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arxiv: 2601.01305 · v2 · submitted 2026-01-04 · ✦ hep-ph

Recognition: 1 theorem link

· Lean Theorem

B_c meson decays into S-wave charmonium plus light meson pairs in the perturbative QCD approach

Jia-Ying Wang (Nanjing Agricultural U.) , Jing Jiang (Nanjing Agricultural U.) , Yu-Jie Liu (Nanjing Agricultural U.) , Da-Cheng Yan (Changzhou U.) , Zhou Rui (Yantai U.) , Zhen-Jun Xiao (Nanjing Normal U.) , Ya Li (Nanjing Normal U.)
Authors on Pith no claims yet

Pith reviewed 2026-05-16 18:37 UTC · model grok-4.3

classification ✦ hep-ph
keywords B_c meson decaysperturbative QCDcharmoniumthree-body decaysbranching ratiospolarization fractionsresonance contributionsLHCb
0
0 comments X

The pith

pQCD calculations of B_c to charmonium plus resonant pion or kaon pairs give a 2π/π branching ratio of 2.67 that matches LHCb data.

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

The paper computes branching ratios and polarization fractions for three-body B_c decays to S-wave charmonium states such as J/ψ or ψ(2S) together with a vector resonance that decays to a pair of light mesons. It employs leading-order perturbative QCD with improved two-meson distribution amplitudes for the collinear meson pair. Longitudinal polarization fractions reach about 90 percent because transverse contributions from the main diagrams are power-suppressed. Direct CP violation is predicted to vanish because only tree-level diagrams enter. A key ratio of branching fractions for the two-pion versus single-pion modes is found to be 2.67 with small theoretical uncertainty, consistent with the LHCb measurement of 2.80.

Core claim

In the perturbative QCD approach at leading order, the P-wave resonance contributions to B_c → Ψ (V→) P1P2 decays are evaluated using improved two-meson distribution amplitudes. The resulting CP-averaged branching ratios, polarization fractions (longitudinal components ~90 percent), and several relative ratios are obtained. In particular, the ratio R^{PQCD}_{2π/π} ≡ B(B_c^+ → J/ψ(ρ→)π^+π^0)/B(B_c^+ → J/ψπ^+) equals 2.67^{+0.21}_{-0.14}, which agrees with the LHCb value 2.80±0.25.

What carries the argument

Improved two-meson distribution amplitudes for the collinear light-meson pair produced via vector resonances such as ρ(770) or K*(892), which encode the non-perturbative dynamics inside the pQCD factorization formula.

If this is right

  • Longitudinal polarization fractions reach approximately 90 percent for B_c → [J/ψ, ψ(2S)](V→)P1P2 decays.
  • Direct CP asymmetries are zero because only tree diagrams contribute.
  • Several other relative ratios among branching fractions are predicted and can be tested experimentally.
  • The framework supplies concrete branching-ratio predictions for modes involving η_c or higher charmonium states.

Where Pith is reading between the lines

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

  • If the two-meson amplitudes remain reliable, the same method can be applied directly to other three-body B decays involving resonances without introducing new non-perturbative inputs.
  • The high longitudinal polarization implies that angular analyses of these modes will be dominated by a single amplitude, simplifying extraction of weak phases.
  • Discrepancies in future data on η_c modes would indicate whether the current two-meson amplitudes need refinement beyond leading order.

Load-bearing premise

The improved two-meson distribution amplitudes determined in earlier work accurately capture the non-perturbative dynamics of the collinear light-meson pair at leading order.

What would settle it

An LHCb or future measurement of the ratio R_{2π/π} lying well outside the interval 2.4 to 2.9 would falsify the central prediction.

Figures

Figures reproduced from arXiv: 2601.01305 by Da-Cheng Yan (Changzhou U.), Jia-Ying Wang (Nanjing Agricultural U.), Jing Jiang (Nanjing Agricultural U.), Ya Li (Nanjing Normal U.), Yu-Jie Liu (Nanjing Agricultural U.), Zhen-Jun Xiao (Nanjing Normal U.), Zhou Rui (Yantai U.).

Figure 1
Figure 1. Figure 1: FIG. 1: Typical leading-order Feynman diagrams for the quas [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2: The overlap of the leading twist distribution amplit [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3: Dependence of the (a) [PITH_FULL_IMAGE:figures/full_fig_p012_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4: Dependence of the ratio [PITH_FULL_IMAGE:figures/full_fig_p013_4.png] view at source ↗
read the original abstract

In this work, we explore the $P$-wave resonance contributions to the three-body charmonium decays of $B_c\to \Psi (V\to) P_1P_2$ using the perturbative QCD formalism at leading order, where $\Psi$ denotes a $S$-wave charmonium state, such as $\eta_c(1S,2S),J/\psi$, and $\psi(2S)$. Here, $P_1P_2$ represents a collinear $\pi\pi$ ($K\pi$) pair in the final state, which was primarily produced through the vector resonance $\rho(770)$ ($K^*(892)$ ). With the improved two-meson distribution amplitudes determined from our previous works, we examined the $CP$-averaged branching ratios and polarization fractions of the considered three-body decays. The longitudinal polarization fractions of the $B_c\to [J/\psi, \psi(2S)] (V\to) P_1P_2$ decays are found to be as large as $\sim 90\%$, since the transverse amplitudes from the dominant factorizable emission diagrams are always power suppressed with respect to the longitudinal ones. The direct $CP$ violations in $B_c\to \Psi (V\to) P_1P_2$ decays are predicted naturally to be zero as they solely receive contributions from tree diagrams. Several interesting relative ratios among the branching fractions of the concerned processes are investigated. In particular, the obtained ratio $R^{\rm PQCD}_{2\pi/\pi}\equiv \mathcal{B}(B^+_c \to J/\psi(\rho\to)\pi^+\pi^0)/{\mathcal{B}(B^+_c \to J/\psi\pi^+)}=2.67^{+0.21}_{-0.14}$ is consistent well with the LHCb measurement $R^{\rm exp}_{2\pi/\pi}=2.80\pm0.25$. Other similar ratios proposed in this work can be tested by LHCb experiments in the near future.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript computes leading-order perturbative QCD amplitudes for B_c decays to S-wave charmonium states (η_c(1S,2S), J/ψ, ψ(2S)) plus collinear ππ or Kπ pairs produced via ρ(770) or K*(892) resonances. Using improved two-meson distribution amplitudes imported from the authors' prior works, the authors obtain CP-averaged branching ratios, polarization fractions (longitudinal fractions ~90% for J/ψ and ψ(2S) modes), and several ratios of branching fractions. The central numerical result is the ratio R^{PQCD}_{2π/π} = 2.67^{+0.21}_{-0.14}, stated to be consistent with the LHCb measurement 2.80 ± 0.25; direct CP violation is predicted to vanish because only tree diagrams contribute.

Significance. If the results hold, the paper supplies a set of concrete, testable predictions for branching-fraction ratios in B_c three-body decays that LHCb can confront in the near future. It also illustrates the utility of the pQCD factorization framework with two-meson DAs for extracting polarization information, and the reported agreement with the single existing LHCb ratio provides a non-trivial consistency check within the stated approximations.

major comments (2)
  1. [Numerical results] Numerical results (around the presentation of R^{PQCD}_{2π/π}): the quoted asymmetric uncertainty (+0.21, -0.14) is not explicitly traced to variations in the parameters of the improved two-meson DAs; because those DAs are taken unchanged from earlier papers and carry fitted parameters, the robustness of the quoted agreement with LHCb data cannot be assessed without a dedicated sensitivity study to the DA parameters.
  2. [Polarization fractions] Polarization analysis (statement that transverse amplitudes are power-suppressed): the claim of ~90% longitudinal polarization rests on the assertion that transverse contributions from the dominant factorizable emission diagrams are power-suppressed, yet no explicit numerical ratio of transverse to longitudinal amplitudes or power-counting estimate is provided for the leading diagrams; this omission weakens the quantitative support for the polarization prediction.
minor comments (2)
  1. [Abstract] Abstract: the resonance notation (ρ→) and the symbol B^+_c are non-standard; replacing them with explicit final-state notation such as B_c^+ → J/ψ(ρ → π^+π^0) would improve immediate readability.
  2. [Introduction] References: the two-meson DA papers are cited but a one-sentence recap of how their parameters were fixed (e.g., from which data sets) would help readers judge the propagation of those uncertainties into the present results.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments and the positive overall assessment of our work. We address each major comment below and will revise the manuscript accordingly to improve clarity and robustness.

read point-by-point responses

  1. Referee: Numerical results (around the presentation of R^{PQCD}_{2π/π}): the quoted asymmetric uncertainty (+0.21, -0.14) is not explicitly traced to variations in the parameters of the improved two-meson DAs; because those DAs are taken unchanged from earlier papers and carry fitted parameters, the robustness of the quoted agreement with LHCb data cannot be assessed without a dedicated sensitivity study to the DA parameters.

    Authors: We agree that explicitly tracing the uncertainty to the two-meson DA parameters would strengthen the result. In the revised manuscript we will add a dedicated paragraph (and a short table) showing the variation of R^{PQCD}_{2π/π} under reasonable changes of the leading Gegenbauer moments and resonance parameters taken from our earlier works. The present asymmetric error is dominated by the B_c decay constant and the choice of hard scale; the DA-parameter contribution will be quantified separately so that the robustness of the agreement with the LHCb value can be assessed directly. revision: yes

  2. Referee: Polarization analysis (statement that transverse amplitudes are power-suppressed): the claim of ~90% longitudinal polarization rests on the assertion that transverse contributions from the dominant factorizable emission diagrams are power-suppressed, yet no explicit numerical ratio of transverse to longitudinal amplitudes or power-counting estimate is provided for the leading diagrams; this omission weakens the quantitative support for the polarization prediction.

    Authors: We acknowledge that an explicit numerical ratio would make the polarization claim more quantitative. In the revised version we will insert a brief paragraph together with a short table that reports the ratio |A_T / A_L| for the leading factorizable emission diagrams of the J/ψ and ψ(2S) modes. We will also add a one-sentence power-counting estimate based on the m_c/m_{B_c} suppression of the transverse helicity amplitudes. This addition will directly support the stated ~90% longitudinal fractions without altering the central results. revision: yes

Circularity Check

1 steps flagged

Branching ratio predictions inherit parameters from prior same-group two-meson DA fits

specific steps
  1. self citation load bearing [Abstract]
    "With the improved two-meson distribution amplitudes determined from our previous works, we examined the CP-averaged branching ratios and polarization fractions of the considered three-body decays. ... the obtained ratio R^{PQCD}_{2π/π}≡B(B_c^+ → J/ψ(ρ→)π^+π^0)/B(B_c^+ → J/ψπ^+)=2.67^{+0.21}_{-0.14} is consistent well with the LHCb measurement"

    The central results are computed by convoluting the hard kernels with the two-meson DAs taken from the authors' prior works. Because those DAs' parameters were already fitted in the cited earlier papers and are not re-fitted or independently validated here, the reported branching ratios and ratio R^{PQCD}_{2π/π} are direct numerical outputs of the imported DA inputs rather than independent first-principles predictions.

full rationale

The paper's numerical results for branching ratios, polarization fractions, and the ratio R^{PQCD}_{2π/π} are obtained by convoluting LO hard kernels with improved two-meson distribution amplitudes that are imported directly from the authors' earlier publications. These DAs are not re-determined or constrained by new data within the present work, so the quoted 'predictions' and their agreement with LHCb data reduce to numerical consequences of the functional forms and fitted parameters chosen in the cited prior papers. The pQCD formalism itself is independent, but the load-bearing non-perturbative inputs create moderate circularity.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

Central results rest on pQCD factorization at leading order and two-meson DAs fitted in prior author papers. No new particles or forces are introduced.

free parameters (1)
  • parameters in improved two-meson DAs
    Determined from previous works; control the non-perturbative input for the P1P2 system.
axioms (1)
  • domain assumption pQCD factorization theorem holds at leading order for these three-body decays
    Invoked throughout the calculation of amplitudes via factorizable emission diagrams.

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