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Natural complex plane for kaon CKM data: framework, status and future

T0 review · 2 major / 5 minor · reviewed 2026-08-16 · deepseek-v4-flash

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Pith's one-line read The paper claims that kaon data alone can fix three of the four CKM parameters in the natural (ds) unitarity-triangle plane, where current measurements leave four islands and upcoming rare-decay data will either single out the…

desk verdict Useful kaon CKM plane, but the 'three of four parameters' claim overstates what the observables actually determine. read the letter →

arxiv 2504.12386 v1 pith:RIKRVQI3 submitted 2025-04-16 hep-ph hep-ex

Avital Dery This is my paper · ORCID
classification hep-phhep-ex
keywords kaonphysicsCKMmatrixunitaritytriangledsraredecaysCPviolationK+topi+nunubarepsilon_K
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

The paper argues that kaon-physics constraints on the CKM matrix should be plotted in a plane spanned by $A^2\lambda^4(1-\hat\rho)$ and $A^2\lambda^4\hat\eta$, the real and imaginary parts of the $(ds)$ unitarity-triangle combination $-R_{ct}e^{-i\theta_{ct}}$. In this plane, every kaon observable considered depends only on these two combinations plus known constants, so results are independent of B-physics inputs and of the poorly known $|V_{cb}|$. Combining the current $B(K^+\to\pi^+\nu\bar\nu)$ and $|\varepsilon_K|$ measurements yields four disjoint allowed regions; adding $B(K_L\to\mu^+\mu^-)$ already disfavors one. If upcoming measurements of the imaginary-part combination agree with the Standard Model, all but the 'SM island' would be excluded, giving a clean cross-sector test of the CKM paradigm; if not, the kaon sector would show new physics.

What carries the argument

The load-bearing object is the normalized $(ds)$ unitarity triangle, characterized by $-R_{ct}e^{-i\theta_{ct}}=A^2\lambda^4[(1-\hat\rho)+i\hat\eta]$. The machinery is the explicit mapping of each kaon observable onto this complex number: $B(K^+\to\pi^+\nu\bar\nu)$ becomes a circle in the plane, $|\varepsilon_K|$ becomes hyperbolic bands, $B(K_L\to\mu^+\mu^-)$ constrains mainly the real part $A^2\lambda^4(1-\hat\rho)$ up to a fourfold discrete ambiguity, and $B(K_S\to\mu^+\mu^-)$ with $A_{CP}(K^0\to\mu^+\mu^-)$ measures the imaginary part $A^2\lambda^4\hat\eta$. Presenting all kaon constraints in this common plane removes the artificial inflation of errors from the parametric uncertainty on $A$ (hence on $|V_{cb}|$) and avoids B-physics input entirely.

What would settle it

A hadronic calculation showing that any kaon observable has a leading CKM dependence that is not proportional to $A^2\lambda^4(1-\hat\rho)$ and $A^2\lambda^4\hat\eta$, for instance a separate dependence on $|V_{ts}|$ or $|V_{cb}|$, would refute the claim that the $(ds)$-triangle plane exhausts the kaon CKM information; likewise, a high-precision $B(K_L\to\pi^0\nu\bar\nu)$ measurement that, combined with the other kaon constraints, excludes all four islands at more than 95% C.L. would show the single-point-in-the-plane picture either fails or requires new physics beyond the Standard Model.

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Extended reading notes

Core claim

The central discovery is that the natural complex plane for kaon CKM information is not the B-physics $(\bar\rho,\bar\eta)$ plane but the $(ds)$-triangle plane, with coordinates $x=A^2\lambda^4(1-\hat\rho)$ and $y=A^2\lambda^4\hat\eta$; these are the real and imaginary parts of the combination $-R_{ct}e^{-i\theta_{ct}}$ defined in Eq. (5). In this plane, $B(K^+\to\pi^+\nu\bar\nu)$, $|\varepsilon_K|$, $B(K_L\to\mu^+\mu^-)$, $B(K_S\to\mu^+\mu^-)$, and the future $B(K_L\to\pi^0\nu\bar\nu)$ and $A_{CP}(K^0\to\mu^+\mu^-)$ all depend on the same two parameter combinations, so plotting them together needs no external value of $A$ or $|V_{cb}|$. The current conjunction of $B(K^+\to\pi^+\nu\bar\nu)$ and $|\varepsilon_K|$ shows four almost disconnected allowed islands, approximately symmetric about $\hat\eta=0$; adding $B(K_L\to\mu^+\mu^-)$ rules out or disfavors the left island depending on the sign of the long-distance amplitude, and future $y$-axis measurements will discriminate the top and bottom islands. The paper concludes that kaon physics alone can determine three of the four CKM parameters (all but $|V_{cb}|$) and that incoming data will either leave only the SM island or expose tension with the Standard Model.

Load-bearing premise

The argument assumes that every measured kaon property that depends on the CKM matrix can be written using only the two combinations $A^2\lambda^4(1-\hat\rho)$ and $A^2\lambda^4\hat\eta$, together with known constants and $\lambda$; the paper checks this for four processes, but does not prove it for all kaon observables or for higher-order corrections.

Editorial extensions

If this is right

  • With $\lambda$ (i.e. $|V_{us}|$) taken from kaon physics, the pair $(A^2(1-\hat\rho), A^2\hat\eta)$ can be determined from kaon observables alone, fixing three of the four CKM parameters without any B-physics input.
  • The current combination of $B(K^+\to\pi^+\nu\bar\nu)$ and $|\varepsilon_K|$ produces four disjoint allowed islands; adding the present $B(K_L\to\mu^+\mu^-)$ constraint already disfavors or excludes the left island at 95% C.L.
  • A future SM-consistent measurement of $B(K_L\to\pi^0\nu\bar\nu)$ or of $A_{CP}(K^0\to\mu^+\mu^-)$ would disfavor the top and bottom islands, leaving only the SM island once theory errors on $B(K_L\to\mu^+\mu^-)$ are reduced.
  • If the future $y$-axis measurements instead deviate from the SM point, the plane would reveal new physics in the kaon sector that is not visible in B physics.
  • Because all plotted constraints share the same two parameter combinations, the confidence regions are not inflated by the $|V_{cb}|$ uncertainty, unlike the traditional $(\bar\rho,\bar\eta)$ presentation.

Reading between the lines

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

  • The same reduction to $(A^2\lambda^4(1-\hat\rho), A^2\lambda^4\hat\eta)$ should hold for any future kaon observable whose short-distance amplitude is controlled by $V_{ts}^*V_{td}$; observables with a different CKM weight would fall outside the plane and could be used to test the framework's completeness.
  • Resolving the sign of the long-distance amplitude in $B(K_L\to\mu^+\mu^-)$, via lattice QCD or dispersion theory, would sharpen the current disfavouring of the left island into a high-confidence exclusion, even before the new y-axis experiments report.
  • If the SM is correct, the framework implies a concrete pattern: the future $B(K_L\to\pi^0\nu\bar\nu)$ and $A_{CP}(K^0\to\mu^+\mu^-)$ measurements must land on the SM island, and their combination with improved $B(K_L\to\mu^+\mu^-)$ theory should shrink the allowed region to one connected domain around the SM point.
  • The same coordinates provide a natural basis for beyond-Standard-Model fits in the kaon sector: new physics that shifts only the real part is cleanly distinguishable, once the islands are resolved, from new physics that shifts the imaginary part.
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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. The paper proposes that the natural complex plane for presenting CKM information from kaon physics is spanned by the combinations (A^2(1−ρ̂), A^2η̂), equivalently the combination −R_ct e^{−iθ_ct} of the (ds) unitarity triangle. Using published inputs for B(K+→π+νν̄) and |εK|, the paper shows that the 95% confidence-level allowed regions form four nearly disconnected islands (the “left”, “top”, “bottom”, and “SM” islands). Adding B(KL→μ+μ−) is shown to disfavor the left island under one sign choice of Re A_LD. The paper argues that future measurements of B(KL→π0νν̄) or A_CP(K→μ+μ−) will narrow the allowed regions and provide a cross-sector test of the CKM paradigm. The abstract and the Discussion claim that kaon physics can determine three of the four CKM parameters independently of B physics.

Significance. The proposed plane is a genuinely useful way to display kaon constraints without the parametric error inflation induced by |Vcb|, and the appendix provides explicit expressions showing that the observables considered depend only on the two coordinates of this plane. The four-island structure and the discussion of which future measurements will discriminate among the islands are valuable and follow from published inputs. The paper is also clear about the sign ambiguities in the KL→μ+μ− interpretation. However, the central counting claim is not supported by the paper's own equations: the kaon observables depend only on two independent combinations of the Wolfenstein parameters, so the statement that kaon physics determines three of the four CKM parameters is an overclaim. The framework itself remains sound once the parameter counting is stated correctly, for example by saying that kaon physics determines two independent combinations (or, if λ is fitted, three constraints including |Vus|).

major comments (2)
  1. [Abstract and Discussion] The statement that kaon physics “can be used to independently determine three out of the four parameters of the CKM matrix” is not supported by the equations in the Appendix. Treating λ as known (as the paper does), every observable in Eqs. (11)–(19) depends only on X = A^2λ^4(1−ρ̂) and Y = A^2λ^4η̂. The map (A, ρ̂, η̂) → (X, Y) has rank 2: for any fixed (X, Y), the one-parameter family ρ̂ = 1 − X/(A^2λ^4), η̂ = Y/(A^2λ^4) with arbitrary A leaves all kaon constraints invariant. Thus kaon observables fix two independent combinations, not three Wolfenstein parameters. The paper’s own Introduction states that “the parameter A ... cannot be determined by kaon physics,” which concedes this degeneracy. Please revise the abstract and the Discussion paragraph beginning “Of the four parameters of the CKM matrix...” to state that kaon physics determines two independent CKM combinations (or, if λ is fitted, three constraints including |Vus|), and that extracting ρ̂ and η̂ separately requires external knowledge of A (e.g., from |Vcb| or a future |Vts| determination).
  2. [Introduction, Eq. (5)] The definition of the plane in Eq. (5) reinforces the same point: the coordinates are A^2λ^4(1−ρ̂) and A^2λ^4η̂, so a measurement of the complex quantity −R_ct e^{−iθ_ct} yields the magnitude and phase of this combination. These are two real numbers. The phase θ_ct is independent of A, but the magnitude R_ct carries an overall A^2 factor, so A cannot be separated from ρ̂ and η̂ without additional input. The text should state explicitly that the plane parameterizes the (ds) unitarity triangle, which fixes two real degrees of freedom, and that this is the maximal information obtainable from the considered kaon observables alone.
minor comments (5)
  1. [Fig. 2] The text says the left and right panels correspond to Re A_LD > 0 and Re A_LD < 0, respectively, but the labels in the figure appear to be reversed; please check that the labels and the text are consistent.
  2. [Eq. (16)] The term “ImA2_LD” in Eq. (16) is ambiguous; it should be written as (Im A_LD)^2 or defined explicitly.
  3. [References] Reference [8] is listed as “To appear” without a preprint number; please update it if a preprint is available.
  4. [Fig. 1] The four islands are named in the caption but not labeled directly in the figure; adding labels would improve readability.
  5. [Discussion] The sentence “We demonstrate, that a different set of three parameters can in principle be determined by kaon physics (excluding |Vcb|)” repeats the counting issue of Major Comment 1 and should be reworded to reflect the two-combination nature of the kaon constraints.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the kaon plane is a coordinate reparametrization of externally measured constraints, and no prediction is fitted from the plotted data.

full rationale

The paper's central step is the definition of a coordinate plane in Eq. (5), spanning the two combinations A^2 lambda^4(1-rho-hat) and A^2 lambda^4 eta-hat that appear in the standard kaon observables. The Appendix reproduces known SM expressions from external sources, and the allowed regions are obtained by overlaying external experimental and theory inputs (NA62, PDG, Hoferichter et al., Brod et al.) rather than by fitting any parameter from the plotted data. The SM reference point is taken from the PDG global fit, which is an external benchmark, not an output of this paper. Self-citations to the author's prior work appear for the A_CP(K->mu+mu-) sign discussion and BSM operator sensitivity, but these do not enter the construction of the plane or the current constraints, so they are not load-bearing. No fitted parameter is renamed as a prediction, and no uniqueness theorem is imported from the author's own work. The claim that kaon physics can determine three CKM parameters is a counting statement; the paper itself acknowledges that A = |Vcb|/lambda^2 cannot be determined by kaon physics, so any reading of the claim as a separate extraction of A, rho-hat, and eta-hat would be a correctness concern rather than a circular reduction. The derivation chain is therefore self-contained with respect to circularity.

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

The paper introduces no free parameters fitted to the plotted data and no new entities. It relies on standard CKM unitarity, the Wolfenstein parameterization, published SM expressions for the observables, and external experimental inputs, all explicitly referenced.

assumptions (4)
  • domain assumption The CKM matrix is unitary and can be parameterized by {A, λ, ρ, η} in the Wolfenstein convention.
    Used throughout as the starting point for the parameterization of the CKM matrix (Section 'Introduction').
  • domain assumption The SM expressions for B(K+→π+νν), |εK|, B(KL→μ+μ−), and B(KS→μ+μ−)ℓ=0, as quoted in the appendix, are correct.
    The paper takes these expressions from Refs. [23,25,2,27] without re-derivation; the conclusions about the allowed islands depend on them.
  • domain assumption λ is well known and is set to the PDG global fit value.
    The paper states it treats λ as well known and substitutes the PDG global fit value (Section 'Introduction' and figure captions).
  • domain assumption The external experimental inputs (NA62 measurement of B(K+→π+νν), the world average for |εK|, and the measured B(KL→μ+μ−)) and their quoted uncertainties are correct.
    The plotted 95% CL regions are derived from these published measurements; if any input were significantly off, the island structure would change.

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Pith. "Pith review of Natural complex plane for kaon CKM data: framework, status and future." pith.science (2026). https://pith.science/paper/RIKRVQI3

@misc{pith2026250412386,
  author       = {Pith},
  title        = {Pith review of: Natural complex plane for kaon CKM data: framework, status and future},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/RIKRVQI3}},
  note         = {Machine review of arXiv:2504.12386}
}
abstract

Kaon physics can be used to independently determine three out of the four parameters of the CKM matrix, without any B physics input. Treating one parameter, $|V_{us}|$, or alternatively Wolfenstein $\lambda$, as well known, we show that the natural plane for the presentation of kaon CKM information is spanned by the combinations $\left(A^2(1-\hat\rho),\, A^2 \hat\eta\right)$. In this way, the use of B physics inputs is avoided, as well as the artificial inflation of errors due to parametric uncertainties, mainly due to $|V_{cb}|$. We show that the current status of kaon CKM constraints, impacted by recent advances in measurement and theory, is characterized by four allowed regions, and find that incoming data will inevitably disfavor a number of them, either confirming the CKM paradigm as dominant, or discovering a departure from the Standard Model.

Figures

Figures reproduced from arXiv: 2504.12386 by the authors.

Figure 1
Figure 1. FIG. 1: Current allowed regions from measurements of [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2: The addition of current [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3: Future projection, assuming 15% uncertainty on the [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗
Figures from the paper (2 more)
Figure 4
Figure 4. Figure 4: FIG. 4: Supplemental Figure: The two kaon constraints, [PITH_FULL_IMAGE:figures/full_fig_p005_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5: Supplemental Figure: Current [PITH_FULL_IMAGE:figures/full_fig_p006_5.png]

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. A CKM blind spot: probing $b$-column rescaling with kaons

    hep-ph 2026-07 accept novelty 6.0 of 10

    A uniform rescaling of the CKM b-column is invisible to B-physics-only fits, and kaon data already bound it to −4%…+4% (2σ), with projections reaching ~2%.

Reference graph

Works this paper leans on

29 extracted references · 8 canonical work pages · cited by 1 Pith paper

  1. [1]

    An overall scaling by a factor of A2λ4, common to all (VtdV ∗ ts)/(VcdV ∗ cs)-dependent kaon observables,

    work page

  2. [2]

    SM-”, “left-

    A transformation of the x-axis, ¯ρ→ (1− ˆρ). In the following, and in all plots, we omit factors of λ and treat it as known, substituting the PDG global fit value [18]. Current Kaon Parameter Space To demonstrate the advantage of the use of the A2(1− ˆρ), A2ˆη plane, we first present the overlap of two leading kaon CKM measurements: the rate of the rare d...

    work page

  3. [3]

    Cortina Gil et al

    E. Cortina Gil et al. (NA62), (2024), arXiv:2412.12015 [hep-ex]

    arXiv 2024

  4. [4]

    Hoferichter, B.-L

    M. Hoferichter, B.-L. Hoid, and J. R. de Elvira, JHEP 04, 071 (2024), arXiv:2310.17689 [hep-ph]

    arXiv 2024

  5. [5]

    D’Ambrosio and T

    G. D’Ambrosio and T. Kitahara, Phys. Rev. Lett. 119, 201802 (2017), arXiv:1707.06999 [hep-ph]

    arXiv 2017

  6. [6]

    A. Dery, M. Ghosh, Y. Grossman, and S. Schacht, JHEP 07, 103 (2021), arXiv:2104.06427 [hep-ph]

    arXiv 2021

  7. [7]

    A. Dery, M. Ghosh, Y. Grossman, T. Kitahara, and S. Schacht, JHEP 03, 014 (2023), arXiv:2211.03804 [hep- ph]

    arXiv 2023

  8. [8]

    Chao and N

    E.-H. Chao and N. Christ, Phys. Rev. D 110, 054514 (2024), arXiv:2406.07447 [hep-ph]

    arXiv 2024

Show all 29 references
  1. [9]

    Fry et al

    J. Fry et al. (KOTO), (2025), arXiv:2501.14827 [hep-ex]

    2025 arXiv

  2. [10]

    A. Dery, Y. Grossman, T. Kitahara, R. Marchevski, D. Martinez Santos, and S. Schacht, (To appear)

  3. [11]

    Wolfenstein, Phys

    L. Wolfenstein, Phys. Rev. Lett. 51, 1945 (1983)

    1983

  4. [12]

    Buchalla and A

    G. Buchalla and A. J. Buras, Phys. Rev. D 54, 6782 (1996), arXiv:hep-ph/9607447

    1996 arXiv

  5. [13]

    A. J. Buras, M. Gorbahn, U. Haisch, and U. Nierste, JHEP 11, 002 (2006), [Erratum: JHEP 11, 167 (2012)], arXiv:hep-ph/0603079

    2006 arXiv

  6. [14]

    Blucher, B

    E. Blucher, B. Winstein, and T. Yamanaka, Prog. Theor. Phys. 122, 81 (2009)

    2009

  7. [15]

    Lehner, E

    C. Lehner, E. Lunghi, and A. Soni, Phys. Lett. B 759, 82 (2016), arXiv:1508.01801 [hep-ph]

    2016 arXiv

  8. [16]

    A. J. Buras and E. Venturini, Acta Phys. Polon. B 53, 6 (2021), arXiv:2109.11032 [hep-ph] . 6 -1.5 -1.0 -0.5 0.0 0.5 1.0-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 A2(1-ρ) A2η 95%C.L -1.5 -1.0 -0.5 0.0 0.5 1.0-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 A2(1-ρ) A2η 95%C.L ACP(K0→μμ)(t=0) >0 -1.5 -1....

    2021 arXiv

  9. [17]

    Lunghi and A

    E. Lunghi and A. Soni, JHEP 12, 097 (2024), arXiv:2408.11190 [hep-ph]

    2024 arXiv

  10. [18]

    A. J. Buras, M. E. Lautenbacher, and G. Ostermaier, Phys. Rev. D 50, 3433 (1994), arXiv:hep-ph/9403384

    1994 arXiv

  11. [19]

    Charles, A

    J. Charles, A. Hocker, H. Lacker, S. Laplace, F. R. Le Diberder, J. Malcles, J. Ocariz, M. Pivk, and L. Roos (CKMfitter Group), Eur. Phys. J. C41, 1 (2005), arXiv:hep-ph/0406184

    2005 arXiv

  12. [20]

    Navas et al

    S. Navas et al. (Particle Data Group), Phys. Rev. D 110, 030001 (2024)

    2024

  13. [21]

    R. F. Lebed, Phys. Rev. D 55, 348 (1997), arXiv:hep- ph/9607305

    1997

  14. [22]

    CKMFitter collaboration, http://ckmfitter.in2p3. fr/

  15. [23]

    Marchevski, J

    R. Marchevski, J. Phys. Conf. Ser. 2446, 012035 (2023), arXiv:2301.06801 [hep-ex]

    2023 arXiv

  16. [24]

    Dery and M

    A. Dery and M. Ghosh, JHEP 03, 048 (2022), arXiv:2112.05801 [hep-ph]

    2022 arXiv

  17. [25]

    A. J. Buras, D. Buttazzo, J. Girrbach-Noe, and R. Kneg- jens, JHEP 11, 033 (2015), arXiv:1503.02693 [hep-ph]

    2015 arXiv

  18. [26]

    Buchalla and A

    G. Buchalla and A. J. Buras, Nucl. Phys. B 548, 309 (1999), arXiv:hep-ph/9901288

    1999 arXiv

  19. [27]

    J. Brod, M. Gorbahn, and E. Stamou, Phys. Rev. Lett. 125, 171803 (2020), arXiv:1911.06822 [hep-ph]

    2020 arXiv

  20. [28]

    Isidori and R

    G. Isidori and R. Unterdorfer, JHEP 01, 009 (2004), arXiv:hep-ph/0311084

    2004 arXiv

  21. [29]

    Brod and E

    J. Brod and E. Stamou, JHEP 05, 155 (2023), arXiv:2209.07445 [hep-ph]

    2023 arXiv

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Reviewed August 16, 2026 · model on record in the stance chip above.