REVIEW
$K_L \to \pi^0 \nu \bar \nu$ Beyond the Standard Model
High Energy Physics - Phenomenology · Submitted 16 Jan 1997 · 25 references
Pith's verdict: The Grossman–Nir paper: a clean sin²θ relation plus a model-independent cap on BR(K_L→π⁰νν̄) that people still quote, with one inverted formula in the leptoquark example.
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Referee report
The paper
The paper claims
A rare kaon decay can be tied to a single CP-violating phase, and capped from data already in hand.
The reading
What carries the argument
The carrier is the quantity $lambda = (q/p)(bar{A}/A)$, the standard mixing-times-decay parameter for a neutral meson decaying to a final state. Its modulus measures whether the final state is a CP eigenstate and its phase measures CP violation in the interference of mixing and decay. The argument turns on two facts about it: for a CP-even final state with negligible absorptive phases, $|lambda| = 1$ so only the phase survives, and the resulting rate ratio is then a pure function of that phase. The isospin relation between the neutral and charged decay amplitudes is the second piece of machinery, letting an unmeasurable $K_S$ rate be replaced by the measurable charged-kaon rate.
Extended reading notes
Core claim
The central claim is a clean, model-independent relation between two rare kaon rates. Defining $lambda = (q/p)(bar{A}/A)$ from the mixing parameters and the $K^0$ and $bar{K}^0$ decay amplitudes, the ratio of rates is $Gamma(K_L to pi^0 nu bar{nu})/Gamma(K_S to pi^0 nu bar{nu}) = (1+|lambda|^2-2 Re lambda)/(1+|lambda|^2+2 Re lambda)$. For a CP-even final state with negligible strong phases, $|lambda| = 1$ to about $10^{-3}$, and the ratio collapses to $tan^2 theta$ with $lambda = e^{2i theta}$. Replacing the $K_S$ denominator by the charged mode through the isospin relation $A(K^0 to pi^0 nu bar{nu})/A(K^+ to pi^+ nu bar{nu}) = 1/sqrt{2}$, corrected by the factor $r_{is} = 0.954$, gives $a_{CP} = r_{is}, Gamma(K_L to pi^0 nu bar{nu})/Gamma(K^+ to pi^+ nu bar{nu}) = sin^2 theta$. Because $sin^2 theta le 1$ and $tau_{K_L}/tau_{K^+} = 4.17$, this yields $BR(K_L to pi^0 nu bar{nu}) < 4.4 times BR(K^+ to pi^+ nu bar{nu})$, and with the then-current 90% CL limit $BR(K^+ to pi^+ nu bar{nu}) < 2.4 times 10^{-9}$ it gives $BR(K_L to pi^0 nu bar{nu}) < 1.1 times 10^{-8}$, far below the direct bound of $5.8 times 10^{-5}$. The paper stresses that the bound uses only isospin relations and does not require the final state to be CP even. It then exhibits two explicit new-physics settings: extra vector-like down-type quarks inducing tree-level $Z$ flavor-changing couplings, where both rates can reach $O(10^{-9})$ and $a_{CP}$ can be $O(1)$; and a scalar leptoquark (equivalently $R$-parity-violating supersymmetry), where $K_L to pi^0 nu_i bar{nu}_j$ with $i ne j$ can proceed even when CP is conserved.
Load-bearing premise
The load-bearing premise is the isospin relation between the neutral and charged amplitudes, $A(K^0 to pi^0 nu bar{nu})/A(K^+ to pi^+ nu bar{nu}) = 1/sqrt{2}$ with a 0.954 correction, together with the assumption that strong phases are negligible so that $|lambda| = 1$ to about $10^{-3}$; the paper itself notes that the bound assumes only isospin relations.
Editorial extensions
If this is right
- The neutral-mode branching ratio is bounded by $4.4$ times the charged-mode branching ratio, giving $BR(K_L to pi^0 nu bar{nu}) < 1.1 times 10^{-8}$ from the charged-kaon limit alone, without assuming any model.
- A measured ratio $a_{CP}$ yields $sin^2 theta$ with no hadronic uncertainty, so the CP-violating phase can be extracted even when the decay amplitude is dominated by new physics of unknown magnitude.
- In $Z$-mediated flavor-changing models with extra vector-like quarks, measuring both rates determines $|U_{ds}|$ from the charged mode and $arg(U_{ds})$ from the neutral mode, with both branching ratios possibly at the $10^{-9}$ level.
- In leptoquark or $R$-parity-violating scenarios, $K_L to pi^0 nu_mu bar{nu}_tau$ and its conjugate can be the dominant channels and can be CP conserving, so a large rate need not indicate CP violation.
- Observing $BR(K_L to pi^0 nu bar{nu}) ge O(10^{-11})$ together with the known $Im,varepsilon ne 0$ would exclude scenarios in which CP violation lives only in $Delta s = 2$ mixing.
- The phase measured this way is not simply the unitarity-triangle angle $beta$, because the charm contribution to the charged mode is comparable to the top one while being negligible in the neutral mode.
Reading between the lines
- The construction is a template: whenever a rare decay amplitude is isospin-related to a mode that has already been bounded, the existing limit converts for free into a bound on the unmeasured partner, so experimental progress on one channel tightens the other proportionally.
- Because the bound scales with the charged-kaon rate, an order-of-magnitude improvement in $BR(K^+ to pi^+ nu bar{nu})$ would push the neutral bound down toward the Standard Model prediction of a few times $10^{-11}$, at which point any excess over the bound would itself be a discovery signal.
- If lepton flavor violation sits near the quoted leptoquark limits, the neutral channel loses its status as a clean CP probe; separating the CP-conserving from the CP-violating part would then require resolving the neutrino flavor content of the final state, for instance through the missing-energy distribution.
- The structural remark that new physics enters mainly through the decay amplitude in the kaon system, while in the $B$ system it enters mainly through mixing, follows from $varepsilon$ pinning the mixing phase to about $10^{-3}$; it suggests looking for new phases in other $Delta s = 1$ processes rather than in kaon mixing.
Original abstract
We analyze the decay $K_L \to \pi^0 \nu \bar \nu$ in a model independent way. If lepton flavor is conserved the final state is (to a good approximation) purely CP even. In that case this decay mode goes mainly through CP violating interference between mixing and decay. Consequently, a theoretically clean relation between the measured rate and electroweak parameters holds in any given model. Specifically, $\Gamma(K_L \to \pi^0 \nu \bar \nu)/\Gamma(K^+ \to \pi^+ \nu \bar \nu)= \sin^2\theta$ (up to known isospin corrections), where $\theta$ is the relative CP violating phase between the $K-\bar K$ mixing amplitude and the $s\to d\nu\bar\nu$ decay amplitude. The experimental bound on $BR(K^+ \to \pi^+ \nu \bar \nu)$ provides a model independent upper bound: $BR(K_L \to \pi^0 \nu \bar \nu) < 1.1 \times 10^{-8}$. In models with lepton flavor violation, the final state is not necessarily a CP eigenstate. Then CP conserving contributions can dominate the decay rate.
The compiled record
The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.
The abstract, sentence by sentence
We analyze the decay $K_L \to \pi^0 \nu \bar \nu$ in a model independent way. If lepton flavor is conserved the final state is (to a good approximation) purely CP even. In that case this decay mode goes mainly through CP violating interference between mixing and decay. Consequently, a theoretically clean relation between the measured rate and electroweak parameters holds in any given model. Specifically, $\Gamma(K_L \to \pi^0 \nu \bar \nu)/\Gamma(K^+ \to \pi^+ \nu \bar \nu)= \sin^2\theta$ (up to known isospin corrections), where $\theta$ is the relative CP violating phase between the $K-\bar K$ mixing amplitude and the $s\to d\nu\bar\nu$ decay amplitude. The experimental bound on $BR(K^+ \to \pi^+ \nu \bar \nu)$ provides a model independent upper bound: $BR(K_L \to \pi^0 \nu \bar \nu) < 1.1 \times 10^{-8}$. In models with lepton flavor violation, the final state is not necessarily a CP eigenstate. Then CP conserving contributions can dominate the decay rate.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | We analyze the decay $K_L \to \pi^0 \nu \bar \nu$ in a model independent way. | contribution 0.90 | 0.92 | 0.07 | asserted 0.97 |
| s1 | If lepton flavor is conserved the final state is (to a good approximation) purely CP even. | result 0.90 | 0.48 | 0.08 | asserted 0.91 |
| s2 | In that case this decay mode goes mainly through CP violating interference between mixing and decay. | interpretation 0.52 | 0.46 | 0.09 | asserted 0.90 |
| s3 | Consequently, a theoretically clean relation between the measured rate and electroweak parameters holds in any given model. | interpretation 0.65 | 0.45 | 0.25 | asserted 0.99 |
| s4 | Specifically, $\Gamma(K_L \to \pi^0 \nu \bar \nu)/\Gamma(K^+ \to \pi^+ \nu \bar \nu)= \sin^2\theta$ (up to known isospin corrections), where $\theta$ is the relative CP violating phase between the $K-\bar K$ mixing amplitude and the $s\to d\nu\bar\nu$ decay amplitude. | result 0.97 | 0.72 | 0.36 | asserted 1.00 |
| s5 | The experimental bound on $BR(K^+ \to \pi^+ \nu \bar \nu)$ provides a model independent upper bound: $BR(K_L \to \pi^0 \nu \bar \nu) < 1.1 \times 10^{-8}$. | result 1.00 | 0.65 | 0.36 | asserted 1.00 |
| s6 | In models with lepton flavor violation, the final state is not necessarily a CP eigenstate. | result 0.43 | 0.34 | 0.04 | hedged possibility 0.60 |
| s7 | Then CP conserving contributions can dominate the decay rate. | result 0.64 | 0.42 | 0.08 | hedged possibility 0.67 |
What was examined, and what was not
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- P abstract
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References
How later papers used it
81 later papers cite it
3 later papers by other authors are listed below: 2 used its method or tool and 1 compared its own results with it.
Uses its method or tool 2 papers
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Used the Grossman-Nir bound from this paper to place an experimental constraint on R(K_L to pi^0 nu nubar).
$K\to\mu^+\mu^-$ beyond the standard model (2021) · 0.96 · paraphrased: the licence does not allow quoting
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Derived a Grossman-Nir bound by means of isospin symmetry following this paper, and used it to give BR(K_L to pi^0 nu nubar)_exp <= 7.7x10^-10.
New physics in $s\to d$ semileptonic transitions: rare hyperon vs. kaon decays (2021) · 0.92 · paraphrased: the licence does not allow quoting
Compares with it 1 of 3 papers with this label
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Observed that in its example 2 the correlation runs along a line parallel to the line representing this paper's Grossman-Nir bound.
The Renaissance of Kaon Flavour Physics (2016) · 0.81 · paraphrased: the licence does not allow quoting
How the literature uses it
76 arXiv papers cite this in 94 sentences
Papers citing it · 80 sentences without a confident label
Uses its method or tool 2 papers
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Then, by using the Grossman–Nir (GN) bound [ref] to place an experimental constraint on $R(K_L→π^0ν̅ν)$,
$K\to\mu^+\mu^-$ beyond the standard model · The relation between $K_S→(μ^+μ^-)_ℓ=0$ and $K_L→π^0ν̅ν$ · 0.96
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On the other hand, following Ref. [ref] one can easily derive by means of isospin symmetry a Grossman-Nir (GN) bound, [equation] Using Eq. ([ref]), the standard GN bound gives $BR(K_L→π^0νν̅)_exp≤7.7×10^-10$.
New physics in $s\to d$ semileptonic transitions: rare hyperon vs. kaon decays · $K→πνν̅$ and $K→ππνν̅$ decays · 0.92
Uses its data or values 3 papers
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The current experimental result, $BR(K^+→π^+νν̅) = 1.73^+1.15_-1.05×10^-10$, obtained at Brookhaven from $K^+$ decays at rest with seven candidate events, [ref] , together with considerations of isospin symmetry [ref] , leads to the model-independent bound $BR(K_L→π^0νν̅) < 1.4×10^-9$.
Physics Beyond Colliders at CERN: Beyond the Standard Model Working Group Report · Experiments sensitive to Flavour Violation · 0.97
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The very recent result for $$ from NA62 [ref] and the $90%$ confidence level (CL) upper bound on $$ from KOTO [ref] read respectively [equation] The four events of $$ reported by KOTO in [ref] , that violated the Grossman and Nir (GN) [ref] upper bound, have been excluded in the 2023 analysis.
Disentangling new physics in $K\rightarrow\pi\bar{\nu}\nu$ and $B\rightarrow K(K^*)\bar{\nu}\nu$ observables · Current experimental limits on $K →πνν̅$ · 1.00
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In small $m_ϕ$ limit, $Br(K_L →π^0 ϕ)/Br(K^±→π^±ϕ) ≃3.7$ [ref] .
Higgs Portal From The Atmosphere To Hyper-K · Introduction · 0.80
Compares with it 3 papers
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This upper limit is greater than the Grossman–Nir bound [ref] .
Electroweak baryogenesis via top-charm mixing · Phenomenological consequences · 0.98
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We observe that in the example 2 the correlation takes place along the line parallel to the line representing Grossman-Nir bound [ref] .
The Renaissance of Kaon Flavour Physics · Lessons on NP Patterns in $Z$ Scenarios · 0.81
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The two-body nature of the decay means that the use of $K_L$ is more sensitive to $X_1(2)X_2$ final states than $K^+$ decays, in contrast with models that modify the effective $d-s-νν$ vertex using the short distance physics [ref] .
Pair production of dark particles in meson decays · Conclusions · 0.87
Cites it as background 6 papers
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The appearance of such a signal is particularly surprising due to the existence of the so-called Grossman-Nir bound [ref] , which is based on a simple relationship between the decay widths of charged and neutral Kaons into charged and neutral pions and neutrinos.
A Light Scalar Explanation of $(g-2)_{\mu}$ and the KOTO Anomaly · Introduction · 0.83
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The two modes are related through the Grossman-Nir bound, which limits $ℬ(K_L →π^0 νν̅)$ based on $ℬ(K^+ →π^+ νν̅)$ [ref] .
Constraints on Dark Photon and Dark $Z$ Model Parameters in the $B$ and $K$ Meson Decays · Kaon decay and mixing · 0.80
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There exist quite a few works trying to explain these intriguing events reported by KOTO [ref] (or constrain particular new physics (NP) models [ref] ) and meanwhile avoid the violation of its relation with the $K^+→π^+ νν̅$ decay, that is the Grossman-Nir bound [ref] .
Implication of $K\to \pi \nu \bar{\nu}$ for generic neutrino interactions in effective field theories · Introduction · 0.88
Discussion
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