---
title: The Eightfold Way and SU(3) Flavor
module: The Quark Model
moduleNumber: 4
lessonNumber: 1
order: 401
summary: >
  Gell-Mann and Ne'eman's classification of the hadrons into geometric
  multiplets, read as representations of an approximate flavor SU(3). The
  fundamental triplet (u, d, s) and its antitriplet combine into the meson
  nonet from 3⊗3̄ = 8⊕1 and the baryon octet and decuplet from 3⊗3⊗3, and the
  empty corner of the decuplet forecast the Ω⁻.
topics: [The Quark Model]
draft: false
sources:
  - book: Griffiths
    ref: "Ch. 5 — Bound States, §5.1 The Quark Model; the Eightfold Way"
  - book: Halzen & Martin
    ref: "Ch. 2 Symmetries and Quarks; Ch. 14 Flavour SU(3)"
  - book: Thomson
    ref: "Ch. 9 — The Quark Model, §9.1–9.3"
  - book: Tong
    ref: "The Standard Model (Cambridge Part III), §3.3 Hadrons"
---

By the early 1960s the list of strongly interacting particles had grown past any
plausible count of elementary constituents. The organizing insight came in two
steps. First, Gell-Mann and Ne'eman noticed in 1961 that the hadrons fall into
geometric families — sets of eight, sets of ten — when plotted against two
additive quantum numbers, and that the families match the multiplets of the
group $SU(3)$.[^gmn] Gell-Mann named the scheme the **Eightfold Way**, after the
eight members of the lightest families and the eight generators of $SU(3)$. Second,
in 1964 Gell-Mann and, independently, Zweig showed that every family follows from
building hadrons out of three fundamental objects, the **quarks** $u$, $d$, $s$,
transforming in the defining representation of that $SU(3)$.[^gz] This lesson
develops the classification: the fundamental triplet, the rules for combining
representations, and the multiplets those rules produce.

The flavor $SU(3)$ used here is an approximate global symmetry of the strong
interaction, distinct from the exact gauged color $SU(3)$ of quantum
chromodynamics. It rotates the three light quark flavors into one another and would
be exact if $u$, $d$, $s$ had equal mass. They do not — the strange quark is about
$90\ \text{MeV}$ heavier than the up and down — so the symmetry is broken at the
$20\%$ level, which is why the multiplets are visible but not mass-degenerate.[^tong-su3]

## Isospin as SU(2)

The lightest instance of a flavor symmetry involves only $u$ and $d$. Heisenberg
observed that the proton and neutron have nearly equal masses ($938$ and
$940\ \text{MeV}$) and identical strong interactions, and proposed treating them as
two states of one particle, the nucleon, distinguished by a two-valued internal
label. That label is **isospin**, and the symmetry rotating the two states is
$SU(2)$, the same group structure as ordinary spin.[^grif-isospin] The nucleon is
an isospin doublet $I=\tfrac12$,

$$
\left|p\right\rangle = \left|\tfrac12,+\tfrac12\right\rangle, \qquad
\left|n\right\rangle = \left|\tfrac12,-\tfrac12\right\rangle,
$$

and the pions form a triplet $I=1$ with $I_3 = +1, 0, -1$ for $\pi^+, \pi^0,
\pi^-$. The building block behind these is the quark doublet: $u$ carries
$I_3 = +\tfrac12$ and $d$ carries $I_3 = -\tfrac12$, and the antiquarks the
opposite. Combining a quark and an antiquark, $\mathbf 2 \otimes \bar{\mathbf 2} =
\mathbf 3 \oplus \mathbf 1$, reproduces the pion triplet and a singlet, exactly the
structure the observed mesons show. The generalization to three flavors replaces
$SU(2)$ with $SU(3)$ and is the subject of the rest of this lesson; isospin
survives inside it as the $SU(2)$ subgroup acting on $u$ and $d$.

## The fundamental triplet

The quark model posits three light flavors carrying the quantum numbers in the
table below. Two additive numbers label a quark completely within flavor $SU(3)$:
the third component of isospin $I_3$, and the **hypercharge** $Y = B + S$, where
$B = \tfrac13$ is the baryon number of a quark and $S$ its strangeness. The
electric charge follows from the **Gell-Mann–Nishijima relation**

$$
Q = I_3 + \tfrac12 Y = I_3 + \tfrac12(B + S).
$$

| Quark | $I_3$ | $S$ | $Y = B+S$ | $Q$ |
| --- | --- | --- | --- | --- |
| $u$ | $+\tfrac12$ | $0$ | $+\tfrac13$ | $+\tfrac23$ |
| $d$ | $-\tfrac12$ | $0$ | $+\tfrac13$ | $-\tfrac13$ |
| $s$ | $0$ | $-1$ | $-\tfrac23$ | $-\tfrac13$ |

Plotting the three flavors on the $(I_3, Y)$ plane places them at the corners of an
equilateral triangle: the **fundamental representation** $\mathbf 3$. The three
antiquarks carry the opposite additive numbers and sit at the inverted triangle,
the **antifundamental** $\bar{\mathbf 3}$. These two triangles are the seeds from
which every hadron multiplet is assembled.

$$
% caption: The quark triplet 3 and antiquark antitriplet 3-bar on the (I_3, Y)
% plane. The up and down quarks share the top edge at hypercharge +1/3; the strange
% quark sits below at -2/3. The antitriplet is the inversion through the origin.
\begin{tikzpicture}[font=\footnotesize, scale=1.0]
  \definecolor{acc}{HTML}{4A6FA5}
  % axes for triplet (left)
  \begin{scope}[shift={(-3.0,0)}]
    \draw[->, black] (-1.6,0) -- (1.6,0) node[right, font=\scriptsize] {$I_3$};
    \draw[->, black] (0,-1.7) -- (0,1.4) node[above, font=\scriptsize] {$Y$};
    \fill (-0.9,0.6) circle (2.2pt) node[above left] {$d$};
    \fill (0.9,0.6) circle (2.2pt) node[above right] {$u$};
    \fill (0,-1.2) circle (2.2pt) node[below] {$s$};
    \draw (-0.9,0.6) -- (0.9,0.6) -- (0,-1.2) -- cycle;
    \node[black, font=\scriptsize] at (0,-1.9) {triplet $\mathbf 3$};
  \end{scope}
  % axes for antitriplet (right)
  \begin{scope}[shift={(3.0,0)}]
    \draw[->, black] (-1.6,0) -- (1.6,0) node[right, font=\scriptsize] {$I_3$};
    \draw[->, black] (0,-1.7) -- (0,1.4) node[above, font=\scriptsize] {$Y$};
    \fill (0.9,-0.6) circle (2.2pt) node[below right] {$\overline{d}$};
    \fill (-0.9,-0.6) circle (2.2pt) node[below left] {$\overline{u}$};
    \fill (0,1.2) circle (2.2pt) node[right=2pt] {$\overline{s}$};
    \draw (0.9,-0.6) -- (-0.9,-0.6) -- (0,1.2) -- cycle;
    \node[black, font=\scriptsize] at (0,-1.9) {antitriplet $\overline{\mathbf 3}$};
  \end{scope}
\end{tikzpicture}
$$

## Combining representations

Hadrons are color-singlet combinations of quarks (the color rule is derived in a
[later lesson](/particle-physics/quark-model/color-confinement-exotics);
here it fixes only which quark counts are allowed). The two simplest are a
quark–antiquark pair and a three-quark bundle. The flavor content of each follows
from decomposing the tensor product of the fundamental representations into
irreducible pieces.

> **Meson combination.** A quark and an antiquark decompose as
>
> $$
> \mathbf 3 \otimes \bar{\mathbf 3} = \mathbf 8 \oplus \mathbf 1.
> $$
>
> The nine $q\bar q$ states split into an octet and a singlet.

> **Baryon combination.** Three quarks decompose in two steps,
>
> $$
> \mathbf 3 \otimes \mathbf 3 = \bar{\mathbf 3} \oplus \mathbf 6, \qquad
> \mathbf 3 \otimes \mathbf 3 \otimes \mathbf 3
>   = \mathbf 1 \oplus \mathbf 8 \oplus \mathbf 8 \oplus \mathbf{10}.
> $$
>
> The twenty-seven $qqq$ states split into a singlet, two octets, and a decuplet.

The dimension count is a useful check: $3\times3 = 8+1$ and
$3\times3\times3 = 1+8+8+10 = 27$. The physical multiplets — which combinations of
the three flavors sit at each point of the weight diagram — are read off by acting
with the $SU(3)$ raising and lowering operators, which change one flavor into
another and move a state by one step along the edges of the diagram. The graphical
rule for a weight diagram of a product is to center a copy of the second factor's
diagram on each weight of the first and collect the results; the multiplets are the
symmetric patterns that emerge.

$$
% caption: Building the meson weight diagram. A copy of the antitriplet (open
% circles) is centered on each vertex of the quark triplet (filled). The nine
% resulting points fall on a hexagon with a doubly-and-triply occupied center: the
% octet plus singlet of 3 x 3-bar = 8 + 1.
\begin{tikzpicture}[font=\footnotesize, scale=1.0]
  \definecolor{acc}{HTML}{4A6FA5}
  \draw[->, black] (-2.4,0) -- (2.4,0) node[right, font=\scriptsize] {$I_3$};
  \draw[->, black] (0,-2.2) -- (0,2.2) node[above, font=\scriptsize] {$Y$};
  % triplet vertices
  \foreach \x/\y in {-0.9/0.6, 0.9/0.6, 0/-1.2}
    \fill (\x,\y) circle (2.3pt);
  % antitriplet copies (open) centered on each triplet vertex
  \foreach \cx/\cy in {-0.9/0.6, 0.9/0.6, 0/-1.2}{
    \foreach \dx/\dy in {0.9/-0.6, -0.9/-0.6, 0/1.2}
      \draw[thin, fill=white] (\cx+\dx,\cy+\dy) circle (1.6pt);
  }
  \node[black, font=\scriptsize, align=center] at (0,-2.6)
    {each open triangle = a copy of $\overline{\mathbf 3}$};
\end{tikzpicture}
$$

## The pseudoscalar meson nonet

The nine lightest mesons — those with spin $0$ and negative parity — occupy the
$\mathbf 8 \oplus \mathbf 1$ exactly. Plotted on the $(I_3, Y)$ plane they form a
hexagon with two states at the center, plus one singlet also at the center. The
outer hexagon carries the four kaons and the two charged pions; the center carries
$\pi^0$, $\eta$, and $\eta'$. The quark content at each site is fixed by the
additive numbers: hypercharge $+1$ requires one $\bar s$, hypercharge $-1$ one $s$,
and the isospin position fixes the up/down content.

$$
% caption: The pseudoscalar meson nonet on the (I_3, Y) plane, labelled by quark
% content (antiquarks carry an overbar). The top row carries an anti-strange quark
% (Y = +1), the bottom row a strange quark (Y = -1). Three neutral states pile up at
% the origin: the octet pi-zero and eta, and the singlet eta-prime.
\begin{tikzpicture}[font=\footnotesize, scale=1.15]
  \definecolor{acc}{HTML}{4A6FA5}
  \draw[->, black] (-2.9,0) -- (3.1,0) node[below right, font=\scriptsize] {$I_3$};
  \draw[->, black] (0,-1.9) -- (0,1.9) node[above, font=\scriptsize] {$Y$};
  \draw (-1,0.9) -- (1,0.9) -- (2,0) -- (1,-0.9) -- (-1,-0.9) -- (-2,0) -- cycle;
  % top row Y=+1
  \fill (-1,0.9) circle (2.3pt) node[above left] {$d\overline{s}$};
  \fill (1,0.9) circle (2.3pt) node[above right] {$u\overline{s}$};
  % middle row Y=0
  \fill (-2,0) circle (2.3pt) node[left] {$d\overline{u}$};
  \fill (2,0) circle (2.3pt) node[right] {$u\overline{d}$};
  \draw[acc, fill=white] (0,0) circle (2.3pt);
  \node[acc, below right, font=\scriptsize] at (0.12,-0.05) {center};
  % bottom row Y=-1
  \fill (-1,-0.9) circle (2.3pt) node[below left] {$s\overline{u}$};
  \fill (1,-0.9) circle (2.3pt) node[below right] {$s\overline{d}$};
\end{tikzpicture}
$$

The three states at the center share the same additive quantum numbers ($I_3 = S =
0$, hence $Q = 0$), so they mix. Group theory selects the octet combinations

$$
\pi^0 = \tfrac{1}{\sqrt 2}\left(u\bar u - d\bar d\right), \qquad
\eta_8 = \tfrac{1}{\sqrt 6}\left(u\bar u + d\bar d - 2 s\bar s\right),
$$

and the singlet

$$
\eta_1 = \tfrac{1}{\sqrt 3}\left(u\bar u + d\bar d + s\bar s\right).
$$

The physical $\eta$ and $\eta'$ are near-mixtures of $\eta_8$ and $\eta_1$; the
[next lesson](/particle-physics/quark-model/meson-spectroscopy)
treats this mixing quantitatively. The scheme is falsifiable: it predicts nine
states with fixed charges and strangeness, and nine are observed. Repeating the
construction with the quark spins aligned rather than anti-aligned gives the vector
nonet $\rho, \omega, K^\ast, \phi$, the same geometric pattern one unit of spin
higher.

## The baryon octet and decuplet

Three quarks give the $\mathbf 1 \oplus \mathbf 8 \oplus \mathbf 8 \oplus
\mathbf{10}$ above. The physical spin-$\tfrac12$ ground-state baryons fill one
octet; the spin-$\tfrac32$ baryons fill the decuplet. Which spin goes with which
flavor multiplet is dictated by the Pauli principle acting on the full
wavefunction; that argument, and the color factor it requires, are the subject of
the [baryon-spectroscopy lesson](/particle-physics/quark-model/baryon-spectroscopy).
The geometry alone is fixed by the additive numbers.

The **octet** has the same hexagonal shape as the meson octet, with two states at
the center. Its corners and edges carry the proton and neutron (the nucleons, $Y =
1$), the three $\Sigma$ and one $\Lambda$ (one unit of strangeness, $Y = 0$), and
the two $\Xi$ cascades (two units, $Y = -1$).

$$
% caption: The baryon octet of spin one-half. Sites are labelled by three-quark
% content. The top edge (Y = 1) carries the nucleons udd and uud; the middle row
% (Y = 0) the Sigma and Lambda baryons, with two states uds at the center; the
% bottom (Y = -1) the two cascade baryons dss and uss.
\begin{tikzpicture}[font=\footnotesize, scale=1.15]
  \definecolor{acc}{HTML}{4A6FA5}
  \draw[->, black] (-2.9,0) -- (3.1,0) node[below right, font=\scriptsize] {$I_3$};
  \draw[->, black] (0,-1.9) -- (0,1.9) node[above, font=\scriptsize] {$Y$};
  \draw (-1,0.9) -- (1,0.9) -- (2,0) -- (1,-0.9) -- (-1,-0.9) -- (-2,0) -- cycle;
  \fill (-1,0.9) circle (2.3pt) node[above left] {$udd$};
  \fill (1,0.9) circle (2.3pt) node[above right] {$uud$};
  \fill (-2,0) circle (2.3pt) node[left] {$dds$};
  \fill (2,0) circle (2.3pt) node[right] {$uus$};
  \draw[acc, fill=white] (0,0) circle (2.3pt);
  \node[acc, below right, font=\scriptsize] at (0.12,-0.05) {$uds$};
  \fill (-1,-0.9) circle (2.3pt) node[below left] {$dss$};
  \fill (1,-0.9) circle (2.3pt) node[below right] {$uss$};
\end{tikzpicture}
$$

The **decuplet** is a larger triangle of ten states, fully symmetric in flavor. Its
top row is the four $\Delta$ resonances ($ddd, udd, uud, uuu$); each lower row adds
a strange quark and shrinks by one, through the $\Sigma^\ast$ and $\Xi^\ast$, down
to a single apex with three strange quarks. When Gell-Mann drew this triangle in
1962 the bottom corner was empty: no baryon of strangeness $-3$ was known. The
pattern forced its existence, charge, and — through the roughly equal mass spacing
between rows — its mass.

> **The Ω⁻ prediction.** The three lower rows of the decuplet are spaced by about
> $145\ \text{MeV}$: $\Delta(1232) \to \Sigma^\ast(1385) \to \Xi^\ast(1533)$. Equal
> spacing places the missing apex at $\Xi^\ast + 145 \approx 1680\ \text{MeV}$, with
> charge $-1$, strangeness $-3$, spin $\tfrac32$. A particle with strangeness $-3$
> cannot decay strongly (no lighter strangeness-$3$ baryon exists to decay to), so
> it must be long-lived and leave a visible track. The $\Omega^-$ was found at
> Brookhaven in 1964 at $1672\ \text{MeV}$, matching every prediction.[^omega]

$$
% caption: The baryon decuplet of spin three-halves, a flavor-symmetric triangle
% of ten states labelled by three-quark content. Each row down adds one strange
% quark. The bottom apex sss was empty when the pattern was drawn; the equal row
% spacing forecast the Omega-minus mass, confirmed in 1964.
\begin{tikzpicture}[font=\footnotesize, scale=1.1]
  \definecolor{acc}{HTML}{4A6FA5}
  \draw[->, black] (-2.9,0.4) -- (2.9,0.4) node[right, font=\scriptsize] {$I_3$};
  \draw[->, black] (0,-2.6) -- (0,1.4) node[above, font=\scriptsize] {$Y$};
  % row Y=+1 : ddd udd uud uuu
  \foreach \x/\lab in {-2.1/ddd, -0.7/udd, 0.7/uud, 2.1/uuu}
    \fill (\x,0.6) circle (2.2pt) node[above, font=\scriptsize] {\lab};
  % row Y=0 : dds dus uus
  \foreach \x/\lab in {-1.4/dds, 0/dus, 1.4/uus}
    \fill (\x,-0.4) circle (2.2pt) node[right=1pt, font=\scriptsize] {\lab};
  % row Y=-1 : dss uss
  \foreach \x/\lab in {-0.7/dss, 0.7/uss}
    \fill (\x,-1.4) circle (2.2pt) node[right=1pt, font=\scriptsize] {\lab};
  % apex Y=-2 : sss (predicted)
  \draw[acc, very thick, fill=acc!12] (0,-2.4) circle (2.6pt);
  \node[acc, below right, font=\scriptsize] at (0.15,-2.45) {$sss$ (predicted)};
  \draw (-2.1,0.6) -- (2.1,0.6);
\end{tikzpicture}
$$

## Why the pattern works

The Eightfold Way succeeds because $SU(3)$ flavor is a real, if approximate,
symmetry of the strong force. Three facts make the multiplets more than numerology.

- **Charges are additive and integer or fractional in a fixed pattern.** Every
  observed hadron charge is reproduced by $Q = I_3 + \tfrac12 Y$ with the quark
  assignments above; no state requires a fourth light flavor or a different charge
  rule.
- **The mass splitting within a multiplet tracks strangeness.** Each strange quark
  adds a roughly constant amount to the mass — about $145\ \text{MeV}$ per unit of
  strangeness in the decuplet — because the symmetry is broken only by the
  strange-quark mass. The **Gell-Mann–Okubo mass formula** makes this quantitative;
  for the octet it reads $2(m_N + m_\Xi) = 3 m_\Lambda + m_\Sigma$, satisfied to a
  few percent.[^gmo]
- **The scheme predicts, not just fits.** The empty decuplet corner and the
  $\Omega^-$ are the decisive case: a classification that only catalogued known
  particles could not have forecast a new one's mass and quantum numbers.

The success also poses the question the rest of the module answers. The quarks
carry fractional charge and were never seen free, so at first the model was read as
a bookkeeping device. Deep inelastic scattering later gave the quarks physical
reality as pointlike constituents, and the requirement that only certain quark
combinations appear — $q\bar q$ and $qqq$, never $qq$ or a lone quark — turns out to
follow from a second, exact $SU(3)$: the color symmetry of QCD, developed in the
[final lesson of this module](/particle-physics/quark-model/color-confinement-exotics)
and gauged in the QCD module. The next two lessons work out the meson and baryon
spectra in detail, including the spin structure that the geometric picture leaves
open.

[^gmn]: Griffiths, _Introduction to Elementary Particles_, §5.1, presents the Eightfold Way and the $SU(3)$ multiplets; Gell-Mann and Ne'eman proposed the scheme independently in 1961.

[^gz]: Griffiths, §5.1; the quark model of 1964 is due to Gell-Mann and, independently, Zweig (who called the constituents "aces").

[^tong-su3]: Tong, _The Standard Model_ (Cambridge Part III), §3.3, treats flavor $SU(3)_V$ as the surviving vector symmetry of QCD with equal-mass quarks and stresses that it is distinct from the gauged color $SU(3)$. Quark masses (current masses) from the Particle Data Group, [pdg.lbl.gov](https://pdg.lbl.gov).

[^grif-isospin]: Griffiths, §4.5 (isospin), and Halzen & Martin, Ch. 2, develop isospin as an $SU(2)$ flavor symmetry with the nucleon doublet and pion triplet.

[^omega]: Griffiths, §5.1; the $\Omega^-$ was discovered at Brookhaven National Laboratory in 1964 at the mass Gell-Mann had predicted from the decuplet spacing. Masses from the Particle Data Group, [pdg.lbl.gov](https://pdg.lbl.gov).

[^gmo]: Griffiths, §5.5, and Tong, §3.3, give the Gell-Mann–Okubo relations for the octet and decuplet; measured masses from the Particle Data Group, [pdg.lbl.gov](https://pdg.lbl.gov).