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REVIEW 3 major objections 5 minor 31 references

Update of HPQCD $B_c\to J/\psi$ Form Factors

T0 review · 3 major / 5 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read This lattice QCD update determines the Bc to J/psi semileptonic form factors across the full kinematical range with roughly twice the previous precision, and adds tensor form factors for the first time.

desk verdict Solid proceedings update with genuinely new tensor form factors and an ultra-fine ensemble, but the factor-of-two precision gain rests on a heuristic mass interpolation that carries no assigned systematic error. read the letter →

arxiv 2501.15180 v2 pith:4QNZQVY3 submitted 2025-01-25 hep-lat

Judd Harrison This is my paper · ORCID
classification hep-lat
keywords latticeQCDB_cmesonsemileptonicformfactorstensorJ/psiz-expansionparameterisationdispersiveboundsheavyquarkphysics
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 proceedings paper updates a lattice QCD calculation of the $B_c\to J/\psi$ semileptonic form factors and adds the tensor form factors for the first time. Two new ensembles enter the analysis: one with physical up and down quark masses at lattice spacing $a\approx 0.06\,\mathrm{fm}$, and one at $a\approx 0.03\,\mathrm{fm}$ on which the physical bottom quark mass can be simulated directly. With these data, the full set of form factors is determined over the whole kinematical range of the decay using a full dispersive parameterisation. The paper reports that the new Standard Model results agree with the earlier calculation and are roughly twice as precise. If correct, this provides the sharpest lattice QCD input yet for $B_c\to J/\psi\ell\nu$ observables, including the new-physics-sensitive ratio $R(J/\psi)$.

What carries the argument

The machinery is the full dispersive $z$-expansion parameterisation of the form factors, $F_Y(z)=[P_Y(z)\phi_Y(z)]^{-1}\sum_n a^Y_n p_n(z)$, where $P_Y$ are Blaschke factors built from the masses of single-particle $\bar h c$ states below the pair-production threshold, $\phi_Y$ are outer functions depending on continuum susceptibilities, and $p_n$ are orthonormal polynomials on the unit circle. Around this sits the physical-continuum extrapolation model: heavy-quark mass dependence enters through heuristic linear interpolation formulas for masses not computed on the lattice, $M_{\mathrm{pole},i}=M_{\mathrm{pole},i}^{\mathrm{phys}}+M_{H_c}^{\mathrm{latt}}-M_{B_c}^{\mathrm{phys}}$ and $M_H=M_B^{\mathrm{phys}}+M_{H_c}^{\mathrm{latt}}-M_{B_c}^{\mathrm{phys}}$, with the $z$-expansion coefficients truncated at $O(z^4)$ under uniform priors, together with quark-mass mistuning and discretisation terms. The same machinery converts the lattice matrix elements into the QCD-basis form factors $V,A_0,A_1,A_{12},T_1,T_2,T_{23}$ and carries them to the physical continuum.

What would settle it

Repeat the physical-continuum extrapolation with an alternative heavy-quark mass dependence, for example adding a curvature term to the interpolation formulas or replacing $M_H$ by $1/M_H$ scaling, and check whether the physical-point form factors move by more than the quoted uncertainties; a particularly direct test is whether the new $a\approx 0.03\,\mathrm{fm}$, $am_h\approx 0.625$ ensemble point, which sits near the physical bottom mass, is reproduced by the fitted $m_h$ trend within errors.

Watch

Extended reading notes

Core claim

The central claim is that lattice QCD, using the Highly Improved Staggered Quark action for all valence quarks and nonperturbatively renormalised current operators, now determines the $B_c\to J/\psi$ vector and axial-vector form factors and, newly, the tensor form factors across the full $q^2$ range with roughly twice the precision of the previous calculation and consistent with it. The improvement comes from adding a physical-light-quark ensemble and a very fine ensemble that reaches the physical bottom quark mass, so the chiral-continuum extrapolation is anchored directly at the physical heavy-quark point. The extrapolation uses the full dispersive parameterisation in the QCD basis, with Blaschke factors built from sub-threshold pole masses and outer functions determined from continuum susceptibilities computed on the lattice. The tensor form factors are presented as essential for constraining possible new physics in this decay mode.

Load-bearing premise

The load-bearing assumption is that the heavy-quark mass dependence of the meson masses not computed on the lattice is captured by the simple linear interpolation formulas $M_{\mathrm{pole},i}=M_{\mathrm{pole},i}^{\mathrm{phys}}+M_{H_c}^{\mathrm{latt}}-M_{B_c}^{\mathrm{phys}}$ and $M_H=M_B^{\mathrm{phys}}+M_{H_c}^{\mathrm{latt}}-M_{B_c}^{\mathrm{phys}}$; if that linear dependence is wrong, the pole and outer-function factors in the $z$-expansion are wrong and the extrapolated form factors are biased.

Editorial extensions

If this is right

  • The Standard Model vector and axial-vector form factors are consistent with the earlier calculation and roughly twice as precise, so $B_c\to J/\psi\ell\nu$ rate predictions inherit a smaller uncertainty.
  • The new tensor form factors complete the lattice description of the decay's operator basis, giving a direct handle on new-physics contributions in $B_c\to J/\psi\tau^+\nu_\tau$ versus $B_c\to J/\psi\mu^+\nu_\mu$.
  • With data on two ensembles spanning the full kinematical range, the analysis can test how much the choice of parameterisation and truncation order changes integrated observables, addressing a known source of underestimated uncertainty in $B\to D^*\ell\nu$ analyses.
  • A sharper Standard Model prediction for $R(J/\psi)$ follows if the quoted precision is confirmed, making the existing experimental measurements a more discriminating lepton-flavour-universality test.

Reading between the lines

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

  • The paper's heuristic mass-interpolation formulas could be checked by adding a third heavy-quark mass value on the fine ensemble and assigning a systematic uncertainty; until then, the factor-of-two gain in precision may understate the true model dependence.
  • The same full dispersive parameterisation could be applied to other heavy-to-heavy exclusive decays, such as $B_s\to D_s^*\ell\nu$, where parameterisation-related tensions with $B\to D^*$ have been reported.
  • If the observed disagreement between the lattice axial-tensor susceptibility and $O(\alpha_s)$ perturbation theory persists, the new tensor form factors offer a clean observable in which to test whether that discrepancy is perturbative or physical.
  • A testable extension is to compare the new $z$-expansion shapes for $T_1,T_2,T_{23}$ against future differential decay-rate measurements in $B_c\to J/\psi\ell\nu$, checking whether truncation at $O(z^4)$ is adequate.
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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 / 5 minor

Summary. This proceedings paper reports an update of the lattice QCD calculation of the B_c -> J/psi semileptonic form factors (vector, axial-vector, and new tensor) using the HISQ action on the MILC n_f=2+1+1 ensembles. Two new ensembles are used: one with near-physical light quarks at a≈0.06 fm and one at a≈0.03 fm where the physical b quark can be simulated. The correlation functions are fit with a spectral decomposition, the currents are renormalised nonperturbatively (with tensor renormalisation in RI-SMOM matched to MS), and the physical-continuum extrapolation uses a full dispersive parameterisation with Blaschke factors, outer functions, and lattice-computed susceptibilities. The paper claims that the new SM form factors are consistent with the previous HPQCD result and roughly a factor of two more precise.

Significance. If substantiated, this update would provide a more precise determination of the B_c -> J/psi form factors across the full kinematic range, relevant for R(J/psi) and new-physics searches. The addition of the physical-light-quark ensemble and the fine ensemble reaching the physical b mass is a clear strength, as is the use of nonperturbative renormalisation and the full dispersive parameterisation with lattice susceptibilities. The new tensor form factors are a valuable addition. However, the paper's quantitative claims are not backed by numerical tables or a systematic error budget, and the treatment of the heavy-quark mass interpolation in the extrapolation is heuristic. The significance is therefore conditional on the details promised for the full paper.

major comments (3)
  1. [Sec. 2.2, mass interpolation formulas] The heuristic relations M_pole,i = M_phys_pole,i + M_latt_Hc - M_phys_Bc and M_H = M_phys_B + M_latt_Hc - M_phys_Bc are introduced without derivation or an assigned systematic uncertainty. These masses enter the Blaschke factors P_Y(z,t+,t0), the outer functions phi_Y, and the conformal variable z through t+ and t0. Because the lattice heavy-quark masses differ from the physical b mass by several hundred MeV on the coarser ensembles, the omitted O(Lambda_QCD/m_h) corrections to the linear shift could bias the central values at the percent level, comparable to the claimed factor-of-two improvement in precision. The authors should either justify the heuristic with a lattice calculation of the relevant pole masses or assign and propagate a systematic uncertainty.
  2. [Sec. 3, Conclusions] The claim that the new SM form-factor results are 'consistent with our previous calculation, though roughly a factor of 2 more precise' is not supported by any numerical evidence in the paper. No table of the physical-continuum form factors with uncertainties is provided, and the only comparison with the previous result [3] is the visual overlay in Fig. 1. To substantiate the precision claim and to make the update usable, the paper should include a table of the final form factors (or the z-expansion coefficients) with total uncertainties, along with a quantitative comparison of the new uncertainties with those of [3] (e.g., the ratio of uncertainty bands at selected q^2 values).
  3. [Sec. 2.2, O(z^4) truncation and priors] The z-expansion is truncated at O(z^4) and the coefficients are assigned uniform priors |a_Y^n|<=1, but no variation of the truncation order or prior width is reported. The conclusions acknowledge that 'This will allow for a detailed study of systematic uncertainties associated with the choice of kinematical parameterisation,' which indicates the present paper does not yet assess that model dependence. Since the claimed improvement in precision is the central message, the authors should demonstrate stability of the extrapolated form factors under, for example, increasing the truncation to O(z^5) or changing the prior range, and include the spread as a systematic uncertainty.
minor comments (5)
  1. [Eq. (11)] The symbol N_n is used before it is defined in Eq. (13); the definition should be moved forward or the notation should be made consistent (e.g., N_Y(s)_n).
  2. [Sec. 2.2, after Eq. (10)] The text says the coefficients 'satisfy the bounds sum_{Y->Gamma} sum_n |a_Y^n|^2' but does not state the actual bound value; the later imposition of the weaker bound |a_Y^n|<=1 should be introduced more explicitly as a choice.
  3. [Fig. 1 caption] The red dashed lines are described as the '±1σ confidence interval' of the previous result; in a Bayesian context this is usually a credible interval, so the wording should be harmonised with the statistical framework used.
  4. [General] The paper would benefit from a short summary of the fit quality (e.g., chi^2/dof) for both the correlator fits and the physical-continuum extrapolation fit, since Fig. 1 alone does not quantify the goodness of fit.
  5. [Abstract and Sec. 2] The abstract states the a≈0.03 fm ensemble 'reaches the physical bottom quark mass', while Sec. 2 specifies that this ensemble has heavier-than-physical light quarks; this is consistent but should be made explicit in the abstract to avoid confusion.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the form factors are extracted from independent lattice three-point correlator data, and the cited prior results (renormalization factors, susceptibilities) are independent external inputs rather than fitted outputs.

full rationale

The paper's derivation chain starts from lattice two- and three-point correlation functions (Sec. 2, Eqs. (1)-(3)) and extracts matrix elements that are then converted to form factors via Eqs. (4)-(6). The physical-continuum extrapolation in Sec. 2.2 uses the dispersive z-expansion of Eq. (10), with Blaschke factors and outer functions. The outer functions depend on susceptibilities taken from Ref. [30], a separate lattice QCD calculation of current-current correlators by the same author; this is an independent input, not the B_c -> J/psi form-factor data being predicted, and it is externally falsifiable. The renormalization factors from Refs. [27,23] are likewise independent calculations. The heuristic mass formulas M_pole,i = M_phys_pole,i + M_latt_Hc - M_phys_Bc and M_H = M_phys_B + M_latt_Hc - M_phys_Bc are admittedly not derived and carry no explicit systematic uncertainty, but they are modeling assumptions used to set inputs, not quantities derived from the fit; they do not make any output equal to an input by construction. The consistency statement comparing to the previous HPQCD result [3] is a comparison, not a fitted input. No equation in the paper reduces to another by definition, and no fitted parameter is renamed as a prediction. The claimed factor-of-two improvement in precision is therefore a genuine result of new lattice data and the stated fit procedure, subject to the unquantified model dependence noted for the heavy-quark mass interpolation, which is a correctness risk rather than circularity.

Assumptions & free parameters 3 free parameters · 6 assumptions · 0 invented entities

The central claim rests on standard lattice QCD assumptions (valid discretization, continuum extrapolation, renormalization), on the dispersive parameterisation of Ref. [26], and on two paper-specific modeling choices: the heuristic heavy-quark mass interpolation formulas and the O(z^4) truncation. The susceptibilities from Ref. [30] are an additional input with a noted perturbative disagreement. No new particles or entities are introduced.

free parameters (3)
  • aY_n z-expansion coefficients = not reported
    Coefficients of the dispersive parameterisation in Eq. (10) for each form factor, fitted to the lattice matrix elements with uniform priors in [-1,1]; numerical values are not listed in the proceedings text.
  • mistuning coefficients AY_n, BY_n, CY_n, DY_n = not reported
    Coefficients in Eq. (13) that correct for valence charm mistuning, sea charm mistuning, sea strange mistuning, and pion mass mistuning; fitted to data, values not reported.
  • discretization coefficients c(nu,Gamma),jkl_n = not reported
    Coefficients in Eq. (17) that model lattice-spacing and heavy-quark mass dependent discretization effects; fitted to data, values not reported.
assumptions (6)
  • domain assumption The HISQ action on the second-generation MILC n_f=2+1+1 gluon field configurations is a valid discretization of QCD, and the continuum limit is described by the extrapolation ansatz in Eq. (17).
    This is the standard working assumption of every lattice QCD calculation; the paper provides no independent derivation of the continuum limit.
  • standard math The dispersive parameterisation of Ref. [26], including Blaschke factors, outer functions, and unitarity bounds, correctly describes the continuum form factors over the full kinematic range.
    The z-expansion follows from analyticity and unitarity of QCD correlation functions; the paper cites [26] and applies it in Eq. (10).
  • ad hoc to paper The heuristic heavy-quark mass interpolation formulas M_pole,i = M_phys_pole,i + M_latt_Hc - M_phys_Bc and M_H = M_phys_B + M_latt_Hc - M_phys_Bc correctly model the m_h dependence of masses not computed on the lattice.
    These forms are introduced after Eq. (12) without derivation or an assigned systematic error; they enforce the correct physical limit but leave the intermediate m_h dependence as a modeling choice.
  • ad hoc to paper Truncating the z-expansion at O(z^4) with uniform priors |aY_n| <= 1 is sufficient for the full kinematic range.
    The text states polynomials up to O(z^4) are included and the weaker bound |aY_n| <= 1 is imposed; no test of higher-order truncation error is displayed.
  • domain assumption The RI-SMOM renormalization factors for the vector, axial-vector, and tensor currents are correct, and the 3-loop running of the tensor factors to m_h(m_h) is valid.
    The paper relies on Refs. [27], [23], and [29] for renormalization and running; no independent check is performed here.
  • domain assumption The heavy-light susceptibilities computed in Ref. [30] are correct enough to normalize the outer functions in Eq. (10), despite the noted disagreement with O(alpha_s) perturbation theory for the tensor susceptibility.
    The paper explicitly notes disagreement for the (axial-)tensor susceptibility but does not propagate a separate uncertainty for it; the dispersive bounds depend on these inputs.

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Pith. "Pith review of Update of HPQCD $B_c\to J/\psi$ Form Factors." pith.science (2026). https://pith.science/paper/4QNZQVY3

@misc{pith2026250115180,
  author       = {Pith},
  title        = {Pith review of: Update of HPQCD $B_c\to J/\psi$ Form Factors},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/4QNZQVY3}},
  note         = {Machine review of arXiv:2501.15180}
}
abstract

We present an update of our lattice QCD determination of the $B_c\to J/\psi$ vector and axial-vector form factors, including new results for the tensor form factors. We use the Highly Improved Staggered Quark action for all valence quarks, together with the second generation MILC $n_f=2+1+1$ HISQ gluon field configurations. This calculation includes two additional ensembles, one with physically light up and down quarks and $a\approx 0.06 \mathrm{fm}$ and one with $a\approx 0.03\mathrm{fm}$ on which we are able to reach the physical bottom quark mass. Our calculation uses nonperturbatively renormalised current operators and covers the full kinematical range of the decay. Our physical-continuum extrapolation utilises the full dispersive parameterisation for $B_c\to J/\psi$.

Figures

Figures reproduced from arXiv: 2501.15180 by the authors.

Figure 1
Figure 1. Lattice data for all SM and tensor form factors together with the result of our physical continuum extrapolation described above, shown as the blue line and error band. The data points shown here have been corrected the posteriors, using eq. (18). We also multiply the form factors by the Blaschke factors and outer functions to better demonstrate the polynomial dependence on 𝑧. The red dashed lines drawn for the SM f… view at source ↗

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