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The rare decay $B^+ \to K^+\ell^+\ell^-(\nu\bar{\nu})$ under the QCD sum rules approach

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Pith's one-line read This paper predicts the Standard Model rates for $B^+\to K^+\ell^+\ell^-$ and $B^+\to K^+\nu\bar{\nu}$ using $B\to K$ transition form factors from QCD light-cone sum rules with kaon twist-2 and twist-3 light-cone distribution amplitudes.

desk verdict Workmanlike LCSR update with new B→K form factors and branching fractions that agree with lattice, but the error budget omits the dominant kaon-LCDA uncertainty. read the letter →

arxiv 2411.12141 v1 pith:JUFKYYLJ submitted 2024-11-19 hep-ph

Hai-Jiang Tian , Hai-Bing Fu , Tao Zhong , Ya-Xiong Wang , Xing-Gang Wu This is my paper · ORCID
classification hep-ph PACS 13.25.Hw11.55.Hx12.38.Aw14.40.Be
keywords Bmesonraredecaysflavor-changingneutralcurrentsQCDlight-conesumrulestoKtransitionformfactorskaondistributionamplitudesleptonuniversalityB+K+nunubarl+l
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 computes the Standard Model rates for the rare decays $B^+\to K^+\ell^+\ell^-$ ($\ell=e,\mu,\tau$) and $B^+\to K^+\nu\bar{\nu}$ from a QCD light-cone sum-rule calculation of the $B\to K$ transition form factors. It finds $f_+^{BK}(0)=f_0^{BK}(0)=0.328^{+0.032}_{-0.028}$ and $f_T^{BK}(0)=0.277^{+0.028}_{-0.024}$, then extrapolates the form factors over the full $q^2$ range with a simplified $z$-series expansion and feeds them into the effective weak Hamiltonian. The resulting branching fractions are about $6.6\times 10^{-7}$ for the electron and muon channels, about $1.8\times 10^{-7}$ for the tau channel, and about $4.1\times 10^{-6}$ for the neutrino channel. These numbers agree with other Standard Model predictions and provide an independent sum-rule cross-check of the hadronic input that controls the decays.

What carries the argument

The central object is the set of $B\to K$ transition form factors $f_+(q^2)$, $f_0(q^2)$, and $f_T(q^2)$, obtained from a two-point QCD light-cone sum rule for the vacuum-to-kaon correlation function. The kaon side is encoded in light-cone distribution amplitudes (LCDAs) of twist 2, 3, and 4; the paper constructs the leading and twist-3 LCDAs by combining the light-cone harmonic-oscillator model with background-field QCD sum rules and evolving them to a scale near 3 GeV. The $B$-meson ground-state pole is isolated by Borel transformation and continuum subtraction, and the invariant amplitudes $F_{0(1)}$, $\widetilde{F}_{0(1)}$, and $F^T_{0(1)}$ carry the leading and next-to-leading QCD corrections. A simplified $z(q^2)$-series expansion then extrapolates the LCSR results from $0\le q^2\le 10~\mathrm{GeV}^2$ to the full physical region, with fit-quality deviations below one percent.

What would settle it

A precise measurement of the $B^+\to K^+\nu\bar{\nu}$ branching fraction would settle the central claim: if the true value stays near the current evidence-level excess around $2.3\times 10^{-5}$ instead of falling to the predicted $4.135\times 10^{-6}$, the form-factor normalization used here is wrong.

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Extended reading notes

Core claim

The paper establishes that the kaon twist-2 and twist-3 light-cone distribution amplitudes, built from the light-cone harmonic-oscillator model and QCD background-field sum rules, determine the $B\to K$ vector, scalar, and tensor form factors at next-to-leading order in QCD. At $q^2=0$ the vector and scalar form factors coincide at $0.328^{+0.032}_{-0.028}$, while the tensor form factor is $0.277^{+0.028}_{-0.024}$; the scalar form factor inherits its small-$q^2$ value from the vector one through the standard relation, and the simplified $z$-series expansion carries all three to the full physical region. With these form factors and the Standard Model Wilson coefficients, the paper predicts $\mathcal{B}(B^+\to K^+ e^+e^-)=6.633^{+1.341}_{-1.070}\times 10^{-7}$, $\mathcal{B}(B^+\to K^+ \mu^+\mu^-)=6.620^{+1.323}_{-1.056}\times 10^{-7}$, $\mathcal{B}(B^+\to K^+ \tau^+\tau^-)=1.760^{+0.241}_{-0.197}\times 10^{-7}$, and $\mathcal{B}(B^+\to K^+ \nu\bar{\nu})=4.135^{+0.820}_{-0.655}\times 10^{-6}$. The lepton-universality ratio $R_K$ is $0.995^{+0.021}_{-0.020}$ in the low-$q^2$ window and $1.001^{+0.003}_{-0.003}$ in the central window, and the flat term $F_H^\tau$ is approximately $0.89$ while $F_H^\mu$ is approximately $0.021$.

Load-bearing premise

The calculation rests on the assumption that the kaon twist-2 and twist-3 light-cone distribution amplitudes used in the sum rule are the true kaon distributions at the 3 GeV scale, since their normalization and shape are not included in the quoted uncertainties.

Editorial extensions

If this is right

  • The $B^+\to K^+\nu\bar{\nu}$ branching fraction is fixed at $4.135^{+0.820}_{-0.655}\times 10^{-6}$, giving a definite Standard Model target for the ongoing B-factory search.
  • The $B^+\to K^+\tau^+\tau^-$ rate is predicted at $1.760^{+0.241}_{-0.197}\times 10^{-7}$, far below the current experimental upper limit, so an observation of this channel would be a clear new-physics signal.
  • $R_K$ stays near unity in both $q^2$ windows, confirming lepton universality in $B\to K\ell^+\ell^-$ within the Standard Model.
  • The flat term $F_H^\tau \approx 0.89$ makes the tau-channel angular distribution almost flat, a distinctive signature of the heavy-lepton phase space.

Reading between the lines

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

  • If the kaon distribution-amplitude parameters were included in the error budget, the quoted branching-fraction uncertainties would likely grow, because the same LCDA model shifts all three form factors coherently.
  • The predicted ratio $f_T(0)/f_+(0)\approx 0.84$ offers a clean cross-check: a lattice-QCD computation of the tensor form factor at low $q^2$ would either confirm the tensor suppression or expose a problem in the twist-3 input.
  • A high-statistics measurement of the $B^+\to K^+\mu^+\mu^-$ differential distribution over the full $q^2$ range would test the entire LCSR-plus-$z$-series curve, not just the integrated rate.
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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

2 major / 5 minor

Summary. This manuscript computes the B→K vector, scalar, and tensor transition form factors using QCD light-cone sum rules at next-to-leading order, with kaon twist-2 and twist-3 distribution amplitudes modeled by the authors' LHCO/BFTSR approach. The form factors are extrapolated to the full kinematic range with a simplified z-series, and are then used to predict the Standard Model branching fractions of B+→K+ℓ+ℓ− (ℓ=e,μ,τ) and B+→K+νν, together with the lepton-universality ratio RK and the flat term FH. The central outputs are f_+^{BK}(0)=f_0^{BK}(0)=0.328^{+0.032}_{-0.028}, f_T^{BK}(0)=0.277^{+0.028}_{-0.024}, B(B+→K+e+e−)=6.633^{+1.341}_{-1.070}×10^{-7}, B(B+→K+μ+μ−)=6.620^{+1.323}_{-1.056}×10^{-7}, B(B+→K+τ+τ−)=1.760^{+0.241}_{-0.197}×10^{-7}, and B(B+→K+νν)=4.135^{+0.820}_{-0.655}×10^{-6}. These values are compared extensively with earlier LCSR, pQCD, lattice QCD, and experimental results.

Significance. If the calculation is taken at face value, it provides an independent LCSR determination of the B→K form factors and derived rare-decay observables that is broadly consistent with the HPQCD lattice results and with other LCSR determinations. The paper's strengths are its use of a specific kaon-LCDA model, the inclusion of NLO QCD corrections in the LCSR, and the unusually complete comparison tables for the branching fractions, RK, and FH. The central values therefore support the Standard Model and are useful as a cross-check of lattice results. The main weakness is that the uncertainty budget is not complete: the dominant kaon-LCDA input is quoted without propagated uncertainties, so the quantitative error bars and the claimed precision are conditional on an unverified input.

major comments (2)
minor comments (5)

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the branching fractions are derived from LCSR form factors and standard SM formulas; the self-cited kaon LCDA model is an input-dependence caveat, not a constructional reduction.

full rationale

Walking the derivation chain: the TFFs f_+(0), f_0(0), and f_T(0) are obtained from the LCSR dispersion relations, Eqs. (32)-(35), with inputs s0, M^2, fB, mb, and the kaon LCDAs; the branching fractions then follow from the standard SM effective-Hamiltonian expressions, Eqs. (13)-(28), using the z-series extrapolation of Eq. (39). None of the final observables (branching fractions, R_K, F_H) is used as a constraint anywhere in the extraction: Table IV's z-series coefficients are fitted to the LCSR TFF curves, not to experimental data, and the comparisons with HPQCD, LHCb, Belle, etc. are checks, not inputs. The strongest candidate for circularity is the kaon twist-2/twist-3 LCDA model of Eqs. (37)-(38) and Table II, whose parameters are said to have been determined in the same group's previous works and which dominate the TFFs; this is a genuine dependency on self-cited prior analysis, and the omission of LCDA-parameter uncertainties from Table III is a robustness concern. However, the prior determinations are BFTSR/QCD-sum-rule results whose stated assumptions do not include the B->K TFFs or branching fractions, so the present prediction is not equivalent by construction to its input. The equality f_+(0)=f_0(0) is the kinematic identity following from Eq. (35), not a fitted equality. The NLO invariant amplitudes are imported, with a declared cross-check, from the non-overlapping Refs. [89,90], not from a self-citation chain. No step in the paper reduces a claimed prediction to its own fitted input by construction.

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

The paper's central numbers are produced by combining an external effective Hamiltonian, LCSR NLO kernels from earlier references, and the authors' own kaon LCDA model. The only quantities adjusted inside this paper are the Borel window, the continuum threshold, and the z-series fit coefficients. No new particles or new conserved quantities are introduced. The lack of independent LCDA validation and the absence of code mean the calculation is a reuse or extension of a known framework rather than a self-contained derivation.

free parameters (7)
  • Borel mass M2 = 22 +/- 1 GeV^2
    Chosen by LCSR criteria: continuum contribution below 30% and twist-4 contribution below 5%; affects the TFF normalization.
  • Continuum threshold s0^B = 34 +/- 1 GeV^2
    Set near the first B-meson resonance; controls the continuum subtraction in the sum rules.
  • Kaon twist-2 LCDA set (A2;K, alpha2;K, Bhat2;K, beta2;K) = At mu_k=3 GeV: 10.20, 0.003, 0.008, 1.117
    Shape and normalization of the leading-twist kaon distribution amplitude; inherited from prior work and recalculated here without quoted uncertainties.
  • Kaon twist-3 pseudoscalar LCDA set (Ap3;K, Bp3;K, Cp3;K, betap3;K) = At mu_k=3 GeV: 138.4, 0.037, 1.521, 0.538
    Twist-3 pseudoscalar component; enters the sum rules and has no uncertainty propagated in Table III.
  • Kaon twist-3 tensor LCDA set (Asigma3;K, Bsigma3;K, Csigma3;K, betasigma3;K) = At mu_k=3 GeV: 172.5, 0.027, 0.181, 0.455
    Twist-3 tensor component; affects the tensor form factor and is taken from prior model work.
  • Constituent quark masses mhat_q and mhat_s in the LHCO wave functions = Not given in this paper
    Appear in Eqs. (37) and (38); the paper refers to Refs. [59,60] for the values, so they are hidden inputs.
  • z-series coefficients beta_k for f_+, f_0, f_T = Table IV, e.g. beta+ = 0.325, -0.966, -1.533
    Fit to the LCSR points up to q^2=10 GeV^2 to extrapolate the TFFs to the full physical region; the quality Delta < 1% is the only justification presented.
assumptions (6)
  • domain assumption The B to K LCSR factorization holds and the correlation function is dominated by light-cone distances, expanding in kaon LCDAs.
    Invoked in Section II around Eq. (29); this is the standard LCSR framework.
  • domain assumption The NLO invariant amplitudes from Refs. [89,90] are valid for B to K after suitable substitutions, as asserted in Section II after Eq. (35).
    The paper states that a cross-check was performed but does not show the formulas or the transformation.
  • domain assumption The LHCO/BFTSR kaon LCDA parameters of Refs. [59,60] are accurate and can be evolved to mu_k=3 GeV without additional uncertainty.
    Section III and Table II inherit these parameters without independent validation or propagated errors.
  • domain assumption Long-distance nonlocal effects beyond the TFFs, such as charm loops and weak annihilation, are either negligible or captured by the adopted effective coefficients C7eff and C9eff.
    The paper mentions these effects in the introduction but does not compute them, relying on Refs. [30,31,77-83].
  • domain assumption The three-coefficient simplified z-series expansion reliably describes the TFFs over the full q^2 range.
    Section III uses Eq. (39) and validates it only by the fit-quality measure Delta < 1%.
  • domain assumption The Standard Model with no new physics is the correct description; Wilson coefficients from Refs. [75,87] are used.
    The calculation is explicitly a Standard Model prediction; beyond-SM interpretations are not considered.

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Pith. "Pith review of The rare decay $B^+ \to K^+\ell^+\ell^-(\nu\bar{\nu})$ under the QCD sum rules approach." pith.science (2026). https://pith.science/paper/JUFKYYLJ

@misc{pith2026241112141,
  author       = {Pith},
  title        = {Pith review of: The rare decay $B^+ \to K^+\ell^+\ell^-(\nu\bar\nu)$ under the QCD sum rules approach},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/JUFKYYLJ}},
  note         = {Machine review of arXiv:2411.12141}
}
abstract

In the paper, we conduct a detailed investigation of the rare decay processes of charged meson, specifically $B^+ \to K^+\ell^+\ell^-$ with $\ell=(e,\mu,\tau)$ and $B^+ \to K^+\nu\bar{\nu}$. These processes involve flavor-changing-neutral-current (FCNC) transitions, namely $b\to s\ell^+\ell^-$ and $b\to s\nu\bar{\nu}$. The essential components $B\to K$ scalar, vector and tensor transition form factors (TFFs) are calculated by using the QCD light-cone sum rules approach up to next-to-leading order QCD corrections. In which, the kaon twist-2 and twist-3 light-cone distribution amplitudes are calculated from both the QCD sum rules within the framework of background field theory and the light-cone harmonic oscillator model. The TFFs at large recoil point are $f_+^{BK}(0)=f_0^{BK}(0) =0.328_{-0.028}^{+0.032}$ and $f_{\rm T}^{BK}(0)=0.277_{-0.024}^{+0.028}$, respectively. To achieve the behavior of those TFFs in the whole $q^2$-region, we extrapolate them by utilizing the simplified $z(q^2)$-series expansion. Furthermore, we compute the differential branching fractions with respect to the squared dilepton invariant mass for the two different decay channels and present the corresponding curves. Our predictions of total branching fraction are ${\cal B}(B^+\to K^+ e^+ e^-)=6.633_{-1.070}^{+1.341}\times 10^{-7}$, ${\cal B}(B^+\to K^+ \mu^+ \mu^-)=6.620_{-1.056}^{+1.323}\times 10^{-7}$, ${\cal B}(B^+\to K^+ \tau^+ \tau^-)=1.760_{-0.197}^{+0.241}\times 10^{-7}$, and ${\cal B}(B^+\to K^+ \nu\bar{\nu})=4.135_{-0.655}^{+0.820}\times 10^{-6}$, respectively. Lastly, the observables such as the lepton universality $\mathcal{R}_{K}$ and the angular distribution `flat term' $F_{\rm H}^\ell$ are given, which show good agreement with the theoretical and experimental predictions.

Figures

Figures reproduced from arXiv: 2411.12141 by the authors.

Figure 1
Figure 1. FIG. 1. Plotting of [PITH_FULL_IMAGE:figures/full_fig_p013_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Branching fraction of [PITH_FULL_IMAGE:figures/full_fig_p016_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Branching fractions of [PITH_FULL_IMAGE:figures/full_fig_p017_3.png] view at source ↗
Figures from the paper (4 more)
Figure 4
Figure 4. Figure 4: FIG. 4. Differential branching fraction for [PITH_FULL_IMAGE:figures/full_fig_p017_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5. Differential branching fraction of [PITH_FULL_IMAGE:figures/full_fig_p018_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6. Differential branching fraction for [PITH_FULL_IMAGE:figures/full_fig_p018_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7. Flat terms [PITH_FULL_IMAGE:figures/full_fig_p020_7.png]

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