pith. sign in

arxiv: 2605.22727 · v1 · pith:GBRZ5YLNnew · submitted 2026-05-21 · ✦ hep-ph · hep-lat

Rare kaon decays K^- to ell^- bar{ν}_ell ell'⁺ ell'⁻: Standard Model predictions from lattice QCD

R. Di Palma , R. Frezzotti , G. Gagliardi , V. Lubicz , G. Martinelli , C.T. Sachrajda , F. Sanfilippo , S. Simula
show 1 more author
N. Tantalo
This is my paper

Pith reviewed 2026-05-22 04:37 UTC · model grok-4.3

classification ✦ hep-ph hep-lat
keywords rare kaon decayslattice QCDStandard Model predictionsbranching fractionsV_us extractionstructure-dependent form factorslepton pair
0
0 comments X

The pith

Lattice QCD now delivers Standard Model branching fractions for rare kaon decays with controlled 2-7% uncertainties.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper computes Standard Model predictions for the four channels of the rare decay K minus to lepton antineutrino lepton plus lepton minus by folding the first complete lattice QCD results for the structure-dependent form factors into the decay amplitude. These decays occur at order G_F squared times alpha_em squared and are sensitive to the Cabibbo angle V_us as well as potential new physics. Using the PDG value of V_us the authors obtain branching fractions whose uncertainties range from 2 to 7 percent and that agree with all existing measurements, including a 1.4 sigma match to the preliminary NA62 result for the mu mu channel. The same predictions can be inverted to extract V_us directly from the measured rates, producing a 1.8 percent determination from the two best-measured modes.

Core claim

Incorporating the lattice QCD structure-dependent form factors into the Standard Model matrix element yields branching-ratio predictions for all four K_ell2ellprime channels that carry total uncertainties between 2 and 7 percent; these predictions reproduce the three published experimental rates and agree at 1.4 sigma with the preliminary NA62 result for K_mu2mu, while the measured rates combined with the lattice inputs give an independent extraction of |V_us| equal to 0.2283 with a 42 error in the last digits.

What carries the argument

The structure-dependent form factors of the hadronic matrix element for the four-body decay, computed non-perturbatively on the lattice in a companion paper.

If this is right

  • The three channels with published data agree with the lattice-based predictions within uncertainties.
  • The K_mu2mu prediction matches the preliminary NA62 result at the 1.4 sigma level.
  • A weighted average of the two most precise channels supplies |V_us| = 0.2283(42), a 1.8 percent determination.
  • These decays can now be used as independent precision probes of the Standard Model flavour structure.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Further reduction of lattice uncertainties would make these modes competitive with the leading |V_us| extractions from semileptonic and leptonic kaon decays.
  • Persistent tension between the lattice prediction and a future high-precision measurement in any channel could indicate contributions from physics beyond the Standard Model.
  • The same lattice framework could be applied to analogous rare decays of charged pions or charmed mesons to test flavour universality.

Load-bearing premise

The lattice calculation of the structure-dependent form factors fully captures the relevant non-perturbative QCD effects inside the quoted systematic uncertainties.

What would settle it

A future experimental branching fraction for any of the four channels that lies many standard deviations outside the predicted interval while the lattice uncertainties remain unchanged would falsify the central claim.

Figures

Figures reproduced from arXiv: 2605.22727 by C.T. Sachrajda, F. Sanfilippo, G. Gagliardi, G. Martinelli, N. Tantalo, R. Di Palma, R. Frezzotti, S. Simula, V. Lubicz.

Figure 1
Figure 1. Figure 1: FIG. 1. Leading-order diagrams contributing to [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Lattice data for the real (open squares) and imag [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Comparison of the SM predictions for the [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Differential branching fractions for [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Given the preliminary nature of the experimental result for the purely muonic channel, we represent the corresponding data point in gray. The black data point, corresponding to |Vus| = 0.2283(42), is the weighted av￾erage of the |Vus| values from Kµ2e and Kµ2µ, taking into account their correlation. CONCLUSIONS In this Letter, together with the companion paper [1], we have presented the first complete SM p… view at source ↗
read the original abstract

Weak decays of charged kaons with an additional lepton-antilepton pair, $K^- \to \ell^- \bar{\nu}_\ell \ell'^{+} \ell'^{-}$ ($K_{\ell2\ell'}$), are suppressed at order $O(G_{F}^{2}\alpha_{\rm em}^{2})$ in the Standard Model (SM) and provide sensitive probes of its flavour structure, as well as independent determinations of the Cabibbo angle $|V_{us}|$. In this Letter we present the SM predictions for all four channels with $\ell,\ell' =e,\mu$, based on the first complete lattice QCD calculation of the structure-dependent form factors reported in a companion paper [1]. Using the PDG value [2] $|V_{us}|^{\rm PDG}=0.22431(85)$, we obtain branching fractions with controlled uncertainties and precisions ranging from $2\%$ to $7\%$, depending on the channel. For the three modes with published measurements, our results agree with experiment. For the $K_{\mu2\mu}$ mode, for which no published experimental result is available, we compare our prediction with the preliminary NA62 result, finding agreement at the $1.4\sigma$ level. Conversely, the measured decay rates can be used together with our results to extract $|V_{us}|$ from these modes. A weighted average over the two most precise channels, $K_{\mu2e}$ and $K_{\mu2\mu}$, yields $|V_{us}|=0.2283(42)$, corresponding to a $1.8\%$ determination. These results pave the way for using $K_{\ell2\ell'}$ decays as precision probes of the SM.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 1 minor

Summary. The manuscript presents Standard Model predictions for the branching fractions of the rare kaon decays K^- → ℓ^- ν̄_ℓ ℓ'^+ ℓ'^- in all four channels (ℓ, ℓ' = e, μ). These are obtained from the first complete lattice QCD calculation of the structure-dependent form factors reported in a companion paper, combined with the PDG value |V_us|^PDG = 0.22431(85). The resulting branching fractions have quoted precisions of 2–7% depending on the channel. The predictions agree with the three published experimental measurements and with the preliminary NA62 result for K_μ2μ at 1.4σ. The measured rates are also used to extract |V_us| = 0.2283(42) from the two most precise channels.

Significance. If the lattice form-factor results hold, the work supplies the first SM predictions for these O(G_F² α_em²) decays with controlled uncertainties from first-principles QCD. This enables their use as precision flavor probes and yields an independent 1.8% determination of |V_us|. The explicit use of lattice QCD for the structure-dependent contributions, rather than model-dependent estimates, is a clear methodological strength that supports the quoted precisions.

major comments (1)
  1. The branching-fraction predictions (2–7% precision) and the |V_us| extraction rest entirely on the structure-dependent form factors from the companion paper [1]. This manuscript provides no summary of the lattice ensembles, chiral extrapolation, discretization effects, or full systematic error budget for those form factors, so the claim of 'controlled uncertainties' cannot be verified independently within the present Letter.
minor comments (1)
  1. The abstract and main text should include the arXiv identifier or publication status of the companion paper [1] to allow readers to locate the underlying lattice details without ambiguity.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their supportive assessment of the manuscript and for the constructive major comment. We address it point by point below.

read point-by-point responses

  1. Referee: The branching-fraction predictions (2–7% precision) and the |V_us| extraction rest entirely on the structure-dependent form factors from the companion paper [1]. This manuscript provides no summary of the lattice ensembles, chiral extrapolation, discretization effects, or full systematic error budget for those form factors, so the claim of 'controlled uncertainties' cannot be verified independently within the present Letter.

    Authors: We agree that a concise overview of the lattice methodology would strengthen the Letter's self-contained character. While the full technical details, including the specific ensembles, chiral and continuum extrapolations, discretization effects, and complete systematic error budget, are necessarily contained in the companion paper [1] (as is standard for a short Letter format), we will add a brief paragraph in the revised manuscript. This paragraph will summarize the key lattice setup, the main sources of uncertainty, and how they combine to yield the quoted 2–7% precisions on the branching fractions, thereby allowing readers to assess the 'controlled uncertainties' claim without immediately consulting the companion work. revision: yes

Circularity Check

0 steps flagged

No significant circularity; predictions derive from independent lattice QCD inputs

full rationale

The paper's derivation chain starts from structure-dependent form factors computed via lattice QCD in the companion reference [1], combined with the external PDG value of |V_us| to obtain branching fractions through standard matrix-element formulas. These form factors are obtained from first-principles simulations on gauge ensembles with stated chiral, continuum, and finite-volume extrapolations; they are not defined in terms of the branching fractions or fitted to the decay rates under discussion. The resulting predictions are compared directly to experimental data and used to extract |V_us|, providing an independent cross-check rather than a tautology. No equation or step reduces the output to the input by construction, and the lattice computation is externally falsifiable against other observables.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The central predictions rest on the lattice QCD form-factor calculation (companion paper) together with the external PDG |V_us| input. No new particles or forces are introduced. Lattice-specific parameters such as spacing, quark masses, and renormalization factors are inherited from the underlying simulation rather than fitted to the decay data itself.

free parameters (1)
  • lattice spacing and quark-mass tuning parameters
    Standard lattice QCD inputs that control discretization and chiral extrapolation errors; values are set by prior simulations and not refitted to the rare-decay observables.
axioms (1)
  • domain assumption Lattice QCD with the chosen action and ensembles reproduces the non-perturbative QCD dynamics relevant to the structure-dependent form factors
    Invoked when the form factors computed on the lattice are taken as the physical values after extrapolation.

pith-pipeline@v0.9.0 · 5909 in / 1347 out tokens · 40884 ms · 2026-05-22T04:37:50.961028+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

29 extracted references · 29 canonical work pages · 6 internal anchors

  1. [1]

    The LFU ratios benefit from partial cancellation of correlated un- certainties in the SD form factors between numerator and denominator

    ,(13) where, as before, fore +e− final-states the branching frac- tions are evaluated with the cutx (′) k ≥0.284. The LFU ratios benefit from partial cancellation of correlated un- certainties in the SD form factors between numerator and denominator. Our results for the six LFU ratios are re- ported in Table I together with the experimental values. Finall...

    work page 2022

  2. [2]

    Di Palma, R

    R. Di Palma, R. Frezzotti, G. Gagliardi, V. Lubicz, G. Martinelli, C. T. Sachrajda, F. Sanfilippo, S. Sim- ula, and N. Tantalo, Complete lattice QCD calculation ofK − →ℓ −¯νℓℓ ′+ℓ ′− form factors (2026)

    work page 2026

  3. [3]

    Navas et al

    S. Navas et al. (Particle Data Group), Review of particle physics, Phys. Rev. D110, 030001 (2024)

    work page 2024

  4. [4]

    S. L. Glashow, J. Iliopoulos, and L. Maiani, Weak Inter- actions with Lepton-Hadron Symmetry, Phys. Rev. D2, 1285 (1970)

    work page 1970

  5. [5]

    J. H. Christenson, J. W. Cronin, V. L. Fitch, and R. Turlay, Evidence for the 2πdecay of thek 0 2 meson, Phys. Rev. Lett.13, 138 (1964)

    work page 1964

  6. [6]

    Aebischer et al., Kaon physics: a cornerstone for future discoveries, J

    J. Aebischer et al., Kaon physics: a cornerstone for future discoveries, J. Phys. G52, 100501 (2025), arXiv:2503.22256 [hep-ph]

    work page arXiv 2025

  7. [7]

    Cabibbo, Unitary Symmetry and Leptonic Decays, Phys

    N. Cabibbo, Unitary Symmetry and Leptonic Decays, Phys. Rev. Lett.10, 531 (1963)

    work page 1963

  8. [8]

    FLAG Review 2024

    Y. Aoki et al. (Flavour Lattice Averaging Group (FLAG)), FLAG review 2024, Phys. Rev. D113, 014508 (2026), arXiv:2411.04268 [hep-lat]. 7

    work page internal anchor Pith review Pith/arXiv arXiv 2024

  9. [9]

    First lattice calculation of the QED corrections to leptonic decay rates

    D. Giusti, V. Lubicz, G. Martinelli, C. T. Sachrajda, F. Sanfilippo, S. Simula, N. Tantalo, and C. Tarantino, First lattice calculation of the QED corrections to lep- tonic decay rates, Phys. Rev. Lett.120, 072001 (2018), arXiv:1711.06537 [hep-lat]

    work page internal anchor Pith review Pith/arXiv arXiv 2018

  10. [10]

    J. C. Hardy and I. S. Towner, Superallowed 0 + →0 + nuclearβdecays: 2020 critical survey, with implications for V ud and CKM unitarity, Phys. Rev. C102, 045501 (2020)

    work page 2020

  11. [11]

    Alexandrou et al

    C. Alexandrou et al. (Extended Twisted Mass), Inclusive Hadronic Decay Rate of theτLepton from Lattice QCD: The u¯s Flavor Channel and the Cabibbo Angle, Phys. Rev. Lett.132, 261901 (2024), arXiv:2403.05404 [hep- lat]

    work page arXiv 2024

  12. [12]

    Desiderio et al., First lattice calculation of radiative leptonic decay rates of pseudoscalar mesons, Phys

    A. Desiderio et al., First lattice calculation of radiative leptonic decay rates of pseudoscalar mesons, Phys. Rev. D103, 014502 (2021), arXiv:2006.05358 [hep-lat]

    work page arXiv 2021

  13. [13]

    Di Palma, R

    R. Di Palma, R. Frezzotti, G. Gagliardi, V. Lubicz, G. Martinelli, C. T. Sachrajda, F. Sanfilippo, S. Sim- ula, and N. Tantalo, Kaon radiative leptonic decay rates from lattice QCD simulations at the physical point, Phys. Rev. D111, 114523 (2025), arXiv:2504.08680 [hep-lat]

    work page arXiv 2025

  14. [14]

    N. H. Christ, X. Feng, T. Izubuchi, L. Jin, C. T. Sachra- jda, and X.-Y. Tuo, Lattice Calculation of Light Me- son Radiative Leptonic Decays (2025), arXiv:2510.26993 [hep-lat]

    work page internal anchor Pith review Pith/arXiv arXiv 2025

  15. [15]

    Goudzovski et al., New physics searches at kaon and hyperon factories, Rept

    E. Goudzovski et al., New physics searches at kaon and hyperon factories, Rept. Prog. Phys.86, 016201 (2023), arXiv:2201.07805 [hep-ph]

    work page arXiv 2023

  16. [16]

    Krnjaic, G

    G. Krnjaic, G. Marques-Tavares, D. Redigolo, and K. To- bioka, Probing Muonphilic Force Carriers and Dark Mat- ter at Kaon Factories, Phys. Rev. Lett.124, 041802 (2020), arXiv:1902.07715 [hep-ph]

    work page arXiv 2020

  17. [17]

    A. A. Poblaguev et al., Experimental study of the ra- diative decays K+ —>mu+ nu e+ e- and K+ —>e+ nu e+ e-, Phys. Rev. Lett.89, 061803 (2002), arXiv:hep- ex/0204006

    work page arXiv 2002

  18. [18]

    First observation of the decay K+ -> e+ nu mu+ mu-

    H. Ma et al., First observation of the decayK + → e+nu(e)mu+mu−, Phys. Rev. D73, 037101 (2006), arXiv:hep-ex/0505011

    work page internal anchor Pith review Pith/arXiv arXiv 2006

  19. [19]

    Fiorenza et al

    R. Fiorenza et al. (NA62), Status and future prospects for rare kaon decay measurements at NA62, PoSHQL2025, 065 (2026)

    work page 2026

  20. [20]

    Radiative Semileptonic Kaon Decays

    J. Bijnens, G. Ecker, and J. Gasser, Radiative semilep- tonic kaon decays, Nucl. Phys. B396, 81 (1993), arXiv:hep-ph/9209261

    work page internal anchor Pith review Pith/arXiv arXiv 1993

  21. [21]

    X.-Y. Tuo, X. Feng, L.-C. Jin, and T. Wang, Lattice QCD calculation of K→ℓνℓℓ’+ℓ’- decay width, Phys. Rev. D 105, 054518 (2022), arXiv:2103.11331 [hep-lat]

    work page arXiv 2022

  22. [22]

    Gagliardi, F

    G. Gagliardi, F. Sanfilippo, S. Simula, V. Lubicz, F. Mazzetti, G. Martinelli, C. T. Sachrajda, and N. Tan- talo, Virtual photon emission in leptonic decays of charged pseudoscalar mesons, Phys. Rev. D105, 114507 (2022), arXiv:2202.03833 [hep-lat]

    work page arXiv 2022

  23. [23]

    Frezzotti, N

    R. Frezzotti, N. Tantalo, G. Gagliardi, F. Sanfil- ippo, S. Simula, and V. Lubicz, Spectral-function de- termination of complex electroweak amplitudes with lattice QCD, Phys. Rev. D108, 074510 (2023), arXiv:2306.07228 [hep-lat]

    work page arXiv 2023

  24. [24]

    On the extraction of spectral densities from lattice correlators

    M. Hansen, A. Lupo, and N. Tantalo, Extraction of spec- tral densities from lattice correlators, Phys. Rev. D99, 094508 (2019), arXiv:1903.06476 [hep-lat]

    work page internal anchor Pith review Pith/arXiv arXiv 2019

  25. [25]

    Alexandrou et al

    C. Alexandrou et al. (Extended Twisted Mass), Ratio of kaon and pion leptonic decay constants with Nf=2+1+1 Wilson-clover twisted-mass fermions, Phys. Rev. D104, 074520 (2021), arXiv:2104.06747 [hep-lat]

    work page arXiv 2021

  26. [26]

    G. P. Lepage, A New Algorithm for Adaptive Multidi- mensional Integration, J. Comput. Phys.27, 192 (1978)

    work page 1978

  27. [27]

    M. S. Atiya, I.-H. Chiang, J. S. Frank, J. S. Haggerty, M. M. Ito, T. F. Kycia, K. K. Li, L. S. Littenberg, A. Stevens, R. C. Strand, W. C. Louis, D. S. Akerib, M. Hildreth, D. R. Marlow, P. D. Meyers, M. A. Se- len, F. C. Shoemaker, A. J. S. Smith, G. Azuelos, E. W. Blackmore, D. A. Bryman, L. Felawka, P. Kitching, Y. Kuno, J. A. Macdonald, T. Numao, P. ...

    work page 1989

  28. [28]

    Y. S. Amhis et al. (HFLAV), Averages of b-hadron, c- hadron, andτ-lepton properties as of 2021, Phys. Rev. D 107, 052008 (2023), arXiv:2206.07501 [hep-ex]

    work page arXiv 2021

  29. [29]

    Bacchio and A

    S. Bacchio and A. Konstantinou, Study of the Λ→pℓν¯ℓ Semileptonic Decay in Lattice QCD, Phys. Rev. Lett. 135, 231901 (2025), arXiv:2507.09970 [hep-lat]. SUPPLEMENTAL MATERIAL Details of the decay amplitude In this section we provide explicit expressions for the quantities entering the final-state-radiation contribution, MFSR(Pf), and the hadronic contribu...

    work page arXiv 2025