---
title: Isospin, SU(2), and Flavor SU(3)
module: Symmetries and Conservation Laws
moduleNumber: 3
lessonNumber: 4
order: 304
summary: >
  The near-equal masses of the proton and neutron, and of the three pions, signal
  a continuous internal symmetry of the strong force: isospin, an SU(2) whose
  ladder operators move between the members of a multiplet. Adding strangeness
  enlarges it to an approximate SU(3) flavor symmetry, and the Gell-Mann–Nishijima
  relation Q = I3 + Y/2 places every hadron on a weight diagram in the
  isospin–hypercharge plane — the language in which the quark model is written.
topics: [Symmetries and Conservation Laws]
draft: false
sources:
  - book: Griffiths
    ref: "Ch. 4 — Symmetries, §4.2–4.3; Ch. 5 §5.1"
  - book: Halzen & Martin
    ref: "Ch. 2; Ch. 14 — Isospin and SU(3) flavor"
  - book: Thomson
    ref: "Ch. 9 — The quark model, isospin and flavor symmetry"
  - book: Tong
    ref: "The Standard Model (Cambridge Part III), §2.1, §3.1"
---

The proton and neutron differ in mass by little more than a tenth of a percent,
and the three pions by a few percent. To the strong force, which supplies almost
all of a hadron's mass, these differences are negligible: it treats proton and
neutron as two states of one object, and the three pions as three states of
another. That statement is a continuous internal symmetry — **isospin**, an
$SU(2)$ acting not on space but on an abstract "charge space" — and it comes with
the full apparatus of a non-Abelian symmetry: multiplets, ladder operators, and
conserved quantum numbers. Enlarging it to include strangeness gives the
approximate **flavor $SU(3)$** whose weight diagrams are the stage on which the
quark model is built. This lesson develops that machinery operationally, from the
nucleon doublet to the isospin–hypercharge plane.

## Isospin as an SU(2) symmetry

Heisenberg observed in 1932 that the strong interaction is
**charge-independent**: the nuclear force between two protons, two neutrons, or a
proton and neutron is the same once the electromagnetic contribution is removed.
The proton and neutron are then two states of a single **nucleon**, distinguished
by a label that behaves exactly like the projection of a spin-$\tfrac12$ angular
momentum.[^tong-isospin] This label is **isospin** $I$ — the name borrows the
formalism of spin while having nothing to do with rotations in space.

The isospin operators satisfy the same algebra as angular momentum, the Lie
algebra of $SU(2)$,

$$
[I_i, I_j] = i\,\epsilon_{ijk}\,I_k ,
$$

so a multiplet of total isospin $I$ contains $2I + 1$ states labeled by the third
component $I_3 = -I, \dots, +I$. The nucleon is an isospin doublet, $I = \tfrac12$:

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

The pions form an isospin triplet, $I = 1$, with $\pi^+$, $\pi^0$, $\pi^-$ carrying
$I_3 = +1, 0, -1$. Because isospin is an exact symmetry of the strong force alone,
it is conserved in strong reactions and broken — by the electromagnetic force,
which _does_ see charge, and by the up–down quark mass difference — only at the
percent level.

$$
% caption: The nucleon isospin doublet (two states, I equals one half) and the
% pion isospin triplet (three states, I equals one) arranged on the third-component
% axis. Members of a multiplet share a nearly equal mass because the strong force
% does not distinguish them.
\begin{tikzpicture}[>=stealth, font=\footnotesize, scale=1.0]
  \definecolor{acc}{HTML}{4A6FA5}
  % nucleon doublet
  \node[font=\scriptsize] at (-3.4,1.6) {nucleon doublet};
  \draw[->, black] (-5.0,0) -- (-1.6,0) node[right, font=\scriptsize] {third comp.};
  \fill (-4.2,0) circle (2.6pt); \node[above, font=\scriptsize] at (-4.2,0.15) {n};
  \fill (-2.8,0) circle (2.6pt); \node[above, font=\scriptsize] at (-2.8,0.15) {p};
  % pion triplet
  \node[font=\scriptsize] at (2.2,1.6) {pion triplet};
  \draw[->, black] (0.4,0) -- (4.4,0) node[right, font=\scriptsize] {third comp.};
  \fill (1.0,0) circle (2.6pt); \node[above, font=\scriptsize] at (1.0,0.15) {pion (-)};
  \fill (2.4,0) circle (2.6pt); \node[above, font=\scriptsize] at (2.4,0.15) {pion (0)};
  \fill (3.8,0) circle (2.6pt); \node[above, font=\scriptsize] at (3.8,0.15) {pion (+)};
\end{tikzpicture}
$$

## Ladder operators and multiplet structure

The non-Abelian structure gives more than a counting rule: it connects the members
of a multiplet. Define the **raising and lowering operators**

$$
I_\pm = I_1 \pm i\,I_2 ,
$$

which shift $I_3$ by one unit,

$$
I_\pm \left\lvert I, I_3 \right\rangle
   = \sqrt{I(I+1) - I_3(I_3 \pm 1)}\;\left\lvert I, I_3 \pm 1 \right\rangle .
$$

For the nucleon, $I_+$ turns a neutron into a proton and $I_-$ does the reverse;
for the pion triplet the ladder walks $\pi^- \to \pi^0 \to \pi^+$. Because these
operators commute with the strong Hamiltonian, states connected by the ladder are
degenerate up to isospin-breaking corrections — this is precisely why the multiplet
members share a mass.

$$
% caption: The SU(2) ladder operators acting within a multiplet. The raising
% operator moves up the third-component axis (neutron to proton, and along the pion
% triplet); the lowering operator moves down. States linked by the ladder are
% degenerate under the strong force.
\begin{tikzpicture}[>=stealth, font=\footnotesize, scale=1.0]
  \definecolor{acc}{HTML}{4A6FA5}
  \draw[->, black] (-0.6,0) -- (5.2,0) node[right, font=\scriptsize] {third comp.};
  \fill (0.6,0) circle (2.6pt); \node[below, font=\scriptsize] at (0.6,-0.2) {pion (-)};
  \fill (2.3,0) circle (2.6pt); \node[below, font=\scriptsize] at (2.3,-0.2) {pion (0)};
  \fill (4.0,0) circle (2.6pt); \node[below, font=\scriptsize] at (4.0,-0.2) {pion (+)};
  \draw[->, acc, thick] (0.7,0.35) to[bend left=35] (2.2,0.35);
  \node[acc, font=\scriptsize, above] at (1.45,0.7) {raising};
  \draw[->, acc, thick] (2.4,0.35) to[bend left=35] (3.9,0.35);
  \draw[->, black, thick] (3.9,-0.55) to[bend left=35] (2.4,-0.55);
  \node[black, font=\scriptsize, below] at (3.15,-0.85) {lowering};
\end{tikzpicture}
$$

Isospin conservation makes quantitative predictions about strong-interaction rates.
Because the initial and final states must be decomposed into total-isospin
eigenstates using Clebsch–Gordan coefficients, the cross sections for related
reactions stand in fixed ratios. The classic case is pion–nucleon scattering
through the $\Delta$ resonance ($I = \tfrac32$): at the resonance the ratios

$$
\sigma(\pi^+ p \to \pi^+ p) : \sigma(\pi^- p \to \pi^0 n) :
\sigma(\pi^- p \to \pi^- p) = 9 : 2 : 1
$$

follow purely from isospin, and are confirmed by experiment — direct evidence that
the symmetry is real and not merely bookkeeping.[^griffiths-isospin]

## From SU(2) to flavor SU(3)

The discovery of strange particles added a new quantum number, strangeness $S$,
conserved by the strong force. The natural move is to enlarge the symmetry: if the
strong interaction is nearly blind to the difference between up and down quarks
(isospin $SU(2)$), it is also — though less exactly, because the strange quark is
heavier — blind to the difference among up, down, and strange. That approximate
symmetry is **flavor $SU(3)$**, observed independently by Gell-Mann and Ne'eman in
1961 and named the **eightfold way**.[^tong-su3]

$SU(3)$ has rank two: two of its generators commute simultaneously and can be
diagonalized together. One is the isospin third component $I_3$; the other is the
**hypercharge** $Y = B + S$ (for the light hadrons). A hadron is therefore labeled
by the pair $(I_3, Y)$, and a multiplet becomes a two-dimensional pattern — a
**weight diagram** — rather than a one-dimensional ladder. The single ladder of
$SU(2)$ is replaced by three sets of raising and lowering operators, one for each
$SU(2)$ subgroup embedded in $SU(3)$, and these move a state along three directions
in the $(I_3, Y)$ plane.

> **Definition (Hypercharge).** The hypercharge of a hadron is $Y = B + S$ for the
> light ($u,d,s$) sector, generalizing to $Y = B + S + C + \mathcal{B} + T$ when
> the heavy flavors are included. It is twice the average electric charge of an
> isospin multiplet and labels the horizontal rows of an $SU(3)$ weight diagram.

## The Gell-Mann–Nishijima relation

The link between these internal labels and the observable electric charge is the
**Gell-Mann–Nishijima relation**,

$$
Q = I_3 + \frac{Y}{2} = I_3 + \frac{B + S}{2} ,
$$

which holds for every hadron. It was first found empirically as a way to organize
the strange particles, and it fixes the electric charge of each
node on a weight diagram: the isospin third component sets the horizontal position,
the hypercharge sets the vertical, and their combination gives the charge. Sloping
lines of constant $Q$ cut across the diagram at a fixed angle.

$$
% caption: The Gell-Mann–Nishijima relation as a grid. The horizontal axis is the
% isospin third component and the vertical axis is the hypercharge. Electric charge
% equals the third component plus half the hypercharge, so lines of constant charge
% run diagonally across the plane. Reading off any node gives its charge.
\begin{tikzpicture}[>=stealth, font=\footnotesize, scale=1.1]
  \definecolor{acc}{HTML}{4A6FA5}
  \draw[->, black] (-2.2,0) -- (2.2,0) node[right, font=\scriptsize] {third comp.};
  \draw[->, black] (0,-1.7) -- (0,1.9) node[above, font=\scriptsize] {hypercharge};
  % constant-charge diagonals
  \draw[acc!55, dashed] (-1.3,1.6) -- (1.1,-1.4);
  \draw[acc!55, dashed] (-0.4,1.6) -- (2.0,-1.4);
  \node[font=\scriptsize, left] at (-1.35,1.55) {charge 0};
  \node[font=\scriptsize, right] at (2.05,-1.35) {charge 1};
  % a few nodes
  \fill (-0.5,1.0) circle (2.2pt);
  \fill (0.5,1.0) circle (2.2pt);
  \fill (-1.0,0.0) circle (2.2pt);
  \fill (1.0,0.0) circle (2.2pt);
  \node[black, font=\scriptsize, below] at (1.0,-0.2) {node};
\end{tikzpicture}
$$

Read the other way, the relation is a constraint: given the electric charge and the
isospin assignment of a candidate state, its hypercharge — and so its strangeness —
is fixed. This same relation determines the quark charges themselves. The up quark
sits at $(I_3, Y) = (+\tfrac12, +\tfrac13)$ giving $Q = +\tfrac23$, the down at
$(-\tfrac12, +\tfrac13)$ giving $Q = -\tfrac13$, and the strange at $(0, -\tfrac23)$
giving $Q = -\tfrac13$ — the fundamental triplet of flavor $SU(3)$.

## Weight diagrams: the language of the quark model

The payoff is that the entire hadron spectrum is organized by these plots. The
three light quarks form the fundamental $SU(3)$ triplet at the corners of a small
triangle in the $(I_3, Y)$ plane; the antiquarks form the inverted antitriplet.
Combining a quark with an antiquark, or three quarks together, builds the meson and
baryon multiplets as larger patterns in the same plane — the hexagons, triangles,
and central multiplicities that the next module derives in full.

$$
% caption: The isospin–hypercharge plane as the stage for flavor SU(3). The three
% light quarks occupy the corners of a triangle (the fundamental triplet); the
% antiquarks form the inverted triangle. Every hadron multiplet is a pattern of
% nodes in this plane, generated by combining these building blocks.
\begin{tikzpicture}[>=stealth, font=\footnotesize, scale=1.1]
  \definecolor{acc}{HTML}{4A6FA5}
  \draw[->, black] (-2.2,0) -- (2.2,0) node[right, font=\scriptsize] {third comp.};
  \draw[->, black] (0,-1.7) -- (0,1.7) node[above, font=\scriptsize] {hypercharge};
  % quark triplet (upright)
  \fill (-0.9,0.55) circle (2.4pt); \node[above left, font=\scriptsize] at (-0.9,0.55) {down};
  \fill (0.9,0.55) circle (2.4pt);  \node[above right, font=\scriptsize] at (0.9,0.55) {up};
  \fill (0,-1.1) circle (2.4pt);    \node[below, font=\scriptsize] at (0,-1.2) {strange};
  \draw[acc, thick] (-0.9,0.55) -- (0.9,0.55) -- (0,-1.1) -- cycle;
  % antiquark triplet (inverted), muted
  \draw[black, thick, densely dotted] (-0.9,-0.55) -- (0.9,-0.55) -- (0,1.1) -- cycle;
  \node[black, font=\scriptsize, above] at (0,1.15) {antiquarks};
\end{tikzpicture}
$$

The logic runs in one direction throughout. A symmetry of the strong force — exact
for isospin $SU(2)$, approximate for flavor $SU(3)$ — organizes the hadrons into
multiplets; the multiplets are patterns on a weight diagram; and the regularity of
those patterns, together with a gap where the $\Omega^-$ was later found, is what
demanded that hadrons be built from a few fundamental constituents. Isospin and its
$SU(3)$ enlargement are thus not just conservation laws but the direct predecessors
of the quark model, whose construction is the subject of the next module.

[^tong-isospin]: Tong, _The Standard Model_ (Cambridge Part III), §2.1 — isospin as the $SU(2)_V$ symmetry noted by Heisenberg from the near-identical strong interactions of the proton and neutron. http://www.damtp.cam.ac.uk/user/tong/standardmodel.html
[^griffiths-isospin]: Griffiths, _Introduction to Elementary Particles_, 2nd ed., §4.2–4.3 — isospin, Clebsch–Gordan decomposition, and the 9:2:1 pion–nucleon cross-section ratios at the $\Delta$ resonance.
[^tong-su3]: Tong, _The Standard Model_ (Cambridge Part III), §2.1 — the approximate $SU(3)_V$ flavor symmetry (the eightfold way) of Gell-Mann and Ne'eman, distinct from the QCD gauge $SU(3)$. http://www.damtp.cam.ac.uk/user/tong/standardmodel.html