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KAON (Karlsruhe ontology)[1] is an ontology infrastructure developed by the University of Karlsruhe and the Research Center for Information Technologies in Karlsruhe. Its first incarnation was developed in 2002 and supported an enhanced version of RDF ontologies. Several tools like the graphical ontology editor OIModeler or the KAON Server were based on KAON.

There are ontology learning companion tools which take non-annotated natural language text as input: TextToOnto (KAON-based) and Text2Onto (KAON2-based). Text2Onto is based on the Probabilistic Ontology Model (POM).[2]

In 2005, the first version of KAON2 was released, offering fast reasoning support for OWL ontologies. KAON2 is not backward-compatible with KAON. KAON2 is developed as a joint effort of the Information Process Engineering (IPE) at the Research Center for Information Technologies (FZI), the Institute of Applied Informatics and Formal Description Methods (AIFB) at the University of Karlsruhe, and the Information Management Group (IMG) at the University of Manchester.[3]

KAON, TextToOnto, and Text2Onto are open source, based on Java. KAON2 is not open source,[4] but the executable can be downloaded from the KAON2 site.

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KAON

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A kaon, also known as a K meson, is a type of consisting of a bound to an up or down antiquark (or the corresponding antiquarks), characterized by the conserved of (S=±1S = \pm 1) in strong and electromagnetic interactions. The four fundamental kaon states are the positively charged K+K^+ (usˉu\bar{s}), the negatively charged K−K^- (uˉs\bar{u}s), the neutral K0K^0 (dsˉd\bar{s}), and the antineutral Kˉ0\bar{K}^0 (dˉs\bar{d}s). The neutral kaons undergo mixing via the , forming two distinct mass eigenstates: the short-lived, nearly CP-even KS0K_S^0 and the long-lived, nearly CP-odd KL0K_L^0. Discovered in 1947 through experiments using cloud chambers, kaons revealed the existence of particles that decayed slowly despite being produced rapidly, challenging existing theories and leading to the concept of . Kaons are unstable subatomic particles that decay predominantly through the weak , with no modes due to flavor-changing processes. The charged kaons have a of 493.677±0.013493.677 \pm 0.013 MeV/c2c^2 and a mean lifetime of (1.2379±0.0021)×10−8(1.2379 \pm 0.0021) \times 10^{-8} s, with principal decay modes including K±→μ±νμK^\pm \to \mu^\pm \nu_\mu (branching fraction 63.56±0.11%63.56 \pm 0.11\%), K±→π±π0K^\pm \to \pi^\pm \pi^0 (20.67±0.08%20.67 \pm 0.08\%), and K±→π±π+π−K^\pm \to \pi^\pm \pi^+ \pi^- (5.583±0.024%5.583 \pm 0.024\%). The neutral K0K^0 has a of 497.611±0.013497.611 \pm 0.013 MeV/c2c^2, while KS0K_S^0 and KL0K_L^0 have es of 497.611±0.013497.611 \pm 0.013 MeV/c2c^2 and 497.611±0.013497.611 \pm 0.013 MeV/c2c^2, respectively, with a difference Δm=(5.293±0.004)×109 ℏ\Delta m = (5.293 \pm 0.004) \times 10^{9} \, \hbar/s. The KS0K_S^0 lifetime is (0.8954±0.0004)×10−10(0.8954 \pm 0.0004) \times 10^{-10} s, primarily decaying to two pions (π+π−\pi^+ \pi^- or π0π0\pi^0 \pi^0, branching fraction ∼100%\sim 100\% excluding CP-violating modes), whereas the KL0K_L^0 lifetime is (5.116±0.021)×10−8(5.116 \pm 0.021) \times 10^{-8} s, with dominant decays to three pions (∼40%\sim 40\%) and semileptonic modes like π±e∓νˉe\pi^\pm e^\mp \bar{\nu}_e or π∓e±νe\pi^\mp e^\pm \nu_e (∼40%\sim 40\%). The study of kaons has profoundly influenced , introducing as a multiplicative to explain their production and decay patterns under the strong force while allowing weak decays to change it. Their discovery prompted the development of the by and in the 1960s, organizing hadrons into multiplets based on flavor symmetry. Notably, the 1964 observation of KL0→π+π−K_L^0 \to \pi^+ \pi^- decays, forbidden under exact CP symmetry, provided the first experimental evidence of , a key mechanism for explaining the in the universe and testing the Cabibbo-Kobayashi-Maskawa matrix in the . Ongoing experiments, such as those at CERN's NA62 and Fermilab's KOTO, utilize kaon decays to probe rare processes, search for , and precisely determine fundamental parameters like the Cabibbo angle.

Overview and Discovery

Definition and Classification

Kaons are pseudoscalar mesons in the Standard Model of particle physics, each consisting of a strange quark bound to an up or down antiquark, or the corresponding antiquark combinations. They form a quartet of particles distinguished by charge and strangeness: the positively charged kaon K+K^+, the negatively charged kaon K−K^-, the neutral kaon K0K^0, and its antiparticle Kˉ0\bar{K}^0. In the , the compositions are K+=usˉK^+ = u\bar{s}, K0=dsˉK^0 = d\bar{s}, K−=uˉsK^- = \bar{u}s, and Kˉ0=dˉs\bar{K}^0 = \bar{d}s, where uu and dd denote up and down quarks, respectively, and ss the . These mesons have total spin 0 and negative intrinsic parity (JP=0−J^P = 0^-), placing them in the category alongside lighter mesons like pions. As strange mesons, kaons carry the conserved of SS, defined such that particles containing a strange antiquark (sˉ\bar{s}) have S=+1S = +1 (for K+K^+ and K0K^0), while those with a (ss) have S=−1S = -1 (for K−K^- and Kˉ0\bar{K}^0). This distinguishes kaons from non-strange mesons such as pions, which are composed only of up and down quarks/antiquarks and lack this ; the inclusion of the heavier in kaons results in greater mass and altered stability compared to pions. Kaons play a key role in weak interactions, where processes can change their .

Historical Context

The discovery of kaons occurred in 1947 when British physicists George D. Rochester and Clifford C. Butler observed unusual forked tracks in cosmic ray experiments using a at the . These tracks indicated the decay of previously unknown unstable particles, dubbed V-particles (V⁰ for neutral and V⁺ for charged), with estimated masses approximately 900–1000 times that of the and lifetimes around 10⁻¹⁰ seconds. Later analyses confirmed these as the neutral and charged kaons, marking the first observation of particles with . In the early 1950s, observations of particles termed θ (decaying to two pions) and τ (decaying to three pions) revealed a puzzle, as they shared similar masses (around 494 MeV/c²) and lifetimes (about 10⁻¹⁰ seconds) but exhibited decay modes with opposite parity, suggesting they could not be the same particle under the assumption of parity conservation in weak interactions. This θ-τ puzzle prompted theoretical advancements; in , proposed "associated production," where strange particles are created in pairs via strong interactions to conserve a new additive . Independently in 1953, and Kazuhiko Nishijima formalized this quantum number as (S), assigning S = +1 to K⁺ and K⁰, and S = -1 to their antiparticles, while resolving the puzzle by linking the θ and τ to the same kaon particle whose decays violate parity. The term "kaon" originated from "K-meson," with the letter K selected in the early 1950s to denote particles carrying non-zero , distinguishing them from lighter π-mesons and heavier hyperons, and reflecting the conservation of strangeness in associated production processes governed by strong interactions. A key experimental milestone came in 1953 at the , where cosmic ray emulsion studies by Robert W. Birge and collaborators measured the masses of positive K-mesons, confirming the identities of K⁺ (mass 493.7 MeV/c²) and neutral K⁰ through decay kinematics. Further confirmation of strangeness properties followed in 1956, when and Chen-Ning Yang proposed parity non-conservation in weak interactions to fully resolve the θ-τ puzzle, a prediction experimentally verified in kaon decays and supported by Chien-Shiung Wu's 1957 experiment demonstrating parity violation.

Physical Properties

Basic Characteristics

Kaons are mesons consisting of a and a light quark (up or down). The charged kaons, K⁺ (u \bar{s}) and K⁻ (\bar{u} s), have electric charges of +1e and -1e, respectively, while the neutral kaons, K⁰ (d \bar{s}) and \bar{K}⁰ (\bar{d} s), are electrically neutral. The es of the kaons are well-measured, with the charged kaons having a of 493.677 ± 0.013 MeV/c² for both K⁺ and K⁻, while the flavor eigenstates neutral kaons have a of 497.611 ± 0.013 MeV/c² for both K⁰ and \bar{K}⁰. The physical neutral eigenstates are K_S^0 with 497.614 ± 0.013 MeV/c² and K_L^0 with 497.978 ± 0.013 MeV/c². These es exhibit negligible differences between particles and their antiparticles due to CPT invariance. The mean lifetimes of kaons vary significantly between charged and neutral species. For the charged kaons, both K⁺ and K⁻ have a mean lifetime of (1.2379 ± 0.0021) × 10^{-8} s. The neutral kaon system mixes to form the short-lived K_S with a mean lifetime of (0.8954 ± 0.0004) × 10^{-10} s and the long-lived K_L with (5.116 ± 0.021) × 10^{-8} s. Key electromagnetic properties of kaons include the decay constant, which parametrizes their leptonic decay rates and matrix elements. The charged kaon decay constant is f_{K^+} = 155.7 ± 0.3 MeV, determined from averages.
PropertyK⁺ / K⁻K⁰ / \bar{K}⁰K_S^0K_L^0
Charge (e)+1 / -1000
Mass (MeV/c²)493.677 ± 0.013497.611 ± 0.013497.614 ± 0.013497.978 ± 0.013
Mean Lifetime (s)(1.2379 ± 0.0021) × 10^{-8}—(0.8954 ± 0.0004) × 10^{-10}(5.116 ± 0.021) × 10^{-8}
Decay Constant (MeV)155.7 ± 0.3 (f_{K^+})Not applicableNot applicableNot applicable

Strangeness and Quantum Numbers

Kaons are characterized by the quantum number SS, which arises from the presence of the or antiquark in their composition. The K+K^+ (usˉ\bar{s}) and K0K^0 (dsˉ\bar{s}) mesons have S=+1S = +1, while the K−K^- (uˉ\bar{u}s) and Kˉ0\bar{K}^0 (dˉ\bar{d}s) have S=−1S = -1. This quantum number is strictly conserved in processes mediated by the strong and electromagnetic interactions, reflecting the approximate flavor symmetry of (QCD) at low energies, but it is violated in weak interactions, allowing kaons to decay into non-strange final states. The following table summarizes the key quantum numbers for the kaon states:
ParticleQuark ContentStrangeness SS III3I_3 YYG-Parity
K+K^+usˉ\bar{s}+11/2+1/2+1−
K0K^0dsˉ\bar{s}+11/2−1/2+1−
K−K^-uˉ\bar{u}s−11/2−1/2−1−
Kˉ0\bar{K}^0dˉ\bar{d}s−11/2+1/2−1−
These assignments are derived from the quark model and experimental observations. In the isospin formalism, which treats the up and down quarks as an approximate SU(2) symmetry doublet, the kaons form two isospin doublets: (K+,K0)(K^+, K^0) with total isospin I=1/2I = 1/2 and third component I3=+1/2I_3 = +1/2 for K+K^+ and I3=−1/2I_3 = -1/2 for K0K^0; the antiparticles (Kˉ0,K−)(\bar{K}^0, K^-) form the conjugate doublet. This structure parallels the nucleon doublet (proton, neutron) and ensures that strong interaction processes respect isospin symmetry, leading to equal production rates for K+K^+ and K0K^0 in the absence of electromagnetic corrections. The YY is defined as Y=B+SY = B + S, where BB is the (zero for mesons), so Y=SY = S for kaons, yielding Y=+1Y = +1 for K+K^+ and K0K^0, and Y=−1Y = -1 for K−K^- and Kˉ0\bar{K}^0. , along with , forms the basis of the SU(3) flavor , under which kaons transform as part of the fundamental representations. G-parity, an extension of charge conjugation combined with a 180-degree rotation, is assigned as negative for the kaon multiplets, consistent with their pseudoscalar nature and odd intrinsic parity. Within the , kaons exemplify the SU(3) flavor symmetry, where the light quarks (u, d, s) transform under the fundamental representation 3 of SU(3). The pseudoscalar mesons, including the kaon isodoublets, pions (I=1 triplet), and (I=0 singlet-octet mixture), fill the 8 (octet) of SU(3), as predicted by the eightfold way. This octet structure arises from the quark-antiquark combinations qˉq′\bar{q} q', with the content distinguishing kaons from non-strange mesons, and it successfully organizes the observed spectrum and decay patterns under approximate flavor symmetry due to the mass.

Decay Processes

Semileptonic Decays

Semileptonic decays of kaons involve the emission of a lepton and a neutrino alongside a hadronic system, primarily probing flavor-changing charged-current weak interactions mediated by the s→us \to u quark transition. These processes are characterized by the presence of both leptonic and hadronic currents, with the hadronic part described by form factors that encode non-perturbative QCD effects. The dominant semileptonic modes for the charged kaon K+K^+ are K+→π0e+νeK^+ \to \pi^0 e^+ \nu_e with a branching ratio of (5.07±0.04)%(5.07 \pm 0.04)\% and K+→π0μ+νμK^+ \to \pi^0 \mu^+ \nu_\mu with (3.352±0.034)%(3.352 \pm 0.034)\%, while the purely leptonic decay K+→μ+νμK^+ \to \mu^+ \nu_\mu dominates overall charged kaon weak decays at (63.56±0.11)%(63.56 \pm 0.11)\%. These branching ratios reflect the suppression of strangeness-changing transitions relative to non-strange ones, governed by the Cabibbo-Kobayashi-Maskawa (CKM) matrix element ∣Vus∣|V_{us}|. The rate for these decays is proportional to ∣Vus∣2sin⁡2θC|V_{us}|^2 \sin^2 \theta_C, where θC\theta_C is the Cabibbo angle, with sin⁡θC≈0.22\sin \theta_C \approx 0.22 determined precisely from kaon semileptonic branching ratios combined with lifetime and form factor inputs. Early extractions of sin⁡θC\sin \theta_C from kaon decays, such as those by Cabibbo in 1963, established the universality of weak interactions across quark generations. Modern analyses yield ∣Vus∣=0.2238±0.0010|V_{us}| = 0.2238 \pm 0.0010 from Kl3K_{l3} modes (l=e,μl = e, \mu), achieving sub-percent precision and serving as a benchmark for CKM unitarity tests. The hadronic matrix element for K→πlνK \to \pi l \nu is parameterized by vector and scalar form factors f+(q2)f_+(q^2) and f−(q2)f_-(q^2), where q2q^2 is the momentum transfer to the pair: ⟨π(pπ)∣Vμ∣K(pK)⟩=f+(q2)[(pK+pπ)μ−mK2−mπ2q2qμ]+f−(q2)mK2−mπ2q2qμ,\langle \pi(p_\pi) | V^\mu | K(p_K) \rangle = f_+(q^2) \left[ (p_K + p_\pi)^\mu - \frac{m_K^2 - m_\pi^2}{q^2} q^\mu \right] + f_-(q^2) \frac{m_K^2 - m_\pi^2}{q^2} q^\mu, with q=pK−pπq = p_K - p_\pi. The f−(q2)f_-(q^2) term contributes negligibly in the electron mode due to the small mass but is more relevant for . Helicity suppression arises in the channel because the massive μ−\mu^- prefers left-handed helicity in the V-A interaction, reducing the overlap with the spin-0 kaon-to-pion transition compared to the massless case; this, combined with phase-space factors, explains the ∼30%\sim 30\% lower branching ratio for K+→π0μ+νμK^+ \to \pi^0 \mu^+ \nu_\mu relative to the electron mode. Form factors are typically expanded linearly as f+(q2)=f+(0)[1+λ+q2/mπ+2]f_+(q^2) = f_+(0) [1 + \lambda_+ q^2 / m_{\pi^+}^2], with λ+=0.02959±0.00025\lambda_+ = 0.02959 \pm 0.00025 and f+(0)≈0.97f_+(0) \approx 0.97 from and experiment. Experimental measurements of these decays began with bubble chamber experiments in the 1960s–1970s, such as those at CERN's Gargamelle and SLAC's 15-foot bubble chamber, which provided early determinations of branching ratios and form factor slopes with ∼5–10%\sim 5–10\% precision by reconstructing decay topologies in neutrino beams or kaon sources. Modern detectors like NA62 at CERN have achieved higher precision, confirming branching ratios at the 1%1\% level and enabling stringent tests of lattice QCD predictions.

Nonleptonic Decays

Nonleptonic decays of kaons are weak processes in which a kaon transitions to a final state consisting solely of hadrons, without the emission of leptons. These decays are mediated by the ΔS=1 part of the weak Hamiltonian and are classified by the change in , ΔI, which can be 1/2 or 3/2. The dominant modes are the two-pion decays, such as K⁺ → π⁺ π⁰ with a branching ratio of (20.67 ± 0.08)% and K_S → π⁺ π⁻ with (69.20 ± 0.05)%. Three-pion modes, like K⁺ → π⁺ π⁰ π⁰, occur at lower rates, with a branching ratio of (1.760 ± 0.023)% for the charged kaon. A key feature of these decays is the ΔI=1/2 rule, which states that the amplitude for ΔI=1/2 transitions is enhanced relative to ΔI=3/2 transitions. In the two-pion final states, the isospin amplitudes A_0 (corresponding to total pion isospin I=0, from ΔI=1/2) and A_2 (I=2, from ΔI=3/2) satisfy |A_0 / A_2| ≈ 22.2, far exceeding the naive SU(3) expectation of unity. This enhancement arises from non-perturbative QCD effects, such as gluon exchange in the , and has been a longstanding puzzle in flavor physics. Isospin analysis of the charged and neutral modes provides relations between the observed rates and these amplitudes, confirming the rule's validity to high precision. Chiral perturbation theory (ChPT), the effective field theory for low-energy QCD, provides a systematic framework for calculating nonleptonic decay amplitudes. At leading order, ChPT reproduces the ΔI=1/2 dominance through the structure of the weak . Higher-order calculations, including loop effects and counterterms, predict decay rates and ratios with accuracies of a few percent; for example, the slope parameter in K → 3π decays and the ππ scattering phases are used to match experimental branching ratios. These predictions align well with data, validating ChPT up to O(p^4) for the dominant modes. Rare nonleptonic modes, such as K⁺ → π⁺ ν ν̄, probe flavor-changing neutral currents (FCNC) and are highly suppressed in the (SM), occurring via Z-penguin and box diagrams. The NA62 experiment observed this decay, first reported in 2024 and published in 2025, with a branching ratio of (13.0_{-3.0}^{+3.3}) × 10^{-11}, consistent with the SM expectation of (8.4 ± 0.6) × 10^{-11} but allowing sensitivity to new physics contributions that could enhance or suppress the rate by up to an . Such modes provide clean tests of short-distance weak interactions due to minimal hadronic uncertainties.

Parity and CP Violation

Parity Violation in Decays

The θ-τ puzzle arose from observations in the 1950s that the charged kaon appeared to decay via two distinct modes: the θ⁺ to two pions (a CP-even final state with even parity) and the τ⁺ to three pions (a CP-odd final state with odd parity), despite sharing the same mass and lifetime, suggesting they were the same particle. This contradiction implied a violation of parity conservation in weak interactions, as proposed by and Yang in their seminal analysis of available weak decay data. The puzzle was resolved by recognizing that the weak decays of kaons do not obey parity invariance, allowing a single particle to access final states of opposite intrinsic parity. The proposal of parity non-conservation was experimentally verified shortly thereafter through the , which demonstrated asymmetric electron emission in the of polarized ^{60}Co nuclei, confirming maximal parity violation in weak interactions. This result directly paralleled the resolution of the θ-τ puzzle in kaon decays, as both processes are mediated by the weak force, where parity is violated to the extent that left-handed fermions couple preferentially. In nonleptonic charged kaon decays such as K⁺ → π⁺ π⁰ π⁰, parity violation manifests as asymmetric angular distributions of the pions, observable in the Dalitz plot representation of the decay kinematics. The linear slope parameter g, which quantifies the asymmetry along the energy of one pion, is measured to be g = 0.58 ± 0.01, a non-zero value arising from interference between parity-conserving and parity-violating amplitudes in the weak decay matrix element. The fundamental origin of this parity violation lies in the vector-axial vector (V-A) structure of the charged weak current, where only left-handed and fields participate, producing parity-odd correlations between spin and momentum in decay products. This V-A form, established from analyses of and decay spectra, applies universally to semi-leptonic and non-leptonic weak processes, including those of kaons.

CP Violation in Neutral Kaons

The discovery of CP violation occurred in 1964 when and Val Fitch observed the decay KL→π+π−K_L \to \pi^+ \pi^-, a process expected to be forbidden under CP conservation, with a branching ratio of approximately 0.17%0.17\%. This unexpected result, obtained using neutral kaons produced at the Alternating Gradient Synchrotron at , demonstrated that the combined symmetry of charge conjugation (C) and parity (P) is not conserved in weak interactions. The challenged the prevailing understanding of symmetries in and earned Cronin and Fitch the 1980 . Indirect CP violation in neutral kaons arises primarily from the complex phase in the Cabibbo-Kobayashi-Maskawa (CKM) matrix, which parametrizes mixing in the . This phase leads to a small admixture of the CP-even state K1K_1 into the CP-odd KLK_L , quantified by the parameter ε\varepsilon, measured as ∣ε∣=(2.228±0.011)×10−3|\varepsilon| = (2.228 \pm 0.011) \times 10^{-3} from the decay amplitude ratio η+−=A(KL→π+π−)/A(KS→π+π−)\eta_{+-} = A(K_L \to \pi^+ \pi^-)/A(K_S \to \pi^+ \pi^-). The value of ε\varepsilon is determined through precise fits to kaon decay rates and lifetimes, incorporating corrections for CPT invariance and breaking effects. In the , ε\varepsilon is proportional to the imaginary part of (VtdVts∗)2(V_{td} V_{ts}^*)^2, directly linking it to the CKM phase and providing a key test of the model's CP-violating sector. Direct CP violation manifests as a difference in the decay amplitudes for CP-conjugate processes, independent of mixing, and is parametrized by ε′\varepsilon'. The ratio Re⁡(ε′/ε)\operatorname{Re}(\varepsilon'/\varepsilon) measures this effect relative to indirect violation, with the world average value (1.66±0.23)×10−3(1.66 \pm 0.23) \times 10^{-3} establishing its nonzero nature and confirming direct CP violation in KL→ππK_L \to \pi\pi decays. This result stems from high-precision measurements by the NA48 experiment at , which reported Re⁡(ε′/ε)=(1.42±0.34)×10−3\operatorname{Re}(\varepsilon'/\varepsilon) = (1.42 \pm 0.34) \times 10^{-3} using simultaneous KSK_S and KLK_L beams, and the KTeV experiment at , which obtained (1.67±0.18)×10−3(1.67 \pm 0.18) \times 10^{-3} from decay rate ratios with controlled systematics. In the , ε′\varepsilon' arises from gluonic and electroweak penguin diagrams, with its phase aligned to the CKM arg(VtdVts∗V_{td} V_{ts}^*), though hadronic uncertainties limit its predictive power. Measurements of ε\varepsilon and ε′\varepsilon' impose significant constraints on the CKM unitarity triangle, a geometric representation of quark mixing unitarity where the apex coordinates (ρˉ,ηˉ)(\bar{\rho}, \bar{\eta}) encode the . The ε\varepsilon parameter primarily bounds the ηˉ\bar{\eta}, yielding ηˉ≈0.35\bar{\eta} \approx 0.35 with inputs for bag parameters, while ε′\varepsilon' provides a complementary limit on ηˉ>0\bar{\eta} > 0 through its sensitivity to the same CKM phase, albeit with larger theoretical errors from . These kaon-derived bounds, when combined with B-meson data, overconstrain the triangle and validate the Standard Model's single-phase origin of , with tensions below 1% in global fits.

Neutral Kaon System

Mixing and Oscillation

The neutral kaon system consists of two flavor eigenstates, denoted as ∣K0⟩|K^0\rangle (with S=+1S = +1) and ∣K‾0⟩|\overline{K}^0\rangle (with S=−1S = -1), which are not eigenstates of and electromagnetic interactions but mix through second-order weak interactions involving ΔS=2\Delta S = 2 transitions. These transitions arise from virtual intermediate states, such as intermediate pions or other hadrons, mediated by the weak Hamiltonian HwH_w, leading to oscillations between the two states over time. This mixing results in two distinct mass eigenstates: the short-lived KSK_S (primarily CP-even) and the long-lived KLK_L (primarily CP-odd), which are superpositions of ∣K0⟩|K^0\rangle and ∣K‾0⟩|\overline{K}^0\rangle. In the absence of CP violation, ∣KS⟩=12(∣K0⟩+∣K‾0⟩)|K_S\rangle = \frac{1}{\sqrt{2}} \left( |K^0\rangle + |\overline{K}^0\rangle \right)
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