Light-Cone Sum Rules for $B\to K\pi$ Form Factors and Applications to Rare Decays

Alexander Khodjamirian; Javier Virto; S\'ebastien Descotes-Genon

REVIEW 2 major objections 3 minor 6 cited by

Light-Cone Sum Rules for $B\to K\pi$ Form Factors and Applications to Rare Decays

T0 review · 2 major / 3 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read The K* width rescales every B→K* form factor by a universal 9 percent.

desk verdict Solid LCSR paper with a real universal finite-width effect; the 20% rate enhancement is directionally right but the quoted ±0.01 on W_K* undersells the line-shape model uncertainty. read the letter →

arxiv 1908.02267 v2 pith:6WR7AUZ7 submitted 2019-08-06 hep-ph

Sébastien Descotes-Genon , Alexander Khodjamirian , Javier Virto This is my paper · ORCID

classification hep-ph

keywords light-conesumrulesB→K*formfactorsfinite-widtheffectsKπvectorfactorrareBdecaysB-mesondistributionamplitudesz-expansionB→Kπℓℓ

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 derives light-cone sum rules that determine the P-wave hadronic form factors for $B\to K\pi$ transitions directly from the $B$-meson light-cone distribution amplitudes, replacing the usual assumption that the $K^*$ is a stable particle. The central result is a universal finite-width correction: all seven $B\to K^*$ form factors extracted in the narrow-width limit must be multiplied by $W_{K^*}=1.09\pm0.01$, independent of helicity and $q^2$. Because the factorizable part of any $B\to K^*X$ rate depends on the square of form factors, this raises the predicted rates by about 20% relative to the narrow-width prediction. That shift goes in the direction of increasing the tension between Standard Model expectations and measured $B\to K^*\mu\mu$ branching fractions, while leaving form-factor-ratio observables such as $P'_5$ unchanged.

What carries the argument

The carrying object is the light-cone sum rule for the correlation function of a $\bar d\gamma_\mu s$ interpolating current with a $b\to s$ (axial)vector or tensor current, written in terms of $B$-meson light-cone distribution amplitudes (the momentum-fraction distributions of the light quark in the $B$ meson on the light cone). Projecting out the $P$-wave gives equations of the form $\int_{s_{\rm th}}^{s_0} ds\, e^{-s/M^2}\,\omega_i(s,q^2)\, f_+^*(s)\, F_i^{(\ell=1)}(s,q^2)=P_i^{\rm OPE}(q^2,\sigma_0,M^2)$ for each form factor. The finite-width result comes from the ratio $W_{K^*} = 2m_{K^*} f_{K^*}\, e^{-m_{K^*}^2/M^2}/I_{K^*}(s_0,M^2)$, where $I_{K^*}$ is the integral over the $K\pi$ invariant mass of $|f_+(s)|^2$ weighted by the Breit-Wigner line shape of the $K^*(892)$. Universality holds because $I_{K^*}$ multiplies every form factor with the same coefficient, so its finite-width correction factorizes and cancels in the ratio to the narrow-width limit; the $z$-expansion then provides the $q^2$ dependence.

What would settle it

Take the measured $\tau\to K\pi\nu$ spectrum (or a future higher-statistics version), extract $|f_+(s)|^2$ directly, recompute the integral $I_{K^*}(s_0,M^2)$, and compare the resulting $W_{K^*}$ with $1.09\pm0.01$; a deviation beyond the propagation of the $s_0$ and $M^2$ uncertainty would invalidate the claimed universality. Alternatively, a lattice QCD calculation of the P-wave $B\to K\pi$ form factor integrated over the $K^*(892)$ window, compared against the narrow-width form factor, would settle the ratio directly.

Watch

Extended reading notes

Core claim

The paper claims that a non-vanishing $K^*(892)$ width produces a universal correction to every $B\to K^*$ form factor obtained from the new sum rules: $W_{K^*}=1.09\pm0.01$ (Eq. 99), independent of the form-factor helicity and of $q^2$. Equivalently, the factorizable parts of $B\to K^*X$ rates are increased by about 20% relative to the narrow-width approximation (Eq. 113). The paper further claims that its $B\to K\pi$ sum rules reduce analytically to the known $B\to K^*$ sum rules in the narrow-width limit, and that the updated operator-product expansion, with two- and three-particle higher-twist $B$-meson distribution amplitudes, yields $B\to K^*$ form factors with lower central values than earlier determinations, mainly because of the twist-four two-particle contribution $g_+(\omega)$. It also obtains a lower effective threshold $s_0$ from a two-point sum rule whose hadronic side uses the measured $K\pi$ spectral density.

Load-bearing premise

The numerical value $W_{K^*}=1.09\pm0.01$ rests on the hadronic model for the $K\pi$ vector form factor $f_+(s)$ in the $K^*(892)$ resonance region, whose line shape is taken from fits to the $\tau\to K\pi\nu$ spectrum; if that line shape is wrong, the universal factor changes.

Editorial extensions

If this is right

Factorizable rates for $B\to K^*\ell\ell$, $B\to K^*\nu\nu$, $B\to K^*\gamma$, and related modes computed from narrow-width LCSR form factors should be multiplied by $|W_{K^*}|^2\simeq 1.20$.
Form-factor-ratio observables, such as $P'_5$ in $B\to K^*\mu\mu$, are insensitive to the finite-width correction because the factor cancels.
Updated narrow-width predictions for the $B\to K^*$ form factors at $q^2=0$ are lower than previous LCSR and lattice central values, with the twist-four $g_+(\omega)$ contribution identified as the main cause.
Measurements of $B\to K\pi\mu\mu$ angular moments in the $K^*(1410)$ window constrain the relative $K^*(1410)$ contribution $\alpha$; current data give $\alpha\lesssim 4$--$5$, which caps any $K^*(1410)$-driven suppression of the $B\to K^*$ form factors at about ten percent.

Reading between the lines

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

Editorial extension: the same ratio construction should apply to the $\rho$ channel, where the paper notes a similar width effect for $B\to\rho$ form factors; a quantitative $W_\rho$ from the pion form factor would shift $B\to\rho\ell\ell$ and $B\to\rho\gamma$ predictions in the same direction.
Editorial extension: because $W_{K^*}$ is $q^2$- and helicity-independent, global fits of $b\to s\ell\ell$ data that normalise to branching fractions will have their New Physics pull weakened by roughly the 20% rate increase, a consequence the paper does not quantify.
Editorial extension: a precision extraction of $f_+(s)$ from future $\tau\to K\pi\nu$ data could replace the resonance model used here in $I_{K^*}$, turning Eq. (99) into a first-principles prediction with a smaller model error.

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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

Dear X,

What you should know: this is the real B→Kπ LCSR paper, generalizing the earlier B→ππ work to unequal masses and adding tensor form factors. The sum rules in Eq. (31) are new, carefully derived, and demonstrably reduce to the known B→K* sum rules in the narrow-width limit. The paper is worth reading for that derivation alone.

The main phenomenological claim is the finite-width factor W_K* = 1.09 ± 0.01, universal for all B→K* form factors and q², implying a 20% enhancement of factorizable B→K*X rates. That is a genuinely important observation: it directly affects the comparison of SM predictions to LHCb b→sμμ branching ratios, in the direction of increasing tension. I think the effect is real. The derivation starts from a resonance model for the Kπ vector form factor anchored to Belle τ data, and the O(Γ) linearized cross-check supports the size and sign. So the central result is solid.

The soft spot is precisely the quoted uncertainty. The stress-test note has it right: W_K* inherits the line-shape model for f+(s). The O(Γ) formula is not independent, since ρ′(m²) comes from the same width parametrization. The two Belle models give a spread near the edge of the stated error, and other plausible line shapes with barrier factors or coupled channels could move W by more than ±0.01. I would not call the 20% enhancement wrong, but the honest statement is W_K* ≈ 1.09 with an unquantified line-shape uncertainty of at least the quoted size, probably ±0.02–0.03. That should be fixed before publication, or at least flagged loudly.

Other soft spots are minor and mostly acknowledged: no NLO QCD corrections, a sometimes large twist-4 g+ contribution that pulls the B→K* form factors lower than previous determinations. Those are fitting choices and data-availability issues rather than errors. The large g+ effect is worth watching, but they provide a transparent deconstruction in Table 7.

Bottom line: the paper is serious, the math holds up, and the finite-width effect is a real contribution to the B→K* literature. Who is it for? Anyone working on b→s transitions, form factors, or LCSRs. It deserves a serious referee and, after a moderate revision that addresses the line-shape uncertainty, publication.

Referee Report

2 major / 3 minor

Summary. The paper derives light-cone sum rules (LCSRs) for the P-wave B→Kπ form factors using B-meson light-cone distribution amplitudes, covering vector, axial-vector, timelike-helicity, and tensor currents. The sum rules are shown to reduce analytically to the known B→K* LCSRs in the narrow-width limit, and the OPE side is updated with a more complete set of two- and three-particle twist-4 contributions. The numerical analysis determines the B→K* form factors in the narrow-width limit, finding central values consistent with but lower than previous determinations, and introduces a universal finite-width factor W_K* = 1.09 ± 0.01 that increases factorizable B→K*X rates by about 20%. Applications include the angular distribution of B→Kπℓℓ and constraints on the K*(1410) contribution from LHCb data.

Significance. If the central claim holds, the paper provides a systematic way to go beyond the narrow-width approximation in LCSR determinations of B→K* form factors, with a concrete phenomenological consequence for b→sℓℓ rate predictions. The analytic derivation is careful: the narrow-width reduction is shown explicitly, the OPE coefficients are documented, the numerical inputs are propagated through Gaussian scans, and the results are cross-checked against earlier LCSR determinations. The use of Belle τ→Kπν data to constrain the Kπ vector form factor and the threshold parameter is a useful step. The main weakness is that the headline finite-width correction W_K* is assigned an uncertainty that does not yet include the model dependence of the Kπ line shape, which is directly relevant to the claimed 20% rate enhancement.

major comments (2)

[Section 5.6, Eqs. (45) and (99)] The quoted result W_K* = 1.09 ± 0.01 propagates only variations of the sum-rule parameters (m_K*, Γ_K*, s0, M²) through the integral I_K*(s0,M²), but not the uncertainty in the Breit-Wigner model of the Kπ vector form factor f+(s), Eq. (40), from which I_K* is computed. The difference between Belle Model 1 and Model 2 in Table 8 already shifts W_K* by about 0.005, i.e. half the quoted error, and plausible alternative line shapes (barrier factors, coupled channels, different s-dependent widths) are not probed. Since the 20% rate enhancement is the paper's headline phenomenological claim, the uncertainty in W_K* needs to include the f+(s) model dependence or be stated as a model-dependent estimate.
[Section 5.6, Eq. (100) and Appendix F.2] The linearized expression W_K* = 1 − (Γ_K*/m_K*) Δ_K*(s0,M²) is presented as a robustness check that is 'less dependent on the details of the hadronic model', but it is not an independent cross-check: its O(Γ) coefficient inherits the assumed s-dependent width through ρ'(m_R²) in Eq. (59). For a fixed-width Breit-Wigner, ρ'(m_R²) would be 1/(2m_R²) ≈ 0.62 GeV⁻², whereas Eq. (59) gives ρ'(m_K*²) ≈ 3.5 GeV⁻², changing the sign and size of the logarithmic term in Δ_R, Eq. (56). The claimed robustness of the linearized check is therefore not established by the derivation.

minor comments (3)

[Section 7, Conclusions] The text refers to 'the K∗(980) and the K∗(1410)' in the resonance model and later to 'the K∗(890) contribution'; these should be K∗(892) (and, where intended, K*(800) for the scalar), since the vector P-wave Kπ resonance under discussion is the K∗(892).
[Figure 2 caption] The caption states that the red vertical band corresponds to the physical (PDG) width, while the numerical input in Table 2 uses the Belle fit values m_K* = 895.4(2) MeV and Γ_K* = 46.1(6) MeV; the caption should identify the source of the band consistently.
[Section 5.4 and Appendix E] The 42×42 correlation matrix is said to be available from the authors upon request; for reproducibility it would be preferable to include it as an ancillary file with the submission, as is done for the OPE coefficients.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the finite-width correction is computed from external inputs; self-citations are rederived and non-load-bearing.

full rationale

The paper's central new result, the universal finite-width factor W_K* = 1.09 ± 0.01 in Eq. (99), is not fitted to B->K* data and is not equivalent to any input by construction. It is defined in Eq. (97) as the ratio of the finite-width integral I_K*(s0,M^2) to its narrow-width limit; the OPE side of the sum rules cancels in this ratio, and the numerical value is obtained by evaluating the hadronic integral with the externally measured Belle tau->K pi nu K pi vector form factor (Eq. (40), Section 5.2), PDG masses, and the SVZ-determined threshold s0. The universality in helicity and q^2 follows algebraically from the one-resonance form of the sum rule (Eq. (96)), where the width-dependent integral I_K* is common to all form-factor helicities; this is a transparent derivation, not a self-fulfilling definition. The narrow-width limit is rederived analytically (Section 4.2, Appendix A) rather than imported, and the OPE side is recalculated from B-meson LCDAs with stated inputs. Citations to the authors' earlier Refs. [12,19] supply the method and the phase-unitarity condition, but the text re-derives the dispersion-relation framework and the NWL reduction explicitly, so these self-citations are not load-bearing in an unverified way. The skeptic's concern that W_K* inherits the Breit-Wigner line-shape model in Eq. (36) is a legitimate model-uncertainty issue, but it is not circularity: the model parameters come from external Belle data, and different line shapes would change the estimate without making the derivation tautological. No step reduces a claimed prediction to a fitted parameter renamed as a prediction, and no uniqueness theorem or ansatz is smuggled in solely via self-citation.

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

The central numerical results depend on four input parameters (λB, R, s0, M²) and on the B-meson LCDA models and the resonance model for f+. The formal LCSR derivation itself uses only standard QCD and LCSR assumptions. No new particles or entities are introduced.

free parameters (6)

λB (inverse moment of B-meson LCDA) = 460 ± 110 MeV
Central input controlling the OPE side of all LCSRs; taken from QCD sum rules [45] and used as the default in Ref. [19]. The paper notes alternative estimates around 358 MeV, so this choice is a significant source of uncertainty.
R (ratio of LCDA moments λE²/λH²) = 0.4 (+0.5, -0.3)
Parameter defining the three models for higher-twist B-meson LCDAs from Ref. [24]; obtained by averaging two QCD sum rule determinations that are only marginally compatible.
Effective threshold s0 = 1.26 ± 0.18 GeV² (M² = 1 GeV²); 1.31 ± 0.12 (M² = 1.25); 1.35 ± 0.09 (M² = 1.5)
Fitted in Section 5.3 by matching the SVZ two-point sum rule with the Belle Kπ spectral density for each value of the Borel mass. Substantially lower than the previously used 1.7 GeV², which lowers the B→K* form factor central values.
Borel mass M² = 1.0, 1.25, 1.5 GeV²
Chosen by hand in the standard LCSR window; results are quoted for three values and averaged, and s0 is correlated with M².
f_K* (K* decay constant) = 206 MeV (Model 1), 203 MeV (Model 2)
Extracted from the Belle τ→Kπν spectrum through the f+ normalization and used in the narrow-width LCSRs and in the finite-width factor W_K*. Lower than the SVZ sum rule value 217(5) MeV.
f_K*(1410) (excited resonance decay constant) = 85 MeV
Extracted from the Belle fit (Model 2) for the second resonance; its presence in the sum rules is suppressed by I_{1410}/I_{892} ≈ 0.03.

assumptions (7)

domain assumption Quark-hadron duality: the hadronic spectral function integrated up to s0 equals the OPE dispersion integral after Borel transformation.
Standard LCSR assumption, stated in Section 3.1, Eq. (17). The paper checks that the continuum above s0 is at most 40% of the total.
domain assumption Light-cone OPE of the correlation function in terms of B-meson LCDAs up to twist-4, neglecting NLO alpha_s corrections.
Used throughout Appendices C and D; the paper explicitly says NLO corrections are not included, so lambda_B is not renormalized.
domain assumption The three models for B-meson LCDAs (Model I, IIA, IIB) from Ref. [24], with EOM constraints, describe the two- and three-particle DAs.
The LCDA functional forms in Appendix B.2 are taken from external theory input Ref. [24]; model dependence is estimated from the spread among the models.
domain assumption The Kpi vector form factor f+(s) is a sum of Breit-Wigner resonances with energy-dependent widths, as in Eq. (40), with parameters from the Belle tau → K pi nu fit.
Used to compute the hadronic integral I_R and the resonance model for B→Kpi form factors; only two Belle-compatible models are considered.
ad hoc to paper The phase convention Eq. (47) tan(delta_Kpi - phi_R) = sqrt(s) Gamma_R/(m_R^2 - s) enforces Im[F_i^(1) f+^*] = 0 per resonance.
This is an ansatz that keeps form factors real below threshold and lets the resonance model satisfy unitarity; a coupled-channel analysis is deferred.
domain assumption Isospin symmetry: the b→s current is an isosinglet and the I = 3/2 component of Kpi does not contribute, giving the 3/2 factor in Eq. (21).
Standard QCD symmetry, used to relate K- pi+ and K0 pi0 matrix elements.
ad hoc to paper For the K*(1410) constraints (Section 6.4), non-local hadronic contributions H are approximated by form factors times effective Wilson coefficients at leading order.
The authors state this is a leading-order OPE approximation and that the bounds are ballpark estimates only.

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Pith. "Pith review of Light-Cone Sum Rules for $B\to K\pi$ Form Factors and Applications to Rare Decays." pith.science (2026). https://pith.science/paper/6WR7AUZ7

@misc{pith2026190802267,
  author       = {Pith},
  title        = {Pith review of: Light-Cone Sum Rules for $B\to K\pi$ Form Factors and Applications to Rare Decays},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/6WR7AUZ7}},
  note         = {Machine review of arXiv:1908.02267}
}

abstract

We derive a set of light-cone sum rules relating the hadronic form factors relevant for $B\to K\pi\ell^+\ell^-$ decays to the $B$-meson light-cone distribution amplitudes (LCDAs). We obtain the sum rule relations for all $B\to K\pi$ form factors of (axial)vector and (pseudo)tensor $b\to s$ currents with a $P$-wave $K\pi$ system. Our results reduce to the known light-cone sum rules for $B\to K^*$ form factors in the limit of a single narrow-width resonance. We update the operator-product expansion for the underlying correlation function by including a more complete set of $B$-meson LCDAs with higher twists, and produce numerical results for all $B\to K^*$ form factors in the narrow-width limit. We then use the new sum rules to estimate the effect of a non-vanishing $K^*$ width in $B\to K^*$ transitions, and find that this effect is universal and increases the factorizable part of the rate of $B\to K^*X$ decays by a factor of $20\%$. This effect, by itself, goes in the direction of increasing the current tension in the differential $B\to K^*\mu\mu$ branching fractions. We also discuss $B\to K\pi$ transitions outside the $K^*$ window, and explain how measurements of $B\to K\pi\ell\ell$ observables above the $K^*$ region can be used to further constrain the $B\to K^*$ form factors.

Figures

Figures reproduced from arXiv: 1908.02267 by the authors.

**Figure 1.** Left: τ → KSπ −ντ spectrum from Belle [36], and the corresponding curves from Model 1 and Model 2 (two solid lines) as well as the isolated contribution from the K∗ (892) (dashed). Right: Normalized form factor fe+(s) in the two models that fit well the spectrum. with the omitted parameters set to zero. The spectrum of events given in Eq. (73) does not depend on the normalization of the rate Nτ . In [PITH_FULL_IMAG… view at source ↗

**Figure 2.** Ratio WK∗ quantifying the finite-width correction to the B → K∗ form factors, as a function of the K∗ width ΓK∗ , for M2 = 1 GeV2 . The red vertical band corresponds to the physical width (PDG). This finite-width correction is universal and q 2 -independent. numerical inputs in [PITH_FULL_IMAGE:figures/full_fig_p034_2.png] view at source ↗

**Figure 3.** Form factor F (`=1) ⊥ (s, 0) at q 2 = 0, as a function of the Kπ invariant mass, for three different values of the parameter α, describing the relative size of K∗ (892) and K∗ (1410) contributions. All the curves are consistent with the Light-Cone Sum Rule. The vertical band indicates the region of the measurements in Ref. [68]. there is a competition between both contributions. Higher values of α suppress the B → K… view at source ↗

Figures from the paper (2 more)

**Figure 4.** Figure 4: Left: The moment hMki differentially in mKπ for three values of α = {1, 10, 20}, compared to the LHCb measurement in the bin m2 Kπ ∈ [1.332 , 1.532 ] GeV2 . For easy comparison, the LHCb binned measurement has been divided by the bin size. Right: The integrated moment…

**Figure 5.** Figure 5: Two- and three-particle LCDAs within the three different models considered: Model I (solid), Model IIA (dashed) and Model IIIA (dotted). These plots are obtained fixing λB and R to their central values, and ω2 = 0.5 GeV. with generic Dirac structures Γa,b. Contracting …

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