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Studying the $B^{0} \to J/\psi h_{1}$ decays with $h_{1}(1170)-h_{1}(1415)$ mixing in the perturbative QCD approach

T0 review · 2 major / 3 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read The $B^0\to J/\psi h_1$ decays are predicted to be sizable and sharply sensitive to the $h_1$ mixing angle $\theta$.

desk verdict A genuine first PQCD calculation for B0→J/ψh1 decays with useful predictions, but the 'exact' amplitude-independent ratio claim is overstated and should be fixed before publication. read the letter →

arxiv 2501.01075 v3 pith:34TIKQLF submitted 2025-01-02 hep-ph

Qin Chang , De-Hua Yao , Xin Liu This is my paper · ORCID
classification hep-ph PACS 13.25.Hw12.38.Bx14.40.Be
keywords BmesondecaysJ/psiaxial-vectormesonsh1(1170)h1(1415)mixingperturbativeQCDbranchingfractions
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 presents the first perturbative-QCD calculation of the color-suppressed decays $B_d^0\to J/\psi h_1(1170)$, $B_d^0\to J/\psi h_1(1415)$, $B_s^0\to J/\psi h_1(1170)$, and $B_s^0\to J/\psi h_1(1415)$, including next-to-leading-order vertex corrections and the charmonium Sudakov factor. It treats $h_1(1170)$ and $h_1(1415)$ as mixtures of singlet and octet axial-vector states with mixing angle $\theta$, taking $\theta=29.5^\circ$ as the default. The predicted branching fractions lie in the range $10^{-6}$ to $10^{-3}$, large enough for LHCb and Belle II to observe, and they move strongly with $\theta$. The paper argues that ratios of pairs of branching fractions, $R_d^{SO}$ and $R_s^{SO}$, cancel most hadronic uncertainties and therefore give tight, nearly model-independent constraints on $\theta$. It also predicts longitudinal polarization dominance with $f_L>80\%$, except near $\theta\approx35^\circ$ and $-55^\circ$, and direct $CP$ asymmetries too small to measure soon.

What carries the argument

The engine is the perturbative QCD factorization formula for color-suppressed $B\to J/\psi$ plus light axial-vector decays, built from factorizable and non-factorizable emission amplitudes with NLO vertex corrections absorbed into effective Wilson coefficients and the Sudakov factor for charmonia included in the hard kernel. The $h_1$ mixing is imposed through the singlet-octet rotation, so the physical amplitudes are linear combinations of $A(B^0\to J/\psi\tilde h_1)$ and $A(B^0\to J/\psi\tilde h_8)$ with coefficients such as $(1/\sqrt{3})\cos\theta$ and $(1/\sqrt{6})\sin\theta$. The near equality of the two singlet amplitudes makes the ratios $R_d^{SO}$ and $R_s^{SO}$ nearly pure functions of $\theta$, and the Gell-Mann-Okubo relation fixes $\theta=29.5^\circ$ from the averaged $K_1$ mixing angle $\theta_K\approx39^\circ$. This machinery yields the branching ratios, polarization fractions, relative phases, and direct $CP$ asymmetries presented in the paper.

What would settle it

Measure the four branching fractions; in particular, a precise LHCb or Belle II measurement of R_s^QF = B(Bs0 -> J/psi h1(1415))/B(Bs0 -> J/psi h1(1170)) that disagrees with the predicted phase-space factor times $cot^{2}$ $\alpha$ would falsify the two-state mixing hypothesis, since this ratio is independent of decay amplitudes. Alternatively, observing f_L well below 80% at $\theta$ values away from the predicted dips near 35 degrees and -55 degrees would contradict the PQCD polarization prediction.

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Extended reading notes

Core claim

The central claim is that the four $B^0\to J/\psi h_1$ channels have branching fractions that are both large and strongly dependent on $\theta$: $\mathcal{B}(B_d^0\to J/\psi h_1(1170))\approx 9.5\times10^{-5}$, $\mathcal{B}(B_d^0\to J/\psi h_1(1415))\approx 1.3\times10^{-6}$, $\mathcal{B}(B_s^0\to J/\psi h_1(1170))\approx 1.5\times10^{-5}$, and $\mathcal{B}(B_s^0\to J/\psi h_1(1415))\approx 1.7\times10^{-3}$ at NLO. Because the two singlet-octet amplitudes $A(B^0\to J/\psi\tilde h_1)$ and $A(B^0\to J/\psi\tilde h_8)$ are nearly equal, the ratios $R_d^{SO}$ and $R_s^{SO}$ reduce largely to trigonometric functions of $\theta$, and in the quark-flavour basis the analogous ratios are exactly phase space times $\cot^2\alpha$. A measurement of $R_d^{SO}\approx72$ or $R_s^{SO}\approx117$ would therefore fix $\theta$ with much smaller theoretical error than any single branching fraction. The polarization fractions are mostly flat in $\theta$ except for sharp dips near $\theta\approx35^\circ$ and $-55^\circ$, where $f_L$ can fall well below 80%, so an anomalously small longitudinal fraction would itself be a $\theta$ signal. Direct $CP$ asymmetries are predicted to be of order $10^{-4}$ to $10^{-2}$ and unlikely to be observed soon.

Load-bearing premise

The calculation assumes that h1(1170) and h1(1415) are pure two-component mixtures of one octet and one singlet axial-vector state, with the mixing angle fixed by the Gell-Mann-Okubo relation from a chosen K1 mixing angle; any extra glueball, continuum, or nonet-mixing component shifts every theta-dependent prediction.

Editorial extensions

If this is right

  • If the predicted branching fractions are confirmed, these four modes become the first measured probes of $h_1$ mixing through $B$-meson decays.
  • A measurement of $R_d^{SO}\approx72$ or $R_s^{SO}\approx117$ would pin $\theta$ to a narrow range, since these ratios nearly eliminate the dominant hadronic uncertainties.
  • The $h_1(1415)$ contribution to $B_s^0\to J/\psi K_S^0 K^\pm\pi^\mp$ is estimated at about $0.48\times10^{-3}$, comparable to the $f_1(1420)$ contribution, so future amplitude analyses of that final state can separate the two resonances.
  • Observing $f_L<80\%$ in any of the four channels would select $\theta\approx35^\circ$ or $-55^\circ$ instead of the default $29.5^\circ$.
  • The NLO vertex corrections substantially increase the branching fractions relative to LO, so these decays also test the color-suppressed decay mechanism itself.

Reading between the lines

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

  • The same ratio construction could be applied to $B^0\to J/\psi f_1(1285), f_1(1420)$ decays, cross-checking the $f_1$ analogue of $\theta$ with identical error cancellation.
  • A future high-statistics measurement of $R_s^{QF}$, being independent of the decay amplitudes, would give one of the cleanest determinations of the quark-flavour mixing angle $\alpha$ and indirectly of $\theta_K$.
  • If $B(B_s^0\to J/\psi h_1(1415))$ is found to deviate significantly from $1.7\times10^{-3}$, the simple two-state mixing hypothesis would have to be extended with glueball or continuum components in $h_1(1415)$.
  • The same NLO machinery can be carried over to $B^0\to J/\psi b_1(1235)$ and $B^0\to J/\psi K_1$ modes, where similar singlet-octet or flavour-basis ratios might provide parameter-free extractions of the corresponding mixing angles.
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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 / 3 minor

Summary. This paper computes for the first time the B0d,s → J/ψ h1(1170), h1(1415) decays in the perturbative QCD approach at NLO accuracy, treating h1(1170) and h1(1415) as mixtures of the SU(3) singlet and octet axial-vector states. The authors predict branching fractions, polarization fractions, relative phases, and direct CP asymmetries, and propose several ratios of branching fractions as sensitive probes of the h1 mixing angle θ. The central quantitative claims are that the branching fractions are large (O(10^-6–10^-3)), that the observables are strongly dependent on θ, and that certain ratios (notably R^QF_d and R^QF_s in Eqs. (37)–(38)) are "exactly derived" and independent of the decay amplitudes, thereby providing much stronger constraints on θ.

Significance. If the predictions are reliable, they constitute the first PQCD-level study of these modes and give concrete, testable observables for LHCb and Belle-II, including an interesting consistency check with the measured B0s → J/ψ K0S K+π− rate through the h1(1415) and f1(1420) resonances. The paper makes good use of existing NLO machinery, includes Sudakov factors for charmonia, and provides a detailed error budget from hadronic inputs. The proposed ratios, if properly qualified, could indeed reduce hadronic uncertainties. However, the advertised "exact" amplitude-independent QF-basis ratios are not exact as stated, and the printed relation between θ and α is internally inconsistent with the paper's own Table II; these issues affect the strength of the paper's main phenomenological punchline and require revision.

major comments (2)
  1. [Section II.A, Eq. (2)] The claim that R^QF_d and R^QF_s are "exactly derived" and "independent of the decay amplitudes" is not correct. Starting from Eqs. (21)–(24), each physical amplitude is a linear combination of A(B→J/ψh1~) and A(B→J/ψh8~) with coefficients involving 1/√3 and 1/√6; the reduction to phase-space × cot²α requires an additional assumption, such as A(h1~)=A(h8~) (or, in the QF basis, equality of the relevant flavor amplitudes) and neglect of mass-dependent hard kernels. The paper itself states immediately before Eq. (36) that this equality is only "approximately valid", and Table VI shows it is not exact (e.g., for B0s the longitudinal amplitudes are 0.292+i2.384 and 0.300+i2.454 in units of 10^-3 GeV^3). More importantly, the formula is numerically inconsistent with the paper's own results: using α=-5.8° from Table II and the phase-space factor in Eq. (37) gives R^QF_d ≈ (1.11) × cot²(5.8°) ≈ 107, whereas the branching fractions in Table IV give R^SO_d = 72.27. The authors should either provide the correct amplitude-dependent expression for the QF-basis ratios or clearly label Eqs. (37)–(38) as approximate, and quantify the resulting systematic uncertainty in the θ-extraction claim.
  2. [Eq. (2)] The relation θ = α − arctan√2 is inconsistent with Table II and with the state definitions in Eq. (1). For every row of Table II one obtains θ − α = 35.3°, which equals arctan(1/√2) (i.e., 90° − arctan√2). The printed equation would give θ ≈ −60° for α ≈ −5.8°, not the quoted θ = 29.5°. Since Eq. (2) is used in the derivation of the QF-basis ratios in Eqs. (37)–(38), this error propagates directly to the paper's central "exact" formula. The correct relation appears to be θ = α + arctan(1/√2) (equivalently θ = α + π/2 − arctan√2), and Eq. (2) should be corrected accordingly.
minor comments (3)
  1. [Table II] The header of Table II appears corrupted: the first entry under α is shown as "α − 90◦ 3.2◦", which is ambiguous. Please clarify the notation and ensure the table lists the numerical values cleanly.
  2. [Throughout] There are minor language issues, e.g., "independent on the decay amplitudes" should be "independent of the decay amplitudes", and "can be exactly derived" should be qualified if the result is approximate.
  3. [Section III, sentence before Eq. (36)] The statement that the SO ratios are approximately model-independent if A(B0→J/ψh1~)=A(B0→J/ψh8~) deserves a quantitative qualification: because the denominator in Eq. (34) is small for θ near 35°, even a few-percent difference between A(h1~) and A(h8~) can produce a much larger relative error in the ratio R^SO_d. A numerical estimate of this systematic effect would strengthen the paper.

Circularity Check

1 steps flagged · score 4.0 of 10

Core PQCD predictions are self-contained, but the 'exact, amplitude-independent' QF-basis ratio claim reduces to the input A(h1tilde)=A(h8tilde) approximation.

  1. self definitional [Section III, Eqs. (34)-(38), discussion following Eq. (36)]
    "It can be found that, these ratios can be used to test or extract the values of mixing angle θ approximately in a model independent way if A(B0 → J/ψ˜h1) = A(B0 → J/ψ˜h8), which is approximately valid as has been shown through the numerical results given in Table VI. ... one can employ a more convenient and intuitive form by extending the ratios to the QF basis, which can be exactly derived as ... where the ratios RQF d and RQF s are independent on the decay amplitudes."

    Starting from Eqs. (21)-(24), the physical ratio is |cosα Ahn + sinα Ahs|^2 / |−sinα Ahn + cosα Ahs|^2 (times phase space), not cot2α. Eq. (37) reduces to the stated cot2α form only when the underlying SO-basis amplitudes are set equal, A(B0→J/ψ˜h1)=A(B0→J/ψ˜h8), or satisfy a special complex relation. That equality is precisely the approximation the authors invoked one paragraph earlier and label 'approximately valid' via Table VI. The claim that RQF d and RQF s are 'independent on the decay amplitudes' is therefore equivalent, by construction, to the input amplitude equality rather than a consequence of the full PQCD amplitudes.

full rationale

The paper's main branching-fraction, polarization, and CP-asymmetry predictions are genuine PQCD computations: no parameter is fitted to the B0→J/ψh1 observables, and the mixing angle θ is taken from an independent Gell-Mann-Okubo analysis using external θK inputs and PDG masses. The NLO vertex corrections, Wilson coefficients, Sudakov factor, and distribution amplitudes are adopted from earlier published calculations (several by the same group), but these are technical inputs with stated assumptions and are not determined by the target data; the self-citations are therefore not load-bearing in a circular sense. The one constructional step is the QF-basis ratio: Eqs. (37)-(38) present cot2α as 'exactly derived' and 'independent on the decay amplitudes', but the exact amplitude expressions in Eqs. (21)-(24) do not reduce to that form unless the h1tilde and h8tilde amplitudes are assumed equal. Since that equality is the paper's own approximation, the amplitude-independent constraint is partially the input mixing scheme in disguise. The amplitude-resolved results, such as RSO d=72.27 and RSO s=116.52 in Eq. (36), are computed with the full amplitudes and remain independent predictions; hence the circularity is partial rather than total.

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

The central predictions rest on a chain of inputs: PQCD factorization, the h1 mixing scheme, GMO-determined theta, QCD sum-rule LCDAs and decay constants, and NLO vertex corrections from earlier papers. None of these are fitted to the B0 to J/psi h1 observables themselves, so the ledger is not a fitted model, but the default theta is an input rather than an output.

free parameters (6)
  • h1 mixing angle theta = 29.5 degrees (default), scanned over [-90, 90] degrees
    Set by Eq. (5) from the GMO relation with average theta_K approximately 39 degrees; not measured directly, and all branching-fraction predictions depend strongly on it (Table II, Fig. 4).
  • Gegenbauer moments a1_parallel and a2_perp for h1 LCDAs = a1_parallel(h1) = -2.00 +/- 0.35, a1_parallel(h8) = -1.95 +/- 0.35, a2_perp = 0.18 +/- 0.22 and 0.14 +/- 0.22
    Hadronic shape parameters of the h1 light-cone distribution amplitudes taken from QCD sum rules (Eq. B7, Ref. [24]); they are a major source of the quoted branching-fraction errors.
  • Decay constants f_h1 and f_h8 = f_h1 = 0.180 +/- 0.012 GeV, f_h8 = 0.190 +/- 0.010 GeV
    Inputs from Ref. [24]; they set the overall normalization of the factorizable amplitudes.
  • B-meson shape parameters omega_Bd and omega_Bs = omega_Bd = 0.40 +/- 0.04 GeV, omega_Bs = 0.50 +/- 0.05 GeV
    Parameters of the B-meson distribution amplitude; listed as the first error source in Table IV.
  • Charm quark mass m_c = 1.50 +/- 0.15 GeV
    Short-distance input for the hard kernels; cited as one of the error sources in Table IV.
  • Hard-scale variation factor a_t = 1.0 +/- 0.2
    Ad hoc variation of the maximum hard scale by 20% to estimate higher-order corrections; the paper reports this reduces the scale error from 30% to 5% at NLO.
assumptions (6)
  • domain assumption PQCD k_T factorization is valid for color-suppressed B to J/psi light-hadron decays, including Sudakov suppression and the charmonium Sudakov factor.
    The entire amplitude construction in Eqs. (12)-(24) assumes this factorization; it is inherited from Refs. [38,43,44,45,53] and not re-derived in this paper.
  • domain assumption h1(1170) and h1(1415) are pure two-state mixtures of the SU(3) singlet h1 and octet h8 with a single mixing angle theta.
    Eq. (1) defines the physical states this way, and Appendix A diagonalizes a 2x2 mass matrix; no glueball, continuum, or additional nonet-mixing component is included.
  • domain assumption The Gell-Mann-Okubo mass relation m^2_b1 + 3 m^2_h8 = 4 m^2_K1B determines the octet mass and hence theta.
    Used in Eqs. (4)-(5) and Appendix A to convert theta_K and PDG masses into theta = 29.5 degrees; GMO relations are known to be approximate.
  • domain assumption The h1 light-cone distribution amplitudes from QCD sum rules up to twist-3 are reliable.
    Eqs. (B1)-(B7) take the LCDAs and Gegenbauer moments from Ref. [24]; uncertainties in these dominate the branching-fraction errors.
  • domain assumption NLO vertex corrections can be absorbed into effective helicity-dependent Wilson coefficients with the same structure as at leading order.
    This is the standard treatment in Refs. [25,30,31] and is used in Eq. (20) to define the effective coefficients a2, a3, a5, a7, a9.
  • domain assumption For the LHCb comparison, h1(1415) to K0S K+ pi- proceeds through K* Kbar with isospin conservation and no interference with f1(1420).
    Eq. (28) derives a secondary branching fraction from B(h1(1415) to K* Kbar) = 0.415 +/- 0.085, and the consistency check with B_s to J/psi K0S K+ pi- assumes the two resonances add without unknown interference, which the paper itself flags.

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Pith. "Pith review of Studying the $B^{0} \to J/\psi h_{1}$ decays with $h_{1}(1170)-h_{1}(1415)$ mixing in the perturbative QCD approach." pith.science (2026). https://pith.science/paper/34TIKQLF

@misc{pith2026250101075,
  author       = {Pith},
  title        = {Pith review of: Studying the $B^0 \to J/\psi h_1$ decays with $h_1(1170)-h_1(1415)$ mixing in the perturbative QCD approach},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/34TIKQLF}},
  note         = {Machine review of arXiv:2501.01075}
}
abstract

In this paper, we study the $B^{0} \to J/\psi h_{1}$ decays for the first time by using perturbative QCD approach up to the presently known next-to-leading order accuracy. The vertex corrections present significant contribution to the amplitude. In the calculation, the mixing between two light axial-vector mesons $h_{1}(1170)$ and $h_{1}(1415)$ are also studied in detail. The observables including the branching ratios, polarization fractions and $CP$ asymmetries are predicted and discussed explicitly. It is found that the $B^{0} \to J/\psi h_{1}$ decays have relatively large branching fractions, which are generally at the order of ${\cal O}(10^{-6}\sim10^{-3})$, and thus are possible to be observed by the LHCb and Belle-II experiments in the near future. Moreover, they are very sensitive to the mixing angle $\theta$ and can be used to test the values of $\theta$. In addition, some ratios between the branching fractions of $B^{0} \to J/\psi h_{1}$ decays can provide much stronger constraints on $\theta$ due to their relatively small theoretical errors. The $B^{0} \to J/\psi h_{1}$ decays are generally dominated by the longitudinal polarization contributions, specifically, $f_{L}(B^{0} \to J/\psi h_{1})>80\%$, except for the case that $\theta\sim 35^\circ$ and $-55^\circ$. Unfortunately, the direct $CP$ asymmetries of $B^{0} \to J/\psi h_{1}$ decays are too small to be observed soon even if the effect of $\theta$ is considered. The future precise measurements on $B^{0} \to J/\psi h_{1}$ decays are expected for testing these theoretical findings and exploring the interesting nature of $h_{1}(1170)$ and $h_{1}(1415)$.

Figures

Figures reproduced from arXiv: 2501.01075 by the authors.

Figure 1
Figure 1. FIG. 1. (Color online) Leading quark-level Feynman diagrams for [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. (Color online) Leading order Feynman diagrams for [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. (Color online) Vertex corrections to [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figures from the paper (3 more)
Figure 4
Figure 4. Figure 4: FIG. 4. (Color online) The dependence of [PITH_FULL_IMAGE:figures/full_fig_p012_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5. (Color online) The dependence of [PITH_FULL_IMAGE:figures/full_fig_p017_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6. (Color online) The dependence of [PITH_FULL_IMAGE:figures/full_fig_p017_6.png]

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Reviewed August 10, 2026 · model on record in the stance chip above.