Subatomic particle

A subatomic particle is a constituent or excitation studied at the level below ordinary atomic structure. The category includes elementary particles, such as the electron, and composite hadrons, such as the proton and neutron.

Calling something a particle does not require a tiny classical solid sphere. Quantum mechanics describes localized detections and wave-like propagation, while high-energy scattering probes whether internal structure is resolved. A particle detector usually infers a particle from transferred energy, emitted light or a trajectory reconstructed from many interactions. The measured track width and the particle's intrinsic spatial structure are therefore different quantities.

Particle decay is a spontaneous transformation of an unstable particle into allowed final particles. Conservation of energy, momentum, electric charge and the relevant interaction's other conserved quantities constrain the possibilities. A decay can be energetically allowed yet absent or weak because its transition amplitude is suppressed.

For an isolated state with an approximately constant rest-frame decay rate, the surviving population follows an exponential in proper time. The mean lifetime $\tau$ and energy decay width $\Gamma$ obey $\Gamma =\hslash /\tau$. A moving particle's laboratory flight-time distribution also includes time dilation.

Experimentally, detectors reconstruct decay products and sometimes a separated decay vertex. Correcting for acceptance and reconstruction efficiency is necessary before interpreting their counts as a lifetime or branching fraction. Different decay channels compete to make the total width.

Decay of a massive parent into exactly two photons. In its rest frame, momentum conservation requires opposite photon momenta of equal magnitude, so each photon has energy $M{c}^2/2$. A boost along their common axis gives laboratory photon energies $\gamma M{c}^2(1\pm \beta )/2$. Their product stays $(M{c}^2/2{)}^2$ in that collinear configuration; other emission angles require the corresponding angular Lorentz transformation.

A parent of rest mass $M$ can decay to on-shell daughters of rest masses $m_i$ only if $M\ge {\sum }_i{m}_i$. In the parent rest frame, each daughter energy is at least its rest energy, so four-momentum conservation proves necessity. The excess energy controls the available Lorentz-invariant phase space. A two-body decay threshold is the two-daughter special case; spin and other selection rules can still forbid an energetically allowed channel.

For a parent moving at $\beta c$ and decaying into two photons of equal laboratory energies, momentum conservation gives photon directions symmetric about the parent direction with $\cos \theta =\beta$. The opening angle is $2\arccos \beta$. In the parent rest frame the photons are opposite and perpendicular to the boost direction.

An approximation to a heavy-hadron decay in which one heavy quark undergoes a weak decay and the other constituents are treated as spectators. The inclusive leading width is approximated by a free-quark width, summed over allowed final states. confinement, final-state masses, strong-interaction corrections and interactions with spectator constituents limit quantitative lifetime predictions.

In a massless Fermi interaction approximation, a charged-current decay channel contributes a common mass-to-the-fifth width multiplied by its mixing and final-state multiplicity factors. A final quark-antiquark pair has three color choices, while a lepton pair has one. An initial quark does not acquire a factor of three in its lifetime merely from its possible initial colors.

A particle decay into three daughter particles subject to four-momentum conservation and the relativistic energy-momentum relation. The daughter energies generally vary continuously even in the parent's rest frame. Grouping two daughters into a recoil system of variable invariant mass turns the remaining kinematics into a relativistic two-body decay.

A boundary of phase space approached as a massless outgoing particle's energy tends to zero. A zero four-vector is not a physical positive-energy one-particle state. An energy bound requiring such a particle is a supremum or infimum, even when the closed kinematic interval is conventionally described using minimum and maximum. This distinction can matter when applying three-body formulas at mixed massive-massless recoil thresholds.

Use natural units $c=1$ and assume open phase space, $M>m_1+m_2+m_3$. In the parent's rest frame, define $s_{23}=(p_2+p_3{)}^2$. Four-momentum conservation givesThe upper recoil-mass limit follows from $E_1\ge m_1$. For positive $m_2,m_3$, the lower recoil-mass limit follows in their pair rest frame and is attained when their relative momentum vanishes. If both recoil particles are massless, $s_{23}=2E_2{E}_3(1-\cos {\theta }_{23})$ reaches zero with nonzero parallel momenta. If exactly one is massless, $s_{23}$ approaches the nonzero daughter's squared mass only when the massless particle becomes soft. Thus that upper $E_1$ bound is a supremum for a genuine three-body event with all massless particles of positive energy. If $m_1=0$, the lower endpoint is likewise a soft limit. Multiplying the mass expressions for $E_1$ by $c^2$ restores energy units.

A weak interaction decay of a baryon into final hadrons without final leptons. A weak charged current changes quark flavour, while spectator quarks and the produced quarks undergo hadronization. A quark-flow Feynman diagram specifies a possible weak topology; it does not calculate the full hadronic decay amplitude.

A nonleptonic meson decay contains final hadrons and no final leptons. Weak quark transitions set flavour factors, but strong hadronization and additional hadronic matrix elements are needed to determine exclusive rates.

For Ds meson spectator decays, $c\to s{W}^{+}$ followed by $W^{+}\to u\bar{d}$ carries CKM magnitude $|V_{cs}{V}_{ud}|$ and can form either $\varphi {\pi }^{+}$ or $K^{+}{\bar{K}}^0$. Replacing $c\to s$ by $c\to d$ carries $|V_{cd}{V}_{ud}|$ and can form $K^{+}{\pi }^0$ through the $-d\bar{d}/\sqrt{2}$ pion component. With otherwise equal reduced hadronic strengths, the last channel's rate is suppressed relative to the neutral-antikaon channel by $|V_{cd}/V_{cs}{|}^2/2$. Actual branching fractions also depend on phase space, colour contractions and strong final-state interactions; the CKM factors do not determine exact exclusive rates.

A leptonic decay has only leptons in its non-radiative final state. For example, a charged pion can decay as ${\pi }^{+}\to {\mu }^{+}{\nu }_{\mu }$. A semileptonic decay instead retains at least one hadron alongside the leptons. This classification concerns the final particles, not whether the initial particle is a hadron or a lepton.

At tree level in a local Fermi interaction with negligible neutrino masses, the charged daughter mass modifies the integrated three-body V-A interaction rate by $f(m_a^2/m_l^2)$. The function tends to one at zero daughter mass and to zero at threshold. It includes both the phase space restriction and the momentum dependence of the matrix element.

A particle decay cascade follows successive unstable daughters to longer-lived or stable particles. Probabilities of specified independent decay paths multiply their branching fractions. Waiting times are governed by each particle's mean lifetime, with time dilation relating proper and laboratory times. A particle which appears stable on one observation timescale can be an intermediate on another.

A semileptonic decay contains both final-state hadrons and leptons, typically through a weak charged current. This distinguishes it from a purely leptonic decay.

Pion beta decay changes a charged pion into a neutral pion and a lepton pair through the weak charged current. The small charged-neutral pion mass gap severely restricts phase space, giving an approximately fifth-power dependence on that gap. It does not have the charged-lepton mass suppression of direct leptonic pion decay.

A Dalitz decay replaces a radiated real photon with a virtual photon converting to a charged-lepton pair. The neutral-pion example is ${\pi }^0\to \gamma e^{+}{e}^{-}$. Double Dalitz decay converts both virtual photons in the two-photon mode.

Double Dalitz decay converts both virtual photons from a two-photon hadronic transition into charged-lepton pairs. The neutral-pion four-electron final state avoids the specific two-lepton helicity suppression of ${\pi }^0\to e^{+}{e}^{-}$, despite its more restricted phase space.

A decay channel specifies a possible final state of a particle decay, such as an electron, a neutrino and an antineutrino in a specified muon decay. Different channels can have different interaction amplitudes, available phase space and experimental visibility.

For disjoint final-state classes, the partial decay widths add: ${\Gamma }_{\mathrm{total}}={\sum }_i{\Gamma }_i$. The branching fraction is ${\Gamma }_i/{\Gamma }_{\mathrm{total}}$, so increasing one partial width changes the fractions assigned to the others even if their own amplitudes are unchanged. Intermediate resonances leading to the same final state can interfere; they cannot always be treated as mutually exclusive counting bins.

An elementary particle has no resolved constituent substructure within the physical theory and experiments under discussion. In the Standard Model, examples include electrons, quarks and photons. A proton is instead a composite hadron, even though it often acts as one particle in a detector or an atomic model.

The distinction is tested through interactions rather than by picturing a smaller material ball inside a larger one. Scattering at increasing momentum transfer probes shorter spatial scales and can reveal internal response. A fundamental quantum field describes the particle's excitations; being elementary does not mean that all measured electromagnetic properties reduce to those of a classical point charge.

A negative squared mass in a relativistic field spectrum signals instability of the background about which the field is expanded. The bosonic string mass spectrum contains such an unexcited state.

A hadron is a strongly interacting composite particle described by quantum chromodynamics. Familiar hadrons contain quarks and gluons in an overall color-neutral state. The proton and neutron are baryons; the pion is a meson.

The quark model labels ordinary hadrons by their valence content, but this does not list every dynamical constituent. Gluons and additional quark–antiquark contributions affect masses and interactions. Distinguishing a composite hadron from an elementary particle is useful when comparing low-energy models with scattering experiments that resolve internal structure.

A hadronic form factor parametrizes a current matrix element between hadron states, separating allowed Lorentz covariant tensor structures from scalar-valued functions of Lorentz invariant combinations of momentum transfer. These functions encode composite structure and strong interaction dynamics. Equal kinematic phase space does not imply equal form factors for different weak interaction transitions.

Ordinary baryons have three valence quarks, and ordinary mesons have a quark and an antiquark. Additive quantum numbers such as electric charge, baryon number and strangeness are sums of constituent values. This valence description organizes flavor symmetry multiplets while allowing sea constituents and gluons.

The quark model makes a large catalogue manageable: the same few quark flavors generate many different hadrons. It predicts relations between electric charge, strangeness and isospin, rather than assigning these independently to each new particle. It is especially useful for asking which combinations are possible and which additional measurements could distinguish them. A constituent list alone does not determine a mass, a decay rate or whether a proposed combination forms a bound state.

Four quark flavours in the defining representation of flavour SU(4) and their antiquarks give sixteen meson flavour states. The trace of $M^i{}_j=q^i{\bar{q}}_j$ is the trivial representation; its traceless part is the Adjoint representation. Under light-flavour SU(3), the full content is $8\oplus 3\oplus \overline{3}\oplus 21$. Relative to the light-flavour nonet, the new states are $u\bar{c},d\bar{c},s\bar{c},c\bar{u},c\bar{d},c\bar{s},c\bar{c}$. Inside the traceless fifteen-dimensional multiplet the additional singlet is instead $(u\bar{u}+d\bar{d}+s\bar{s}-3c\bar{c})/\sqrt{12}$. The pure $c\bar{c}$ state is a linear combination of this state and the full SU(4) singlet. The classification concerns flavour; it does not imply degenerate physical masses or fix spin.

For three light flavours and two spin states, an approximate $SU(6)$ spin-flavour symmetry acts on the six-dimensional single-quark spin-flavour space. A spatial ground-state three-quark colour singlet requires the symmetric representation of dimension $56$. Under flavour symmetry and ordinary spin rotations, it decomposes as $(10,4)\oplus (8,2)$: the baryon decuplet has spin $3/2$ and the baryon octet spin $1/2$. The product of spin and flavour factors can require a correlated sum, rather than a single unentangled product.

The constituent-quark model describes a low-energy hadron using effective massive quarks and phenomenological interactions. Its constituent quark masses include strong-interaction dressing and differ from short-distance running quark masses.

This effective baryon mass model adds a colour-magnetic spin-spin interaction to a sum of constituent quark masses. The common coefficient $A^{\prime }$ absorbs a spatial contact density, colour factors, and the strong coupling. $A^{\prime }$ has dimensions of energy cubed in natural units. A shared coefficient is an approximation because different flavour configurations can have different spatial wavefunctions.

Use dimensionless spin angular momentum operators with $\hslash =1$ and the common-coefficient baryon hyperfine mass formula. The identical pair has total spin one, so its pair expectation is $1/4$. For total baryon spin $1/2$, the two mixed-flavour pair expectations sum to $-1$; for total spin $3/2$, they sum to $1/2$. Their difference is $3/2$, giving the title formula for either repeated-flavour choice. The equality assumes the same effective coefficient $A^{\prime }$ in the two flavour configurations; different spatial overlaps can change it.

For a three-light-flavor example, take up quarks, down quarks and strange quarks with the following additive quantum numbers. Here $Q$ denotes electric charge in units of the positive elementary charge $e$, $B$ is baryon number, $S$ is strangeness, $I_3$ is the third component of isospin, and $Y=B+S$ is flavor hypercharge. An antiquark reverses all five displayed values. These are flavor coordinates, not spatial coordinates or electroweak hypercharge.

Adding these columns immediately explains familiar hadrons. A proton with valence quarks $uud$ has $Q=2/3+2/3-1/3=1$ and $B=1$; a neutron with $udd$ has $Q=0$ and $B=1$. A positive kaon with $u\bar{s}$ has $Q=2/3+1/3=1$, $B=0$ and $S=+1$. Thus the same constituent rules describe both baryons and mesons.

Two hadrons can have identical additive labels while differing in their internal state. The Lambda baryon and neutral Sigma baryon both have valence content $uds$, yet their total isospin differs. Likewise, the neutral pion flavor direction $(u\bar{u}-d\bar{d})/\sqrt{2}$ and the Eta octet state $(u\bar{u}+d\bar{d}-2s\bar{s})/\sqrt{6}$ both have $Q=Y=0$, but their inner product vanishes. Expanding that inner product gives $(1-1+0)/\sqrt{12}=0$. Constituent counting must therefore be supplemented by quantum superposition and the symmetry of the state.

A particularly economical example is the baryon decuplet. Restrict to fully symmetric three-quark flavor states. The nonnegative occupation numbers satisfy $n_u+n_d+n_s=3$, giving $\left(\genfrac{}{}{0ex}{}{3+3-1}{3}\right)=10$ possibilities by stars and bars. Their additive labels areThese formulas follow directly from the constituent table. With fixed $n_s$, there are $4-n_s$ possible values of $n_u$, so the weight diagram has rows of four, three, two and one states. Its outer triangle is therefore a consequence of the constituent construction, rather than an arbitrary drawing convention.

The bottom corner has $n_s=3$, hence $I_3=0$, $Y=-2$ and $Q=-1$: the Omega baryon position. The top-right corner has $uuu$ and $Q=2$, corresponding to the doubly charged Delta baryon. Every point satisfies the Gell-Mann--Nishijima formula $Q=I_3+Y/2$.

The flavor construction is only part of the state. The Pauli constraint on three-quark flavour multiplets also involves spin angular momentum, the spatial wavefunction and color charge. In particular, three identical fermions cannot be assigned a completely symmetric total state merely because the flavor diagram is symmetric. Nor does the same flavor diagram determine the hadron masses: an approximate flavor symmetry can organize a spectrum without making all its members degenerate.

In his 1964 paper, Murray Gell-Mann proposed fractional-charge quarks, constructing baryons and mesons from them. He also distinguished a mathematical constituent model from the hypothesis of finite-mass physical particles. The successful bookkeeping did not itself settle that distinction.^[ref]

George Zweig's 1964 CERN model used three constituents called aces, with fractional electric charge and baryon number $1/3$. Its abstract also proposed an experimental search.^[ref] The theoretical construction and the experiment that might discriminate between interpretations are separate parts of the argument. The Omega baryon discovery experiment illustrates the power of testing an incomplete particle pattern against new observations.

A correlated pair of quarks. Its color, flavor, spin and orbital states are separate degrees of freedom and must collectively satisfy Fermi statistics. In flavor SU(3), the antisymmetric pair space is ${\Lambda }^{2}3\cong \overline{3}$; contraction with the invariant Levi-Civita symbol realizes the isomorphism. The antisymmetric $ud$ flavor pair is an isospin singlet with flavor hypercharge $2/3$. Combining two such flavor-antitriplet pairs with a strange antiquark permits the highest weight of the pentaquark antidecuplet; this flavor construction does not itself establish a bound state.

A meson is a hadron with integer spin and zero baryon number. In the ordinary valence quark model, it contains one quark and one antiquark. Their spins and relative orbital angular momentum combine to determine total angular momentum and parity.

Ground-state pions and kaons are examples of pseudoscalar mesons, with spin-parity $0^{-}$. A meson need not be electrically neutral: the constituent quark charges can sum to a nonzero value. The valence label is a classification of its quantum numbers, not a claim that its exact interacting state consists of only two particles at all times.

In a simple constituent quark model, a spin-dependent contribution $A{\mathbf{S}}_1\cdot {\mathbf{S}}_2/(m_1{m}_2)$ splits a spin-zero and spin-one meson having the same spatial configuration. For two spin-half constituents, addition of angular momentum gives ${\mathbf{S}}_1\cdot {\mathbf{S}}_2=[S(S+1)-3/2]/2$, equal to $-3/4$ for $S=0$ and $+1/4$ for $S=1$. ThusThe spin-averaged mass is $(M_0+3M_1)/4=M_{\mathrm{base}}$. Positive $A$ places the vector state above the pseudoscalar. Determining $A$ from just $M_0$ needs an additional assumption about $M_{\mathrm{base}}$; setting it equal to $m_1+m_2$ neglects all other spin-independent binding terms. A splitting below the pion mass closes a one-pion strong decay to the lower meson, often making the allowed electromagnetic transition with photon emission dominant.

A quarkonium is a bound state of a heavy quark and its own antiquark. Examples are charmonium and the bottom-antibottom states called Upsilon mesons. Spectroscopic labels $n{}^{2S+1}{L}_J$ distinguish radial excitation, total quark spin, relative orbital angular momentum and total angular momentum.

A hidden-bottom pseudoscalar meson, with leading quark model content $b\bar{b}$. Its lowest configuration has $L=S=J=0$ and $J^{PC}=0^{-+}$. Its net bottom flavor is zero, so annihilation into gluons is possible.

The ground-state pseudoscalar meson with quark model content $c\bar{c}$ is denoted ${\eta }_c(1S)$. It is a hidden-charm quarkonium state, with zero net charm, rather than an open-charm D meson.

Single-photon electron-positron annihilation can directly form quarkonium with $J^{PC}=1^{--}$. A vector S-wave quarkonium can annihilate back through a virtual photon into a charged-lepton pair. At leading nonrelativistic order, its electronic partial decay width is proportional to ${\alpha }^2{Q}_q^2|\psi (0){|}^2/M^2$, where the wavefunction at the origin describes the quark-antiquark annihilation overlap.

Below the threshold for a pair of heavy-light mesons, hadronic decays require annihilation, primarily into three gluons for a color-singlet $1^{--}$ state. One gluon cannot carry a color-singlet initial state, and two gluons have the wrong charge-conjugation parity. Above the heavy-light pair threshold, a strong interaction can instead separate the heavy quark and antiquark into different mesons after creating a light pair. This can greatly increase the total decay width without comparably changing the short-distance electronic partial width. A specific example is the contrast between the first three Upsilon mesons and the $4S$ state. Higher radial states can additionally undergo hadronic or radiative transitions to lower quarkonium states.

A meson containing a bottom quark or its antiquark partner. The neutral pair $B^0=d\bar{b}$ and ${\bar{B}}^0=\bar{d}b$ undergoes neutral meson mixing.

A neutral strange B meson has valence content $\bar{b}s$ in the conventional $B_s^0$ assignment; its antiparticle has $b\bar{s}$. In its charged-current spectator amplitude for $J/\psi \varphi$, $\bar{b}\to \bar{c}{W}^{+}$ and $W^{+}\to c\bar{s}$, while the original strange quark is carried into the final strange pair. This is one Standard-Model contribution, not a complete treatment of mixing, loop amplitudes or hadronization.

A bottom quark-antiquark bound state. Its $4S$ resonance provides a source of B meson pairs.

Charmonium is a family of bound states of a charm quark and its antiquark, with zero net charm. It differs from open-charm D mesons, which carry a charm quark together with a light antiquark.

The constituent spins and relative orbital angular momentum distinguish states such as the J/psi meson, eta c meson and chi c mesons. Their level separations and radiative transitions test the interaction between heavy quarks. Decay can proceed through annihilation of the charm–anticharm pair, so net charm conservation does not imply that a charmonium state is stable. A valence quark label organizes the spectrum while additional gluonic and quark-pair contributions remain part of the interacting state.

The spin-triplet P-wave charmonium multiplet $1{}^3{P}_J$, with $J=0,1,2$ and $J^{PC}=0^{++},1^{++},2^{++}$. Radiative cascades between S-wave vector mesons and these states generate bands in three-body Dalitz plots.

The J/psi meson is the ground spin-triplet S-wave vector state of charmonium, with leading valence content $c\bar{c}$ and quantum numbers $J^{PC}=1^{--}$. It has zero net charm and can decay into hadrons or charged-lepton pairs.^[ref]

Its comparatively narrow decay width makes it a conspicuous resonance in reconstructed mass distributions. An electron–Positron mass peak in a fixed-target experiment and an enhanced event rate during an energy scan in an electron-positron collider were complementary observations in the Richter-Ting independent J/psi discovery experiments of 1974.^[ref][ref] A peak's measured width also includes detector resolution or beam-energy spread, so it must not automatically be identified with the intrinsic lifetime width.

An isospin-zero quark $h$ paired with $\overline{u}$ or $\overline{d}$ gives a meson isospin doublet of charges $q_h-2/3,q_h+1/3$. Spin addition at zero orbital angular momentum gives both pseudoscalar mesons and vector mesons. Pairing the new antiquark with $u,d$ gives the charge-conjugate doublets. For strange or charm flavour these are the kaon or D-meson families and their vector partners.

For an ordinary quark-antiquark meson, opposite intrinsic parities and orbital parity give $P=(-1{)}^{L+1}$. For a self-conjugate flavor state, charge conjugation exchanges the two constituents. Reversal of their relative momentum gives $(-1{)}^L$, spin exchange gives $(-1{)}^{S+1}$, and reordering the two fermionic creation operators gives one extra minus sign. Their product is $C=(-1{)}^{L+S}$. Electrical neutrality alone does not imply that a state is an eigenstate of charge conjugation; charged mesons have no individual C eigenvalue.

The omega meson is an isospin-zero vector meson with charge conjugation value minus one and consequently G parity minus one. Three-pion decay is compatible with the isospin-symmetric strong interaction, whereas two-pion decay violates its G-parity rule.

The rho meson is an isospin-one vector meson. Its neutral member has charge conjugation value minus one, so all three charge states have G parity plus one. The two-pion decay is allowed by G parity, whereas a purely strong isospin-conserving three-pion decay is excluded.

A meson with total angular momentum one and negative parity. In the ordinary quark model, a quark-antiquark pair with orbital angular momentum zero and total spin one has these quantum numbers; its neutral self-conjugate states have charge conjugation $C=-1$. The light-flavour tensor product gives an octet and a singlet, forming a vector nonet, just as spin-zero negative-parity pairs form a pseudoscalar meson nonet. Nonzero orbital excitations can also produce vector states.

Mixing of the flavour singlet and isoscalar octet vector states so that the mass eigenstates separate into nonstrange and strange flavour. In the basis $|1⟩=(u\bar{u}+d\bar{d}+s\bar{s})/\sqrt{3}$, $|8_0⟩=(2s\bar{s}-u\bar{u}-d\bar{d})/\sqrt{6}$, the determinant-one rotation to $(\omega ,\varphi )$ has angle $-\arctan (1/\sqrt{2})$. Basis phases change the quoted angle but not the flavour content.

A neutral vector meson of spin-parity $1^{-}$, predominantly $s\bar{s}$ in the ideal quark-flavour description. A pseudoscalar decaying to this meson and a pion requires relative orbital angular momentum one if total angular momentum is zero.

A meson with zero spin angular momentum and negative parity. The pion and kaon are familiar examples. Between two pseudoscalar mesons, parity conservation in the strong interaction forces the axial current matrix element to vanish, while the vector current has pseudoscalar-to-pseudoscalar form factors. The weak decay as a whole need not conserve parity.

The vacuum-to-meson axial-current normalization defined by $⟨0|{\bar{q}}_1{\gamma }^{\mu }{\gamma }^5{q}_2|P(p)⟩=i{f}_P{p}^{\mu }$. It controls exclusive weak annihilation rates. Ratios of tau lepton decays into kaons and pions determine $f_K|V_{us}|/(f_{\pi }|V_{ud}|)$; extracting the Cabibbo-Kobayashi-Maskawa matrix ratio alone requires independent hadronic input.

In the quark model, the light-flavour triplet and its antiquark conjugate give an octet and a singlet. A colour-singlet pair with orbital angular momentum zero and total spin zero has parity minus one, giving a pseudoscalar nonet. Its six outer weights are charged pions and four kaons. At the centre the octet contains the neutral pion and octet eta direction; the additional singlet direction gives a third state at the same weight.

These neutral isospin-zero pseudoscalar mesons have $J^{PC}=0^{-+}$. Their light-quark components mix the Eta octet state and eta singlet state. The singlet axial anomaly contributes to the unusually large eta-prime mass, while flavour breaking also affects both masses and the mixing. The simple orthogonal one-angle flavour model is useful for explaining the nonet; detailed decay-constant descriptions can require more than one parameter.

The heavier physical isoscalar member of the light pseudoscalar meson nonet. It mixes the octet and singlet flavor directions, and is distinct from the lighter eta meson; both occupy the central flavor weight.

Mixing of the Eta octet state and eta singlet state to produce physical neutral pseudoscalar mass eigenstates. It limits mass identities derived by treating the physical eta meson as a pure octet state.

The lighter physical neutral pseudoscalar meson in the eta and eta prime mesons system. The pure Eta octet state is a useful approximation to its flavor content, not its exact physical mass eigenstate.

This flavour combination is the invariant scalar in the octet and singlet in a fundamental SU3 tensor product. Its isospin and flavour hypercharge vanish. Although it shares the central weight with the Eta octet state, it lies in a different irreducible representation. Physical eta mixing and the singlet axial anomaly distinguish the mass eigenstates from the pure flavour directions.

Between spin-zero pseudoscalars, Lorentz covariance decomposes a vector current matrix element into $(p+k{)}^{\mu }{f}_{+}(q^2)+(p-k{)}^{\mu }{f}_{-}(q^2)$, with $q=p-k$. Strong interaction parity symmetry in quantum field theory makes the corresponding axial current matrix element vanish.

Transitions between a neutral meson carrying flavour and its antiparticle. Decays are described on this two-state subspace by an effective Hamiltonian operator $M-i\Gamma /2$ with Hermitian matrices $M,\Gamma$.

If a wrong-sign amplitude is $r{e}^{-i\delta }{g}_{+}+g_{-}$, the same final state's right-sign amplitude is $g_{+}+r{e}^{-i\delta }{g}_{-}$. Their ratio is not exactly obtained by replacing the right-sign rate with a pure exponential. For series $N=n_0+n_{1}t+n_2{t}^2$ and $D=1+d_{1}t+d_2{t}^2$, division gives $N/D=n_0+(n_1-n_0{d}_1)t+[n_2-n_1{d}_1+n_0(d_1^2-d_2)]t^2+O(t^3)$.

For ${\lambda }_f=(q/p){\bar{A}}_f/A_f$, exchanging both initial flavour and final-state labels defines ${\bar{\lambda }}_{\bar{f}}=(p/q)A_{\bar{f}}/{\bar{A}}_{\bar{f}}$. It equals ${\lambda }_f^{-1}$ exactly when $A_{\bar{f}}/{\bar{A}}_{\bar{f}}=A_f/{\bar{A}}_f$. This holds for a common CP eigenstate final ray. CP symmetry of paired amplitudes alone does not imply it for distinct final states: $p=q$, $A_f={\bar{A}}_{\bar{f}}=2$, ${\bar{A}}_f=A_{\bar{f}}=1$ gives both ratios $1/2$.

For mass eigenstates $p|M⟩\pm q|\bar{M}⟩$, unequal forward and reverse flavour transition probabilities occur when $|q/p|\ne 1$. They are proportional to $|q/p{|}^2|g_{-}{|}^2$ and $|p/q{|}^2|g_{-}{|}^2$. This diagnostic is invariant under rephasing the two flavour eigenstates.

If semileptonic decay tags the decaying flavour, wrong-sign decays require neutral meson mixing. With no direct CP violation and no wrong-sign direct decay amplitudes, their rate asymmetry is $(|p/q{|}^2-|q/p{|}^2)/(|p/q{|}^2+|q/p{|}^2)=(1-|q/p{|}^4)/(1+|q/p{|}^4)$. The common time dependence cancels; contaminating direct amplitudes invalidate that isolation.

Neutral B meson mixing converts the flavor eigenstates $B^0=\bar{b}d$ and ${\bar{B}}^0=b\bar{d}$ into one another through the weak interaction. The propagating mass eigenstates are $|B_L⟩=p|B^0⟩+q|{\bar{B}}^0⟩$ and $|B_H⟩=p|B^0⟩-q|{\bar{B}}^0⟩$, with $|p{|}^2+|q{|}^2=1$.

In units $\hslash =c=1$, neglect their width difference and take $|q/p|=1$. Evolution of an initial $B^0$ at proper time $t$ gives the undecayed flavour probabilitieswhere $\Delta m=m_H-m_L$ and $\Gamma$ is the common decay width. Their sum is the survival probability, not one.

For a common final state, mixing and direct decay interfere through ${\lambda }_f=(q/p)({\bar{A}}_f/A_f)$. In the Standard Model, $f=J/\psi K_S$ combines a J/psi meson with a short-lived neutral kaon. Neglecting subleading weak amplitudes and small kaon CP violation,Here the $V_{ij}$ are CKM matrix elements; the minus sign reflects the approximate negative CP parity of this specified final state. A different charmonium-kaon final state need not have that same sign.^[ref]

A decaying particle-antiparticle pair is described by a two-dimensional effective Hamiltonian for neutral-meson mixing with dispersive matrix $M$ and decay matrix $\Gamma$, both Hermitian matrices. The effective Hamiltonian for neutral-meson mixing itself is generally not a Hermitian matrix. CPT symmetry equates its diagonal entries, by separately equating diagonal masses and decay widths. For $H_{\mathrm{eff}}=\left(\begin{array}{cc}a& b\\ c& a\end{array}\right)$ the eigenvalues are $a\pm \sqrt{bc}$ and generic eigenvectors are $(\sqrt{b},\pm \sqrt{c}{)}^T$, with correlated square-root branches. Complex eigenvalues give masses and decay widths through $\mu =m-i\Gamma /2$.

CPT symmetry equates the diagonal entries of the decay-effective Hamiltonian operator. It does not make this effective matrix Hermitian operator; its imaginary diagonal part describes a total decay width.

Neutral D-meson mixing is the time evolution between the flavor states $D^0=c\bar{u}$ and ${\bar{D}}^0=\bar{c}u$. The states with definite propagation masses and decay widths are generally superpositions of these flavor states, so an initially tagged flavor need not remain unchanged before decay.

A measurement tags the production flavor and compares time-dependent rates into selected final states. Some final states can be reached both directly and after mixing, producing interference; a wrong-flavor candidate is therefore not automatically a pure mixing event. The effective two-state Hamiltonian includes both dispersive and absorptive parts. Separating mixing, direct suppressed decays and detector charge asymmetries is part of the inference.^[ref]

A kaon is a strange meson. The ground-state flavor assignments are $K^{+}=u\bar{s}$ and $K^0=d\bar{s}$, with antikaons $K^{-}=s\bar{u}$ and ${\bar{K}}^0=s\bar{d}$. These ground states have spin-parity $0^{-}$. The anti-strange assignments have strangeness $+1$, with the opposite sign for their antiparticles.^[ref]

Charged and neutral kaons differ in more than electric charge. Through neutral kaon mixing, the neutral flavor states combine into propagation states with markedly different decay behavior. Their time-dependent decays provide sensitive tests of CP violation. Specifying whether a label denotes a flavor state or a propagation state is therefore essential when describing a neutral-kaon experiment.

The antiparticle of a kaon, such as $K^{-}=s\bar{u}$ and ${\bar{K}}^0=s\bar{d}$.

The neutral strange meson with strangeness $-1$ and valence content $s\bar{d}$. It is distinct from the neutral kaon flavour state $K^0=d\bar{s}$, which has strangeness $+1$. These are flavour eigenstates, not the weak-interaction propagation eigenstates of neutral kaon mixing.

The neutral flavour states $d\bar{s}$ and $s\bar{d}$ can mix through second-order weak interactions. Box diagrams with W bosons and internal up-type quarks induce a strangeness-changing-by-two four-quark operator. The GIM mechanism removes its flavour-independent part.

The short-lived neutral kaon $K_S$ is one of the two propagating eigenstates of the effective neutral kaon mixing problem. It is a coherent combination $p|K^0⟩+q|{\overline{K}}^0⟩$, with coefficients depending on the phase convention and normalized so $|p{|}^2+|q{|}^2=1$.

In the limit of exact CP symmetry, it is approximately the CP-even combination and readily decays to two pions; the corresponding long-lived state has different allowed decay channels. Small CP violation makes the exact identification with a CP eigenstate imperfect. The observation of two-pion decays in the nominally long-lived beam was therefore evidence for CP violation, not a redefinition of the short-lived state.^[ref]

In the exchange-positive CP convention for neutral kaon mixing, a coherent source with decay amplitudes $A$ and $B$ for $K^0$ and ${\bar{K}}^0$ has projected amplitudes $(A+B)/\sqrt{2}$ and $(A-B)/\sqrt{2}$. For equal phase space, their decay-rate asymmetry is $\mathcal{R}=2\mathrm{Re}r/(1+|r{|}^2)$, $r=B/A$. Expanding the squared moduli proves the identity. Knowledge of $|r|$ alone cannot fix the asymmetry: $r=i|r|$ gives zero, whereas $r=|r|$ gives constructive interference in the symmetric channel.

The CP eigenstates of neutral kaons depend on a flavor-state phase convention. Using $CP|K^0⟩=-|{\bar{K}}^0⟩$, set $|K_1⟩=(|K^0⟩-|{\bar{K}}^0⟩)/\sqrt{2}$ and $|K_2⟩=(|K^0⟩+|{\bar{K}}^0⟩)/\sqrt{2}$. They are respectively CP even and odd. Reversing the flavor-exchange sign reverses which written sum is even; it does not change the CP eigenvalues. With CP violation, the physical short- and long-lived states need not equal these CP eigenstates; kaon CP mixing parameter and neutral-kaon CP impurity and two-pion decay describe the mixtures.