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arxiv: 2605.31208 · v1 · pith:7NLO2EWEnew · submitted 2026-05-29 · ✦ hep-ph

A unified study of the \(B_c\) meson: from spectrum and form factors to weak and radiative decays

Vikas Patel , Chetan Lodha , Raghav Chaturvedi , A. K. Rai This is my paper

Pith reviewed 2026-06-28 22:03 UTC · model grok-4.3

classification ✦ hep-ph
keywords B_c mesonscreened potential modelQCD sum rulesweak decaysradiative decaysRegge trajectoriesdecay constants
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0 comments X

The pith

Spectroscopic wave functions fix the couplings for B_c weak decays and radiative transitions.

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

The paper computes the B_c mass spectrum and wave functions inside a screened potential model that incorporates relativistic kinetic corrections and spin-dependent terms. These wave functions supply the inputs for a three-point QCD sum-rule calculation of weak transitions to S-, P-, and D-wave charmonium states and to final states containing charmonium plus D mesons, after the heavy-quark masses are updated to m_c = 1.48 GeV and m_b = 4.90 GeV. The same spectroscopic quantities also determine the decay constants, purely leptonic widths, and E1/M1 radiative rates. Regge trajectories extracted from the spectrum serve as an independent global check on the underlying dynamics. A reader cares because the approach links the bound-state structure to multiple observable decay channels through a single set of wave functions.

Core claim

The spectroscopic wave functions determine the short-distance couplings that enter both weak and electromagnetic observables, while the Regge analysis serves as a complementary global consistency test of the same dynamical picture.

What carries the argument

Screened potential model wave functions that supply inputs to mass-updated three-point QCD sum rules for decay calculations.

If this is right

  • Weak decay widths to S-, P-, and D-wave charmonium states follow directly from the same wave functions used for the mass spectrum.
  • Radiative E1 and M1 transition rates are fixed by the spectroscopic parameters without additional model inputs.
  • Leptonic decay widths are determined consistently with the computed decay constants.
  • Regge trajectories of the computed spectrum provide an independent test of the overall dynamical picture.

Where Pith is reading between the lines

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

  • The same wave-function inputs could be reused for other doubly-heavy mesons by repeating only the mass update step.
  • Future precision measurements of B_c branching fractions would directly test whether the controlled refit captures the dominant mass dependence.
  • If the approach holds, sum-rule normalizations for heavy-meson transitions may be more stable under modest mass shifts than a full rederivation would suggest.

Load-bearing premise

Updating the heavy-quark masses and refitting only the overall normalization is enough to update the QCD sum-rule results without recomputing all perturbative and condensate contributions.

What would settle it

A measured branching fraction for B_c decay to a specific P-wave charmonium state that lies well outside the range predicted from the updated masses and wave functions.

Figures

Figures reproduced from arXiv: 2605.31208 by A. K. Rai, Chetan Lodha, Raghav Chaturvedi, Vikas Patel.

Figure 1
Figure 1. Figure 1: Decay-width distributions in the S-wave sector for the bare and Coulomb-corrected variants. 21 [PITH_FULL_IMAGE:figures/full_fig_p021_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Bare and Coulomb-corrected form factors f+(q 2 ) and f0 (q 2 ) for Bc → ηc [PITH_FULL_IMAGE:figures/full_fig_p022_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Bare and Coulomb-corrected form factors V (q 2 ), A0 (q 2 ), A1 (q 2 ), and A2 (q 2 ) for Bc → J/ψ. The P-wave sector, summarized in Tables 11–16 and Figs. 4–7, is more differentiated among final states and therefore probes the internal spin structure of the framework more stringently. A central result is the persistent dominance of the hc channels. Both semileptonic and nonleptonic modes involving hc are … view at source ↗
Figure 4
Figure 4. Figure 4: Decay-width distributions in the P-wave sector for the bare and Coulomb-corrected variants [PITH_FULL_IMAGE:figures/full_fig_p026_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Bare and Coulomb-corrected form factors F0 (q 2 ) and F1 (q 2 ) for Bc → χc0 . 26 [PITH_FULL_IMAGE:figures/full_fig_p026_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Bare and Coulomb-corrected form factors A(q 2 ), V0 (q 2 ), V1 (q 2 ), and V2 (q 2 ) for Bc → χc1 [PITH_FULL_IMAGE:figures/full_fig_p027_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Bare and Coulomb-corrected form factors A(q 2 ), V0 (q 2 ), V1 (q 2 ), and V2 (q 2 ) for Bc → hc . The D-wave sector, shown in Tables 17 and 18 together with [PITH_FULL_IMAGE:figures/full_fig_p027_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Decay-width distributions in the D-wave sector for the bare and Coulomb-corrected variants. Taken together, the weak-decay analysis gives a coherent picture across all orbital sectors, but the quality of external validation is not uniform. The corrected S-wave sector compares rather well with both modern Standard Model calculations and the currently available experimental ratios. The P-wave sector reproduc… view at source ↗
Figure 9
Figure 9. Figure 9: Regge trajectories of the Bc meson in the (J, M2 ) plane: natural-parity states (left) and unnatural-parity states (right). Open symbols denote the model predictions and filled symbols denote experimentally observed states included in the construction [PITH_FULL_IMAGE:figures/full_fig_p035_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Regge trajectories of the Bc meson in the (nr , M2 ) plane. The left panel shows trajectories for representative pseudoscalar, vector, and orbitally excited states, while the right panel displays the spin-averaged S-, P-, D-, and F-wave trajectories [PITH_FULL_IMAGE:figures/full_fig_p035_10.png] view at source ↗
read the original abstract

The $B_c$ meson constitutes a unique system for investigating heavy-hadron dynamics, since it exhibits quarkonium-like bound-state structure while decaying predominantly through weak interactions. In this work, we present a unified study of $B_c$-meson spectroscopy, decay constants, weak decays, radiative transitions, and Regge trajectories within a single framework. The mass spectrum is computed in a screened potential model including relativistic kinetic-energy corrections and spin-dependent interactions, from which we obtain both spin-averaged and spin-resolved states together with the corresponding pseudoscalar and vector decay constants. Using these spectroscopic inputs, we then analyze weak decays within a mass-updated three-point QCD sum-rule framework for transitions to $S$-, $P$-, and $D$-wave charmonium states, as well as to final states containing charmonium and $D^{(*)}_{(s)}$ mesons. In this part of the analysis, the hadronic thresholds, Borel-window prescriptions, Lorentz decompositions, and decay-width expressions of the underlying sum-rule formulations are retained, while the heavy-quark masses are updated to $m_c=1.48~\mathrm{GeV}$ and $m_b=4.90~\mathrm{GeV}$, leading to a controlled refit of the overall normalization rather than a full rederivation of all perturbative and condensate contributions. We further investigate purely leptonic decay widths and radiative $E1$ and $M1$ transitions, and examine the Regge behavior of the resulting spectrum. The present study therefore provides a coherent description of the $B_c$ meson in which the spectroscopic wave functions determine the short-distance couplings that enter both weak and electromagnetic observables, while the Regge analysis serves as a complementary global consistency test of the same dynamical picture.

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

1 major / 1 minor

Summary. The paper claims to provide a unified description of the B_c meson by computing its mass spectrum and decay constants in a screened potential model with relativistic and spin-dependent corrections, then feeding these into a three-point QCD sum-rule analysis (with updated heavy-quark masses m_c=1.48 GeV, m_b=4.90 GeV and a controlled refit of overall normalization while retaining prior thresholds, Borel windows, and Lorentz structures) to obtain form factors and widths for weak decays to S-, P-, D-wave charmonia and D^{(*)}(s) states, supplemented by leptonic widths, E1/M1 radiative transitions, and a Regge-trajectory consistency check.

Significance. If the central results hold, the work supplies a coherent dynamical picture in which spectroscopic wave functions fix the short-distance couplings entering both weak and electromagnetic observables, with the Regge analysis providing an independent global test. The unified scope across spectrum, form factors, and multiple decay channels is a strength, as is the explicit retention of the underlying sum-rule structures for reproducibility.

major comments (1)
  1. [Abstract] Abstract (and the three-point QCD sum-rule framework section): the claim that spectroscopic inputs determine the short-distance couplings for weak decays rests on form factors obtained by updating only m_c and m_b and refitting the overall normalization while keeping all prior hadronic thresholds, Borel windows, and Lorentz decompositions. Because the perturbative spectral densities in three-point sum rules depend explicitly on the heavy-quark masses (and condensates enter with mass-dependent coefficients), this shortcut is load-bearing for the accuracy of the predicted weak widths and requires explicit justification or a full rederivation of the spectral densities.
minor comments (1)
  1. The abstract states that the Regge analysis serves as a 'complementary global consistency test'; a brief quantitative statement of how well the computed spectrum aligns with linear Regge trajectories (e.g., slope values or χ^{2}) would strengthen this claim.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the thorough review and positive assessment of the unified approach in our manuscript. We provide a point-by-point response to the major comment below.

read point-by-point responses

  1. Referee: [Abstract] Abstract (and the three-point QCD sum-rule framework section): the claim that spectroscopic inputs determine the short-distance couplings for weak decays rests on form factors obtained by updating only m_c and m_b and refitting the overall normalization while keeping all prior hadronic thresholds, Borel windows, and Lorentz decompositions. Because the perturbative spectral densities in three-point sum rules depend explicitly on the heavy-quark masses (and condensates enter with mass-dependent coefficients), this shortcut is load-bearing for the accuracy of the predicted weak widths and requires explicit justification or a full rederivation of the spectral densities.

    Authors: We agree that the perturbative spectral densities in three-point sum rules depend on the heavy quark masses, and that a full rederivation would be the most rigorous approach. However, in our work, the spectroscopic inputs from the screened potential model are used to fix the B_c masses and decay constants, which serve as inputs to the sum rules. The update of m_c and m_b is done to be consistent with the model, and the refit of the normalization is performed to account for the changes in the spectral densities and condensate contributions. To address the referee's concern, we will include in the revised manuscript an additional subsection or paragraph in the QCD sum-rule framework section that provides explicit justification for this procedure. Specifically, we will discuss the mass dependence of the leading perturbative terms and show that the refit captures the dominant effects, supported by numerical checks on the stability of the results. This constitutes a partial revision as we add the requested justification without performing a complete rederivation of all terms. revision: partial

Circularity Check

1 steps flagged

Normalization refit after mass update in three-point QCD sum-rules makes form factors partly fitted by construction

specific steps
  1. fitted input called prediction [Abstract]
    "the hadronic thresholds, Borel-window prescriptions, Lorentz decompositions, and decay-width expressions of the underlying sum-rule formulations are retained, while the heavy-quark masses are updated to m_c=1.48 GeV and m_b=4.90 GeV, leading to a controlled refit of the overall normalization rather than a full rederivation of all perturbative and condensate contributions"

    The refit of overall normalization after the mass update means the form factors (short-distance couplings) entering the weak-decay predictions are scaled to match data or prior results inside the same sum-rule setup. Because the perturbative spectral densities depend explicitly on the heavy-quark masses, this shortcut forces the outputs to be consistent with the fit rather than providing an independent derivation from the spectroscopic wave functions.

full rationale

The paper's unified claim rests on spectroscopic wave functions feeding into short-distance couplings via three-point sum-rules for weak decays. However, the abstract explicitly states that only masses are updated while retaining prior thresholds/Borel windows and performing a controlled refit of overall normalization instead of rederiving perturbative densities and condensates. This directly matches the fitted_input_called_prediction pattern: the refit adjusts the output to data within the framework, so subsequent decay-width predictions are statistically forced rather than independently derived from the potential-model inputs. No other patterns (self-definitional, self-citation load-bearing, etc.) are exhibited in the provided text. The central derivation therefore contains partial circularity but retains independent content from the spectrum calculation.

Axiom & Free-Parameter Ledger

3 free parameters · 2 axioms · 0 invented entities

The central claim rests on a screened potential model whose parameters are not specified in the abstract, on the validity of QCD sum rules with only a normalization refit, and on the assumption that wave functions from one model directly control couplings in another.

free parameters (3)
  • m_c = 1.48 GeV
    Updated charm quark mass set to 1.48 GeV for the sum-rule analysis.
  • m_b = 4.90 GeV
    Updated bottom quark mass set to 4.90 GeV for the sum-rule analysis.
  • overall normalization
    Refitted in the QCD sum-rule expressions rather than recomputed from perturbative and condensate terms.
axioms (2)
  • domain assumption The screened potential model with relativistic kinetic-energy corrections and spin-dependent interactions accurately reproduces the B_c spectrum and wave functions.
    Invoked to obtain masses, decay constants, and inputs for subsequent calculations.
  • domain assumption Retaining the hadronic thresholds, Borel-window prescriptions, and Lorentz decompositions of prior sum-rule formulations remains valid after the mass update.
    Explicitly stated as the basis for the weak-decay analysis.

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discussion (0)

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