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REVIEW 3 major objections 4 minor 71 references

$|V_{ub}|$ determination and testing of lepton flavour universality in semileptonic $B_c \rightarrow D^{(\ast)}$ decays

T0 review · 3 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read Semileptonic $B_c\to D^{(*)}\ell\nu$ decays can provide a competitive determination of the CKM element $|V_{ub}|$ with 7.5% theoretical uncertainty, and normalizing to $B_c\to J/\psi\,\mu\bar{\nu}_\mu$ offers a low-uncertainty ratio…

desk verdict New B_c->D(*) form factors and an interesting |Vub| extraction proposal, but the 7.5% uncertainty claim omits the paper's own admitted 20-30% z-series truncation error. read the letter →

arxiv 1909.01213 v3 pith:WA6FUO3C submitted 2019-09-03 hep-ph hep-ex

Domagoj Leljak , Blazenka Melic This is my paper · ORCID
classification hep-phhep-ex
keywords B_cmesondecaysCKMmatrix|Vub|QCDsumrulesleptonflavouruniversalitysemileptonictoDformfactorsz-seriesparametrizationLHCbUpgradeII
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 shows that semileptonic $B_c \to D^{(*)} \ell \nu$ decays, once measured at the planned LHC upgrade, can become a new exclusive probe of the CKM element $|V_{ub}|$. Using three-point QCD sum rules to calculate the $B_c \to D$ and $B_c \to D^*$ form factors, the authors find that the $B_c \to D^0 \mu \bar{\nu}_\mu$ decay rate determines $|V_{ub}|$ with a 7.5 percent theoretical uncertainty. Normalizing this rate to $B_c \to J/\psi\,\mu \bar{\nu}_\mu$ gives the ratio $|V_{ub}|/|V_{cb}|$ with strongly reduced hadronic uncertainty, which is easier to measure. The same form factors yield branching-ratio predictions, $q^2$ distributions, angular observables, and lepton-flavour-universality ratios $R_c(D^0)=0.64\pm0.05$ and $R_c(D^*)=0.55\pm0.05$ that future data can test.

What carries the argument

The load-bearing object is the set of $B_c\to D^{(*)}$ form factors ($f_+, f_0$ for $D$, and $V, A_1, A_2, A_0$ for $D^*$) computed from three-point QCD sum rules at $q^2\le 10$ GeV$^2$ and extrapolated to the full kinematic range with a linear BCL $z$-series, checked against BGL parametrization. These form factors convert a measured $B_c\to D^{(*)}\ell\nu$ rate into $|V_{ub}|$; their uncertainties, largely from effective thresholds calibrated to lattice decay constants, set the 7.5% error in $|V_{ub}|$ from the $D^0$ mode and the roughly 20% error from the $D^*$ mode. For the ratio $|V_{ub}|/|V_{cb}|$, the same machinery is applied to $B_c\to J/\psi$ form factors that reproduce lattice results, so the hadronic uncertainties largely cancel.

What would settle it

Measure the differential rate $d\Gamma(B_c\to D^0\mu\bar{\nu}_\mu)/dq^2$ in the low-$q^2$ bins tabulated in the paper; if the extracted $|V_{ub}|$ from different $q^2$ bins disagrees by more than the quoted errors, or if the measured ratio to $B_c\to J/\psi$ falls outside the predicted band, the sum-rule form factors are falsified. A direct lattice computation of the $B_c\to D$ form factors would also settle the extrapolation question.

Watch

Extended reading notes

Core claim

The central discovery is that the $B_c \to D^0 \mu \bar{\nu}_\mu$ channel is a viable exclusive route to $|V_{ub}|$ once the $B_c \to D^{(*)}$ form factors are computed with the three-point sum-rule method. The paper derives $\zeta_{D^0} = (2.0\pm0.3)\times 10^{-3}$ eV for the reduced decay width and $\zeta_{D^*} = (5\pm2)\times 10^{-3}$ eV, implying a 7.5% theoretical error for the $D^0$ channel and 20% for the $D^*$ channel. By dividing by the $B_c\to J/\psi\,\mu\bar{\nu}_\mu$ width, the ratio $|V_{ub}|/|V_{cb}|$ can be extracted with form-factor uncertainties largely cancelling, which is the most practical route for the expected dataset of roughly 30,000 reconstructed $B_c\to D^0\ell\nu$ events. The paper also establishes that non-perturbative condensate contributions to the sum rules are numerically negligible and that the form factors obey heavy-quark spin symmetry relations, validating the ratio to $B_c\to B_s$ for a future $|V_{ub}|/|V_{cs}|$ determination.

Load-bearing premise

The central assumption is that the leading-order three-point sum-rule form factors, with effective thresholds calibrated to lattice decay constants, remain accurate after a linear $z$-series extrapolation from $q^2\le 10$ GeV$^2$ to the full kinematic range, so that the stated 20-30% truncation error can be left out of the quoted uncertainty budget.

Editorial extensions

If this is right

  • A measurement of $B_c \to D^0 \mu \bar{\nu}_\mu$ at the LHCb Upgrade II would determine $|V_{ub}|$ with about 7.5% theory error, rivalling the $B\to\pi$ exclusive determination.
  • The ratio $B(B_c\to D^0\mu\bar{\nu}_\mu)/B(B_c\to J/\psi\,\mu\bar{\nu}_\mu)$ gives $|V_{ub}|/|V_{cb}|$ with reduced hadronic uncertainty, and the provided binned tables allow direct comparison to future data.
  • The predicted lepton-flavour-universality ratios $R_c(D^0)=0.64\pm0.05$ and $R_c(D^*)=0.55\pm0.05$ provide a new $B_c$-based test of lepton-flavour universality.
  • The $B_c\to D^*\mu\bar{\nu}_\mu$ channel is less suited for $|V_{ub}|$ extraction due to its 20% uncertainty but remains useful as a cross-check.
  • The $q^2$ distributions and angular observables supplied in the paper give future experiments a direct way to test the Standard Model predictions for these decays.

Reading between the lines

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

  • One extension not drawn in the paper: an explicit $\mathcal{O}(\alpha_s)$ correction to the three-point sum rule would replace the effective-threshold calibration and likely reduce the quoted 7.5% uncertainty below its current value.
  • If lattice QCD later computes $B_c\to D^{(*)}$ form factors, the effective-threshold calibration used here can be checked directly, and a lattice-driven $|V_{ub}|$ from this channel would provide an independent handle on the inclusive-exclusive $|V_{ub}|$ puzzle.
  • The integrated ratios $R_c(D^0)$ and $R_c(D^*)$ are relatively form-factor-independent, but their $q^2$ distributions carry sharper discriminating power between the competing form-factor models listed in the paper.
  • The zero-recoil ratio $B_c\to D^0$ over $B_c\to B_s$ could become a practical $|V_{ub}|/|V_{cs}|$ measurement if lattice input supplies the high-$q^2$ extrapolation that the sum rules cannot provide.
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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 / 4 minor

Summary. This paper computes the B_c → D^0 and B_c → D^* semileptonic form factors in the framework of three-point QCD sum rules in the range 0 ≤ q^2 ≤ 10 GeV^2, extrapolates them to q^2_max with BCL and BGL z-expansions, and uses the results to predict branching fractions, the LFU ratios R_c(D^0) and R_c(D^*), angular observables, and binned decay-rate distributions. Based on ζ_{D^0} = (2.0 ± 0.3) × 10^{-3} eV, the authors claim that |V_{ub}| can be extracted from B_c → D^0 μ ν with 7.5% theoretical uncertainty, and that normalizing to B_c → J/ψ μ ν provides a determination of |V_{ub}|/|V_{cb}| with reduced theoretical uncertainty. They also examine |V_{ub}|/|V_{cs}| via the B_c → B_s zero-recoil ratio and obtain R_FF = 0.8 ± 0.3.

Significance. If the headline uncertainty claims hold, the paper would add a new, independent exclusive channel to the inclusive-exclusive |V_{ub}| puzzle, which is valuable for flavor physics. The work is careful in several respects: Borel plateaus are checked, continuum contributions are kept below 50%, lattice decay constants anchor the effective thresholds, the B_c → J/ψ form factors are compared with HPQCD lattice results with good agreement, and a dedicated section examines correlations among pseudo-data points. The tabulated BCL coefficients, covariance matrices, and binned Δζ_{D^0} and R_{D^0J/ψ} distributions are useful for future LHCb analyses. However, the central quantitative claims rest on an unpropagated systematic error and on an inconsistency between the 10% and 20–30% truncation estimates; until these are reconciled and propagated, the 7.5% and 'significantly reduced' statements are not established.

major comments (3)
  1. [Sec. 2.1, App. B.1, Eq. (3.15), Table 9] The headline 7.5% theoretical uncertainty on |V_{ub}| is not supported by the error budget as presented. Appendix B.1 states that the quoted form-factor uncertainties 'do not include the truncation error, which is always of the order of 20-30% in our calculations, since we extrapolate only linearly in z(q^2)', while Sec. 2.1 says that including higher orders in z changes the central value of f_+(q^2_max) by at most ~10% and that this change 'always stays inside the uncertainties of the linear z fit.' These two statements are inconsistent, and neither is propagated into Table 5 or Eq. (3.15). Since approximately 40% of the integrated rate lies above q^2 = 10 GeV^2 (Table 9), a 20–30% error on the extrapolated form factors in that region cannot be neglected in a 7.5% claim. The manuscript must either include this systematic error in the quoted uncertainty or base the central claim on the q^2 ≤ 10 GeV^2 partial rate, where the extrapolation is not needed.
  2. [Sec. 3.2, Fig. 11] The claim that normalizing B_c → D^0 μ ν to B_c → J/ψ μ ν yields a |V_{ub}|/|V_{cb}| determination with significantly reduced theoretical uncertainty is not demonstrated. The text following Fig. 11 says that, in line with Sec. 2.2, 'one should assign a further 10% uncertainty to |V_{ub}|/|V_{cb}| not shown in the plot.' The figure and the surrounding discussion present the ratio as a prediction without this additional uncertainty, and no error budget for R_{D^0J/ψ} is provided in which the cancellation of hadronic uncertainties is quantified. Please show the version of Fig. 11 that includes the 10% correlation uncertainty and state explicitly whether the claimed reduction in theoretical uncertainty survives.
  3. [Sec. 2, Table 1, Eqs. (2.9)-(2.10)] The 3ptSR calculation is performed at leading order in α_s, and all higher-order and higher-dimension corrections are assumed to be absorbed into effective thresholds s0^{eff} calibrated from two-point sum rules against lattice decay constants. This is a reasonable strategy, but for the three-point correlators it is an uncontrolled assumption because the threshold modification is not validated at q^2 > 0. The authors acknowledge this implicitly by calling it a 'hope' in Sec. 2, yet the 7.5% uncertainty contains no explicit component for this systematic effect. I request a sensitivity study in which s0^{eff} for the B_c and D channels is varied by ±10–20% (larger than the two-point fitting uncertainty) and the resulting changes in ζ_{D^0} and R_{D^0J/ψ} are reported. If the sensitivity is small, the statement should be made quantitative; if not, the error budget must incorporate it.
minor comments (4)
  1. [Eq. (2.25)] The chi-square definition in Eq. (2.25) has [σ^2_{Fi}(q^2_j)]^2 in the denominator. Unless this is intentional, a conventional chi-square divides by the variance, not by the variance squared; please correct the typo or clarify the definition.
  2. [Sec. 2.2, Eqs. (2.26)-(2.29)] The Jacobian-based covariance estimate is not fully specified: which six parameters enter the matrix P in Eq. (2.28), and are the derivatives in J analytic or numerical? This matters because the 'extra 10%' uncertainty used in Sec. 3.2 is derived from this procedure.
  3. [Sec. 2.1, Table 4] The choice of t^* is motivated in the text, but since t^* is not the lowest physical threshold, the sensitivity of the BCL coefficients and of the extrapolated f_+(q^2_max) to the choice t^* = (m_B(∗)+m_π)^2 should be quantified, even if that choice produces less stable pole behavior.
  4. [Sec. 3.1, Table 6] Several integrated angular observables have very large uncertainties (for example F_L^{D*,τ} = 0.4 ± 0.3). The statement that these observables are 'relatively independent of the hadronic form factors' would be more persuasive if a form-factor variation study were shown, or if the sentence were restricted to the observables for which the uncertainty is actually small.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the form-factor predictions are calibrated to external lattice inputs and the |Vub| extraction is explicitly prospective, not fitted.

full rationale

The derivation chain is self-contained against external benchmarks. The Bc→D(∗) form factors are obtained from three-point QCD sum rules with effective thresholds fixed to reproduce lattice decay constants (Table 1, Sec. 2); this is calibration of inputs, not a recycled prediction, since the form factors themselves are independent correlation-function outputs. No measured |Vub| is fed into the calculation: Table 5 uses the PDG average only to convert computed Γ/|Vub|^2 into branching-ratio estimates, and Sec. 3.2 explicitly frames ζD0 as a coefficient to be combined with future LHCb data, so the claimed |Vub| extraction is a prospect rather than a fitted result. The Bc→J/ψ form factors used in the |Vub|/|Vcb| ratio are cross-checked against independent HPQCD lattice points (Fig. 12, Ref. [64]), so the self-citation of Ref. [18] is historical and not load-bearing. The manuscript's own stated limitations—Appendix B.1's warning that the quoted uncertainties 'do not include the truncation error, which is always of the order of 20-30% in our calculations, since we extrapolate only linearly in z(q^2)' and Sec. 3.2's note that 'one should assign a further 10% uncertainty to |Vub|/|Vcb| not shown in the plot'—are error-accounting omissions that affect the numerical reliability of the headline uncertainty, but they are not circular reductions of the derivation to its inputs.

Assumptions & free parameters 7 free parameters · 6 assumptions · 0 invented entities

The central calculation relies on standard QCD sum rule machinery plus several paper-specific modeling choices: LO-only perturbation theory with threshold renormalization, linear z-series extrapolation with unpropagated truncation error, and chosen effective thresholds. No new particles or forces are postulated.

free parameters (7)
  • Effective continuum thresholds s0^eff(Bc), s0^eff(D0), s0^eff(D*) = 53-55, 7-7.5, 6-8 GeV^2
    Adjusted so that the two-point LO QCD sum rules reproduce the lattice decay constants in Table 1; all unknown higher-order corrections are absorbed into these thresholds, making the form factors dependent on this calibration.
  • Borel mass windows M^2_Bc, M^2_D0, M^2_D* (3pt) = 60-90, 8-12, 12-16 GeV^2
    Chosen from Borel plateau stability and the less than 50% continuum condition; variations enter the form-factor uncertainties.
  • b-quark mass m_b (PS scheme) = 4.6 +/- 0.1 GeV
    Input potential-subtracted mass, close to Upsilon(1S) and kinetic scheme masses; value taken from prior literature.
  • Z = m_c/m_b = 0.29 +/- 0.1
    Varied around the lattice ratio; authors choose a lower value to approximate pole-mass behavior.
  • m0^2 (average quark virtuality, mixed condensate parameter) = 0.8 - 1.05 GeV^2
    Standard OPE parameter for non-local and mixed quark-gluon condensates; contributes about 1% to the form factors.
  • Gluon condensate <alpha_s/pi G^2> = 0.009 +/- 0.007 GeV^4
    Taken from charmonium sum rules [46]; contributes under 1%.
  • BCL fit coefficients b0, b1 per form factor = Table 4
    Least-squares fit parameters to the 11 QCDSR pseudo-data points; treated as uncorrelated in the central fits, with crude correlation estimates in Sec. 2.2.
assumptions (6)
  • standard math Borel transform and double spectral representation convert the OPE of the three-point function into a sum rule for each form factor.
    Used in Sec. 2, Eqs. (2.9)-(2.12); standard QCD sum rule machinery.
  • ad hoc to paper Quark-hadron duality: the hadronic continuum is represented by effective thresholds s0, and all higher-order QCD corrections are absorbed into these thresholds.
    Authors state the 3-point function is known only to LO and model higher orders by thresholds that reproduce lattice decay constants (Sec. 2, p. 8).
  • ad hoc to paper The BCL/BGL expansion truncated at linear order in z is a sufficient description of the form factors over the full kinematic range.
    Sec. 2.1 and Appendix B.1: truncation error is 20-30% and is not propagated into the quoted uncertainties.
  • ad hoc to paper The chosen thresholds t* = (m_B* + m_eta)^2 and t* = (m_B* + m_rho)^2 avoid sub-threshold branch cuts and yield expected pole behavior.
    Footnote 1 in Sec. 2.1; excludes the lowest B* + pi threshold because of instability.
  • domain assumption Heavy-quark spin symmetry and a common B_c wave function determine the zero-recoil ratio of B_c to D and B_c to B_s form factors.
    Used in Sec. 3.3 following [65] to discuss |Vub|/|Vcs|.
  • domain assumption Lattice QCD results for f_Bc, f_D, f_D*, f_J/psi, and f_Bs are reliable inputs.
    Tables 1 and 7; thresholds and decay constants are taken from lattice, so the whole form-factor calculation inherits lattice uncertainties and any lattice systematics.

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Cite this review

Pith. "Pith review of $|V_{ub}|$ determination and testing of lepton flavour universality in semileptonic $B_c \rightarrow D^{(\ast)}$ decays." pith.science (2026). https://pith.science/paper/WA6FUO3C

@misc{pith2026190901213,
  author       = {Pith},
  title        = {Pith review of: $|V_ub|$ determination and testing of lepton flavour universality in semileptonic $B_c \rightarrow D^(\ast)$ decays},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/WA6FUO3C}},
  note         = {Machine review of arXiv:1909.01213}
}
abstract

In light of prospects for measurements of $B_c \rightarrow D^{(\ast)} l \nu$ decays in the upcoming Upgrade II of the LHC, we show that by using calculated $B_c \rightarrow D^{(\ast)}$ form factors competitive extraction of the $|V_{ub}|$ CKM matrix element from the $B_c \to D \mu \bar{\nu}_{\mu}$ decay might be possible. To minimize experimental and theoretical uncertainties we provide the ratio $|V_{ub}|/|V_{cb}|$ by normalizing the $B_c \rightarrow D^{(\ast)} \mu \bar{\nu}_{\mu}$ to $B_c \to J/\psi \mu \bar{\nu}_{\mu}$ decay. We also briefly examine the suggestion to extract $|V_{ub}|/|V_{cs}|$ from the theoretically interesting ratio of $B_c \rightarrow D^0 e \bar{\nu}_{e}$ and $B_c \rightarrow B_s e \bar{\nu}_{e}$ decay rates in the zero-recoil limit. With the present average value of $|V_{ub}|$, the predicted branching ratios are estimated to be $BR(B_c \to D^0 \mu \bar{\nu}_{\mu}) = (2.4\pm 0.4)\cdot 10^{-5} $ and $BR(B_c \to D^{\ast} \mu \bar{\nu}_{\mu}) = (7\pm3)\cdot 10^{-5} $, and the semileptonic ratios for testing the lepton flavour universality in these $B_c$ decays are $R_c(D^0) =0.64 \pm 0.05$ and $R_c(D^{\ast}) = 0.55 \pm 0.05$. We also provide $q^2$ distributions and various angular observables of $B_c \rightarrow D^{(\ast)} l \nu$ decays.

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