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REVIEW 2 major objections 5 minor 73 references

Rare kaon decay can still beat the Standard Model tenfold despite the new K+ measurement.

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

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2026-08-01 13:32 UTC pith:PQPLZIIK

load-bearing objection A useful Z' existence proof for enhanced K_L->pi0 nu nu after NA62, but the numerical demonstration as written has a consistency hole: it sets Delta_R^sd=0 while relying on Eq. (66), whose whole point is a tuned nonzero Delta_R^sd to keep epsilon_K SM-like. the 2 major comments →

arxiv 2607.19055 v1 pith:PQPLZIIK submitted 2026-07-21 hep-ph hep-exhep-lat

Prospects for K_(L)toπ⁰νbar{ν}, K_Stoμ^+μ^-, K_Ltoπ⁰ ell^+ell^- and varepsilon^(prime)/varepsilon after the new K⁺toπ⁺νbar{ν} result from NA62

Monika Blanke , Andrzej J. Buras , Cristina Lazzeroni , Joel C. Swallow This is my paper
classification hep-ph hep-exhep-lat
keywords rare kaon decaysK_L → π0 ν νbarK+ → π+ ν νbarZ' modelleft-right couplingsCP violationε_Kε'/ε
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The paper asks whether the recent measurement of K+→π+ννbar, which agrees with the Standard Model, also dooms the hope of large new-physics effects in the even rarer K_L→π0ννbar decay. It argues that it does not: if new physics couples to both left- and right-handed quarks and carries a large new CP-violating phase, K_L→π0ννbar can still be enhanced by an order of magnitude while the K+ rate and the CP-violation parameter ε_K stay SM-like. The same scenario simultaneously enhances K_S→μ+μ-, K_L→π0ℓ+ℓ-, and the direct-CP ratio ε'/ε. The demonstration uses a concrete Z' model, a new heavy neutral boson, and shows numerically that all seven observables can meet current constraints at once. This matters because it tells the next round of kaon experiments exactly what patterns to look for.

Core claim

The central claim is that the combination of a left-handed and a small right-handed s̄d coupling to a new Z' boson, together with a large new weak phase β_X ≈ 110° in the combination X_eff that controls K→πννbar amplitudes, can decouple the two kaon-pion-neutrino modes: the K_L→π0ννbar rate can be pushed close to the Grossman-Nir bound (about ten times the Standard Model value) while K+→π+ννbar remains within the recently measured SM-like band. The small right-handed coupling is not a correction; it is the load-bearing ingredient, because the ratio Δ_R^sd/Δ_L^sd ≈ 1/240 cancels the Z' tree-level contribution to K0–K0bar mixing, keeping ε_K and ΔM_K SM-like. The paper then shows that the same

What carries the argument

The argument runs on two pieces. First, the branching ratios of the two K→πννbar decays are expressed through a single complex function X_eff; its imaginary part alone controls K_L→π0ννbar, while both magnitude and phase enter K+→π+ννbar. A large phase β_X ≈ 110° therefore allows K_L to be enhanced without touching K+, and measuring both modes pins down |X_eff| and β_X up to a four-fold ambiguity. Second, in the Z' model, the ratio of right-handed to left-handed s̄d coupling is tuned to Δ_R^sd/Δ_L^sd ≈ 1/240, the value at which the left-left, right-right and left-right contributions to the K0–K0bar mixing amplitude cancel (Eq. 66), so ε_K and ΔM_K stay SM-like while the ΔF=1 decays proceed u

Load-bearing premise

The load-bearing premise is that the hadronic matrix-element ratio κ_sd equals about −120 for M_Z' between 5 and 20 TeV, so that the right-handed s̄d coupling tuned to 1/240 of the left-handed one exactly cancels the Z' contribution to K0–K0bar mixing; if that ratio changes, ε_K and ΔM_K are no longer automatically SM-like.

What would settle it

Measure B(K_L→π0ννbar) and B(K+→π+ννbar) precisely: if K_L stays within a factor of ~2 of the SM while K+ remains SM-like, the large-phase enhancement is falsified. Alternatively, an updated lattice calculation of the hadronic matrix elements entering κ_sd that moves κ_sd noticeably away from −120 would remove the ε_K cancellation, and the benchmark scenario with Δ_R^sd=0 would cease to satisfy the ε_K constraint.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • If K_L→π0ννbar is measured by the proposed next-generation experiment, an order-of-magnitude enhancement over the SM prediction is still a live possibility even if the final K+ result stays SM-like.
  • A discovery of K_L→π0ννbar near the Grossman-Nir bound would, within this scenario, imply strongly enhanced K_S→μ+μ- (up to two orders of magnitude in parts of the parameter space) and enhanced K_L→π0ℓ+ℓ-.
  • The same NP phase structure produces a positive contribution to ε'/ε of order 10^-3, potentially closing the gap between the experimental value and the lower SM estimate.
  • A measurement of both K→πννbar branching ratios would determine the underlying new-physics amplitude |X_eff| and phase β_X, up to a four-fold ambiguity that other observables like K_S→μ+μ- or ε'/ε can resolve.

Where Pith is reading between the lines

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

  • The scenario's viability rests on a precise cancellation: if lattice or other hadronic inputs shift κ_sd from ≈ −120, the required Δ_R^sd/Δ_L^sd ratio changes and the ε_K/ΔM_K suppression weakens, so the 'SM-like regardless of other parameters' claim is not robust to hadronic revisions.
  • Because the cancellation requires a tuned ratio, a fully natural model would need a mechanism that generates such a hierarchy between left- and right-handed couplings; absent that, the scenario is an existence proof rather than a complete theory.
  • The paper's correlation structure suggests a decisive test: if K_L→π0ννbar is found enhanced while K_S→μ+μ- remains SM-like, the mixed-chirality Z' scenario would be disfavoured, pointing instead to models with pure left-handed couplings or to non-Z' mediators.
  • One could extend the same flavour structure to B-meson or top decays involving equivalent chiral couplings; such extensions are not explored here but would sharpen the model's predictivity.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

2 major / 5 minor

Summary. The paper argues that, after the NA62 measurement of B(K+→π+ννbar) consistent with the SM, appreciable NP effects in other rare kaon observables are still possible. A model-independent parametrisation of K→πννbar shows that a large CP-violating phase β_X≈110° can enhance B(KL→π0ννbar) by an order of magnitude while keeping B(K+→π+ννbar) SM-like. The authors then construct a specific Z′ model with both left- and right-handed sbar-d couplings and show, in Sec. 4, that a benchmark with chosen couplings can simultaneously keep ε_K and ΔM_K SM-like, satisfy the KL→μ+μ− bound, enhance KS→μ+μ−, KL→π0ℓ+ℓ−, and ε′/ε, and reach R^0_{ννbar}≈O(10). The numerical demonstration is a parameter scan over the couplings that appear in the observables, not a global fit.

Significance. If the claimed benchmark is valid, the paper provides a useful counterexample to the expectation that a SM-like NA62 result and SM-like ε_K exclude large enhancements of KL→π0ννbar, KS→μ+μ−, KL→π0ℓ+ℓ− and ε′/ε. The general formulae collected in Sec. 2 are standard and clearly presented, and the analysis correctly identifies that both LH and RH sbar-d couplings are needed to evade the ε_K-induced correlation that would otherwise bound KL→π0ννbar. The explicit Z′ scenario with stated coupling choices and constraints is a concrete, reproducible existence proof. The central gap is that the numerical scan sets to zero the very right-handed coupling on which the ε_K/ΔM_K cancellation relies, so the headline claim of simultaneous satisfaction of all seven requirements is not yet demonstrated.

major comments (2)
  1. [Sec. 4, Point 1; Secs. 3.2–3.3] Point 1 of the requirement list is asserted to follow from the choice Δ_R^sd(Z′)≈(1/240)Δ_L^sd(Z′)≈0. However, the numerical analysis in the same section states 'we set Δ_R^sd(Z′)=0'. With Δ_R=0, z_sd in Eq. (65) equals 1, not 0, because the cancellation relies on the 2κ_sd r term with κ_sd≈−120. Consequently the tree-level Z′ contribution to (M_12^*)^sd in Eq. (64) is unsuppressed and proportional to (Δ_L^sd)^2. The benchmarks in Fig. 10 have non-zero Re Δ_L^sd and Im Δ_L^sd, so ε_K and ΔM_K receive an unquantified NP contribution. The statement that requirement 1 is satisfied 'regardless of the values of the other parameters' is therefore not demonstrated by the numerical scan. Including the small tuned value Δ_R^sd=Δ_L^sd/240 in the scan would not affect the plotted rates and should restore the cancellation; this needs to be checked and reported.
  2. [Sec. 3.2, Eq. (66)] The cancellation condition is a fine-tuning between VLL, VRR and LR contributions that depends on the hadronic matrix-element ratio κ_sd≈−120 from Table 5 of Ref. [75]. No uncertainty on κ_sd is propagated, and no tolerance on Δ_R/Δ_L around 1/240 is given. If κ_sd changes by ~20% (or the hadronic matrix-element input is revised), the ratio required for z_sd≈0 shifts and the ε_K suppression is no longer automatic. Since the paper's summary claims ε_K remains SM-like, the authors should quantify the sensitivity, e.g. by varying κ_sd within its expected uncertainty and showing the allowed band of Δ_R/Δ_L, or at least state explicitly that the benchmark is a fine-tuned illustration.
minor comments (5)
  1. [Introduction, Eq. (1) and throughout] Several occurrences of 'π oℓ+ℓ−' should read 'π^0ℓ+ℓ−' (e.g. Eq. (1), Eqs. (4)–(7) region, Sec. 2.4 heading).
  2. [Eq. (65)] The last denominator is missing a closing parenthesis: it should be Δ_L^sd(Z′), not Δ_L^sd Z′).
  3. [Fig. 10 caption] The caption writes 'Δq¯q_L(R′)' where the parameter is Δq¯q_R(Z′). This typo could confuse readers.
  4. [Fig. 10, right panel] The legend entry 'R^S_{μ+μ−}/10' is not explained in the caption; please clarify that the displayed curve is rescaled by a factor 1/10.
  5. [Secs. 2.5 and 3.2] The symbol for ε_K is written as both 'ε_K' and 'ϵ_K'; please standardise the notation.

Circularity Check

0 steps flagged

No significant circularity: the model-independent formulas and Z' benchmark scan are parameter demonstrations, and the noted Delta_R=0 issue is an internal consistency flaw, not a circular derivation.

full rationale

Walking the derivation chain: (i) Sec. 2.1 defines X_eff via Eqs. (8)-(9), and Eq. (27) is a rearrangement of those definitions; requiring beta_X about 110 degrees for a large K_L to pi0 nu nubar rate is an algebraic consequence, not a fitted prediction. (ii) Sec. 3 introduces the Z' parameters and writes the Delta F=1 and Delta F=2 contributions in terms of them. Eq. (66) is not imported as a uniqueness theorem; it is derived in the paper from kappa_sd about -120, quoted from Ref. [75]. That hadronic matrix-element input is external to the paper's target claim (it does not contain the K_L enhancement as an input), so even though [75] shares an author, the self-citation does not by itself make the argument circular. (iii) Sec. 4 is an explicit benchmark scan, not a fit-then-predict exercise: the authors state that they enforce R+_nu nubar = 1, R_L_mu mu < 3, kappa_epsilon' > 0 and then display the correlated rates. Choosing model parameters that satisfy constraints and showing that other observables are enhanced is a standard existence demonstration; the plotted quantities are not equal to the inputs by construction because the correlations are nontrivial functions of the couplings (Table 1). (iv) I checked for renamed known results: the beta_X / |X_eff| parameterization is a re-parametrization of previous kaon formulae, but the paper does not present it as a new derivation. The one serious concern is not circularity: Sec. 4 sets Delta_R^sd(Z')=0 while the derivation of requirement 1 relies on z_sd about 0 from Delta_R^sd / Delta_L^sd about 1/240. At Delta_R=0, Eq. (65) gives z_sd=1, so the tree-level M_12 contribution is not suppressed; this means point 1 is not actually demonstrated by the numerical scan. That is an internal consistency or correctness problem, not a reduction of the prediction to its inputs, so it does not raise the circularity score.

Axiom & Free-Parameter Ledger

6 free parameters · 6 axioms · 1 invented entities

The paper's central claim rests on five hand-set couplings (plus one mass) and on several external inputs. The free parameters are not fitted to data in a statistical sense, but they are chosen exactly to realize the desired enhancement pattern, so the 'prediction' is better understood as a proof of existence. The ε'/ε and ε_K suppression steps rely on hadronic matrix elements from earlier papers by the same group.

free parameters (6)
  • Δ_ℓℓ_L(Z') (leptonic LH coupling) = 0.5
    Set by hand; chosen to give an illustrative NA62-allowed annulus and to keep the Z' contribution in a perturbative range.
  • Δ_ℓℓ_R(Z') (leptonic RH coupling) = 0, 0.3, -1.00, 1.25 (set in different benchmark scenarios)
    Varied to control R^L_μμ and the K_L→π0ℓ+ℓ- pattern; the paper presents several representative values.
  • Δ_R^qq(Z') (flavour-conserving RH quark coupling) = -0.01, -0.015 (benchmarks)
    Chosen to produce κ_ε' = O(1) while respecting the LHC bound of Eq. (77).
  • Re Δ_L^sd(Z') = not given explicitly; constrained to small values by R^L_μμ<3
    Parameter space coordinate; the paper shows contours, not benchmark values.
  • Im Δ_L^sd(Z') = not given explicitly; constrained by R^0_ννbar and κ_ε'
    Parameter space coordinate; effectively set by requiring large β_X and κ_ε'>0.
  • M_Z' = 5 TeV
    Fixed by hand; choosing another mass would rescale the couplings, so this is a light benchmark choice, though the paper cites constraints from [75,84].
axioms (6)
  • domain assumption The SM contributions and input parameters (X(x_t), P_c(X), κ_+, κ_L, κ_μ, Y_0, etc.) from cited literature are correct.
    Used throughout the paper in Eqs. (8)-(10), (14), (30)-(31), etc. If these are wrong, the derived enhancements shift accordingly.
  • domain assumption The BV-strategy CKM determination (|V_ts|, |V_td|, β, β_s, Im λ_t) from Refs. [14,15,24] is adopted.
    Appendix A gives the CKM values; all numerical SM predictions and plots depend on them.
  • standard math The Grossman-Nir bound is valid (SM neutrinos in the final state).
    Used in Eq. (28) and to cap R^0_ννbar at ~10 in the numerical analysis; if violated, the upper end of the benchmark changes.
  • domain assumption The hadronic matrix-element ratio κ_sd ≈ -120 from Ref. [75] is correct.
    Used to derive the fine-tuning condition Eq. (66) Δ_R^sd/Δ_L^sd ≈ 1/240, which is the basis for setting Δ_R^sd = 0 and keeping ε_K SM-like.
  • domain assumption The ε'/ε NP coefficient 14.5 B_8^(3/2) from Ref. [82] captures the dominant Q_8 contribution to 10% precision.
    Used in Eq. (76) to compute κ_ε'; this is a strong factorization/hadronic assumption.
  • domain assumption The 2026 NA62 measurement and KOTO upper bound are quoted correctly.
    The whole motivation rests on the experimental inputs in Eqs. (3) and (4).
invented entities (1)
  • Z' boson with flavour-violating sbar-d couplings and lepton-flavour-universal couplings independent evidence
    purpose: To produce the LH+RH current pattern that enhances K_L→π0ννbar while keeping K+ and ε_K SM-like
    A 5 TeV Z' with O(0.5) couplings would have collider/LEP constraints (partially discussed via Eq. (77) and [84]); the paper does not propose a UV completion and the couplings are ad hoc, but the model is falsifiable through the rare kaon pattern and collider searches.

pith-pipeline@v1.3.0-alltime-deepseek · 24749 in / 9518 out tokens · 75136 ms · 2026-08-01T13:32:37.963934+00:00 · methodology

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read the original abstract

The recently announced NA62 measurement of the $K^{+}\to\pi^{+}\nu\bar{\nu}$ branching ratio, based on 2016--2024 data, is fully consistent with its very accurate Standard Model (SM) prediction. While it is not excluded that the final result based on 2016--2026 data will deviate from the SM prediction, the question arises whether a similar fate awaits the $K_{L}\to\pi^{0}\nu\bar{\nu}$ decay. This decay is currently being searched for by the KOTO experiment, with the present upper bound roughly two orders of magnitude above its very precise SM prediction. The proposed KOTO II experiment aims to provide the first discovery of the $K_{L}\to\pi^{0}\nu\bar{\nu}$ decay and measure its branching ratio. Building on the findings of several previous papers we demonstrate that in the presence of suitably chosen large new complex phases and of a small amount of new right-handed $\bar s d$ couplings, in addition to the left-handed ones, the $K_{L}\to\pi^{0}\nu\bar{\nu}$ branching ratio can still be enhanced by one order of magnitude with respect to the SM prediction while keeping both $K^{+}\to\pi^{+}\nu\bar{\nu}$ and $\varepsilon_K$ SM-like. Simultaneously the decays $K_S\to\mu^+\mu^-$, studied by LHCb, and $K_L\to\pi^0\ell^+\ell^-$, searched for by KOTO II, can be strongly enhanced and the anomaly in the ratio $\varepsilon^{\prime}/\varepsilon$, as claimed by Dual QCD, removed. We illustrate this with an example of a specific $Z^\prime$ scenario.

Figures

Figures reproduced from arXiv: 2607.19055 by Andrzej J. Buras, Cristina Lazzeroni, Joel C. Swallow, Monika Blanke.

Figure 1
Figure 1. Figure 1: Illustrations of common correlations in the B(K+ → π +νν¯) versus B(KL → π 0 νν¯) plane. The expanding red region illustrates the lack of correlation for models with general LH and RH NP couplings. The green region shows the correlation present in models obeying Constrained Minimal Flavour Violation [31–33]. The blue region shows the correlation induced by the constraint from εK if only LH or RH couplings … view at source ↗
Figure 2
Figure 2. Figure 2: Correlation between B(K+ → π +νν¯) and B(KL → π 0 νν¯) for different values of the phase βX. in [49] and updated below, the huge enhancement of B(KL → π 0 νν¯) necessarily requires a large weak phase βX ≈ 110◦ , as B(KL → π 0 νν¯) B(KL → π 0νν¯)SM = [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Correlation between B(K+ → π +νν¯) and B(KL → π 0 νν¯) for different values of |Xeff|. the two branching ratios lie on a line parallel to the Grossman-Nir bound, with the offset determined by the charm contribution to K+ → π +νν¯. Analogously in [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Determination of |Xeff| and βX from the K → πνν¯ decays. The red shaded region shows the 1σ allowed range for B(K+ → π +νν¯) by NA62, while the coloured lines display different hypothetical measurements of B(KL → π 0 νν¯). to KL → µ +µ − can be reliably calculated. Despite this limitation, the constraint on the SD contribution places important bounds on certain NP scenarios. In contrast to K+ → π +νν¯ and … view at source ↗
Figure 5
Figure 5. Figure 5: Feynman diagram for K0 − K¯ 0 mixing in Z ′ models. next section. As we shall see, this simplifies our analysis significantly while allowing ∆sd L (Z ′ ) to be sufficiently large to provide significant NP contributions to the rare kaon decays and the ratio ε ′/ε considered by us. A compendium of the Z ′ contributions to the observables considered by us has been pre￾sented in [64] and can also be found in S… view at source ↗
Figure 6
Figure 6. Figure 6: Feynman diagrams for K+ → π +νν¯ (left) and KL → π 0 νν¯ (right) in Z ′ models. which are defined in eqs. (35) and (36) of [75] [PITH_FULL_IMAGE:figures/full_fig_p017_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Feynman diagrams for KL,S → µ +µ − (left) and KL → π 0 ℓ +ℓ − (right) in Z ′ models. For the leptonic modes KL → µ +µ − and KS → µ +µ − the relevant contribution to the expressions collected in Sections 2.2 and 2.3 is given by ∆YA(K) = ∆ µµ¯ A (Z ′ ) g 2 SMM2 Z′ (∆sd L (Z ′ ) − ∆sd R (Z ′ )) V ∗ tsVtd (70) and therefore Y A eff = λtY0 + 1 g 2 SMM2 Z′ ∆ µµ¯ A [PITH_FULL_IMAGE:figures/full_fig_p018_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Representative Feynman diagram for ε ′/ε in the Z ′ scenario. Within 10% precision it can be written as follows [82]:  ε ′ ε NP =  ε ′ ε L Z′ = 14.5 B (3/2) 8  5 TeV MZ′ 2 Im ∆ sd L (Z ′ )  ∆ qq¯ R (Z ′ ). (76) This accuracy is sufficient for our analysis, not only because of the non-perturbative uncertain￾ties in the SM contribution, but also because of the new coupling parameter ∆qq¯ R (Z ′ ) ente… view at source ↗
Figure 9
Figure 9. Figure 9: left: Constraints on R + νν¯ and R0 νν¯ in the plane of Re ∆sd L (Z ′ )  ,Im ∆sd L (Z ′ )  for different choices of ∆ℓℓ¯ L (Z ′ ). The shaded regions correspond to the allowed 1σ interval of the NA62 measurement. Right: Assuming ∆ℓℓ¯ L (Z ′ ) = 0.5, the NA62 constraint is shown in green and a hypothetical future measurement of R0 νν¯ = 10±1 is shown in orange. When ∆sd L (Z ′ ) = 0 there is no NP contri… view at source ↗
Figure 10
Figure 10. Figure 10: R + νν¯ , R L,S µ+µ− , R0 πℓ+ℓ− and κε ′ as functions of R0 νν¯ for two different choices of the flavour-conserving Z ′ coupling parameters, enforcing R + νν¯ = 1, RL µ+µ− < 3 and κε ′ > 0. For R0 πℓ+ℓ− , the solid (dashed) line show the case of positive (negative) interference between direct and mixing-induced contributions. Left: ∆ℓℓ¯ L (Z ′ ) = 0.5, ∆ℓℓ¯ R(Z ′ ) = 0.3, ∆qq¯ R (Z ′ ) = −0.01. Right: ∆ℓℓ… view at source ↗
Figure 11
Figure 11. Figure 11: Top: comparison of [PITH_FULL_IMAGE:figures/full_fig_p024_11.png] view at source ↗

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Reference graph

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