dataesr/scientific-paragraphs-categorization huggingface dataset:
The NA62 experiment has also established upper limits on B(K + → π + X) with invisible X, at the O(10 -9 ) level in the 110-155 MeV mass range, in the vicinity of the π 0 mass, from a dedicated analysis based on 10% of the Run 1 minimum bias dataset  [110]  (Figs. 1, 3 ). An upper limit on the invisible neutral pion decay rate is also reported: B(π 0 → inv) < 4. 4 × 10 -9 at 90% CL. The sensitivity of this search is limited by the K + → π + π 0 background with two undetected photons from the π 0 → γγ decay. The search resulted in an observation of 12 candidate events, compared to the expected background of 10 +22 -8 events, and was thus consistent with the background-only hypothesis. To improve the sensitivity, the event selection has been optimised to reduce the probability of missing both photons produced in the π 0 decay, where a probability of (2.
8 +5. 9 -2. 1 )×10 -9 has been achieved. The obtained upper limits weaken as a function of |m Xm π 0 | because the same definition of the signal region in terms of the reconstructed missing mass squared m 2 miss (optimized for the π 0 → inv search and centered on the π 0 mass) is used for each m X value. The analysis procedure can be modified to use a signal region centered on m X for each mass hypothesis, which would lead to sensitivity that would be improving as a function of |m Xm π 0 |. The sensitivity to be obtained with larger datasets is potentially limited by the systematic uncertainties on the expected K + → π + π 0 background. A search for the K + → π + X decay in the 260-350 MeV mass range (above the m X values probed by the beam-dump experiments) is in principle possible with the NA62 data.
The KOTO experiment has reported upper limits of B(K L → π 0 X) down to 2. 4 × 10 -9 at 90% CL in the m X range 0-260 MeV with the 2015 dataset, without degradation in the π 0 mass region  [239]. More recently, KOTO has published a search for the K L → π 0 ν ν decay with the 2016-2018 dataset, reporting three signal candidate events for an expected background of 1. 22 ± 0. 26 events  [240]. KOTO has not updated the K L → π 0 X decay limits since no improvement on the 2015 dataset was expected due to the background contribution. However, while still preliminary, the KOTO 2016-2018 analysis was re-interpreted to derive the B(K L → π 0 X) limits  [108]. Applying the same method to the KOTO 2016-2018 result published in Ref. [240] , we obtain the upper limits of O(10 -3 ) on the mixing angle θ presented in Fig.
1. These limits are very similar to those originally reported by KOTO using the 2015 dataset  [239]. KOTO sensitivity is expected to improve in the short term, using datasets with reduced background to be collected with an upgraded setup. Future K L → π 0 ν ν experiments (KOTO step-2, KLEVER) promise further improvements, however the sensitivity will be limited by backgrounds. In particular, the KLEVER sensitivity to B(K L → π 0 X) for m X < 200 MeV is estimated to be 0. 8 × 10 -11 , corresponding to a sensitivity of 4 × 10 -5 on the Higgs portal mixing parameter θ (Fig. 44  of Ref. [362] ). The KOTO step-2 sensitivity can be expected to be similar. Fig. 1  shows that a sensitivity of O(10 -13 ) to B(K → πX) would cover the entire parameter space of the Higgs portal scalar below 200 MeV, down to the BBN bound.
These results are used to set a limit on the axial flavor violating ALP coupling (Section 2. 2. 1. 3). The NA62 Run 1 minimum bias dataset is expected to improve on the ISTRA+ sensitivity by 1-2 orders of magnitude in the above m X range. The sensitivity will depend on the background contamination, in particular from decays with undetected photons: K + → π + π 0 γ (relevant for low m X ) and K + → π + π 0 π 0 (affecting mostly m X hypotheses in the π 0 mass region). The strongest limit on the K L → π 0 π 0 ν ν branching fraction (8. 1 × 10 -7 at 90% CL) has been reported at the KEK E391a experiment  [120]. Upper limits on the K L → π 0 π 0 X branching fraction, also obtained from this analysis, vary from 7×10 -7 for m X = 50 MeV to 4×10 -5 for m X = 200 MeV. The degradation of sensitivity towards larger m X values is caused by the reduction in acceptance.
The K → πγ + inv decays The quark level transition that leads to this signature is s → dγ(ϕ, a, A ), where the photon is radiated either from the initial kaon or final state pion. Other possibilities are: the photon arising from a decay of a dark sector particles, e. g. , s → d(ϕ/a → γA ), or from the decay of a neutral pion, K → π(π 0 → γA ). Phenomenologically the most important example is the dark photon, see Section 2. 4, since the extra γ in the final state allows for the K → πγA transition even in the massless case. A search for K + → π + π 0 , π 0 → γA decay chain involving an invisible dark photon in the mass range 30-130 MeV has been performed by NA62 using a fraction of Run 1 dataset corresponding to fewer than 10 11 kaon decays  [365] ; the trigger chain used for the analysis is not available in the main NA62 Run 1 dataset.
The background from the π 0 → γγ decay with an undetected photon limits the sensitivity for A mass below about 40 MeV, while the search is almost background free for higher m A values. The upper limits on the dark photon coupling parameter ε 2 reach the 10 -7 level, and improve on the previous searches at NA64 and BaBar in the 60-110 MeV mass range. An upper limit of B(π 0 → γν ν) < 1. 9 × 10 -7 is also reported from this study. The NA62 sensitivity to the K + → π + γA decay is yet to be established. The above processes could in principle be studied at future high statistics K + experiments, most likely using minimum bias datasets. The K L → π 0 γA decay is being studied by KOTO with a dataset of 2. 8 × 10 10 pot collected using a trigger requiring three clusters in the calorimeter, already used for the K L → π 0 γ search  [366].
The K → πγγ decays The quark level transition is s → d(ϕ/a → γγ). For Higgs mixed scalar, the ϕ → γγ decay rate is negligibly small (Section 2. 1) but could be important for more general scalars, for instance a dilaton. For ALP, the a → γγ decay can be one of the discovery channels (Section 2. 2. 2. 1). The K → πγγ decay channel is important for the examples of GN violating models for which the violation is only apparent due to experimental loopholes (Section 2. 7). Owing to an experimental signature similar to that of the dominant K + → π + π 0 decay, the K + → π + γγ process is typically studied using donwscaled minimum bias datasets. A sample of K ± → π ± γγ decays obtained from minimum bias datasets collected in 2003-2007, corresponding to 2. 5 × 10 9 kaon decays in total, has been analysed by the NA48/2 and NA62 experiments  [147, 148].
In total, 380 candidates with a 10% background contamination from K ± → π ± π 0 γ and K ± → π ± π 0 π 0 decays are observed in the kinematic range m γγ > 220 MeV. The combined model-dependent  [367]  measurement of the decay rate in the full kinematic range based on the two datasets is reported to be B(K ± → π ± γγ) = (1. 003 ± 0. 056) × 10 -6  [148]. Below the π 0 peak, the BNL E949 experiment has set an upper limit of B(K + → π + γγ) < 8. 3×10 -9 at 90% CL for m γγ < 108. 5 MeV based on the observation of no data events in this mass range, compared to the SM expectation of B = (6. 10 +0. 16  -0. 12 ) × 10 -9  [146]. This result is based on O(10 12 ) stopped kaons collected with a dedicated trigger line. None of the experiments has performed dedicated peak searches on the di-photon mass. Interpretation of these measurements in terms of searches for ALP decays is discussed in Section 2.
2. 2. 1. The NA62 Run 1 minimum bias dataset is estimated to contain about 3 × 10 3 K + → π + γγ candidates in the range m γγ > 220 MeV, which is the world's largest sample by far. Moreover, the m γγ resolution is improved with respect to the earlier datasets thanks to the beam tracker. A dedicated peak search at NA62 would improve the sensitivity to ALP decays substantially, and allow for an extension of the lower limit of the accessible m γγ range below 220 MeV. Due to the absence of a dedicated trigger line, NA62 is currently unable to improve on the E949 limit below the π 0 mass. The larger datasets to be collected at CERN (Sections 3. 1 and 3. 2) would improve the sensitivity further, although the search is already background-dominated. The K L → π 0 γγ decay rate has been measured to a 3% precision by NA48  [368]  and KTeV  [369]  using samples of O(  10 3 ) signal candidates.
2. 2. 2 and 2. 2. 3 (Figs. 10  and 12 ). The K + → π + e + e -decay sample collected by the NA62 experiment during Run 1 using the (downscaled) di-electron trigger is larger than the previous ones  [370] , and is essentially background free in both mass ranges, m ee < 100 MeV and m ee > 140 MeV  [371]. In the vicinity of the π 0 mass in the m ee spectrum, a large contribution from the K + → π + π 0 decay (followed by π 0 → e + e -and possibly π 0 D → γe + e -, π 0 DD → e + e -e + e -decays) is expected. A comprehensive analysis of the m ee spectrum in the full kinematic range is yet to be performed with the NA62 Run 1 dataset, including a π 0 → e + e -measurement competitive with the KTeV one (see Section 4. 7). While the blind spot in the vicinity of the π 0 mass can be avoided in the m ee resonance search, the sensitivity near the π 0 mass would be limited by the enhanced background, and by the uncertainty in the SM prediction of the π 0 → e + e -decay rate  [372].
The NA62 dataset also allows for a systematic search for the K + → π + X, X → e + e -process with displaced X decay vertices, providing sensitivity to X lifetimes of up to 10 ns. Searches for dark photon production and prompt decay K ± → π ± A , A → e + e -(Section 2. 4) are not competitive with the world data due to the suppression of the K ± → π ± A process  [373]. This has been demonstrated by the NA48/2 analysis in the 140-210 MeV mass range  [189] , leading to O(10 -5 ) sensitivity to the mixing parameter ε 2 , to be compared with the limits shown in Fig. 15  (right). The NA48/2 dataset has been used to obtain a sample of 1. 69 × 10 7 fully reconstructed Dalitz decays of the neutral pion, π 0 D → γe + e -decays, produced mainly in the K ± → π ± π 0 process. This sample has been used to search for a promptly decaying dark photon via the π 0 → γA , A → e + e - decay chain  [189].
1) is not competitive with the LHCb limits (Fig. 1 ). A preliminary study of 2. 8 × 10 4 K + → π + µ + µ -candidates obtained from the NA62 Run 1 dimuon dataset has been reported recently  [152]. An extension of this analysis to search for K + → π + X followed by prompt or displaced X → µ + µ -decays is yet to be performed. For the prompt decay, we expect the sensitivity to the product B(K + → π + X) • B(X → µ + µ -) to improve on average by about a factor of four with respect to NA48/2. In contrast with NA48/2, the NA62 trigger logic does not inhibit the displaced vertex search, which is almost background-free and would provide O(10 -10 ) sensitivity for X lifetimes in the 0. 1-10 ns range. This makes NA62 potentially competitive with LHCb in terms of the dark scalar search via the K + → π + ϕ, ϕ → µ + µ -decay chain for m ϕ > 2m µ (Section 2.
10 , are similar to or weaker than those obtained from the K + → π + + -decays. First observation of these decay modes can be expected within the hybrid phase of the future CERN kaon program (Section 3. 2). Unlike dedicated K L → π 0 ν ν experiments, the hybrid phase detector would be equipped with a tracker for measurements of charged particles produced in K L decays, enabling these and other measurements. First observation and study of the K ± → π ± π 0 e + e -decay has been reported by the NA48/2 experiment  [163]. Based on 4919 candidates with a 4. 9% background contamination, the decay branching fraction is measured to be B(K ± → π ± π 0 e + e -) = (4. 24 ± 0. 14) × 10 -6. This result has been used to evaluate the sensitivity to prompt ALP decays (Section 2. 2. 3, Fig. 12 ). The NA62 dataset is not expected to be competitive with NA48/2 due to the highly forward geometry leading to a reduced acceptance, and the downscaling applied to the di-electron trigger chain.
The K S,L → π + π -e + e -decays have been measured by the NA48 and NA48/1 experiments from low-background samples of O(10 4 ) candidates  [376, 377] , leading to PDG averaged branching fractions  [255] B(K S → π + π -e + e -) = (4. 79 ± 0. 15) × 10 -5 , (117) B(K L → π + π -e + e -) = (3. 11 ± 0. 19) × 10 -7. (118) It is expected that the future LHCb Run 3 dataset (Section 3. 4) would allow observation of the K S → π + π -e + e -decay, possibly surpassing the NA48/1 sensitivity  [378]. Improvements on the K L → π + π -e + e -decay are expected within the hybrid phase of the future CERN kaon program (Section 3. 2). Upper limits on the K L → π 0 π 0 + -decay rates have been set by the KTeV experiment from background-free analyses, including a search for resonances in the µ + µ -system at the O(10 -10 ) sensitivity level  [164, 379].
The K → π + α - α + β - β decays The quark level transition is s → dXX, with X decaying visibly, X → 2e, 2µ. In most models the quark transition is due to a decay chain s → dX ( * ) heavy , X ( * ) heavy → 2X. For on-shell X heavy one can search for resonances in 2 and 4 mass spectra. The kinematically allowed decays are K → π4e, K → π2e2µ, where X → e + e -and/or X → µ + µ -. One may also consider the flavor violating options K → π3eµ, K → πe3µ, which would arise from a combination of X → e + e -, X → µ + µ -and X → µ ± e ∓ decays. A related signature is also K → π4 +inv. The models are discussed in Section 2. 8. The NA62 Run 1 di-electron dataset offers an opportunity for the first search for the K + → π + A A decay followed by prompt A → e + e -decays, as well as the prompt decay chain including an intermediate resonance (K + → π + ϕ, ϕ → A A , A → e + e -), and the SM K + → π + e + e -e + e - decay (Section 2.
8. 1). One can expect single event sensitivities to the decay branching fractions at the O(10 -9 ) level. The background is likely to be dominated by the K + → π + π 0 D π 0 D decays (where π 0 D denotes the Dalitz decay, π 0 D → γe + e -) with two undetected soft photons, as well as by coincidences in time of multiple K + → π + π + π -decays and possibly other kaon decays. The backgrounds can be expected to be low but are yet to be evaluated. Further possibilities of multi-track processes with muons (Section 2. 8. 1)  or missing energy (Section 2. 8. 4)  have not been considered by experiments yet, though they could be searched for in principle with the NA62 Run 1 di-lepton datasets.
The K L → γ + inv decays The relevant model is dark photon in Section 2. 4. The K L → γA decay to a massless dark photon A (Section 2. 4. 1) is being searched for by KOTO. A dataset corresponding to 7 × 10 16 pot (i. e. about 0. 1% of the total KOTO exposure to date) has been collected requiring a single cluster in the calorimeter. The expected single event sensitivity is O(10 -6 ), and the main background is caused by the beam neutrons. The K L → γA decay also represents a natural target for KOTO step-2 and KLEVER experiments.
4. 10 The K → πγ and K L → 3γ decays The decays K → πγ and K L → 3γ are forbidden by Lorentz and gauge invariance. Since the observational bounds on the photon mass are very severe, the relevant models induce such decays through Lorentz violation in Section 2. 10. 4. The BNL E949 experiment has set an upper limit B(K + → π + γ) < 2. 3 × 10 -9 at 90% CL, obtained as a by-product of a search for K + → π + γγ near the pion momentum endpoint  [146]. This search is based on O(10 12 ) stopped kaons collected with a dedicated trigger line, and is backgroundfree. In contrast, NA62 has only donwscaled minimum bias datasets available for this search, with an estimated Run 1 single-event sensitivity at the O(10 -9 ) level. A search with the KOTO 2016-2018 dataset has led to an upper limit of B(K L → π 0 γ) < 1.
The K L → γγ + inv decays This is an important channel for the GN violating models discussed in Section 2. 7. The models can be grouped into i) the π 0 impostor decays, K L → X 2 X 1 , X 2 → γγ with X 1 invisible, ii) dipole portal transitions K L → X 2 X 2 , X 2 → X 1 γ, and iii) the non-standard π 0 production, K L → X 2 X 1 , X 2 → X 1 π 0 , as well as K L → X * 2 → X 1 π 0 or K L → X * 2 → X 1 X 1 π 0. Searches for the K + → µ + N decay, with N invisible, have been performed with prescaled minimum bias datasets since a single muon is the most common K + decay final state. The BNL E949 and NA62 (Run 1) experiments have established O(10 -8 ) upper limits on |U µN | 2 in complementary HNL mass ranges of 180-300 MeV and 200-384 MeV, respectively  [171, 389]. The constraints on HNL mass and mixing parameters |U N | 2 established by these searches are shown in Fig.
NA62 is expected to able to perform a search for the K + → e + νX decays with invisible X at the O(10 -6 ) sensitivity level with the Run 1 minimum bias dataset. The current upper limit B(K + → e + νν ν) < 6 × 10 -5  [391]  is also likely to be improved by NA62 by 1-2 orders of magnitude. Similarly to the K + → e + N case, this search would be strongly limited by background.
The K + → + α - β + β + inv decays This signature can arise from K + → + α (N → + β - β ν) decays , where the heavy neutral lepton decays through a non-standard neutral current interaction such as an extra U (1) d (Sections 2. 3 and 2. 8), or from K + → + α ν(a/A → + β - β ) , such as in leptonic force models (Section 2. 5). Out of the four SM four-lepton rare K + decay modes  [392] , all with the exception of K + → µ + µ -µ + ν have been measured by the BNL E865 experiment  [393, 394]. The NA62 Run 1 dilepton dataset offers an opportunity to measure all these processes. Due to the excellent resolution on the vertex position and kinematic variables, NA62 has observed for the first time K + → µ + µ -µ + ν decay with the SM branching fraction of 1. 35 × 10 -8  [392]  affected by the background from the abundant K + → π + π + π -process followed by pion decays in flight  [395].
4. 16 The K + → + α γγ + inv decays This signature can be due to K + → + α ν(a → γγ) decays, if the leptonic or neutrino force mediator, a, decays to photons (Sections 2. 5 and 2. 6). The other possibility is K + → + α N , where the HNL decays via the N → γγν channel. An example is the N → aν, a → γγ decay chain, see Section 2. 3 and also models in Section 2. 8. The two photons may also be a signal of the π 0 decay in the K + → π 0 + α N chain. The K + → µ + νX, X → γγ decay chain, involving both prompt and displaced X decay vertices, allows for a search for the muonic force mediator (Section 2. 5). Reconstruction of a visible resonance improves the sensitivity with respect to the case of an invisible X. The expected single-event sensitivity (i. e. the sensitivity in the assumption of no background) provided by the NA62 Run 1 minimum bias dataset is O(10 -9 ).
Nevertheless, the K + → π 0 e + N decay uniquely provides the sensitivity to |U eN | 2 in the mass gap 135-144 MeV not currently covered by either the K + → e + N search  [169]  (due to the background conditions) or the π + → e + N search  [388]  (due to the kinematic endpoint), as seen in Fig. 14  (left). The estimated sensitivity to |U eN | 2 achievable with the NA62 Run 1 dataset in this mass range is 2 × 10 -6  [174].
Lepton flavor violating kaon decays The decays are K L → µ ± e ± and K + → π + µ ± e ∓ , for instance from s → da/Z quark level transition, followed by the flavor violating decay of ALP (light Z ). The ALP can also be produced off-shell. The restrictions on such models are discussed in Section 2. 10. NA62 has obtained an upper limit of B(K + → π + µ -e + ) < 6. 6 × 10 -11 at 90% CL with the Run 1 di-lepton dataset, under the assumption of uniform phase space distribution  [396]. This analysis has also led to the upper limit of B(π 0 → µ -e + ) < 3. 2 × 10 -10 at 90% CL, using the K + → π + π 0 process as the source of tagged neutral pions. For the K + → π + µ + e -decay, the strongest limit to date of 1. 3×10 -11 at 90% CL comes from the BNL E777/865 experiments  [300]. The NA62 Run 1 sensitivity for this mode is comparable, and a new search with the Run 1 data is not justified.
The sensitivity of NA62 and follow-up experiments (Section 3. 2) is expected to improve almost linearly with the size of the future datasets thereby reaching O(10 -12 ), and searches for displaced X → µ ± e ∓ vertices are possible at NA62 in principle. The LHCb experiment is projected to reach a similar O(10 -12 ) sensitivity to the K + → π + µ ± e ∓ decays in the long term  [397]. The BNL E871 experiment has established an upper limit of B(K L → µ ± e ∓ ) < 4. 7 × 10 -12 at 90% CL  [398]. The KTeV experiment has established the following upper limits at 90% CL: B(K L → π 0 µ ± e ∓ ) < 7. 56 × 10 -11 , B(K L → π 0 π 0 µ ± e ∓ ) < 1. 64 × 10 -10 [399], and B(K L → e ± e ± µ ∓ µ ∓ ) < 4. 1 × 10 -11  [256]. Further improvements on LFV decays of the K L are foreseen at the hybrid phase of the future CERN kaon program (Section 3.
Lepton number violating kaon decays If the heavy neutral lepton is a Majorana fermion, then the quark level transition us → + α (N * → dū + β ) is possible, resulting in the decay K + → π -+ α + β (see Section 2. 3). NA62 has reported the following upper limits at 90% CL of the decay branching fractions using Run 1 di-lepton datasets: B(K + → π -e + e + ) < 5. 3 × 10 -11 and B(K + → π -π 0 e + e + ) < 8. 5 × 10 -10 [370]; B(K + → π -µ + µ + ) < 4. 2 × 10 -11  [371] ; B(K + → π -µ + e + ) < 4. 2 × 10 -11  [396]. These results are based on the full Run 1 dataset, with the exception of the K + → π -µ + µ + bound based on about 30% of the dataset. The results are obtained under the assumption of uniform phase space distributions, which is customary for LNV decay searches. The searches are not currently limited by backgrounds, and the sensitivity is expected to improve in the future almost linearly with the size of the dataset, possibly with the exception of the K + → π -µ + µ + mode.
Rare K S decays Most of the rare K L decays have their K S counterparts, but with τ K S /τ K L = O(10 -3 ) smaller branching ratios, which means that significantly more K S decays are required for a similar reach. The signatures with potentially interesting phenomenological reach are K S → π2 that can arise for Higgs mixed scalar (Section 2. 1), promptly decaying ALP (Section 2. 2) and dark photon (Section 2. 4), as well as K S → 2 2γ (see also Section 4. 12) and K S → 2 (see Section 4. 13). The observable specific to K S is the CP violating amplitude A(K S → µ + µ -) =0 which represents a clean probe for heavy new physics (Section 2. 11. 2). The NA48/1 experiment at CERN has reported the first evidence for the rare decays K S → π 0 + -using the full dataset collected in 2002, measuring their branching fractions to be  [400, 401]  B(K S → π 0 e + e -) = 3.
0 +1. 5 -1. 2 × 10 -9 , B(K S → π 0 µ + µ -) = 2. 9 +1. 5 -1. 2 × 10 -9 , in agreement with the SM expectations. In the former case, the measurement is performed in the accessible kinematic range m ee > 165 MeV/c 2. The LHCb experiment is expected to overtake the NA48/1 sensitivity for K S → π 0 µ + µ -channel with the future Run 3 dataset  [402]. The first observation of the rare K S → π 0 γγ decay reported by the NA48 experiment based on the dataset collected in 2000 is also in agreement with the SM expectation  [403] : B(K S → π 0 γγ) = (4. 9 ± 1. 8) × 10 -8. The LHCb experiment has recently reported an upper limit at 90% CL on the K S → µ + µ - branching fraction using the combined Run 1+2 dataset  [404] : B(K S → µ + µ -) < 2. 1 × 10 -10 , to be compared to the SM prediction of B SM (K S → µ + µ -) = (5.