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Polarization observables in semileptonic $V\to P$ decays of quarkonia

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Pith's one-line read Quarkonium weak decays get first polarization predictions

desk verdict Solid helicity-amplitude formalism with first CCQM predictions for quarkonia weak decays, but the numerical results lean on model form factors with a known 3σ tension with lattice. read the letter →

arxiv 2506.21902 v1 pith:7EW6FUPE submitted 2025-06-27 hep-ph

Chien-Thang Tran , Mikhail A. Ivanov , Ngoc-Dung Nguyen , Quang-Chau Vo This is my paper
classification hep-ph
keywords semileptonicdecaysJ/psiweakUpsilon(1S)polarizationobservablesforward-backwardasymmetryhelicityamplitudescovariantconfinedquarkmodelcharm-taufactory
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 works out the full twofold angular distribution for a semileptonic decay of a vector quarkonium into a pseudoscalar meson plus a charged lepton and neutrino, and uses it to predict the forward-backward asymmetry, the final-lepton longitudinal polarization, and the lepton-side convexity parameter for $J/\psi\to D_{(s)}\ell\bar{\nu}_\ell$ and $\Upsilon(1S)\to B_{(c)}\ell\bar{\nu}_\ell$. The distribution, Eq. (20), is derived with helicity amplitudes and is independent of the hadronic model; the numerical predictions come from taking the quark-model form factors from the authors' earlier work. Most of these predictions are the first in the literature, and they give a concrete Standard Model baseline that a future charm-tau factory can test. Where comparison is possible, the lepton polarization and convexity agree with an earlier quark-model calculation, while the forward-backward asymmetry differs noticeably.

What carries the argument

The carrying object is the helicity-amplitude decomposition of the $V\to P\ell\bar{\nu}_\ell$ amplitude. The paper evaluates the hadronic tensor in the parent-meson rest frame and the leptonic tensor in the $W^*$ rest frame, contracting them through a helicity basis; the nonzero amplitudes are $H_{t0}$, $H_{\pm\mp}$, and $H_{00}$. The resulting normalized angular distribution is a tilted parabola in $\cos\theta$, $\tilde W(\theta)=a+b\cos\theta+c\cos^2\theta$, whose linear and quadratic coefficients map directly onto $A_{FB}$ and $C_F^\ell$, while $P_L^\ell$ follows from the helicity-flip and non-flip parts of the total rate.

What would settle it

Measure the $q^2$-dependent forward-backward asymmetry in $J/\psi\to D_{(s)}\ell\bar{\nu}_\ell$ at a charm-tau factory and compare the $q^2$ average with Table 3; a deviation larger than the quoted uncertainties would disprove either Eq. (20)'s angular structure or the input form factors. Alternatively, a lattice QCD computation of the $J/\psi\to D_s$ form factors that moves them by more than the model's error bars would directly falsify the numerical predictions, since Eq. (20) would still hold but the helicity amplitudes would change.

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Extended reading notes

Core claim

The central claim is that the Standard Model effective Hamiltonian, written in terms of the helicity amplitudes $H_{t0}$, $H_{+-}$, $H_{-+}$, and $H_{00}$ of Eq. (18), produces the exact twofold differential decay rate of Eq. (20). From its angular coefficients the paper defines three observables, $A_{FB}$, $C_F^\ell$, and $P_L^\ell$, and, with the covariant confined quark model form factors, predicts their $q^2$ dependence and $q^2$ averages in Table 3. The paper presents these as the first predictions for the charmonium channels and as a model-dependent Standard Model baseline for the bottomonium ones; the tension with lattice QCD in one channel is reported but not resolved here.

Load-bearing premise

The numbers rest on quark-model hadronic transition form factors taken from the authors' earlier papers without being re-derived here; if those form factors are off—and the paper itself records roughly a $3\sigma$ disagreement with lattice QCD for $J/\psi\to D_s$—the predicted asymmetries and polarizations will move.

Editorial extensions

If this is right

  • The $q^2$-differential and $q^2$-averaged values in Table 3 become testable Standard Model predictions once a charm-tau factory measures $J/\psi\to D_{(s)}\ell\bar{\nu}_\ell$ decays.
  • The near $-1$ lepton polarization for the electron and muon modes, and its sizable deviation for the tau modes, gives a lepton-universality-style check within a single decay.
  • Because Eq. (20) is form-factor independent in shape, any other hadronic model can insert its own form factors and produce competing angular predictions without rederiving the distribution.
  • The reported difference in $\langle A_{FB}\rangle$ between this work and the earlier quark-model calculation means a precise measurement can discriminate between the competing transition form factors.

Reading between the lines

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

  • The angular-distribution coefficients themselves, not just the three named observables, could be extracted as moments from future data and compared directly with the model's $\tilde W(\theta)$.
  • If lattice QCD form factors for $J/\psi\to D_s$ become available and shift the covariant confined quark model ones by the suggested $3\sigma$, the forward-backward asymmetry is the observable most likely to move, since it depends on the interference term $H_{t0}H_{00}$.
  • The same helicity machinery transfers to any vector-to-pseudoscalar semileptonic transition, such as $D^*\to D\ell\bar{\nu}_\ell$, for which no angular predictions exist yet.
  • The lepton-mass effects enter through the helicity-flip factor $\delta_\ell$, so the $\tau$ modes of $\Upsilon(1S)\to B_{(c)}\ell\bar{\nu}_\ell$ are the most sensitive channels for testing that structure.
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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

2 major / 6 minor

Summary. This paper derives a twofold angular decay distribution for semileptonic vector-to-pseudoscalar decays V → P ℓ ν̄ using the helicity amplitude method, and uses it to define and predict the forward-backward asymmetry, lepton-side convexity parameter, and final-lepton longitudinal polarization. The formalism is applied to J/ψ → D_(s) ℓ ν̄ and Υ(1S) → B_(c) ℓ ν̄, with hadronic form factors taken from earlier Covariant Confined Quark Model studies. The general distribution in Eq. (20) is form-factor independent; the numerical predictions in Table 3 are model-dependent and are compared with BSW and NRQCD results where available. The paper validates its helicity formalism by recovering the branching fractions obtained from the standard integration in Eq. (9).

Significance. If the numerical predictions are accepted, the paper provides a useful Standard Model baseline for rare semileptonic quarkonia decays that could be tested at BESIII and a future Super Charm-Tau factory. The central derivation is internally consistent: Eq. (21) follows from integrating Eq. (20), and Eqs. (25)-(27) agree with the coefficients of the normalized angular distribution in Eq. (24). The reproduction of the previously known branching fractions is a strong consistency check. The main weakness is the model dependence of Table 3: the CCQM form factors used for J/ψ → D_s are in about 3σ tension with the recent LQCD calculation in Ref. [18], and the paper does not quantify how this tension propagates into the predicted polarization observables.

major comments (2)
  1. [§3, Table 3 and Eq. (29)] The central numerical predictions in Table 3 inherit the CCQM form factors from Refs. [19,20] and are not cross-checked against an independent form-factor input for the J/ψ channels. Table 1 shows that the CCQM branching fraction for J/ψ → D_s e ν̄, 3.30(50)×10⁻¹⁰, is about 3σ above the LQCD result 1.90(8)×10⁻¹⁰ from Ref. [18]; the agreement is better for J/ψ → D but still not at the 1σ level. Because ⟨A_FB⟩, ⟨C_F^ℓ⟩, and ⟨P_L^ℓ⟩ are q²-weighted ratios of helicity amplitudes, this discrepancy can shift the observables beyond the quoted uncertainties, which reflect only CCQM parameter errors. I request that the authors either recompute the observables using the LQCD form factors of Ref. [18] (or another independent set), or quantify the sensitivity of Table 3 to the form-factor input, and state clearly in the abstract and summary that the numerical predictions are CCQM-model predictions rather than model-independent SM predictions.
  2. [§2, Eq. (3) and Refs. [19,20]] The form factors A0, A+, A−, and V are taken from previous work without providing their explicit q² parametrization or the fitted model parameters. This makes Table 3 and Figures 2–9 not independently reproducible; a reader cannot assess the propagation of uncertainties or test the sensitivity to the model. Please include the explicit form-factor functions (or a supplemental file) used in the numerical evaluation, together with the relevant CCQM parameter values.
minor comments (6)
  1. [General] There are typos throughout: 'seperate' and 'hierachies' in Section 1, 'upcomming' in Section 1, and 'devided' near Eq. (22).
  2. [Eq. (4)] Eq. (4) contains a duplicated trace identity; one of the two identical tr(γ5γμγνγαγβ) lines should be removed or corrected.
  3. [Eq. (20)] The first line of Eq. (20) has a prefactor 1/96 while the second line has 1/36; the relation W(θ) = (3/8){…} should be stated explicitly to avoid an apparent inconsistency.
  4. [Table 3] In the Table 3 caption, 'Res ults form Ref. [16]' should read 'Results from Ref. [16]'.
  5. [References] Reference [19] lists PRD 92 074030 (2015) with arXiv:1701.07377; please verify the arXiv number, as it appears to correspond to a 2017 submission, and correct it if needed.
  6. [Eq. (28)] Eq. (28) uses the notation '−−−→' instead of a proper arrow; please typeset it as a standard arrow.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the angular-distribution derivation is self-contained, and the numerical predictions use prior CCQM form factors as model input rather than as fits to the predicted observables.

full rationale

The central derivation of Eq. (20) is self-contained: the helicity amplitudes in Eq. (18) follow from the Lorentz decomposition in Eq. (3), the polarization vectors in Eq. (16), and the standard W*-frame leptonic tensor. The leptonic tensor in Eq. (19) is quoted from the authors' earlier Ref. [29], but it is a standard perturbative result, and the paper provides an internal cross-check: integrating Eq. (20) over cos(theta) gives Eq. (21), which reproduces the same branching fractions as Eq. (9) in Tables 1 and 2. The numerical predictions in Table 3 are functions of the CCQM form factors A0, A+, A-, and V taken from Refs. [19,20]; those form factors were fixed by other hadronic inputs in earlier work and are not fitted to the forward-backward asymmetry, convexity, or lepton polarization reported here. The paper itself notes the ~3-sigma tension with LQCD for J/psi -> D_s and ~2-sigma agreement for J/psi -> D, which indicates model uncertainty rather than circularity. Self-citations appear, but they are not used as a uniqueness theorem or as a substitute for an independent derivation of the central result, so the derivation chain is not circular.

Assumptions & free parameters 1 free parameters · 5 assumptions · 0 invented entities

The analysis assumes the Standard Model weak Hamiltonian and the validity of the Covariant Confined Quark Model form factors from Refs [19,20]. No new entities are introduced. The only fitted inputs are the CCQM parameters inherited from prior work.

free parameters (1)
  • CCQM parameters (constituent quark masses and infrared confinement cutoff) = not quoted here; set in Refs [19,20]
    These determine the form factors A0, A+, A-, V that enter every numerical prediction in Tables 1-3 and Figures 2-9.
assumptions (5)
  • domain assumption The weak decay is governed by the Standard Model effective Hamiltonian of Eq. (1) with left-handed currents.
    This neglects possible new physics operators and higher-order electroweak corrections; the paper's predictions are SM baselines.
  • standard math The V to P hadronic matrix element has the four-form-factor decomposition of Eq. (3).
    This is the standard Lorentz covariant decomposition for the transition with on-shell mesons, used across the cited literature.
  • domain assumption The CCQM form factors from Refs [19,20] correctly describe the hadronic dynamics.
    All numerical results depend on this input; the paper does not re-derive or refit them here.
  • standard math The helicity basis and completeness relations in Eqs. (12)-(13) hold.
    Standard construction for the W* helicity amplitudes; no physical assumption beyond the chosen frame.
  • domain assumption Parent vector mesons are treated as narrow resonances when converting differential rates to branching fractions using total widths.
    J/psi and Upsilon(1S) have very small widths, so the narrow width approximation is safe.

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Cite this review

Pith. "Pith review of Polarization observables in semileptonic $V\to P$ decays of quarkonia." pith.science (2026). https://pith.science/paper/7EW6FUPE

@misc{pith2026250621902,
  author       = {Pith},
  title        = {Pith review of: Polarization observables in semileptonic $V\to P$ decays of quarkonia},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/7EW6FUPE}},
  note         = {Machine review of arXiv:2506.21902}
}
abstract

We investigate the weak semileptonic decays of the charmonium $J/\psi$ and bottomonium $\Upsilon(1S)$ into a pseudoscalar meson, namely, the $J/\psi\to D_{(s)}\ell\bar{\nu}_\ell$ and $\Upsilon(1S)\to B_{(c)}\ell\bar{\nu}_\ell$ decays. We focus on the polarization observables including the forward-backward asymmetry, final lepton polarization, and lepton-side convexity parameter. In order to define these quantities, we apply the helicity amplitude method and obtain a twofold angular decay distribution for a general $V\to P\ell\bar{\nu}_\ell$ case. The hadronic form factors are taken from our previous work, where they were calculated in the Covariant Confined Quark Model. We provide theoretical predictions for the polarization observables and compare them with the literature.

Figures

Figures reproduced from arXiv: 2506.21902 by the authors.

Figure 1
Figure 1. Feynman diagram for semileptonic decays V → P ℓν¯ℓ. The hadronic matrix element in Eq. (2) is often parametrized as a linear combination of Lorentz structures multiplied by scalar functions, namely, invariant form factors which depend on the momentum transfer squared. For the V → P transition one has hP|q¯2γµ(1 − γ5)q1|V i ≡ ǫ ν 1T V P µν = ǫ ν 1 m1 + m2 [−gµνpqA0(q 2 ) + pµpνA+(q 2 ) + qµpνA−(q 2 ) + iεµναβp α q βV… view at source ↗
Figure 2
Figure 2. J/ψ → D(s) transition: q 2 dependence of the decay rate dΓ/dq2 for e − mode (dashed) and µ − mode (solid). 0 5 10 15 0.0 0.1 0.2 0.3 0 05 0.6 q 2 (GeV2 ) (dΓ/dq 2)×10 -13 Υ (1S) B 0 2 4 6 8 10 0.00  0.10  0.20  q 2 (GeV2 ) (dΓ/dq 2)×10 -10 Υ (1) c [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. Υ(1S) → B(c) transition: q 2 dependence of the decay rate dΓ/dq2 for e − mode (dashed) and τ − mode (solid) [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figures from the paper (6 more)
Figure 4
Figure 4. Figure 4: J/ψ → D(s) transition: q 2 dependence of AF B for e − mode (dashed) and µ − mode (solid). 0 10 [PITH_FULL_IMAGE:figures/full_fig_p007_4.png]
Figure 5
Figure 5. Figure 5: Υ(1S) → B(c) transition: q 2 dependence of AF B for e − mode (dashed) and τ − mode (solid) [PITH_FULL_IMAGE:figures/full_fig_p007_5.png]
Figure 6
Figure 6. Figure 6: J/ψ → D(s) transition: q 2 dependence of the lepton-side convexity parameter C ℓ F for e − mode (dashed) and µ − mode (solid). 0 10 - ! -1.0 - "! 0.0 q 2 (GeV2 ) # % $ Υ (1&)' 0 2 4 6 8 10 - ()* -1.0 - +)* 0.0 q 2 (GeV2 ) , . - Υ (1/)23 [PITH_FULL_IMAGE:figures/full…
Figure 7
Figure 7. Figure 7: Υ(1S) → B(c) transition: q 2 dependence of the lepton-side convexity parameter C ℓ F for e − mode (dashed) and τ − mode (solid). Finally, we consider the longitudinal polarization P ℓ L (q 2 ) of the final lepton. The expression for this observable can be simply obtain…
Figure 8
Figure 8. Figure 8: J/ψ → D(s) transition: q 2 dependence of the longitudinal polarization of the final lepton P ℓ L for e − mode (dashed) and µ − mode (solid). The q 2 dependence of the decay rate and polarization observables allows for probing the dynamics of the decays in great detail.…
Figure 9
Figure 9. Figure 9: Υ(1S) → B(c) transition: q 2 dependence of the longitudinal polarization of the final lepton P ℓ L for e − mode (dashed) and τ − mode (solid). 4 Summary We have studied in great detail the semileptonic decays J/ψ → D(s)ℓν¯ℓ and Υ(1S) → B(c)ℓν¯ℓ with special focus on th…

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