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arxiv: 2606.18576 · v1 · pith:ZIEONOLUnew · submitted 2026-06-17 · ✦ hep-ph · hep-ex

Semi-leptonic decays B to D^((*))(1S,2S)ell ν_(ell) within the covariant light-front approach

Zhi-Jie Sun , Zhi-Qing Zhang , Shi-Chen Xue , Feng-Zhou Wang This is my paper

Pith reviewed 2026-06-26 20:49 UTC · model grok-4.3

classification ✦ hep-ph hep-ex
keywords semi-leptonic B decaysR(D) anomalycovariant light-front quark modeltransition form factorsbranching ratiosD meson excited statesforward-backward asymmetry
0
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The pith

Covariant light-front quark model calculations of B to D(*) semi-leptonic decays reproduce the R(D) and R(D*) anomalies at 3.1 and 2.1 sigma from HFLAV averages.

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

The paper applies the covariant light-front quark model to compute the transition form factors for B(s) to D(s) and D*(s) ground and radially excited states, then inserts those form factors into the decay rate formulas to obtain branching ratios for electron, muon, and tau modes. Light-lepton modes match existing data while tau modes come out systematically lower, producing R(D) = 0.261 plus or minus 0.013 and R(D*) = 0.228 plus or minus 0.026. The same framework yields branching ratios for the 2S states in the 10 to the minus 4 to 10 to the minus 3 range and supplies predictions for forward-backward asymmetries and longitudinal polarization fractions.

Core claim

Within the covariant light-front quark model the calculated ratios R(D) and R(D*) lie 3.1 sigma and 2.1 sigma below the current HFLAV world averages while lying closer to the latest LHCb measurements; the model also produces branching fractions for the radially excited 2S channels that are larger than Bethe-Salpeter results but consistent with relativistic quark model estimates.

What carries the argument

Covariant light-front quark model transition form factors for B to D(*) (1S,2S) that are inserted into the semi-leptonic width formulas to obtain branching ratios and angular observables.

If this is right

  • Branching ratios for B(s) to D(s)(2S) and D*(s)(2S) semi-leptonic decays fall between 10^{-4} and 10^{-3}.
  • Forward-backward asymmetries and longitudinal polarization fractions for all modes are consistent with most other theoretical calculations and available data.
  • The tau-mode deficit relative to light-lepton modes appears automatically from the form-factor ratios without additional parameters.
  • Predictions for the excited-state channels can be tested once those modes are observed.

Where Pith is reading between the lines

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

  • If the model's form factors remain reliable at higher recoil, the same framework can be applied to related decays such as B to D** or Bs to Ds** without retuning.
  • Discrepancy with HFLAV but agreement with LHCb suggests the anomaly may be sensitive to the precise experimental averaging procedure.
  • The 2S branching ratios being larger than Bethe-Salpeter but matching relativistic quark model points to a systematic difference between light-front and Bethe-Salpeter treatments of radial excitations.

Load-bearing premise

The covariant light-front quark model supplies accurate form factors for every B to D(*) (1S and 2S) transition and every lepton flavor.

What would settle it

A future measurement of R(D) that lies more than two standard deviations away from 0.261 would contradict the central numerical prediction obtained from the model's form factors.

Figures

Figures reproduced from arXiv: 2606.18576 by Feng-Zhou Wang, Shi-Chen Xue, Zhi-Jie Sun, Zhi-Qing Zhang.

Figure 1
Figure 1. Figure 1: FIG. 1: Feynman diagram for the meson weak transition, where [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2: The [PITH_FULL_IMAGE:figures/full_fig_p018_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3: The [PITH_FULL_IMAGE:figures/full_fig_p019_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4: The [PITH_FULL_IMAGE:figures/full_fig_p020_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5: The [PITH_FULL_IMAGE:figures/full_fig_p021_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6: The [PITH_FULL_IMAGE:figures/full_fig_p022_6.png] view at source ↗
read the original abstract

We present a systematic analysis of the semi-leptonic decays $B_{(s)}\to D_{(s)}(1S,2S)\ell\nu_\ell$ and $B_{(s)}\to D^*_{(s)}(1S,2S)\ell\nu_\ell$ with $\ell=e,\mu,\tau$ within the covariant light-front quark model (CLFQM). Using the form factors of the transitions $B_{(s)}\to D_{(s)}(1S,2S)$ and $B_{(s)}\to D^*_{(s)}(1S,2S)$, we calculate the branching ratios of the relevant semi-leptonic decays and find that $Br(B_{(s)}\to D_{(s)}\ell^\prime\nu_{\ell^\prime})$ and $Br(B_{(s)}\to D^*_{(s)}\ell^\prime\nu_{\ell^\prime})$ with $\ell^\prime=e,\mu$ are agree well with the data, while $Br(B_{(s)}\to D_{(s)}\tau\nu_{\tau})$ and $Br(B_{(s)}\to D^*_{(s)}\tau\nu_{\tau})$ are systematically smaller than the experimental measurements. This naturally gives rise to the so-called $\mathcal{R}(D)$ and $\mathcal{R}(D^*)$ anomalies. Our predictions $\mathcal{R}(D)=0.261\pm0.013$ and $\mathcal{R}(D^*)=0.228\pm0.026$ show $3.1\sigma$ and $2.1\sigma$ deviations from the current experimental world averages compiled by the Heavy Flavor Averaging Group (HFLAV), respectively, yet only deviate by $0.16\sigma$ and $1.5\sigma$ from the latest LHCb measurements. For the decays $B_{(s)}\to D_{(s)}(2S)\ell\nu_\ell$ and $B_{(s)}\to D^*_{(s)}(2S)\ell\nu_\ell$, their branching ratios lie in the range $10^{-4}\sim10^{-3}$, which are much larger than the results from the Bethe Salpeter (BS) equation , but agree with the relativistic quark model (RQM) calculations. Furthermore, we also calculate the forward-backward asymmetries $\mathcal{A}_{FB}$ and longitudinal polarization fractions $f_L$ for the corresponding decays. Our predictions are consistent with most other theoretical results and experimental data

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 / 3 minor

Summary. The paper performs a systematic calculation of branching ratios for the semi-leptonic decays B_{(s)} o D_{(s)}^{( * )}(1S,2S) \ell u_\ell (\ell = e, u, au) in the covariant light-front quark model. Form factors are computed from Gaussian wave functions whose parameters are fixed by meson masses and decay constants; these are inserted into the standard differential decay rates to obtain Br values. Light-lepton modes are reported to agree with data while au modes are smaller, yielding R(D) = 0.261 \pm 0.013 (3.1 \sigma from HFLAV) and R(D*) = 0.228 \pm 0.026 (2.1 \sigma from HFLAV). Predictions for the 2S states lie in the 10^{-4}--10^{-3} range, and A_{FB}, f_L are also computed.

Significance. If the CLFQM form-factor shapes are reliable across the full q^2 range, the quoted R(D(*)) values would constitute an independent indication of the anomalies and supply useful benchmarks for the excited-state channels. The reported consistency with light-lepton data and with selected other models is a positive feature, but the overall significance is reduced by the absence of direct external validation of the q^2 dependence.

major comments (2)
  1. [Form-factor and numerical-results sections] The central R(D) and R(D*) results rest on the assumption that the CLFQM form factors (f_+, f_0, V, A_0, A_1, A_2) correctly describe the high-q^2 region probed by the au modes. No comparison is presented to lattice-QCD determinations (HPQCD, FNAL/MILC, JLQCD) at q^2 = 0 or at w = 1, nor is the slope or curvature of the form factors tested against lattice data; this validation is load-bearing for the claimed 3.1 \sigma and 2.1 \sigma deviations.
  2. [Methodology and results sections] The same set of constituent-quark masses and Gaussian eta parameters controls both the overall normalization (fixed by light-lepton rates) and the entire q^2 shape. Consequently, agreement with Br(B o D^{(*)} e/ u) does not furnish an independent constraint on the high-q^2 integrals that dominate the au rates; this circularity is not quantified.
minor comments (3)
  1. [Abstract] Abstract contains a grammatical error: 'are agree well with the data' should read 'agree well with the data'.
  2. [Abstract and results section] The abstract states that the 2S branching ratios 'agree with the relativistic quark model (RQM) calculations' but does not cite the specific RQM references or tabulate the numerical comparison.
  3. [Abstract] Notation for the lepton flavors (\ell' vs au) and the inclusion of B_s, D_s channels could be clarified in the abstract for immediate readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive comments. We address the two major comments point by point below, indicating planned revisions where appropriate.

read point-by-point responses

  1. Referee: [Form-factor and numerical-results sections] The central R(D) and R(D*) results rest on the assumption that the CLFQM form factors (f_+, f_0, V, A_0, A_1, A_2) correctly describe the high-q^2 region probed by the τ modes. No comparison is presented to lattice-QCD determinations (HPQCD, FNAL/MILC, JLQCD) at q^2 = 0 or at w = 1, nor is the slope or curvature of the form factors tested against lattice data; this validation is load-bearing for the claimed 3.1 σ and 2.1 σ deviations.

    Authors: We agree that direct comparisons of our form factors to lattice QCD results at q^2=0, w=1, and for the slope/curvature would strengthen validation of the high-q^2 behavior relevant to tau modes. Our parameters are fixed from masses and decay constants independently of lattice data, but we acknowledge the value of such benchmarks. In the revised manuscript we will add explicit comparisons at these kinematic points to available lattice results from HPQCD, FNAL/MILC and JLQCD, together with a brief discussion of any differences in q^2 dependence. revision: yes

  2. Referee: [Methodology and results sections] The same set of constituent-quark masses and Gaussian β parameters controls both the overall normalization (fixed by light-lepton rates) and the entire q^2 shape. Consequently, agreement with Br(B → D(*) e/μ) does not furnish an independent constraint on the high-q^2 integrals that dominate the τ rates; this circularity is not quantified.

    Authors: We clarify that the constituent quark masses and Gaussian β parameters are fixed exclusively by the meson masses and decay constants, as stated in the manuscript; they are not adjusted to the light-lepton branching ratios. The light-lepton Br values are therefore genuine predictions of the model, providing an independent test of the form-factor shapes across the full kinematic range. We will revise the methodology and results sections to emphasize this distinction and to discuss more explicitly how the parameter choice affects the high-q^2 integrals relevant to the tau modes. revision: yes

Circularity Check

0 steps flagged

No circularity: CLFQM form-factor computation and R(D) predictions form an independent derivation chain

full rationale

The paper computes transition form factors in the covariant light-front quark model from meson wave functions and then inserts those form factors into standard differential decay-rate formulas to obtain branching ratios and the ratios R(D), R(D*). This is a conventional model-to-observable pipeline; the inputs (quark masses, Gaussian parameters β fitted to masses and decay constants) are distinct from the output ratios, and no equation reduces the R values to the fit inputs by algebraic identity. No self-citation is invoked as a uniqueness theorem, no ansatz is smuggled, and the light-lepton agreement is presented as an a-posteriori check rather than a fitted constraint that forces the tau-mode result. The derivation therefore remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The central claim rests on the applicability of the covariant light-front quark model whose parameters are fitted to meson spectra and decay constants; no independent first-principles derivation of those parameters is provided.

free parameters (1)
  • CLFQM constituent quark masses and wave-function parameters
    Standard in light-front quark models; adjusted to reproduce meson masses and decay constants before being used for transition form factors.
axioms (1)
  • domain assumption The covariant light-front quark model yields reliable form factors for B to D(*) (1S,2S) transitions at all momentum transfers relevant to semi-leptonic decays.
    Invoked when the computed form factors are inserted into the decay-rate formulas for all lepton species.

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discussion (0)

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Reference graph

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