Prospects for $K_{L}\to\pi^{0}\nu\bar{\nu}$, $K_S\to\mu^+\mu^-$, $K_L\to\pi^0 \ell^+\ell^-$ and $\varepsilon^{\prime}/\varepsilon$ after the new $K^{+}\to\pi^{+}\nu\bar{\nu}$ result from NA62

Andrzej J. Buras; Cristina Lazzeroni; Joel C. Swallow; Monika Blanke

Prospects for $K_{L}\to\pi^{0}\nu\bar{\nu}$, $K_S\to\mu^+\mu^-$, $K_L\to\pi^0 \ell^+\ell^-$ and $\varepsilon^{\prime}/\varepsilon$ after the new $K^{+}\to\pi^{+}\nu\bar{\nu}$ result from NA62

T0 review · 2 major / 5 minor · reviewed 2026-08-01 · deepseek-v4-flash

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Pith's one-line read Rare kaon decay can still beat the Standard Model tenfold despite the new K+ measurement.

desk verdict 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. read the letter →

arxiv 2607.19055 v2 pith:PQPLZIIK submitted 2026-07-21 hep-ph hep-exhep-lat

Monika Blanke , Andrzej J. Buras , Cristina Lazzeroni , Joel C. Swallow This is my paper · ORCID

classification hep-phhep-exhep-lat

keywords rarekaondecaysK_L→π0ννbarK+π+Z'modelleft-rightcouplingsCPviolationε_Kε'/ε

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 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.

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

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.

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Extended reading notes

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

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.

Editorial extensions

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.

Reading between the lines

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.

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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

The new element here is concrete and timely: after the 2026 NA62 result pins K^+->pi^+ nu nu to the SM, the authors show that a Z' with both left- and right-handed sbar-d couplings can still push K_L->pi0 nu nu toward the Grossman-Nir bound while leaving epsilon_K and K^+ SM-like, and can simultaneously boost K_S->mu+mu-, K_L->pi0 l+l-, and epsilon'/epsilon. The model-independent part of the paper is clean, the general formulae are standard and traceable, and the figures showing how NA62 and KOTO II pin down |X_eff| and beta_X are helpful. Credit where due: the LH+RH cancellation insight in K^+ was already in earlier work, but the explicit correlated scan for this specific Z' scenario after the new NA62 measurement is the paper's real contribution.

Now the main soft spot, and it is load-bearing. Section 4 says “we set Delta_R^sd(Z')=0” after arguing that Eq. (66) requires Delta_R^sd/Delta_L^sd ~ 1/240 for the Z' contribution to K0-K0bar mixing to cancel. But Eq. (65) gives z_sd = 1 when Delta_R^sd=0, not z_sd=0. So Eq. (64) does not vanish: the tree-level mixing contribution is proportional to (Delta_L^sd)^2. The benchmarks in Fig. 10 require both Re and Im Delta_L^sd nonzero, so epsilon_K receives an unquantified NP contribution. This is not a hadronic-uncertainty footnote; the numerical demonstration does not satisfy requirement 1 as written. The fix is probably simple—include a small tuned Delta_R^sd at the ratio from Eq. (66), which has negligible effect on the plotted decay rates—and I would ask the authors to rerun the scan with that coupling included and show the epsilon_K constraint explicitly.

The other soft spots are milder. The scan is an existence proof, not a fit or prediction, so “prospects” should be read as “possibility space.” The suppression depends on kappa_sd ~ -120 from Ref. [75], not independently derived, and no uncertainty is assigned to that ratio. The epsilon'/epsilon motivation leans on the Dual QCD estimate, and the paper itself notes the SM estimate is disputed. Also, the interesting discovery pattern for K_L->pi0 e+e- requires negative interference; the positive-interference case is excluded in the large-enhancement region. These are stated honestly, and they do not undermine the conceptual point.

Who gets value: kaon phenomenologists and anyone interpreting KOTO II or LHCb results in the next decade. It is worth a serious referee, but the referee should insist on a corrected numerical section before anything is published in its current form. My verdict is: engage with it, but do not cite the numerics as evidence that all seven constraints are simultaneously satisfied until the tuned Delta_R^sd is actually in the scan.

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)

[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.
[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)

[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).
[Eq. (65)] The last denominator is missing a closing parenthesis: it should be Δ_L^sd(Z′), not Δ_L^sd Z′).
[Fig. 10 caption] The caption writes 'Δq¯q_L(R′)' where the parameter is Δq¯q_R(Z′). This typo could confuse readers.
[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.
[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 · score 0.0 of 10

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.

Assumptions & free parameters 6 free parameters · 6 assumptions · 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].

assumptions (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.

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Pith. "Pith review of Prospects for $K_{L}\to\pi^{0}\nu\bar{\nu}$, $K_S\to\mu^+\mu^-$, $K_L\to\pi^0 \ell^+\ell^-$ and $\varepsilon^{\prime}/\varepsilon$ after the new $K^{+}\to\pi^{+}\nu\bar{\nu}$ result from NA62." pith.science (2026). https://pith.science/paper/PQPLZIIK

@misc{pith2026260719055,
  author       = {Pith},
  title        = {Pith review of: Prospects for $K_L\to\pi^0\nu\bar\nu$, $K_S\to\mu^+\mu^-$, $K_L\to\pi^0 \ell^+\ell^-$ and $\varepsilon^\prime/\varepsilon$ after the new $K^+\to\pi^+\nu\bar\nu$ result from NA62},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/PQPLZIIK}},
  note         = {Machine review of arXiv:2607.19055}
}

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.