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REVIEW 3 major objections 5 minor 29 references

Invisible decays of vector Charmonia and Bottomonia to determine the Weak Mixing Angle at quarkonia scale

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Pith's one-line read Invisible decays of vector quarkonia could determine the weak mixing angle at quarkonium mass scales.

desk verdict A clean, checkable SM calculation with a modestly new angle-extraction proposal, but the alpha(0) choice biases the central numbers at the level of the claimed precision. read the letter →

arxiv 2411.09124 v2 pith:6YB5HXHZ submitted 2024-11-14 hep-ph

G. Hernández-Tomé , C. S. Kim , G. López Castro This is my paper · ORCID
classification hep-ph
keywords invisibledecaysvectorquarkoniaweakmixingangleneutrinopairsJ/psiUpsilon(1S)Dirac-MajoranaconfusionStandardModelpredictions
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 paper sets out to show that the otherwise-unobserved decays of vector quarkonium states into neutrino pairs are clean Standard Model rates that change predictably with the weak vector coupling of the heavy quark, and that measuring them would give the first determination of the weak mixing angle at charmonium and bottomonium mass scales. The key formula, Eq. (8), relates each invisible branching fraction to the measured branching fraction into electron-positron pairs, with the quarkonium annihilation constant cancelling out, so the prediction is essentially free of strong-interaction uncertainty. The paper's numerical predictions are $\mathrm{BR}(J/\psi\to\nu\bar\nu)=2.04\times10^{-8}$ and $\mathrm{BR}(\Upsilon(1S)\to\nu\bar\nu)=1.02\times10^{-5}$, which would require a few-percent measurement to compete with existing low-energy determinations of $\sin^2\theta_W$. It also shows that within the Standard Model the Dirac and Majorana neutrino rates coincide in the massless limit, and that only new neutrino couplings could separate them, at a precision beyond current facilities.

What carries the argument

The load-bearing object is the vector quarkonium annihilation constant $f_{V_Q}$, defined by $\langle 0|\bar Q\gamma_\mu Q|V_Q\rangle = M^2 f_{V_Q}^{-1}\epsilon_\mu$. It is the same hadronic matrix element appearing in $V_Q\to\ell^+\ell^-$ and $V_Q\to\nu\bar\nu$, and because Eq. (8) is the ratio of the two widths, $f_{V_Q}$ cancels exactly. What is left is the weak vector current, so only $g_V^Q$ matters and the axial coupling drops out. The Z propagator is kept with finite quarkonium mass $M$, producing the factor $m_Z^2/(m_Z^2-M^2)$ in the amplitude.

What would settle it

A high-statistics $e^+e^-$ sample that limits $\mathrm{BR}(\Upsilon(1S)\to\text{invisible})$ below about $1\times10^{-5}$ at 90% C.L., with no new invisible particles, would falsify the prediction; equally, a measurement that breaks the predicted scaling across $J/\psi$, $\psi(2S)$, and $\Upsilon(1S)$ would falsify the ratio formula.

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Extended reading notes

Core claim

The central claim is Eq. (8): the invisible branching fraction of a vector quarkonium state is fixed by its measured electromagnetic annihilation rate, $$\mathrm{BR}(V_Q\to\nu\bar\nu)=N_\nu\left[\frac{G_F $M^{2}$}{4\pi\$\alpha$}\left(\frac{g_V^Q}{q_Q}\right)\left(\frac{$m_Z^{2}$}{$m_Z^{2}$-$M^{2}$}\right)\right]^2 \mathrm{BR}(V_Q\to e^+e^-),$$ with $N_\nu=3$ light neutrinos. Because the same decay constant $f_{V_Q}$ enters the photon- and Z-mediated amplitudes, taking the ratio cancels it; the remaining dependence is the heavy-quark weak vector charge $g_V^Q=t_3^Q-2q_Q\sin^2\theta_W$, so an invisible-rate measurement reads $\sin^2\theta_W$ at the quarkonium mass scale. The numerical table gives $2.04\times10^{-8}$, $5.52\times10^{-9}$, $1.02\times10^{-5}$, and $1.05\times10^{-8}$ for $J/\psi$, $\psi(2S)$, $\Upsilon(1S)$, and $\Upsilon(4S)$, respectively. The paper also derives the Majorana version and finds the Standard Model Dirac-Majorana difference is proportional to $m_\nu^2$, vanishing for massless neutrinos; with nonstandard couplings the difference scales with $\epsilon_V-\epsilon_A$ and is too small for current experiments.

Load-bearing premise

The load-bearing premise is that a quarkonium state annihilates into a Z boson exactly as it annihilates into a photon, so that no QCD or electroweak correction shifts the weak process relative to the electromagnetic one.

Editorial extensions

If this is right

  • A 10% measurement of $\mathrm{BR}(J/\psi\to\nu\bar\nu)$ would determine $\sin^2\theta_W$ at the charmonium scale to about $\pm0.007$; a 5% measurement would give $\pm0.003$, comparable to existing low-energy determinations.
  • For $\Upsilon(1S)$, the predicted $1.02\times10^{-5}$ sits one order of magnitude below the current experimental upper limit, so a moderate increase in $e^+e^-$ statistics could either confirm the Standard Model prediction or constrain new invisible decay modes.
  • The cancellation of $f_{V_Q}$ removes the dominant hadronic uncertainty, making these decays a relatively clean electroweak precision observable at low energy scales, complementary to neutrino-scattering determinations.
  • Within the Standard Model, the invisible width of quarkonia cannot distinguish Dirac from Majorana neutrinos; only nonstandard neutrino couplings create a difference, and that difference is smaller than the expected experimental precision.

Reading between the lines

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

  • Beyond the paper, the same ratio formula should apply, with obvious changes of mass and charge, to other $1^{--}$ vector states such as $\Upsilon(2S)$ and $\Upsilon(3S)$ for which $e^+e^-$ branching fractions are measured, giving cross-checks of the extracted $\sin^2\theta_W$ at several nearby scales.
  • Beyond the paper, if QCD or electroweak corrections to the Z-mediated vertex differ from the photon-mediated one, Eq. (8) would carry a correction not visible in the ratio; comparing the extracted $\sin^2\theta_W$ across charmonium and bottomonium states could expose such an effect.
  • A natural testable extension is to push the search for $\Upsilon(1S)\to\nu\bar\nu$ with tagged radiative or dipion transitions at high-luminosity $e^+e^-$ colliders, since an improved upper limit near $10^{-5}$ would either confirm the prediction or require new invisible decay products.
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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

3 major / 5 minor

Summary. The manuscript computes Standard Model branching fractions for invisible decays of vector quarkonia (J/psi, psi(2S), Upsilon(1S), Upsilon(4S)) into neutrino pairs. Using vector-meson dominance, the authors obtain the compact formula of Eq. (8), in which the quarkonium decay constant cancels against the measured e+e- branching ratio. The predicted values are BR(J/psi -> nu nubar) = 2.04e-8 and BR(Upsilon(1S) -> nu nubar) = 1.02e-5. The authors propose that a 5-10% measurement of these invisible branching fractions would determine sin^2 theta_W at the quarkonia mass scale with uncertainties of roughly 0.007 (charmonia) and 0.027 (bottomonia). They also discuss gamma-Z interference in the charged-lepton modes and argue that non-standard neutrino couplings could, in principle, distinguish Dirac and Majorana neutrinos.

Significance. If the central formula is correct, the paper offers a clean, hadron-uncertainty-free ratio method for a low-energy determination of the weak mixing angle, complementary to NuTeV and coherent neutrino scattering. The cancellation of the decay constant in Eq. (8) is elegant, the numerical predictions are falsifiable, and the comparison with the existing BaBar upper limits is informative. The significance is, however, conditional on correcting an unquantified running-alpha systematic, on including the electroweak rho corrections in the extraction, and on a more realistic assessment of experimental backgrounds.

major comments (3)
  1. [Section 2, Eq. (8) and Table I] The numerical predictions use alpha = 1/137.036 instead of the time-like running fine-structure constant at the quarkonium mass scale. Since the ratio in Eq. (8) is built from the theoretical leptonic width in Eq. (2) and then multiplied by the measured BR(V_Q -> e+e-), the alpha that appears is the coupling of the virtual photon at virtuality M^2, not at zero momentum. With alpha(m_c) ~ 1/132 and alpha(m_b) ~ 1/130, the tabulated branching fractions are systematically high by about 7% for charmonia and 11% for bottomonia. For the assumed 10% measurement uncertainty, this shift is comparable to the quoted precision; propagated into the extracted sin^2 theta_W, it moves the central value by roughly +0.005 (J/psi) and +0.03 (Upsilon(1S)), at or beyond the quoted +/-0.007 and +/-0.027. Please replace alpha(0) by the appropriate scale-dependent alpha(M) (including hadronic vacuum polarization and final-state QED corrections) and recompute Table I and the projected sensitivities.
  2. [Section 2, footnote 1 and Eq. (8)] The numerical analysis in Table I uses the tree-level relation g_V^Q = t_3 - 2 q_Q sin^2 theta_hat, while footnote 1 correctly states that electroweak corrections modify this to sqrt(rho_f) (t_3 - 2 q_f sin^2 theta_hat). The rho_f factor shifts g_V^Q by about a percent, which translates into a shift in the extracted sin^2 theta_W of roughly 0.001 for J/psi and 0.003 for Upsilon(1S). These shifts are smaller than the 10% statistical projections but become a non-negligible fraction of the 5% scenario and should be either implemented in the formula or explicitly bounded before the precision statement in the conclusions can be taken at face value.
  3. [Introduction and Conclusions, experimental reach] The paper states that the predicted branching fractions 'could be within the reach of current and future data sets at BES-III and Belle-II' and that with 10^12 J/psi and a 1-10% detection efficiency the weak mixing angle can be determined. No estimate of backgrounds is provided for the invisible decay signature, such as e+e- -> gamma gamma with both photons escaping detection, radiative Bhabha with undetected tracks, or other two-photon processes. Since the signal rates are at the 1e-8 to 1e-5 level and the extraction requires a few-percent measurement of a completely invisible final state, the feasibility claim is not yet supported. Please either add a quantitative background estimate or soften the reach claim accordingly.
minor comments (5)
  1. [Introduction] There is a typo: 'botommonium' should be 'bottomonium'.
  2. [Section on Majorana neutrinos] The section title contains 'DECA Y' and should read 'DECAY'.
  3. [Eqs. (9)-(11)] The notation V_Q is used both for the quarkonium state and, in Eq. (10), for the factor multiplying the electromagnetic amplitude. This is confusing and should be clarified, for example by renaming the factor in Eq. (10) to R_V.
  4. [Table I and text after it] The statement that the predicted Upsilon(1S) rate is 'only one order of magnitude below' the BaBar limit is imprecise: the limit 3.0e-4 is a factor of about 30 above the prediction 1.02e-5, i.e., about 1.5 orders of magnitude.
  5. [Conclusions] The sentence 'Precise measurements of decay fractions would indeed provide a competitive determination at those scales, but even a not-so-precise measurement could give us for the first time indications of the central value' is vague; it would be useful to state the minimum precision required for a meaningful determination.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: the central ratio cancels the hadronic constant and uses external inputs; Ref. [24] is cosmetic.

full rationale

The derivation chain is self-contained. Eq. (2) uses the photon matrix element (1); Eqs. (5)-(7) use the same f_{V_Q} for the weak Z amplitude; the ratio in Eq. (8) cancels f_{V_Q} and depends only on known masses, G_F, alpha, the external PDG leptonic branching fractions, and the running weak-mixing angle taken from Refs. [20-22]. No parameter is fitted to invisible-width data; the invisible branching fractions in Table I are generated from external inputs, and the sin^2 theta_W uncertainty columns are error-propagation sensitivities, not claims to have measured the angle. The only self-citation is Ref. [24] (C.S. Kim), used to recognize the Lorentz/CP form of Eq. (6); that form is already derived from the SM amplitude, so the citation is cosmetic and not load-bearing. The alpha(0) versus alpha(M^2) choice and possible hadronic form-factor differences are numerical-QCD corrections, not circular reductions. Thus no circularity step can be exhibited. The score reflects only the single non-load-bearing self-citation.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

No free parameters are fitted in the central calculation. All numerical inputs (G_F, m_Z, alpha, M_V, BR(V_Q to e+e-), and sin^2 theta_W(mu)) come from the PDG or standard references. The epsilon_{V,A} parameters in Eq. (14) are illustrative assumptions for a beyond-SM scenario, not part of the central SM prediction, and are therefore not listed as free parameters.

assumptions (6)
  • domain assumption The weak neutral current matrix element of the quarkonium state is given by <0|Qbar gamma_mu Q|V_Q> = M^2/f_{V_Q} epsilon_mu (Eq. 1), with the same f_{V_Q} as in the electromagnetic decay.
    Central modeling assumption of vector-meson dominance; used to relate the Z-mediated amplitude to the measured e+e- width.
  • domain assumption The decay V_Q to nu nubar proceeds via single Z-boson exchange; other Standard Model diagrams are neglected.
    Implied by Eq. (5); standard treatment for these rates.
  • domain assumption The values of running sin^2 theta_W in the MS scheme at the quarkonia masses are taken from Refs [20-22] (0.237 for J/psi, 0.236 for psi(2S), 0.233 for Upsilon(1S), 0.232 for Upsilon(4S)).
    These external running values enter directly into g_V^Q and therefore into the predicted branching fractions in Table I.
  • domain assumption Neutrino masses are negligible and N_nu = 3 for the Standard Model prediction.
    Used to simplify Eq. (7) and the ratio in Eq. (8); justified by the known smallness of neutrino masses compared to M_V.
  • ad hoc to paper The non-standard neutrino couplings in Eq. (14) are parametrized as C_{V,A} = 1/2 + epsilon_{V,A}.
    Introduced for the Majorana discussion to illustrate the effect of new physics; not fitted, and not used in the central Standard Model prediction.
  • standard math The Dirac-Majorana confusion theorem (Ref [26]) applies for massless neutrinos with V-A interactions.
    Invoked in the 'Comments on invisible decay width in case of Majorana neutrinos' section to interpret the mass-suppressed difference.

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Cite this review

Pith. "Pith review of Invisible decays of vector Charmonia and Bottomonia to determine the Weak Mixing Angle at quarkonia scale." pith.science (2026). https://pith.science/paper/6YB5HXHZ

@misc{pith2026241109124,
  author       = {Pith},
  title        = {Pith review of: Invisible decays of vector Charmonia and Bottomonia to determine the Weak Mixing Angle at quarkonia scale},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/6YB5HXHZ}},
  note         = {Machine review of arXiv:2411.09124}
}
abstract

We compute the branching fractions of vector quarkonia ($V_Q=J/\psi, \psi', \Upsilon(nS)$) decays into neutrino pairs, considering both Dirac and Majorana types, within the Standard Model (SM) and beyond. The vector nature of quarkonium states yields a decay width in the SM that depends upon the weak vector coupling of the heavy quark, offering the possibility to measure the weak mixing angle at the quarkonia mass scales. If neutrinos have non-standard neutral weak couplings, this could help to distinguish the nature of neutrinos in principle.

Figures

Figures reproduced from arXiv: 2411.09124 by the authors.

Figure 1
Figure 1. FIG. 1: Leptonic decays of heavy [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2: Scale dependence of the weak mixing angle in the [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

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