(1)/EA mechanism and to argue against a large glueball component; they do not enter the numerical fit. Concerns that the relative coefficient 4 in Eq. (7) may be unjustified or SU(3)-broken are model-robustness/correctness objections, not circularity, since those coefficients are imported from an external derivation and not fitted. No self-definitional, fitted-as-prediction, or self-citation-chain reduction is exhibited, so the circularity score is 0.

Assumptions & free parameters 2 free parameters · 5 assumptions · 0 invented entities

The central explanation rests on one fitted quantity (alpha) and the eta-eta' mixing angle (phi_P), plus standard assumptions about flavor symmetry, the relative size of diagrams, and the absence of a glueball component in eta. No new particles or interactions are introduced.

free parameters (2)
  • alpha = A_EA/A_CT = -0.039(10)
    Ratio of the W-exchange annihilation amplitude to the colour-suppressed tree amplitude. Determined from the product of the measured LHCb ratios R_d and R_s using Eq. (8).
  • phi_P = 41.6(1.2) degrees (from R_d/R_s)
    eta-eta' mixing angle extracted from the data in Eq. (9). It is not predicted by the model but is part of the fit.
assumptions (5)
  • domain assumption Flavor SU(3) symmetry is used to relate the amplitudes for B_d and B_s decays into eta and eta'.
    Invoked before Eq. (3) and used throughout; no SU(3)-breaking corrections are estimated.
  • domain assumption The relative weights of A_EA in the four amplitudes are (1, 2, 4) as given in Eq. (7).
    Imported from Ref [3] without derivation in this comment; the central fit depends on these coefficients.
  • domain assumption The ratio alpha = A_EA/A_CT is real.
    Stated after Eq. (8): 'where alpha=A_EA/A_CT is assumed to be real'. A strong phase would change the fit.
  • domain assumption Penguin and penguin-annihilation contributions are negligible.
    Stated after Fig. 1 with citation [13]; these contributions are omitted from the amplitude expressions.
  • domain assumption The eta meson has no glueball component.
    In Eq. (2): 'eta is mainly the flavor octet whose glueball content is temporarily neglected'.

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Cite this review

Pith. "Pith review of Comment on "Improved measurement of $\eta/\eta'$ mixing in $B_{(s)}^0\to J/\psi \eta^{(\prime)}$ decays"." pith.science (2026). https://pith.science/paper/R52GO5TR

@misc{pith2026250719872,
  author       = {Pith},
  title        = {Pith review of: Comment on "Improved measurement of $\eta/\eta'$ mixing in $B_(s)^0\to J/\psi \eta^(\prime)$ decays"},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/R52GO5TR}},
  note         = {Machine review of arXiv:2507.19872}
}
abstract

Instead of the glueball-$\eta'$ mixing ansatz, the latest measured ratios of the branching fractions of $B_{(s)}^0\to J/\psi \eta^{(\prime)}$ decays by LHCb can be understood by including the contribution from the light quark annihilation effect enhanced by the QCD $\mathrm{U}_A(1)$ anomaly for light pseudoscalar mesons.

Figures

Figures reproduced from arXiv: 2507.19872 by the authors.

Figure 1
Figure 1. FIG. 1. The leading topologies contributing to the decays [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

Reference graph

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