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Study of $B_c \to \chi_{cJ}\ (P, V)$ decays in the improved perturbative QCD formalism

T0 review · 3 major / 6 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read The improved perturbative QCD formalism, keeping charm quark mass effects in the Sudakov factor, reproduces the LHCb ratios for B_c^+ → χ_{c2}π^+ and χ_{c1}π^+ decays, and extends to predict χ_{cJ}ρ^+ modes and polarization fractions.

desk verdict First iPQCD application to B_c -> chi_cJ(P,V) yields one solid ratio (chi_c2/Jpsi) and a fragile chi_c1/chi_c2 prediction that rests on unvalidated twist-3 DA normalization. read the letter →

arxiv 2505.20605 v2 pith:O57IJR6J submitted 2025-05-27 hep-ph hep-ex

Xin Liu , Hsiang-nan Li , Zhen-Jun Xiao This is my paper · ORCID
classification hep-phhep-ex PACS 13.25.Hw12.38.Bx14.40.Nd
keywords B_cmesondecaysχ_cJcharmoniaimprovedperturbativeQCDSudakovresummationbranchingratioratioslongitudinalpolarizationLHCbfactorization
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

The paper aims to establish that the improved perturbative QCD (iPQCD) formalism—a version of perturbative QCD that keeps charm quark mass effects in the Sudakov factor—can describe the measured $B_c^+$ decays into $p$-wave charmonia plus a light meson. Its central results are the ratios $R_{\chi_{c2}/J/\psi} = 0.32 \pm 0.05$ and $R_{\chi_{c1}/\chi_{c2}} = 0.11 \pm 0.01$, which agree with LHCb data within uncertainties, supporting the framework's predictive ability. The paper also predicts $R_{\chi_{c0}/J/\psi} = 0.17 \pm 0.02$, lower than the value $1.41^{+0.50}_{-0.45}$ inferred from present data, and attributes this tension to the poorly known $\chi_{c0}$ decay constant. It further computes the $\chi_{c1,c2}\rho^+$ branching ratios, finds nearly unity longitudinal polarization fractions, and identifies a constructive (destructive) twist-2/twist-3 interference in the $\chi_{c1}\rho^+$ ($\chi_{c1}\pi^+$) mode. If correct, these predictions give concrete targets for future precision measurements and probe the internal structure of charmonium states.

What carries the argument

The machinery is the iPQCD factorization formula, $A \sim \int dx_1 dx_2 dx_3\, db_1 db_2 db_3\, \mathrm{Tr}\,[C\, \Phi_{B_c}\Phi_{P,V}\Phi_{\chi_{cJ}} H\, e^{-S}]$, in which the Sudakov factor $e^{-S}$ now includes charm quark mass effects, and $\Phi_{\chi_{cJ}}$ are model distribution amplitudes for the $p$-wave charmonia (twist-2 and twist-3) built from Coulombic wave functions with small relativistic corrections. The symmetry properties of these amplitudes under $x \leftrightarrow 1-x$, together with normalization to the decay constants $f_{\chi_{cJ}}$, control the relative factorizable and nonfactorizable contributions and produce the distinctive ratio pattern, notably the huge $R^{\chi_{c1}}_{\rho/\pi}\approx 34$ and the near-unity longitudinal polarization fractions.

What would settle it

High-precision LHCb measurements of $R_{\chi_{c2}/J/\psi}$ and $R_{\chi_{c1}/\chi_{c2}}$ with errors below about 20% would settle the claim: if the values move outside $0.32\pm0.05$ and $0.11\pm0.01$ respectively by more than the combined uncertainties, the iPQCD prediction is falsified.

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Extended reading notes

Core claim

The paper's central discovery is that, at leading order in the strong coupling $\alpha_s$, the improved perturbative QCD formalism reproduces the two LHCb ratios for $p$-wave charmonium production in $B_c^+$ decays: $R_{\chi_{c2}/J/\psi} = 0.32 \pm 0.05$ against the measured $0.37 \pm 0.06$, and $R_{\chi_{c1}/\chi_{c2}} = 0.11 \pm 0.01$, which satisfies the experimental upper bound $0.49$. The framework also predicts $R_{\chi_{c0}/J/\psi} = 0.17 \pm 0.02$, well below the value $1.41^{+0.50}_{-0.45}$ inferred from LHCb evidence, and the paper traces this tension to the uncertain $\chi_{c0}$ decay constant: a value around $0.3$ GeV would boost the $\chi_{c0}\pi$ branching ratio by a factor of ten. For the vector modes, the paper predicts longitudinal polarization fractions of about $95\%$ ($\chi_{c1}\rho^+$) and $93\%$ ($\chi_{c2}\rho^+$), and identifies a twist-2/twist-3 interference that is constructive in $B_c^+\to\chi_{c1}\rho^+$ and destructive in $B_c^+\to\chi_{c1}\pi^+$, a pattern absent in the $\chi_{c0}$ and $\chi_{c2}$ channels.

Load-bearing premise

The load-bearing premise is that the model distribution amplitudes for the $\chi_{cJ}$ mesons—especially the $\chi_{c0}$ amplitude with decay constant $f_{\chi_{c0}}=0.093$ GeV—are accurate enough, since the paper shows that changing $f_{\chi_{c0}}$ to about $0.3$ GeV multiplies the $\chi_{c0}\pi$ branching ratio by ten.

Editorial extensions

If this is right

  • The measured ratios $R_{\chi_{c2}/J/\psi}$ and $R_{\chi_{c1}/\chi_{c2}}$ become tests of iPQCD: future LHCb data with smaller uncertainties will either confirm or shift the calculation.
  • The predicted $R_{\chi_{c0}/J/\psi}=0.17\pm0.02$ implies the present LHCb evidence for $B_c^+\to\chi_{c0}\pi^+$ is in tension with the framework unless $f_{\chi_{c0}}$ is near $0.3$ GeV, which would raise the rate tenfold.
  • Longitudinal polarization fractions $f_L\approx95\%$ ($\chi_{c1}\rho^+$) and $93\%$ ($\chi_{c2}\rho^+$) mean angular analyses should find these modes dominated by the longitudinal component.
  • The predicted multibody branching ratios for $B_c^+\to\chi_{cJ}(\to\pi^+\pi^-,\, K^+K^-,\, 2\pi^+\pi^-,\, \pi^+\pi^-K^+K^-)\pi^+/\rho^+$ give concrete, searchable final states at LHCb.

Reading between the lines

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

  • A lattice or sum-rule determination of $f_{\chi_{c0}}$ and of the $\chi_{c0}$ twist-2/twist-3 distribution amplitudes would resolve the $\chi_{c0}$ tension; the paper's reliance on the model amplitude is the fragile link.
  • If future data confirm $R^{\chi_{c1}}_{\rho/\pi}\approx34$ (far above the $J/\psi$ value around 3.15), that would corroborate the identified constructive/destructive twist interference; a value near the $J/\psi$ ratio would call for revised $\chi_{c1}$ distribution amplitudes.
  • The same iPQCD framework could be applied to semileptonic $B_c\to\chi_{cJ}\ell\nu$ decays or other two-body $B_c$ modes as a consistency check of the Sudakov treatment of charm quark mass.
  • Because the ratios are computed at leading order in $\alpha_s$, a next-to-leading-order calculation would test the stability of the agreement; the current match could in principle be a truncation accident.
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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 / 6 minor

Summary. The authors study the two-body weak decays B_c^+ -> chi_cJ (P,V)^+ (J=0,1,2; P=pi,K; V=rho,K*) in the improved perturbative QCD (iPQCD) formalism at leading order in alpha_s. They employ model distribution amplitudes for the chi_cJ mesons taken from earlier work, compute branching ratios, the ratios R_chi_c2/Jpsi and R_chi_c1/chi_c2, longitudinal polarization fractions, and translate BESIII strong-decay branching ratios into predictions for multibody final states under the narrow-width approximation. The headline results are R_chi_c2/Jpsi = 0.32 ± 0.05 and R_chi_c1/chi_c2 = 0.11 ± 0.01, which they claim agree with recent LHCb data; they also report R_chi_c0/Jpsi = 0.17 ± 0.02, which they acknowledge deviates from the LHCb-inferred value.

Significance. If the iPQCD framework is validated, these calculations provide a coherent set of predictions for B_c -> chi_cJ transitions and useful guidance for LHCb searches, including multibody chains. The paper is thorough in comparing with the existing theoretical literature and in exploiting the BESIII measurements. However, the validation is partial: one headline ratio is an upper limit, the chi_c0 ratio fails by a factor of about eight, and the agreement for chi_c1 rests on an interference that is sensitive to an unconstrained twist-3 distribution amplitude. The paper's own suggestion that f_chi_c0 ~ 0.3 GeV would repair the chi_c0 discrepancy shows that the current DA models are not predictive for that channel.

major comments (3)
  1. [Sec. III.A, Eq. (39)] The statement that R_chi_c1/chi_c2 = 0.11 ± 0.01 'agree well with the data within uncertainties' overstates the test: Eq. (2) is a 90% confidence upper limit, not a measured value. A prediction below an upper limit is consistent but cannot validate the framework at the claimed precision. Moreover, the ±0.01 error includes only variations of the inputs beta_Bc, decay constants, Gegenbauer moments, and CKM elements at fixed leading order; no renormalization-scale variation or next-to-leading-order estimate is provided. Please rephrase the comparison as 'satisfies the current upper limit' and add an estimate of the perturbative uncertainty.
  2. [Sec. II, Eqs. (15)-(18); Sec. III.A, Eq. (37)] The smallness of BR(B_c -> chi_c1 pi+) = (4.16^{+0.68}_{-0.62}) x 10^{-5} is produced by a destructive interference between twist-2 and twist-3 contributions, as the paper itself emphasizes. The relative normalization of the twist-3 amplitude phi^t_chi_c1 is fixed by the model coefficient 23.16 in Eq. (17) together with the Coulomb-potential factor C(x) with v^2 = 0.3; the paper cites no independent lattice or QCD sum-rule determination of this amplitude. Since a difference of two comparable terms is hypersensitive to their relative normalization, the prediction R_chi_c1/chi_c2 = 0.11 is not robust. The uncertainty analysis in Eqs. (32)-(38) rescales all chi_cJ decay constants by a common ±10%, so it does not test this relative normalization. Please provide a sensitivity scan over the twist-3 coefficient or v^2 and report the resulting range of R_chi_c1/chi_c2.
  3. [Sec. III.A, Eqs. (43)-(44)] The paper explicitly acknowledges that R_chi_c0/Jpsi = 0.17 ± 0.02 lies below the LHCb-inferred value 1.41^{+0.50}_{-0.45} by roughly a factor of eight, and then states that f_chi_c0 around 0.3 GeV would amplify BR(B_c -> chi_c0 pi+) by a factor of ten. Because f_chi_c0 is an input, not predicted, this sensitivity means the framework currently makes no robust prediction for the chi_c0 mode; the range 0.09-0.3 GeV for f_chi_c0 changes the central result by an order of magnitude. The paper should either carry f_chi_c0 as a varying parameter with a prior from independent determinations, or present the chi_c0 discrepancy as evidence that the DA model in Eqs. (7)-(9) needs revision, rather than leaving it as a post hoc fix.
minor comments (6)
  1. [Sec. I, last paragraph] The phrase 'the relative ratios among the B_c -> (J/psi, chi_c1,c2) pi+ BRs from the iPQCD also match the LHCb measurements' is inaccurate because the chi_c1 constraint is only an upper limit; please adjust the wording.
  2. [Eq. (10)] The formula for C(x) is typeset in a confusing way; please clarify the power of the numerator, the domain of x, and the role of v^2 = 0.3.
  3. [Eq. (28)] Please specify whether m_b and m_c are pole masses or MSbar masses, since the decay constants are evolved using Eq. (31) and the hard kernels depend on the mass scheme.
  4. [Sec. III.B, Eq. (83)] The ratio R_chi_c1_K*/rho is quoted as (1.78 ± 0.04) x 10^{-2} with a relative uncertainty of about 2%, yet the text contrasts it with the naive expectation; the uncertainty estimate appears to exclude the dominant DA-model uncertainties. Please state which errors are included.
  5. [Ref. [36] and Eq. (30)] Ref. [36] provides QCD sum-rule estimates of the chi_cJ decay constants; it would be helpful to also quote lattice determinations or at least mention that no lattice results are used, given the 10% uncertainty in Eq. (30) is an assumption.
  6. [Sec. III.A, Eqs. (32)-(38)] The notation f_cbar c is shorthand for the set of chi_cJ decay constants but is not defined; please define it at first use.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the iPQCD predictions are independent calculations compared with LHCb data, not fitted to it.

full rationale

The paper's central predictions are obtained by convolving hard-scattering kernels with meson distribution amplitudes in Eq. (6). The chi_cJ distribution amplitudes (Eqs. (7)-(24)) and decay constants (Eq. (30)) are adopted from Refs. [13,31,32] and [36], respectively, i.e., external or earlier work, not fitted to the LHCb ratios being predicted. The iPQCD formalism and the B_c -> J/psi pi+ BR used in the denominator of R_chi_c2/J/psi come from the authors' own prior papers ([19]-[21]), but those are published, self-contained calculations used as inputs; the new numerators B_c -> chi_cJ pi+ are computed here for the first time in this framework, and no parameter is adjusted to enforce agreement with the quoted LHCb results. The paper's explicit discrepancy for R_chi_c0/J/psi (Eq. (44) vs Eq. (43)) further demonstrates that the framework is not tuned to the data. The post-hoc observation that a larger f_chi_c0 would improve the chi_c0 mode is a statement of input sensitivity, not a retrofitted prediction. The multibody branching ratios in Eqs. (54)-(67) and (88)-(97) are simple products of computed single-particle BRs and measured strong-decay BRs under the narrow-width approximation, and are not presented as first-principles derivations. No step in the derivation is equivalent by construction to its inputs.

Assumptions & free parameters 4 free parameters · 4 assumptions · 0 invented entities

The central results depend on an externally supplied factorization framework, model charmonium distribution amplitudes, and several nonperturbative inputs. The most fragile input is f_chi_c0, which the paper itself proposes to enlarge to reconcile the chi_c0 ratio. No new entities are introduced.

free parameters (4)
  • chi_c0 decay constant f_chi_c0 = 0.093 +/- 0.009 GeV (central); ~0.3 GeV (alternative)
    Central value from QCD sum rules [36]; the paper proposes a value around 0.3 GeV to amplify BR(B_c -> chi_c0 pi+) by a factor of 10 and bring R_chi_c0/Jpsi closer to the LHCb-inferred value. This is a data-driven lever arm on the central ratio.
  • Other chi_cJ decay constants (f_chi_c1, f_chi_c1_perp, f_chi_c2, f_chi_c2_perp) = 0.185, 0.090, 0.181, 0.131 GeV at mu=m_c
    External inputs from QCD sum rules [36], varied by 10%; they dominate the quoted uncertainties in the branching ratios. Not fitted to target data, but their values directly set the predicted rates.
  • B_c wave function shape parameter beta_Bc = not quoted in the paper
    Taken from Ref. [19]; it is the largest single source of uncertainty in Eqs. (32)-(38) and (75)-(78), but the value is not given, so the predictions cannot be reproduced without consulting the earlier paper.
  • Relativistic correction v^2 in charmonium distribution amplitude C(x) = 0.3
    Chosen from Ref. [32] to account for small relativistic corrections to the Coulomb wave functions; enters every chi_cJ distribution amplitude through Eq. (10).
assumptions (4)
  • domain assumption The k_T factorization formula in Eq. (6), with the Sudakov factor from Refs. [19-21, 29], is valid for B_c -> chi_cJ (P,V) decays at leading order in alpha_s.
    The paper inherits the entire iPQCD framework from previous work and does not re-derive or test its factorization theorem for these new final states.
  • ad hoc to paper The chi_cJ distribution amplitudes take the specific model forms in Eqs. (7)-(24), with coefficients fixed by normalization conditions and C(x) by a Coulomb-potential model.
    These models come from Refs. [13, 31, 32] and are stated, not derived; the chi_c0 case is explicitly fragile because f_chi_c0 is poorly known.
  • domain assumption Higher-order corrections in alpha_s and scale choices do not alter the LO predictions beyond the quoted parametric uncertainties.
    All results are leading order; no scale variation or NLO estimate is provided.
  • domain assumption The narrow-width approximation is valid for secondary decays chi_cJ -> pi pi, K K, and four-body channels.
    Used in Eqs. (54)-(67) and (88)-(97) to multiply branching ratios; ignores off-shell effects and interference.

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Pith. "Pith review of Study of $B_c \to \chi_{cJ}\ (P, V)$ decays in the improved perturbative QCD formalism." pith.science (2026). https://pith.science/paper/O57IJR6J

@misc{pith2026250520605,
  author       = {Pith},
  title        = {Pith review of: Study of $B_c \to \chi_cJ\ (P, V)$ decays in the improved perturbative QCD formalism},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/O57IJR6J}},
  note         = {Machine review of arXiv:2505.20605}
}
abstract

Motivated by the recent LHCb measurements of the ratios between the branching ratios (BRs), $R_{\chi_{c2}/J/\psi} \equiv {\rm BR}(B_c^+ \to \chi_{c2} \pi^+)/{\rm BR}(B_c^+ \to J/\psi \pi^+)$ and $R_{\chi_{c1}/\chi_{c2}} \equiv {\rm BR}(B_c^+ \to \chi_{c1} \pi^+)/{\rm BR}(B_c^+ \to \chi_{c2} \pi^+)$, we analyze the decays $B_c^+ \to \chi_{cJ} (P, V)^+$ in the improved perturbative QCD (iPQCD) formalism, where $\chi_{cJ}$ denotes the $p$-wave charmonia with $J=0, 1, 2$, and $P (V)$ denotes the pseudoscalars $\pi$ and $K$ (the vectors $\rho$ and $K^*$). Our results $R_{\chi_{c2}/J/\psi} = 0.32 \pm 0.05$ and $R_{\chi_{c1}/\chi_{c2}}= 0.11 \pm 0.01$ at leading order in the strong coupling $\alpha_s$ agree well with the data within uncertainties. The result $R_{\chi_{c0}/J/\psi} \equiv {\rm BR}(B_c^+ \to \chi_{c0} \pi^+)/{\rm BR}(B_c^+ \to J/\psi \pi^+) = 0.17 \pm 0.02$ is lower than $1.41^{+0.50}_{-0.45}$ inferred by the $B_c^+ \to \chi_{c0} \pi^+$ measurement. The large longitudinal polarization fractions imply that these components dominate the $B_c^+ \to \chi_{c1,c2} \rho^+$ BRs. A peculiar constructive (destructive) interference between the twist-2 and twist-3 contributions is identified in the $B_c^+\to \chi_{c1} \rho^+$ ($B_c^+\to \chi_{c1} \pi^+$) mode, but not in the corresponding ones with $\chi_{c0,c2}$ mesons. Inputting the BRs of the strong decays $\chi_{c0,c2} \to \pi^+ \pi^- / K^+ K^-$ and $\chi_{cJ} \to \pi^+\pi^-(\pi^+ \pi^-/K^+K^-)$ reported by the BESIII Collaboration, we obtain those of $B_c^+ \to \chi_{cJ} (\rho, \pi)^+$ followed by secondary decay chains under the narrow-width approximation. All the above iPQCD predictions can be confronted by more precise experiments in the future to deepen our understanding of QCD dynamics in $B_c\to\chi_{cJ}$ transitions.

Figures

Figures reproduced from arXiv: 2505.20605 by the authors.

Figure 1
Figure 1. FIG. 1. Leading-order Feynman diagrams for the decays [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗

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Forward citations

Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Quasi-two-body decays $B_c^+ \to \chi_{c0,c1} [\rho(K^*) \to] \pi\pi(K\pi)$ in the PQCD approach

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    A PQCD calculation predicts B(B_c→χc0ππ)=3.24×10⁻³, B(B_c→χc1ππ)=4.19×10⁻³ and R_{χc1/χc0}≈1.30 for rho-mediated decays.

  2. Systematic analysis of the transition form factors of $B_c$ to $D$-wave charmonia and corresponding semileptonic decays

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Reviewed August 7, 2026 · model on record in the stance chip above.