---
title: Meson Multiplets and Quantum Numbers
module: The Quark Model
moduleNumber: 4
lessonNumber: 2
order: 402
summary: >
  Mesons as quark–antiquark bound states. The spin singlet and triplet, orbital
  excitations, and the assignment of J^PC from the quark spins and orbital angular
  momentum, giving the pseudoscalar and vector nonets. The η–η' and ω–φ mixing
  problems, and the charmonium and bottomonium spectra read as heavy-quark
  positronium.
topics: [The Quark Model]
draft: false
sources:
  - book: Griffiths
    ref: "Ch. 5 — Bound States, §5.5–5.6 Mesons and J^PC"
  - book: Thomson
    ref: "Ch. 9 — The Quark Model, §9.4"
  - book: Tong
    ref: "The Standard Model (Cambridge Part III), §3.3.1 Mesons"
---

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](/particle-physics/quark-model/eightfold-way-su3);
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.[^grif-mesons]

## Spin, parity, and charge conjugation

The two constituents each carry spin $\tfrac12$, 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

$$
\vec J = \vec L + \vec S, \qquad J = |L - S|, \dots, L + S.
$$

$$
% caption: The two quark spins combine into total spin S = 0 (anti-aligned) or
% S = 1 (aligned); together with the orbital angular momentum L this fixes the
% meson spin J. The ground states have L = 0, giving J = 0 and J = 1.
\begin{tikzpicture}[>=stealth, font=\footnotesize, scale=1.0]
  \definecolor{acc}{HTML}{4A6FA5}
  % S=0 case
  \begin{scope}[shift={(-3.2,0)}]
    \fill (-0.7,0) circle (2pt);
    \fill (0.7,0) circle (2pt);
    \draw[very thick, ->] (-0.7,-0.35) -- (-0.7,0.45);
    \draw[very thick, ->] (0.7,0.45) -- (0.7,-0.35);
    \node[black, font=\scriptsize] at (0,-0.9) {spins anti-aligned};
    \node[font=\scriptsize] at (0,-1.3) {$S = 0$};
  \end{scope}
  % S=1 case
  \begin{scope}[shift={(3.2,0)}]
    \fill (-0.7,0) circle (2pt);
    \fill (0.7,0) circle (2pt);
    \draw[very thick, ->] (-0.7,-0.35) -- (-0.7,0.45);
    \draw[very thick, ->] (0.7,-0.35) -- (0.7,0.45);
    \node[black, font=\scriptsize] at (0,-0.9) {spins aligned};
    \node[font=\scriptsize] at (0,-1.3) {$S = 1$};
  \end{scope}
\end{tikzpicture}
$$

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

$$
P = -(-1)^L = (-1)^{L+1}.
$$

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

$$
C = (-1)^{L+S}.
$$

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

> **Ground-state nonets.** With no orbital motion, $L = 0$:
>
> - $S = 0$ gives $J = 0$, $P = (-1)^{1} = -1$, $C = (-1)^{0} = +1$: the
>   **pseudoscalar** nonet $J^{PC} = 0^{-+}$ ($\pi$, $K$, $\eta$, $\eta'$).
> - $S = 1$ gives $J = 1$, $P = -1$, $C = -1$: the **vector** nonet
>   $J^{PC} = 1^{--}$ ($\rho$, $K^\ast$, $\omega$, $\phi$).

The two nonets share the geometry of the [meson octet-plus-singlet](/particle-physics/quark-model/eightfold-way-su3):
the same nine sites on the $(I_3, Y)$ plane, one unit of spin apart.

$$
% caption: The two ground-state nonets share one hexagonal weight pattern, labelled
% here by quark content. Anti-aligned quark spins give the spin-0 pseudoscalar nonet
% (J^PC = 0^-+); aligned spins give the spin-1 vector nonet (J^PC = 1^--). Only the
% spin differs; the flavor geometry is identical.
\begin{tikzpicture}[font=\footnotesize, scale=0.92]
  \definecolor{acc}{HTML}{4A6FA5}
  \foreach \sh/\ttl in {-3.4/{spin 0}, 3.4/{spin 1}}{
    \begin{scope}[shift={(\sh,0)}]
      \draw (-1,0.9) -- (1,0.9) -- (2,0) -- (1,-0.9) -- (-1,-0.9) -- (-2,0) -- cycle;
      \fill (-1,0.9) circle (2pt) node[above left, font=\scriptsize] {$d\overline{s}$};
      \fill (1,0.9) circle (2pt) node[above right, font=\scriptsize] {$u\overline{s}$};
      \fill (-2,0) circle (2pt) node[left, font=\scriptsize] {$d\overline{u}$};
      \fill (2,0) circle (2pt) node[right, font=\scriptsize] {$u\overline{d}$};
      \draw[fill=white] (0,0) circle (2pt);
      \fill (-1,-0.9) circle (2pt) node[below left, font=\scriptsize] {$s\overline{u}$};
      \fill (1,-0.9) circle (2pt) node[below right, font=\scriptsize] {$s\overline{d}$};
      \node[font=\scriptsize] at (0,-1.5) {\ttl};
    \end{scope}
  }
\end{tikzpicture}
$$

## 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

$$
{}^1P_1: 1^{+-}, \qquad {}^3P_0: 0^{++}, \qquad {}^3P_1: 1^{++},
\qquad {}^3P_2: 2^{++},
$$

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](/particle-physics/quark-model/color-confinement-exotics).

## 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 $\eta'$ are rotations of the
$SU(3)$ octet state $\eta_8$ and singlet state $\eta_1$ by a **pseudoscalar mixing
angle** $\theta_P$,

$$
\begin{aligned}
\eta &= \eta_8 \cos\theta_P - \eta_1 \sin\theta_P, \\
\eta' &= \eta_8 \sin\theta_P + \eta_1 \cos\theta_P,
\end{aligned}
$$

with $\theta_P \approx -11^\circ$ to $-24^\circ$ from the measured masses.[^mixing]
The mixing is small, so the physical $\eta$ is nearly the octet state and $\eta'$
nearly the singlet. That $\eta'$ 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.[^tong-etaprime]

$$
% caption: Flavor mixing as a rotation. The octet (eta-8) and singlet (eta-1) basis
% states, degenerate in the symmetry limit, mix through the quark-mass and anomaly
% terms; the physical eta and eta-prime are the rotated eigenstates. For the vector
% nonet the analogous rotation is near-ideal, so phi is almost pure strange-antistrange.
\begin{tikzpicture}[>=stealth, font=\footnotesize, scale=1.0]
  \definecolor{acc}{HTML}{4A6FA5}
  % basis axes
  \draw[->, black] (0,0) -- (3.2,0) node[right, font=\scriptsize] {octet};
  \draw[->, black] (0,0) -- (0,3.2) node[above, font=\scriptsize] {singlet};
  % rotated axes
  \draw[acc, thick, ->] (0,0) -- (2.7,1.0) node[right, acc, font=\scriptsize] {eta};
  \draw[acc, thick, ->] (0,0) -- (-1.0,2.7) node[left, acc, font=\scriptsize] {eta prime};
  % angle arc
  \draw[black] (1.3,0) arc (0:21:1.3);
  \node[black, font=\scriptsize] at (1.7,0.28) {mixing angle};
\end{tikzpicture}
$$

For the vector nonet the mixing is near **ideal**: the physical $\phi$ is almost
pure $s\bar s$ and the $\omega$ almost pure $\tfrac{1}{\sqrt 2}(u\bar u + d\bar d)$.
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.[^tong-mixing]

## 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,

$$
V(r) = -\frac{4}{3}\frac{\alpha_s}{r} + k\,r,
$$

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

$$
% caption: The charmonium (c-cbar) spectrum, a positronium-like ladder. Levels are
% labelled by spectroscopic term n^(2S+1)L_J and mass in MeV. The 1S states are the
% pseudoscalar eta_c (spin 0) and the vector J/psi (spin 1); the 1P triplet is the
% chi_c states; the 2S vector is the psi-prime. The dashed line marks the open-charm
% threshold above which decay to two D mesons opens.
\begin{tikzpicture}[font=\footnotesize, scale=1.0]
  \definecolor{acc}{HTML}{4A6FA5}
  \draw[->, black] (0,0) -- (0,4.4) node[above, font=\scriptsize] {mass (MeV)};
  \foreach \y/\lab in {0.6/2980, 1.5/3100, 3.0/3500, 3.7/3686}
    \draw[black] (-0.12,\y) -- (0.12,\y) node[left=2pt, black, font=\scriptsize] {\lab};
  % 1S: eta_c (S=0) and J/psi (S=1)
  \draw[very thick] (0.8,0.6) -- (1.8,0.6) node[right, font=\scriptsize] {1S spin 0};
  \draw[very thick] (0.8,1.5) -- (1.8,1.5) node[right, font=\scriptsize] {1S spin 1};
  % 1P triplet chi_c
  \draw[very thick] (3.2,2.8) -- (4.2,2.8);
  \draw[very thick] (3.2,3.0) -- (4.2,3.0);
  \draw[very thick] (3.2,3.2) -- (4.2,3.2) node[right, font=\scriptsize] {1P triplet};
  % 2S psi'
  \draw[very thick] (0.8,3.7) -- (1.8,3.7) node[right, font=\scriptsize] {2S spin 1};
  % open charm threshold
  \draw[black, dashed] (0.4,4.0) -- (5.6,4.0) node[right, black, font=\scriptsize] {open charm};
\end{tikzpicture}
$$

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
$\psi'$ ($3686\ \text{MeV}$).[^charmonium] 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 $\Upsilon(1S)$ 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 $\tau = \hbar/\Gamma$, with $\hbar = 6.58\times10^{-22}\
\text{MeV·s}$, so a $150\ \text{MeV}$ width corresponds to $\tau \approx
4\times10^{-24}\ \text{s}$.[^tong-lifetimes] 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.

[^grif-mesons]: Griffiths, _Introduction to Elementary Particles_, §5.5–5.6, derives the $J^{PC}$ assignments for $q\bar q$ mesons and lists the pseudoscalar and vector nonets. Quantum numbers and masses follow the Particle Data Group, [pdg.lbl.gov](https://pdg.lbl.gov).

[^mixing]: Griffiths, §5.6, and Thomson, §9.4, give the $\eta$–$\eta'$ mixing angle; the value depends on whether a linear or quadratic mass formula is used. Masses from the Particle Data Group, [pdg.lbl.gov](https://pdg.lbl.gov).

[^tong-etaprime]: Tong, _The Standard Model_ (Cambridge Part III), §3.3.1, identifies the octet pseudoscalars as pseudo-Goldstone bosons and attributes the anomalously large $\eta'$ mass to the axial $U(1)$ anomaly.

[^tong-mixing]: Tong, §3.3.1, contrasts the pseudoscalar and vector nonets: the pseudoscalars are dominated by $SU(3)$ (octet/singlet content), the vectors by the strange-quark mass (near-ideal $\phi \approx s\bar s$, $\omega \approx \tfrac{1}{\sqrt2}(u\bar u + d\bar d)$).

[^cornell]: Griffiths, §5.3, motivates the quark–antiquark potential with a short-range Coulomb term from one-gluon exchange and a linear confining term; the color factor $4/3$ is derived in the QCD module.

[^charmonium]: Griffiths, §5.6, treats charmonium as heavy-quark positronium; masses and $J^{PC}$ from the Particle Data Group, [pdg.lbl.gov](https://pdg.lbl.gov).

[^tong-lifetimes]: Tong, §3.3.2, tabulates meson lifetimes and gives the width–time rule of thumb ($100\ \text{MeV} \leftrightarrow 10^{-23}\ \text{s}$) and the characteristic time scales of the strong, electromagnetic, and weak decays.