Meson

A hadron with integer spin. An ordinary meson has valence content consisting of a quark and an antiquark.

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.

A bound state of a charm quark and its antiquark.

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.

A charmonium state independently discovered in 1974 through the Richter-Ting independent J/psi discovery experiments.

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.

Weak box diagrams change neutral bottom-meson flavour. The propagating eigenstates are combinations $p|B^0⟩\pm q|{\bar{B}}^0⟩$. For a shared final state, interference is governed by ${\lambda }_f=(q/p)({\bar{A}}_f/A_f)$; in the dominant charmonium-kaon mode this yields the CKM angle through ${\lambda }_f\simeq -e^{-2i\beta }$, with stated meson and kaon phase conventions.

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.

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.

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.

In the flavor convention $K_1=(K^0-{\bar{K}}^0)/\sqrt{2}$ and $K_2=(K^0+{\bar{K}}^0)/\sqrt{2}$, the parameter $ϵ=(a-b)/(a+b)$ writes normalized mass states as $(K_1+ϵK_2)/\sqrt{1+|ϵ{|}^2}$ and $(K_2+ϵK_1)/\sqrt{1+|ϵ{|}^2}$. In the Hermitian mass-only approximation with correlated roots, $ϵ=i\tan (\phi /2)$ and the states are orthogonal. For the full non-Hermitian decay matrix the real part need not vanish. The parameter depends on the flavor-state phase convention; observable CP violation involves convention-invariant combinations with decay amplitudes.

In the phase convention $CP|K^0⟩=|{\bar{K}}^0⟩$, the symmetric flavor combination is CP even and the antisymmetric one CP odd. For an equal-diagonal effective matrix with entries b,c off diagonal, choose correlated roots r,s with $r^2=b$, $s^2=c$ and eigenvalues $a\pm rs$. Its eigenvectors are $(r,\pm s{)}^T$. Writing $ϵ=(r-s)/(r+s)$ gives the displayed CP-basis mixtures, up to an overall phase. Their deviation is small only if $|ϵ|\ll 1$; noninvariance alone does not guarantee that. This convention must not be mixed with one in which flavor exchange carries a minus sign.

With CPT symmetry, a Hermitian kaon mass matrix is $M=\left(\begin{array}{cc}{r}_0& z\\ z^{\ast }& r_0\end{array}\right)$. Its eigenvalues are $r_0\pm |z|$. For correlated roots $a^2=z$, $b^2=z^{\ast }$ and $ab=|z|$, the normalized lower and upper eigenvectors are $(a,-b)/\sqrt{2|z|}$ and $(a,b)/\sqrt{2|z|}$. They are orthogonal. This is a mass-only result; including decay requires the generally non-Hermitian effective matrix $M-i\Gamma /2$.

In a Hermitian neutral-kaon mass matrix, independently choosing square-root signs of $z$ and $z^{\ast }$ can interchange the eigenvalue labels. Choose $z=r{e}^{i\phi }$, $a=\sqrt{r}{e}^{i\phi /2}$ and $b=\sqrt{r}{e}^{-i\phi /2}$, so $ab=r>0$, continuously from the desired CP-conserving limit. Then $(a,-b)$ is the lower-mass eigenvector. At a degenerate off-diagonal zero the mass eigenbasis is not fixed by diagonalization alone.

A Standard Model charged-current box diagram converts $d\bar{s}$ into $s\bar{d}$. Internal quarks run over $u,c,t$, and the flavor sum contains ${\lambda }_i{\lambda }_j$ with ${\lambda }_i=V_{is}^{\ast }{V}_{id}$. The loop function depends on internal masses; CKM matrix unitarity gives ${\sum }_i{\lambda }_i=0$, producing the GIM mechanism cancellation of flavor-independent terms. The amplitude is of order $g^4$ and second order in the weak interaction.

For relativistically normalized kaon states and $Q=(\bar{s}{\gamma }_{\mu }(1-{\gamma }^5)d)(\bar{s}{\gamma }^{\mu }(1-{\gamma }^5)d)$, define $⟨{\bar{K}}^0|Q|K^0⟩=(8/3)f_K^2{m}_K^2{B}_K$. The dimensionless kaon bag parameter measures strong interaction corrections to the vacuum-insertion normalization. The coefficient and matrix element must be defined in the same renormalization scheme and at the same scale. Dividing by $2m_K$ gives the corresponding two-state mixing-Hamiltonian normalization.

A charmed strange meson. Its ground-state charged particle has valence content $c\bar{s}$ and spin-parity $0^{-}$. Its antiparticle has the conjugate valence content. Nonleptonic decays are governed by weak charged currents and hadronic transition matrix elements.