Towards more accurate $B_{(s)}\rightarrow\pi(K)$ and $D_{(s)}\rightarrow\pi(K)$ form factors

Chris Bouchard; Logan Roberts; Olmo Francesconi; Will Parrott

REVIEW 2 major objections 8 minor 22 references

Towards more accurate $B_{(s)}\rightarrow\pi(K)$ and $D_{(s)}\rightarrow\pi(K)$ form factors

T0 review · 2 major / 8 minor · reviewed 2026-08-09 · deepseek-v4-flash

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Pith's one-line read The paper claims that the heavy-HISQ lattice QCD approach, with heavy quark masses from charm to near bottom, covers nearly the full kinematic range for the four decays $B\to\pi$, $B_s\to K$, $D\to\pi$, and $D_s\to K$, and that a chained…

desk verdict A competent, honest status report from a group extending a proven method to new channels; the main open question is whether the chained-fit split preserves all needed correlations. read the letter →

arxiv 2501.18586 v1 pith:5YDQ7BBF submitted 2025-01-30 hep-lat hep-ph

Logan Roberts , Chris Bouchard , Olmo Francesconi , Will Parrott This is my paper · ORCID

classification hep-lathep-ph MSC 81T2581V05 PACS 12.38.Gc13.20.He13.20.Fc

keywords latticeQCDheavy-HISQsemileptonicformfactorsBtopidecayDCKMmatrixelementVubVcdchainedfits

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 is a progress report on a lattice QCD calculation of scalar, vector, and tensor form factors for the semileptonic decays $B\to\pi$, $B_s\to K$, $D\to\pi$, and $D_s\to K$. The central claim is that using the fully relativistic heavy-HISQ action with heavy quark masses spanning from the charm quark mass to near the bottom quark mass provides nearly complete coverage of the kinematic $q^2$ range for all four channels. If this holds, the same simulation data can produce accurate form factors across the full kinematic range, which are needed to extract the CKM matrix elements $|V_{ub}|$ and $|V_{cd}|$ from experimental decay rates. The paper also shows that a chained fitting procedure, which splits the large correlator dataset into weakly correlated subsets, can successfully model ground, excited, and oscillating states, yielding preliminary form factor results on one ensemble.

What carries the argument

The central machinery is the heavy-HISQ formalism: fully relativistic highly improved staggered quarks for the heavy quark, with local scalar, vector, and tensor currents. The analysis relies on global correlator fits with Bayesian priors, implemented with gvar, lsqfit, and corrfitter, and uses a chained fitting procedure to handle the large amount of correlator data, splitting parameters into subsets based on their statistical correlations. The final extrapolation to the continuum, physical quark masses, and full $q^2$ range uses the modified $z$-expansion.

What would settle it

Compute the same form factor matrix elements on the f-5 ensemble using an independent renormalization procedure, such as the conserved vector current or a momentum-subtraction scheme, and compare with the results using the adopted $Z$ factors; a difference larger than the quoted uncertainties would indicate the renormalization assumption fails. Alternatively, test the kinematic identity $f_+(0)=f_0(0)$ at $q^2=0$ after the $z$-expansion, which must hold exactly; a violation beyond errors would signal a problem with the fitting or renormalization.

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Extended reading notes

Core claim

The paper advances the claim that the heavy-HISQ approach, combined with HISQ valence quarks on MILC ensembles, can determine the form factors for $H\to\pi$ and $H_s\to K$ over nearly the entire kinematic range, where $H$ is a generic heavy meson whose heavy quark mass is tuned from charm to near bottom. The key evidence is the successful chained global fit to two- and three-point correlators, which models non-oscillating and oscillating states, and the resulting preliminary $f_0$, $f_+$, and $f_T$ results for $H\to\pi$ on the f-5 ensemble. The paper argues that this method, already demonstrated to give precise $B\to K$ and $D\to K$ form factors, will extend the same precision to the pion and kaon final states needed for $|V_{ub}|$ and $|V_{cd}|$.

Load-bearing premise

The renormalization factors for the scalar, vector, and tensor currents, taken from earlier calculations, are assumed to be correct for all heavy-quark masses and for the pion and kaon final states; if they are not, every form factor carries an unknown systematic shift.

Editorial extensions

If this is right

If the chained fits succeed on all ensembles, the project will produce continuum-limit, physical-mass form factors for $B\to\pi$, $B_s\to K$, $D\to\pi$, and $D_s\to K$ over the full $q^2$ range.
These form factors, combined with experimental decay rates, will yield exclusive determinations of $|V_{ub}|$ and $|V_{cd}|$.
The nearly full kinematic coverage reduces the model dependence of the $z$-expansion extrapolation compared to calculations that only cover a limited $q^2$ range.
The success of the chained fitting procedure demonstrates a practical way to handle the large correlator datasets required for multi-channel, multi-mass lattice calculations.

Reading between the lines

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

If the anticipated precision materializes, the exclusive $|V_{ub}|$ derived here could sharpen the comparison with inclusive determinations and shed light on the long-standing inclusive-exclusive tension.
The observed correlation grouping of the four currents suggests that scalar and temporal-vector parameters share much of their statistical fluctuation, so the renormalization uncertainties of these currents may be partially correlated; a combined analysis could exploit this.
The method is naturally extensible to other light final states, such as $K^*$ or heavier resonances, as long as the correlator signals remain controllable, potentially broadening the set of decays accessible to the same formalism.
The physical-mass ensembles included in the ongoing work will provide a direct check of the light-quark mass dependence, testing whether the 5:1 strange-to-light ratio results extrapolate consistently.

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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

This is a solid, honest proceedings status report from a group that has already demonstrated the heavy-HISQ method for B->K and D->K. The genuinely new content is the extension to pion final states and the heavy-strange channels, plus the chained-fit machinery needed to handle the larger correlator data sets. The preliminary H->pi form factors on the f-5 ensemble (Fig. 5) look sensible, and the reconstruction check in Fig. 4 shows the fits capture the correlator behavior well. The paper does not overclaim; it explicitly says the global fitting methodology is not finalized, and the priors are deliberately widened to at least ten times the posterior width, which reduces circularity concerns.

The main soft spot is the chained-fit split. The authors split the data into two groups (scalar plus temporal vector versus spatial vector plus tensor) based on a representative correlation matrix, and assert that this preserves correlations between subsets. That is a plausible practical choice, but they do not validate it against a full global fit or demonstrate that cross-link correlations are genuinely negligible. Since all four currents share the same H and pi energies and amplitudes, ignoring cross-link correlations could bias or underestimate uncertainties on the reconstructed amplitudes that feed into the preliminary form factors. For a status report this is acceptable, but it becomes load-bearing if the final analysis relies on this split without further checks.

The renormalization factors from refs [12-15] are taken from other calculations. Again, standard practice, but for a final result one would want a justification that they hold for these channels and over the full heavy-mass range. The paper also shows only one ensemble's preliminary results, so the continuum and chiral extrapolation are untouched.

The citation pattern is appropriate; the authors cite their own prior work and the renormalization papers, which is not a problem given the established nature of the method. The writing is clear and the mathematics is well laid out.

Overall, this is a useful status report for the lattice QCD and flavor physics community. It deserves peer review in the context of the Lattice proceedings, where a careful progress report is a legitimate contribution. I would accept it, while flagging to the authors that the chained-fit validation should be addressed in the eventual full paper.

Referee Report

2 major / 8 minor

Summary. This proceedings paper reports work in progress toward lattice QCD calculations of scalar, vector, and tensor form factors for B→π, B_s→K, D→π, and D_s→K. The calculation uses HISQ valence quarks on MILC ensembles at several lattice spacings, including 0.044 fm, with heavy-HISQ heavy quarks ranging from charm to near bottom. The paper describes the two- and three-point correlator fit ansatz, the prior construction procedure, and a chained-fit strategy in which the data are split by current type. It shows a representative reconstruction check and preliminary f-5 ensemble form-factor plots. Section 3 explicitly states that the global fitting methodology is not yet finalized, so the numerical results are presented as preliminary.

Significance. If the program succeeds, it would provide form factors over nearly the full q^2 range for four phenomenologically important decays and could improve determinations of |V_ub| and |V_cd|. The paper's strengths are the broad ensemble coverage (including physical and 5:1 light masses, and lattice spacings down to 0.044 fm), the systematic use of the established heavy-HISQ framework of refs. [5,6], the explicit description of prior construction including Gaussian Bayes factor optimization, and the transparent presentation of preliminary reconstruction checks. Because the fitting methodology is not finalized (Section 3) and no continuum or physical-mass extrapolation is presented, the contribution should be read as a status report rather than as a source of final numerical values.

major comments (2)

[Section 2.2, Fig. 1] The split of the chained fit into two groups (scalar plus temporal-vector, spatial-vector plus tensor) is justified only by the representative correlation matrix in Fig. 1. That figure shows that within-group correlations are high, but it does not demonstrate that cross-link correlations are negligible. Because all four currents are connected through the same H and pi energies and amplitudes in Eq. (3), the two groups are not obviously independent. The statement that this procedure 'preserves correlations between the various subsets of data' is therefore not established. Please quantify the cross-link correlations, and compare the chained-fit posterior with a full simultaneous fit on at least one ensemble (or a reduced data set), reporting differences in J^{nn}_{00} and its uncertainty. Without such a check, the reconstruction in Fig. 4 and the preliminary form factors in Fig. 5 cannot be taken as evidence that the method is unbiased.
[Section 2.1, Eqs. (4)-(7)] The renormalization factors Z_disc, Z_V, and Z_T(μ) are taken from refs. [12-15], which were computed in other heavy-HISQ calculations. The manuscript does not state whether these factors are applicable to H→π and H_s→K for the full range of heavy masses used here, nor does it give an estimate of the systematic error if they are not. Since these factors enter every matrix element in Eq. (4), please state their provenance for these channels and masses, or provide an estimate of the induced uncertainty. This may be acceptable for a proceedings paper if the final publication will address it, but it should be made explicit.

minor comments (8)

[Section 2, first paragraph] The phrase 'global (one fit per ensemble)' appears to conflict with the chained-fit procedure described in Section 2.2; please clarify whether 'global' refers to all data in an ensemble or to a single link.
[Section 2.2, Fig. 1 caption] The caption lists only two twists (θ=0.0, 0.4281) while the text mentions five twists; please explain why only these are shown.
[Figure 5 caption] The axes and the definition of the error bands are not given in the caption; please add them so the plot can be interpreted independently.
[Section 2.3] The text 'denote the prior central value and uncertainty ... as \tilde{P}_i + \tilde{\sigma}_{P_i}' should read '\tilde{P}_i \pm \tilde{\sigma}_{P_i}' or 'with uncertainty \tilde{\sigma}_{P_i}'.
[Throughout] There are several typos, including 'York Uiversity' in the author affiliation, 'apprach' in Section 3, and 'spacial' in the discussion of Eq. (6); these should be corrected.
[Table 1] The column header for the separation widths is only 'T'; please define the unit (e.g., lattice time slices or physical time) in the table caption or a footnote.
[Section 2.1, Eqs. (5)-(7)] The symbols M_H, M_π, p^μ, and the renormalization scale μ are not all defined with their lattice versus continuum conventions; a short definition sentence would improve readability.
[Section 3] The explicit limitation that the global fitting methodology is not finalized is stated only near the end; the abstract and introduction should carry a similar caveat so that the preliminary nature of the results is clear from the outset.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity found: the form-factor extraction is a direct fit to lattice correlator data with broad priors and externally computed renormalization constants.

full rationale

The paper is a progress report whose central quantitative outputs (Figs. 4 and 5) are obtained by fitting two- and three-point correlators to the spectral decompositions in Eqs. (2) and (3), then converting the fitted ground-state current amplitude J_nn^00 into form factors via Eqs. (4)-(7). The fit parameters are determined by the correlator data themselves; the only external inputs are the renormalization factors Z_disc, Z_V, and Z_T(mu) taken from refs. [12-15], which are independent published quantities not derived in this paper and not computed by the present authors. The prior methodology in Sec. 2.3 is explicitly designed not to constrain the answers: prior widths are set to at least ten times the corresponding posterior widths, and the physics-based priors concern pion/kaon masses and dispersion relations rather than the target form factors. The chained-fit grouping in Sec. 2.2 is a statistical approximation based on a correlation-matrix inspection; it could affect the reliability of uncertainties if cross-link correlations are non-negligible, but it does not reduce any predicted quantity to an input by construction. The abstract's claim of 'nearly full coverage of the kinematic range' is a direct property of the chosen heavy-quark masses and simulated daughter momenta, not a fitted prediction. No uniqueness theorem, ansatz, or known result is imported from the authors' prior work in a load-bearing way; refs. [5,6] are ordinary methodological citations. Accordingly, no circular step is exhibited.

Assumptions & free parameters 2 free parameters · 4 assumptions · 0 invented entities

The central claim is a progress report on an established lattice method; it rests on standard lattice QCD assumptions and renormalization factors from prior work, plus two hand-tuned fitting parameters (prior widths and N_exp). No new entities are introduced.

free parameters (2)

Prior width inflation factor = at least 10x posterior uncertainty
Section 2.3: the authors set prior uncertainties at least ten times the posterior uncertainty to avoid over-constraining; this hand-chosen factor shapes all fit results.
Number of exponential states N_exp = 4
Section 2.2 and Fig. 2: tested on one ensemble (sf-phys) and found sufficient, but not verified for all ensembles and channels.

assumptions (4)

domain assumption HISQ action on MILC Nf=2+1+1 ensembles correctly describes QCD at the simulated lattice spacings.
Foundation of the calculation, cited to refs [2-4] and used throughout.
domain assumption The renormalization factors Z_disc, Z_V, Z_T from refs [12-15] are valid for the local currents used with these heavy quark masses and channels.
Used in Eqs. (4)-(7) to convert bare lattice matrix elements to continuum form factors; not re-derived in this paper.
standard math The modified z-expansion (ref [8]) is model-independent and will control the extrapolation to the full kinematic range.
To be used in future analysis (Section 3) to extrapolate in q^2, quark masses, and lattice spacing.
domain assumption The chained fit method [10] with the chosen grouping preserves the relevant statistical correlations.
Section 2.2: the split into subsets is based on a representative correlation matrix (Fig. 1) and is asserted to preserve correlations, but is not exhaustively validated.

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Cite this review

Pith. "Pith review of Towards more accurate $B_{(s)}\rightarrow\pi(K)$ and $D_{(s)}\rightarrow\pi(K)$ form factors." pith.science (2026). https://pith.science/paper/5YDQ7BBF

@misc{pith2026250118586,
  author       = {Pith},
  title        = {Pith review of: Towards more accurate $B_(s)\rightarrow\pi(K)$ and $D_(s)\rightarrow\pi(K)$ form factors},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/5YDQ7BBF}},
  note         = {Machine review of arXiv:2501.18586}
}

abstract

We present progress on the calculation of scalar, vector, and tensor form factors for the following meson decays: $B\rightarrow\pi$, $B_s\rightarrow K$, $D\rightarrow\pi$ and $D_s\rightarrow K$. This calculation uses the MILC HISQ gluon field ensembles with HISQ valence quarks. We generate ensembles of correlator data with varying lattice spacings, some as small as 0.044 fm. Some ensembles have a strange-to-light quark mass ratio of 5:1 and others use the physical light quark mass. The fully-relativistic, heavy-HISQ approach is used for the heavy quark, with simulation masses ranging from the charm to near the bottom. This heavy-HISQ approach provides nearly full coverage of the kinematic range.

Figures

Figures reproduced from arXiv: 2501.18586 by the authors.

**Figure 1.** Representative sample three point amplitude correlation matrix for 𝐻 → 𝜋 on the f-5 ensemble. Each row and column denotes a distinct fit parameter. The amplitude tags 𝑆𝑉𝑛𝑛, 𝑇𝑉𝑛𝑛, 𝑉𝑉𝑛𝑛, and 𝑋𝑉𝑛𝑛 denote a scalar, tensor, temporal vector, or spacial vector current respectively. Parameters shown cover heavy quark masses of 𝑎𝑚ℎ = 0.450, 0.55, and twists 𝜃 = 0.0, 0.4281. (such as figure 3) can be constructed via equations… view at source ↗

**Figure 2.** Sample 𝑡min and 𝑁exp plot on the sf-phys ensemble. The mother meson’s rest mass, which contains a heavy valence quark ℎ of mass 𝑎𝑚ℎ = 0.2585, is the fit posterior whose central value and uncertainty is tracked with changing values for 𝑡min and 𝑁exp [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗

**Figure 3.** f-5 ensemble 𝐻 → 𝜋 scalar current three-point effective amplitude plot for all mass 𝑎𝑚ℎ and twist 𝜃 options. Different heavy quark mass options for a given 𝑡/𝑎 are offset along x-axis for visual aid. The five different twists are represented by the five, typically vertically separated, clusters of 𝐽eff(𝑡, 𝑇) for a single 𝑡/𝑎. 6 [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗

Figures from the paper (2 more)

**Figure 4.** Figure 4: f-5 ensemble 𝐻 → 𝜋 prior vs. posterior sample comparison plot. Data shown is for a three point temporal vector current with a heavy quark mass of 𝑎𝑚ℎ = 0.675, a daughter meson twist of 𝜃 = 1.1282, and a mother-daughter separation width of 𝑇 = 24. A reconstructed effect…

**Figure 5.** Figure 5: Preliminary 𝐻 → 𝜋 form factor results for all current insertions on the f-5 ensemble. Results are shown for all heavy quark masses 𝑎𝑚ℎ and currents. Note that 𝑎𝑚ℎ = 0.450 corresponds to the tuned charm quark mass. over-constrains nor under-constrains the fit [20]. A pr…

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Pith tools

Reviewed August 9, 2026 · model on record in the stance chip above.

[1] [1]

B Meson Semileptonic Form Factors from Unquenched Lattice QCD

HPQCDCollaboration, E. Gulez, A. Gray, M. Wingate, C. T. H. Davies, G. P. Lepage, and J. Shigemitsu,𝐵 meson semileptonic form factors from unquenched lattice QCD, Phys. Rev. D 73(2006) 074502, [hep-lat/0601021]

work page Pith review arXiv 2006

[2] [2]

Follana, Q

E. Follana, Q. Mason, C. Davies, K. Hornbostel, G. P. Lepage, J. Shigemitsu, H. Trottier, and K. Wong,Highly improved staggered quarks on the lattice with applications to charm physics, Phys. Rev. D75 (2007) 054502, [hep-lat/0610092]

arXiv 2007

[3] [3]

Bazavov, C

A. Bazavov, C. Bernard, C. DeTar, W. Freeman, S. Gottlieb, U. M. Heller, J. E. Hetrick, J.Laiho,L.Levkova, M.Oktay,J.Osborn,R.L.Sugar, D.Toussaint,andR.S.VandeWater, Scaling studies of QCD with the dynamical highly improved staggered quark action, Phys. Rev. D82(2010) 074501, [1004.0342]

arXiv 2010

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Bazavov, C

A. Bazavov, C. Bernard, J. Komĳani, C. DeTar, L. Levkova, W. Freeman, S. Gottlieb, R. Zhou, U. M. Heller, J. E. Hetrick, J. Laiho, J. Osborn, R. L. Sugar, D. Toussaint, and R. S. Van de Water,Lattice QCD ensembles with four flavors of highly improved staggered quarks, Phys. Rev. D87(2013) 054505, [1212.4768]

arXiv 2013

[5] [5]

W. G. Parrott, C. Bouchard, C. T. H. Davies, and D. Hatton,Toward accurate form factors for 𝐵-to-light meson decay from lattice QCD, Phys. Rev. D103 (2021) 094506, [2010.07980]

arXiv 2021

[6] [6]

W. G. Parrott, C. Bouchard, and C. T. H. Davies,𝐵→𝐾 and𝐷→𝐾 form factors from fully relativistic lattice QCD, Phys. Rev. D107 (2023) 014510, [2207.12468]

arXiv 2023

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Navas et al.,Review of particle physics,Phys

Particle Data GroupCollaboration, S. Navas et al.,Review of particle physics,Phys. Rev. D 110 (2024) 030001

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Bourrely, L

C. Bourrely, L. Lellouch, and I. Caprini,Model-independent description of𝐵→𝜋𝑙𝜈 decays and a determination of|𝑉𝑢𝑏|,Phys. Rev. D79(2009) 013008, [0807.2722]

arXiv 2009