Improved covariant analysis of $B_c^+ \to \chi_{c1}(nP)$ decays and implications for the nature of $\chi_{c1}(3872)$

Chao-Hsi Chang; Guo-Li Wang; Jia-Yi Zheng; Shi-Hang Zhang; Su-Yan Pei; Tianhong Wang; Wei Li; Zhi-Hui Wang

REVIEW 3 major objections 5 minor 113 references

Improved covariant analysis of $B_c^+ \to \chi_{c1}(nP)$ decays and implications for the nature of $\chi_{c1}(3872)$

T0 review · 3 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read The paper predicts that $B_c^+\to\chi_{c1}(3872)\mu^+\nu_\mu$ is the most accessible weak-decay probe of $\chi_{c1}(3872)$'s structure.

desk verdict Solid Salpeter update for Bc to chi_c1(nP) with an honest charmonium assumption, but the 20x and 2x luminosity claims outrun the model uncertainty the paper itself documents. read the letter →

arxiv 2608.04693 v1 pith:TSPO724F submitted 2026-08-05 hep-ph

Shi-Hang Zhang , Jia-Yi Zheng , Wei Li , Su-Yan Pei , Tianhong Wang , Zhi-Hui Wang , Chao-Hsi Chang , Guo-Li Wang This is my paper · ORCID

classification hep-ph

keywords B_cmesonweakdecaysχc1(3872)charmonium2PstateBethe–SalpeterequationSalpeterwavefunctionssemileptonicdecaynonleptonicrelativisticcorrections

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 argues that $\chi_{c1}(3872)$ remains a viable conventional charmonium state, the $\chi_{c1}(2P)$, and that weak decays of the $B_c$ meson provide a sharp way to test that idea. The authors compute $B_c^+\to\chi_{c1}(nP)\ell^+\nu_\ell$ and $B_c^+\to\chi_{c1}(nP)X$ in the Bethe–Salpeter formalism, using an upgraded transition amplitude that evaluates the final-state wave function in its rest frame. Their central prediction is $R_{\chi_{c1}(3872)/\psi(2S)}=0.0024$, consistent with the experimental upper limit, which they read as a luminosity shortfall rather than evidence against the charmonium picture: roughly twenty times the current $B_c$ sample would be needed for the $\pi^+$ mode. By contrast, they predict $B_c^+\to\chi_{c1}(3872)\mu^+\nu_\mu$ should be observable with about twice the existing data, making it the most practical channel to probe the state's structure.

What carries the argument

The load-bearing object is the covariant hadronic transition amplitude $M^B_\mu$ (Eq. 18), which expresses $\langle\chi_{c1}|J_\mu|B_c\rangle$ as an overlap of the $B_c$ Salpeter wave function and the final-state Salpeter wave function evaluated in its own rest frame, with a factor $(M_f-\tilde{\omega}_1-\tilde{\omega}_2)/(E_f-\omega'_1-\omega'_2)$ from the quark propagators. This replaces the older amplitude $M^A_\mu$, which parameterized the final-state momentum in the initial-state rest frame and handled large recoil poorly. The wave functions are constructed by $J^P$ rather than by $^{2S+1}L_J$, and the Cornell potential with screening is used; the radial wave functions are obtained by solving the Salpeter equation.

What would settle it

Use twice the current $B_c$ sample to search for $B_c^+\to\chi_{c1}(3872)\mu^+\nu_\mu$ and, if it is found, measure its branching fraction; a rate far below the predicted $\sim7.4\times10^{-5}$, or a clear absence at twenty times the sample for the $\pi^+$ mode, would break the $\chi_{c1}(2P)$ assumption that carries the calculation.

Watch

Extended reading notes

Core claim

Under the assignment $\chi_{c1}(3872)=\chi_{c1}(2P)$, the paper claims that all $B_c\to\chi_{c1}(nP)$ widths shrink sharply with $n$ because the 2P and 3P wave functions have radial nodes that cancel much of the overlap integral. The semileptonic width $B_c^+\to\chi_{c1}(3872)e^+\nu_e$ is predicted as $(0.96^{+0.61}_{-0.42})\times10^{-16}\ \mathrm{GeV}$, about an order of magnitude below the 1P channel, and many nonleptonic channels are even smaller. Using its own $B_c\to\psi(2S)\pi^+$ branching fraction for normalization, the paper obtains $R_{\chi_{c1}(3872)/\psi(2S)}=0.0024$ and infers that the observed non-observation of $B_c^+\to\chi_{c1}(3872)\pi^+$ is expected from statistics, not from an exotic structure. The semileptonic mode is singled out as the decisive channel, since $R(\pi^+)=0.085$ makes the pion mode about twelve times harder to see.

Load-bearing premise

The calculation assumes that $\chi_{c1}(3872)$ really is the ordinary $\chi_{c1}(2P)$ charmonium state and that moving the model's predicted mass from 3928.7 MeV to the observed 3871.6 MeV by tuning the free potential parameter $V_0$ yields a trustworthy wave function; if the state is a tetraquark, molecule, hybrid, or mixture, the predicted branching fractions do not describe it.

Editorial extensions

If this is right

The $\pi^+$ channel is not a failure of the charmonium picture; observing it requires roughly twenty times the current $B_c$ data.
The semileptonic muon channel should be the experimental priority, needing only about twice the current data.
The ratio $R_{\chi_{c1}(3872)/\psi(2S)}=0.0024$ gives a concrete normalization target for future searches.
Nonleptonic channels with $\rho^+$ and $D_s^{*+}$ have larger branching fractions but are harder to access experimentally, so the semileptonic mode remains the clean probe.
A measured value of $R(\pi^+)$ near 0.085 would support the conventional charmonium assignment.

Reading between the lines

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

Extension beyond the paper: repeating the same $M^B_\mu$ calculation with $\chi_{c1}(3872)$ wave functions from molecular or tetraquark models would test whether the ratio $R(X)$ is more discriminating than any single width.
Extension beyond the paper: the mass-tuning step is the fragile point, so a sensitivity study of $R_{\chi_{c1}(3872)/\psi(2S)}$ to $V_0$ within the model's own uncertainty would sharpen the twenty-times luminosity estimate.
Extension beyond the paper: if the semileptonic rate is confirmed near prediction, it would indirectly validate the large relativistic corrections, which enter at the 43% level for the 2P state.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The paper is a legitimate extension of the authors' Bethe-Salpeter program. The new covariant amplitude M^B_mu evaluates the final-state wave function in its own rest frame, a genuine improvement for large-recoil transitions, and it yields semileptonic widths for chi_c1(1P) that agree well with several independent quark models. The new chi_c1(3P) predictions and the explicit treatment of the 2P nodal suppression are useful. The paper is also transparent about its key assumption: chi_c1(3872) is taken as conventional charmonium, and the model mass is moved to the physical mass by tuning V0. That is a defensible strategy, and the resulting R = 0.0024 is consistent with the LHCb upper limit.

The soft spots are concentrated in the experimental feasibility claims. The paper quotes uncertainties of only a few percent on B(B_c -> chi_c1(3872) pi^+) from parametric variation, but Table II shows a factor of roughly 40 spread for this same width across published models. Adopting the largest prediction would make R exceed the LHCb bound, so the statement that the non-observation is simply a luminosity problem is not robust. The ratio R also mixes the new M^B amplitude for the numerator with an older M^A branching fraction for B_c -> psi(2S)pi from Ref. [108]; that is an internal inconsistency that should be fixed or justified. Finally, the 2x luminosity estimate for the semileptonic channel is a simple scaling argument and needs a real sensitivity study to be credible.

None of this undermines the core calculation. The amplitude derivation is plausible, the comparisons are honest, and the paper is careful to say it is probing the charmonium hypothesis rather than settling the nature of X(3872). The citation pattern is appropriate, and the self-citations are to the group's own prior work, which is relevant here. It is a paper for hadron phenomenologists and LHCb analysts, and it deserves a serious referee. My recommendation: send to peer review, and in the report push the authors to add a model-uncertainty band, use a consistent amplitude for both numerator and denominator, and temper or qualify the luminosity claims.

Referee Report

3 major / 5 minor

Summary. This paper calculates semileptonic and color-favored nonleptonic B_c^+ decays to chi_c1(nP) (n=1,2,3) in the Bethe-Salpeter/Salpeter formalism, treating chi_c1(3872) as the conventional chi_c1(2P) charmonium state. The authors introduce a new covariant hadronic amplitude M_B that evaluates the final-state wave function in its rest frame, replacing the older M_A amplitude in large-recoil processes. Using this amplitude they compute decay widths and branching fractions, predict the ratio R_{chi_c1(3872)/psi(2S)} = 0.0024, and conclude that about 20 times the current B_c sample is needed to observe B_c^+ -> chi_c1(3872)pi^+, while B_c^+ -> chi_c1(3872)mu^+nu_mu could be accessible with roughly twice the existing data.

Significance. The methodological upgrade from M_A to M_B is a genuine step forward for large-recoil transitions within this model class, and the semileptonic channel B_c^+ -> chi_c1(3872)mu^+nu_mu as a more promising probe is a concrete, falsifiable suggestion for LHCb. The prediction R = 0.0024 is consistent with the existing upper limit and provides a useful target. However, the quantitative luminosity claims are based on a single model's branching fraction with only parametric error bars, while Table II shows a factor of about 40 spread among published model predictions for the same channel. The paper's central conclusions are therefore conditional on model assumptions that are acknowledged but not propagated into the stated uncertainties.

major comments (3)

[Section V.C, Tables II and III] The central phenomenological statement of the paper, R_{chi_c1(3872)/psi(2S)} = 0.0024 and the inference that roughly 20 times the current B_c sample is needed, rests on B(B_c^+ -> chi_c1(3872)pi^+) = (6.31^{+0.97}_{-1.24}) x 10^-6, whose quoted uncertainty is generated only by the +/-5% scan of quark masses and potential parameters described in Section V.A. Table II, however, documents a factor of about 40 spread for the same channel among published calculations: 0.81 x 10^-17 GeV in this work, 1.4 x 10^-17 GeV in Ref. [74], 24 x 10^-17 GeV in Ref. [107], and 32.3 x 10^-17 GeV in Ref. [76]. If a prediction in the upper part of this range were adopted, the corresponding R would exceed the LHCb upper limit of 0.05, and the 'insufficient luminosity' explanation would cease to hold. The feasibility claim therefore needs a sensitivity study over the existing model predictions, or at least an explicit statement that it is conditional on the authors' model.
[Section V.C, Eqs. (15) and (18), Ref. [108]] The predicted ratio R combines a numerator computed with the new amplitude M_B (Eq. 18) with a denominator B(B_c^+ -> psi(2S)pi^+) = 0.0266% taken from Ref. [108], which is based on the older amplitude M_A (Eq. 15). The two amplitudes are not equivalent at the recoil points relevant here: Table I shows M_B versus M_A differences of 12.9 vs 15.2 (x 10^-16 GeV) for B_c^+ -> chi_c1(1P)e^+nu_e and 0.96 vs 1.53 for B_c^+ -> chi_c1(3872)e^+nu_e. Recomputing the psi(2S) branching ratio with M_B would change R by an amount that is not negligible relative to the precision with which R = 0.0024 is quoted. I ask the authors to supply an M_B-consistent denominator or to estimate the sensitivity of R to this choice.
[Section III, Eqs. (16)-(18)] The derivation of M_B proceeds by inserting positive-energy projectors for the final-state quarks and neglecting the negative-energy contributions in the third equality of Eq. (16), together with the spectator condition p'_1 = p_1. For chi_c1(2P), the paper itself reports v^2 = 0.39 (Section I), so the negative-energy components are not obviously small. No estimate of the truncation error is given. Because M_B is the basis of every numerical result in Tables I-IV, the authors should either quantify this error, for example by retaining the negative-energy terms in the residue integration or by comparing with a different spectator prescription, or state this as a systematic limitation in the error budget.

minor comments (5)

[References] Reference [53] is labeled 'A. M. Sirunyan et al. (CDF Collaboration)'; Sirunyan is the CMS collaboration, so the collaboration label appears to be wrong. Please verify the citation.
[Table III] Table III lists 'B_c^+ -> chi_c1(1P)tau^+tau_l'; this should presumably be tau^+nu_tau, since the decay is semileptonic.
[Introduction] The Introduction contains the typo 'corraboration' for 'collaboration'.
[Section II, Eq. (3)] The abbreviation 'epsilon*_mu epsilon PP_f = epsilon*_{mu alpha beta gamma} epsilon_alpha P_beta P_gamma^f' has an index mismatch: epsilon*_mu has one free Lorentz index while the right-hand side involves an additional polarization vector epsilon_alpha. Please re-express the K_4 term with consistent indices.
[Section V, Eq. (24) and Table I] The ratio in Eq. (24), R_{chi_c1(nP)}, is a lepton-universality ratio for tau versus electron final states, but the same symbol R is used for the LHCb ratio R_{chi_c1(3872)/psi(2S)} in Eq. (1). Using different notation for these two quantities would reduce confusion.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the central R prediction is computed from a new amplitude and an independently benchmarked previous result, not fitted to the LHCb limit.

full rationale

The paper's central prediction, R_{χc1(3872)/ψ(2S)} = 0.0024, is obtained by combining a newly computed branching fraction B(B_c^+→χ_{c1}(3872)π^+) = 6.31×10^{-6} (Table III) with B(B_c^+→ψ(2S)π^+) = 0.0266% taken from the authors' Ref. [108]. Although Ref. [108] is a self-citation, the paper explicitly shows that the relevant ratios from that work, B_{ψ(2S)π}/B_{ψ(1S)π} = 0.240 and B_{ψ(1S)K}/B_{ψ(1S)π} = 0.0763, agree with LHCb measurements 0.254 and 0.079, so the input carries independent external support. The model mass of χ_{c1}(3872) is adjusted through the free parameter V0, but this fit is to the observed mass, not to the LHCb upper limit or to the predicted branching ratio; the resulting R value lies below the limit rather than being constructed to do so. The large spread among other model predictions in Table II is a genuine model-uncertainty concern, but it is not a circularity: the paper's own calculation is not defined in terms of those other results, and no equation reduces the prediction to an input. The use of a new amplitude MB for the numerator and an older MA-based value for the denominator is an internal-consistency caveat, but not a demonstration that the result is forced by construction.

Assumptions & free parameters 8 free parameters · 5 assumptions · 0 invented entities

The central predictions rest on a phenomenological quark model with several fitted parameters (most notably V0) and on truncation assumptions (positive-energy dominance, spectator mapping) that are plausible but not systematically validated. No new particles or forces are introduced.

free parameters (8)

V0 = tuned to shift χc1(2P) mass from 3928.7 MeV to 3871.6 MeV
Free constant in the Cornell potential, adjusted to reproduce the χc1(3872) mass (Sec. V).
λ (string tension) = 0.21 GeV^2
Cornell potential parameter, set from previous hadron spectroscopy fits (Appendix A).
α (screening parameter) = 0.06 GeV
Exponential screening added to avoid divergence; value from prior literature (Appendix A, Refs. [109,110]).
Λ_QCD = 0.27 GeV
Confinement scale in the running coupling (Appendix A).
a = e = 2.7183
Constant in the running coupling logarithm (Appendix A).
m_b and m_c = 4.96 GeV and 1.62 GeV
Quark masses used as model inputs (Sec. V).
a1 = 1.14
Effective Wilson coefficient for nonleptonic decays in naive factorization (Sec. V.B).
fπ, fρ, fK, fK*, fD, fD*, fDs, fDs* = 0.130, 0.2085, 0.156, 0.217, 0.214, 0.27, 0.251, 0.3 GeV
Decay constants of final-state mesons, taken as inputs from experiment/lattice (Sec. V.B).

assumptions (5)

domain assumption The instantaneous approximation is valid for heavy quark mesons, reducing the BS equation to the Salpeter equation.
The paper assumes the interaction is instantaneous; the quark relative velocity squared for χc1(2P) is 0.39 (quoted in Sec. I from Ref. [56]), which is not very small, making this approximation a potential concern.
domain assumption Positive-energy wave function components dominate; negative-energy components are ignored in the transition amplitude.
In deriving Eq. (18), the paper states 'we impose the condition that the positive-energy wave function dominates' and later 'ignoring the contribution from negative energy wave functions' (Sec. III).
domain assumption Spectator assumption: the quark that does not participate in the weak decay has the same momentum in initial and final mesons (p'_1 = p_1).
Used to connect the final-state internal momentum q_f to the initial q (Eq. (20), Sec. III). The paper notes this is a standard prescription but acknowledges earlier ambiguities with the final-state rest frame.
domain assumption The revised Cornell potential (linear confinement plus screened Coulomb) is the correct quark-antiquark interaction kernel.
The potential is a phenomenological model, not derived from QCD; the paper states it is 'the revised Cornell potential' (Appendix A).
domain assumption The PDG mass of χc1(3872) is an input, and the model's own mass prediction is shifted by varying V0.
The mass of χc1(3872) is an experimental input, while the model prediction of 3928.7 MeV is moved to 3871.6 MeV by varying V0 (Sec. V).

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Pith. "Pith review of Improved covariant analysis of $B_c^+ \to \chi_{c1}(nP)$ decays and implications for the nature of $\chi_{c1}(3872)$." pith.science (2026). https://pith.science/paper/TSPO724F

@misc{pith2026260804693,
  author       = {Pith},
  title        = {Pith review of: Improved covariant analysis of $B_c^+ \to \chi_c1(nP)$ decays and implications for the nature of $\chi_c1(3872)$},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/TSPO724F}},
  note         = {Machine review of arXiv:2608.04693}
}

abstract

We study the weak decays $B_c^+\to\chi_{c1}(nP)\ell^+ \nu_{\ell}$ and $B_c^+\to\chi_{c1}(nP)X$ (n=1,2,3) within the Bethe-Salpeter formalism, treating $\chi_{c1}(3872)$ as the conventional $\chi_{c1}(2P)$ charmonium. We upgrade the covariant hadronic transition amplitude to consistently evaluate the final-state wave function in its rest frame, enabling a reliable description of large-recoil processes and sizable relativistic corrections pertinent to highly excited charmonium. Our results provide a novel platform to probe the internal structure of $\chi_{c1}(3872)$ via $B_c$ decays. While the LHCb search for $B_c^+\to\chi_{c1}(3872)\pi^+$ yielded only an upper limit, we demonstrate that this is primarily due to insufficient luminosity, approximately 20 times the current $B_c$ data sample would be required for a definitive observation. In contrast, we identify the semileptonic mode $B_c^+\to\chi_{c1}(3872)\mu^+\nu_{\mu}$ as a far more promising channel, which could become accessible with merely twice the existing data set. Our predictions offer concrete guidance for upcoming LHCb analyses and complement ongoing efforts to resolve the nature of $\chi_{c1}(3872)$.

Figures

Figures reproduced from arXiv: 2608.04693 by the authors.

**Figure 2.** FIG. 2: Feynman diagram for [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗

**Figure 3.** FIG. 3: Form Factors of [PITH_FULL_IMAGE:figures/full_fig_p013_3.png] view at source ↗

Figures from the paper (3 more)

**Figure 4.** Figure 4: FIG. 4: Form Factors of [PITH_FULL_IMAGE:figures/full_fig_p013_4.png]

**Figure 5.** Figure 5: FIG. 5: Differential decay width [PITH_FULL_IMAGE:figures/full_fig_p013_5.png]

**Figure 6.** Figure 6: FIG. 6: Differential decay width [PITH_FULL_IMAGE:figures/full_fig_p014_6.png]

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As a spectator, p′ 1 =p1

/Pf Mf eΛ+ 2 (p′ 2⊤)¯η(qf ⊤)eΛ+ 1 (p′ 1⊤) /Pf Mf S1(p1)η(q⊥)S2(−p2)γµ(1 −γ5)] # , where qf ⊤ ≡qf − Pf ·qf M 2 f Pf has been defined. As a spectator, p′ 1 =p1. In the second equality above, we have applied the relation 1 = /Pf Mf /Pf Mf = /Pf Mf eΛ+ 2 (p′ 2⊤) + eΛ− 2 (p′ 2⊤) = eΛ+ 1 (p′ 1⊤) + eΛ− 1 (p′ 1⊤) /Pf Mf , 9 where eΛ± i (p′ i⊤) = 1 2eωi /Pf Mf eωi...

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Substituting these propagator representations into Eq

= Λ+ 2 (p′ 2⊥) p′ 2P −ω2f +iϵ + Λ− 2 (p′ 2⊥) p′ 2P +ω2f −iϵ, (17) where piP ≡ pi ·P M =αiM +JiqP , p′ 2P =α′ 2PfP −qfP = −qP −α1M +Ef. Substituting these propagator representations into Eq. (16), performing the qP integration via the residue theorem, and ignoring the contribution from negative energy wave functions, then we derive a new expression for the...