Meson Multiplets and Quantum Numbers

A meson is a bound state of a quark and an antiquark. The flavor content and the $SU(3)$ multiplet structure were fixed in the previous lesson; this one adds the spin and space degrees of freedom, which turn each flavor nonet into a tower of states labelled by their spin, parity, and charge conjugation. The labelling scheme, $J^{PC}$, is the language the Particle Data Group uses for every meson, and it follows entirely from the two quark spins and their relative orbital motion.¹

Spin, parity, and charge conjugation

The two constituents each carry spin $\frac{1}{2}$, so their spins combine into a total quark spin $S=0$ (anti-aligned, the singlet) or $S=1$ (aligned, the triplet). The pair also has an orbital angular momentum $L$ about their common center. The meson's total angular momentum is the vector sum

Parity and charge conjugation are multiplicative quantum numbers built from the intrinsic parities of the constituents and the spatial and exchange properties of the wavefunction. A fermion and its antifermion carry opposite intrinsic parity, so the pair contributes a factor $-1$; the orbital wavefunction contributes $(-1{)}^L$. The result is

Charge conjugation is defined only for the flavor-neutral mesons (those equal to their own antiparticle). Interchanging the quark and antiquark exchanges their positions, spins, and charges; collecting the factors gives

These two rules generate the whole meson spectrum from the allowed $(L,S)$.

The two nonets share the geometry of the meson octet-plus-singlet: the same nine sites on the $(I_3,Y)$ plane, one unit of spin apart.

Orbital excitations

Raising $L$ builds heavier multiplets on the same flavor nonets. The rules above give their quantum numbers directly. For $L=1$ the parity is $P=(-1{)}^2=+1$ and the four $(L,S)$-combinations produce

in spectroscopic notation ${}^{2S+1}{L}_J$. These are the $b_1$, $a_0$, $a_1$, $a_2$ families near $1200\text{–}1300\text{MeV}$. Some combinations of $J^{PC}$ that a $q\bar{q}$ pair cannot produce — $0^{--}$, $0^{+-}$, $1^{-+}$, $2^{+-}$ — are called exotic; a meson observed with such quantum numbers cannot be a simple quark and antiquark, a signature pursued in the final lesson.

Flavor mixing at the center

The three flavor-neutral pseudoscalars share $I_3=S=0$ and therefore mix, as noted in the previous lesson. The physical $\eta$ and ${\eta }^{\prime }$ are rotations of the $SU(3)$ octet state ${\eta }_8$ and singlet state ${\eta }_1$ by a pseudoscalar mixing angle ${\theta }_P$,

with ${\theta }_P\approx -11^{\circ }$ to $-24^{\circ }$ from the measured masses.² The mixing is small, so the physical $\eta$ is nearly the octet state and ${\eta }^{\prime }$ nearly the singlet. That ${\eta }^{\prime }$ is anomalously heavy ($958\text{MeV}$, against $548\text{MeV}$ for $\eta$) is a separate effect — the singlet is not a would-be Goldstone boson of chiral symmetry, and the axial $U(1)$ anomaly lifts its mass — treated in the QCD module.³

For the vector nonet the mixing is near ideal: the physical $\varphi$ is almost pure $s\bar{s}$ and the $\omega$ almost pure $\frac{1}{\sqrt{2}}(u\bar{u}+d\bar{d})$. The difference between the pseudoscalar and vector cases is a competition between the $SU(3)$ symmetry, which favors the octet/singlet combinations, and the strange-quark mass, which favors the pure $q\bar{q}$ states; the symmetry wins for the light pseudoscalars, the mass for the heavier vectors.⁴

Heavy quarkonia

A bound state of a heavy quark and its antiquark — charm $c\bar{c}$ (charmonium) or bottom $b\bar{b}$ (bottomonium) — is the closest thing in the strong interaction to a hydrogen atom. The large quark mass makes the constituents move slowly, so the states are approximately non-relativistic and organize into a discrete level spectrum labelled by radial and orbital quantum numbers, in direct analogy to positronium. The heavy mass also sets the size below the confinement scale, where the potential is Coulomb-like from single-gluon exchange plus a linear confining term,

the Cornell potential.⁵ The short-distance Coulomb piece produces the hydrogen-like level ordering; the linear piece pushes the higher levels up and eventually confines.

The charmonium levels are the classic case. The $1S$ states are the pseudoscalar ${\eta }_c$ ($2980\text{MeV}$, $J^{PC}=0^{-+}$) and the vector $J/\psi$ ($3097\text{MeV}$, $1^{--}$); the discovery of the sharp $J/\psi$ resonance in 1974 established the charm quark. The $1P$ states are the ${\chi }_c$ triplet ($0^{++},1^{++},2^{++}$) near $3.5\text{GeV}$, and the $2S$ vector is the ${\psi }^{\prime }$ ($3686\text{MeV}$).⁶ All sit below the open-charm threshold at $2m_D\approx 3730\text{MeV}$; above it, a $c\bar{c}$ state can fall apart into two $D$ mesons and the levels broaden sharply. Bottomonium repeats the pattern at higher mass — the ${\rm Y}(1S)$ vector at $9460\text{MeV}$ heads a similar ladder — with the levels more nearly Coulombic because the heavier $b$ quark probes shorter distances. These spectra are the strongest evidence that the quark model describes real dynamical bound states, not merely a labelling of the light hadrons.

Lifetimes and widths

The wide range of meson lifetimes reflects which interaction drives the decay. A meson that can decay through the strong force does so in $\sim 10^{-23}\text{s}$; these are the resonances, seen as broad bumps in a cross section, such as the $\rho \to \pi \pi$ ($\Gamma \approx 150\text{MeV}$). Mesons for which the strong channel is closed decay electromagnetically (${\pi }^0\to \gamma \gamma$, $\sim 10^{-16}\text{s}$) or weakly (${\pi }^{\pm }\to {\mu }^{\pm }\nu$, $\sim 10^{-8}\text{s}$), and are correspondingly narrow. The conversion between a width and a lifetime is the uncertainty relation $\tau =\hslash /\Gamma$, with $\hslash =6.58\times 10^{-22}\text{MeV⋅s}$, so a $150\text{MeV}$ width corresponds to $\tau \approx 4\times 10^{-24}\text{s}$.⁷ The spread — fourteen orders of magnitude across the meson tables — is not a property of the quark content but of the strongest interaction the particular decay can use.