---
title: The Standard Model
module: Electroweak Unification and the Higgs
moduleNumber: 9
lessonNumber: 5
order: 905
summary: >
  The Standard Model combines the quark model, quantum chromodynamics, and the
  electroweak theory. SU(3) symmetry sorts the hadrons and predicted the omega;
  color explains why only colorless quark combinations exist; QCD gives
  asymptotic freedom and confinement; and spontaneous symmetry breaking through
  the Higgs field gives the weak bosons their mass.
topics: [Electroweak Unification and the Higgs]
draft: false
sources:
  - book: Tipler & Llewellyn
    ref: "Ch. 12 — Particle Physics; §12-4 The Standard Model"
  - book: Tipler & Mosca
    ref: "Ch. 40 — Nuclear Physics and Elementary Particles"
---

The Standard Model is the accepted theory of elementary particles. It combines
three parts: the quark model of hadron structure, the electroweak theory unifying
the electromagnetic and weak interactions, and quantum chromodynamics (QCD) for
the strong interaction. It is second only to QED in the precision of its
predictions. This lesson assembles the three parts: how SU(3) symmetry organizes
the hadrons and led to quarks, how color and QCD confine them, and how
spontaneous symmetry breaking gives the weak bosons their mass.

## The eightfold way and quarks

The accelerators of the 1950s produced a flood of hadrons. The most successful
classification, the **eightfold way**, was proposed by Gell-Mann and Ne'eman in
1961. Hadrons of the same spin and parity are grouped into **supermultiplets**
plotted in the [hypercharge-isospin plane](/particle-physics/symmetries/conservation-laws-symmetries).
The scheme rests on the Lie group **SU(3)**, whose lowest multiplets contain
$1, 3, 8, 10, \dots$ members; the eight-member octets give the method its name.

The eightfold way predicted a gap. In the spin-$\tfrac{3}{2}$ baryon **decuplet**,
nine of ten members were known in 1961; the tenth, the $\Omega^-$, was predicted
with definite mass and spin and discovered in 1964, an exact parallel to
Mendeleev's periodic-table gaps.

$$
% caption: The spin-3/2 baryon decuplet in the hypercharge-isospin plane, labeled
% by quark content. Each step down adds a strange quark; the apex sss is the
% omega, predicted before it was found.
\begin{tikzpicture}[>=stealth, font=\footnotesize, scale=1.05]
  \definecolor{acc}{HTML}{4A6FA5}
  \draw[->, black] (-2.9,0) -- (2.9,0) node[right, font=\scriptsize] {$I_3$};
  \draw[->, black] (0,-3.4) -- (0,0.9) node[above, font=\scriptsize] {$Y$};
  % top row Y=1 (four Delta)
  \foreach \x/\lab in {-2.4/ddd, -0.8/udd, 0.8/uud, 2.4/uuu}
    {\fill[acc] (\x,0) circle (2.2pt); \node[above, font=\scriptsize] at (\x,0.1) {$\lab$};}
  % row Y=0 (three Sigma*)
  \foreach \x/\lab in {-1.6/dds, 0/uds, 1.6/uus}
    {\fill[acc] (\x,-1.1) circle (2.2pt); \node[below, font=\scriptsize] at (\x,-1.2) {$\lab$};}
  % row Y=-1 (two Xi*)
  \foreach \x/\lab in {-0.8/dss, 0.8/uss}
    {\fill[acc] (\x,-2.2) circle (2.2pt); \node[below, font=\scriptsize] at (\x,-2.3) {$\lab$};}
  % apex Y=-2 (Omega)
  \fill[acc] (0,-3.3) circle (2.6pt); \node[right, font=\scriptsize, acc] at (0.15,-3.3) {$sss$ (omega)};
  % triangle outline
  \draw[acc!55] (-2.4,0) -- (2.4,0) -- (0,-3.3) -- cycle;
\end{tikzpicture}
$$

The regularities and the missing SU(3) triplet led Gell-Mann and Zweig in 1964 to
propose **quarks** as the constituents of all hadrons.

> **Definition (Quark model).** Every baryon is three quarks (antibaryons three
> antiquarks); every meson is a quark-antiquark pair. The additive quantum
> numbers of a hadron are the sums of those of its constituent quarks.

Because the quantum numbers add, a hadron's properties are arithmetic. The $uds$
combination has charge $\tfrac{2}{3} - \tfrac{1}{3} - \tfrac{1}{3} = 0$ and baryon
number $\tfrac{1}{3} + \tfrac{1}{3} + \tfrac{1}{3} = 1$. Every hadron in the
tables is built from the six quarks and their antiquarks.

| Baryon | Quarks | Meson | Quarks |
| --- | --- | --- | --- |
| $p$ | $uud$ | $\pi^+$ | $u\bar{d}$ |
| $n$ | $udd$ | $\pi^-$ | $\bar{u}d$ |
| $\Lambda^0$ | $uds$ | $K^+$ | $u\bar{s}$ |
| $\Sigma^+$ | $uus$ | $K^0$ | $d\bar{s}$ |
| $\Omega^-$ | $sss$ | $J/\psi$ | $c\bar{c}$ |

$$
% caption: Quark content of the proton, neutron, and the positive pion. Baryons
% hold three quarks; a meson holds a quark and an antiquark.
\begin{tikzpicture}[>=stealth, font=\footnotesize, scale=1.0,
  qk/.style={circle, draw=acc, fill=acc!12, minimum size=8mm, font=\scriptsize}]
  \definecolor{acc}{HTML}{4A6FA5}
  % proton
  \node[qk] (p1) at (0,0.6) {$u$};
  \node[qk] (p2) at (0.7,-0.6) {$u$};
  \node[qk] (p3) at (-0.7,-0.6) {$d$};
  \draw[acc] (p1)--(p2) (p2)--(p3) (p3)--(p1);
  \node[font=\scriptsize] at (0,-1.5) {proton $uud$};
  % neutron
  \begin{scope}[xshift=3.6cm]
    \node[qk] (n1) at (0,0.6) {$u$};
    \node[qk] (n2) at (0.7,-0.6) {$d$};
    \node[qk] (n3) at (-0.7,-0.6) {$d$};
    \draw[acc] (n1)--(n2) (n2)--(n3) (n3)--(n1);
    \node[font=\scriptsize] at (0,-1.5) {neutron $udd$};
  \end{scope}
  % pion
  \begin{scope}[xshift=7.0cm]
    \node[qk] (m1) at (-0.5,0) {$u$};
    \node[qk] (m2) at (0.5,0) {$\bar{d}$};
    \draw[acc] (m1)--(m2);
    \node[font=\scriptsize] at (0,-1.5) {pion $u\bar{d}$};
  \end{scope}
\end{tikzpicture}
$$

Direct evidence for quarks comes from **deep inelastic scattering**: electrons or
muons of $15$–$200$ GeV strike a nucleon and scatter at large angles off pointlike
spin-$\tfrac{1}{2}$ constituents much smaller than the nucleon, exactly as
Rutherford's alpha scattering
revealed the nucleus. The same experiments raised the **proton spin crisis**:
polarized scattering shows the three valence quarks account for only 20 to 30
percent of the proton's spin, with the rest carried by quark orbital angular
momentum, gluon spin, and a polarized sea of virtual quark-antiquark pairs. The
naive picture in which three quark spins add to $\tfrac{1}{2}$ is incomplete, and
the origin of the proton's spin remains unsettled.

The charm quark arrived in 1974 with the $J/\psi$ meson ($c\bar{c}$), whose
lifetime of $10^{-20}$ s was extraordinarily long for its mass. The tau lepton
(1975), the upsilon ($b\bar{b}$, giving the bottom quark), and the top quark
(1995, at $176\ \text{GeV}/c^2$ the heaviest known particle) completed three
matched generations of quarks and leptons.

## Color and QCD

The early quark model had two problems: no free quarks were found, and some
baryons violated the [exclusion principle](/atomic-physics/many-electron-atoms/periodic-table-atomic-spectra).
The $\Delta^{++}$ has spin $\tfrac{3}{2}$ and content $uuu$, three identical
fermions in the same state. Greenberg's solution gave each quark flavor a new
threefold **color charge** (red, blue, green), so the three $u$ quarks differ in
color and the exclusion principle is respected.

> **Invariant (Color confinement).** Every particle observed in nature is
> colorless: either the total color charge is zero (equal amounts of red, blue,
> and green, as in a baryon), or a color pairs with its anticolor (as in a
> meson).

QCD is the theory of the strong interaction between quarks and gluons, built on
the model of QED. The mediator is the **gluon**, massless and spin-1 like the
photon, but with a decisive difference: gluons carry color (one color and one
anticolor), so unlike the neutral photon they interact with one another. Gluon
self-interaction produces two effects on the running coupling $\alpha_s$:

- **Asymptotic freedom.** Gluon loops dilute the color charge at short distance,
  so $\alpha_s$ decreases as quarks approach. Inside a nucleon the quarks move
  nearly freely, as deep inelastic scattering confirms.
- **Confinement.** At large separation the potential grows without bound.

One model potential captures both:

$$
V_{\text{QCD}}(r) = -\frac{4}{3}\frac{\alpha_s}{r} + kr,
$$

with $k \approx 1\ \text{GeV/fm}$. The Coulomb-like first term dominates at small
$r$; the linear term dominates at large $r$, so the force
$F = -\nabla V \to k$ (constant) rather than falling to zero.

$$
% caption: The QCD quark potential. The Coulomb-like term rules at short range;
% the linear term rises without bound at long range, giving a constant restoring
% force that confines quarks.
\begin{tikzpicture}[>=stealth, font=\footnotesize, scale=1.0]
  \definecolor{acc}{HTML}{4A6FA5}
  \draw[->, black] (0,-1.4) -- (0,3.2) node[above, font=\scriptsize] {$V(r)$};
  \draw[->, black] (0,0) -- (6.2,0) node[right, font=\scriptsize] {separation $r$};
  % V = -a/r + k r ; choose a=0.6, k=0.7, shift for display
  \draw[acc, very thick] plot[domain=0.28:5.6, samples=90] (\x, {0.7*\x - 0.6/\x});
  \node[acc, right, font=\scriptsize] at (4.6,2.7) {$V \sim kr$ (confining)};
  \node[black, font=\scriptsize, align=left] at (1.6,-1.05) {Coulomb-like\\at small $r$};
\end{tikzpicture}
$$

Confinement explains why free quarks are never seen. Pulling a quark from a
hadron adds energy to the color field; when enough accumulates, a
quark-antiquark pair is created rather than a lone quark liberated. The new quark
and antiquark bind with the fragments, producing hadrons and never a free quark.
This is the origin of the virtual pions in the
[Yukawa picture](/nuclear-physics/nuclear-force-deuteron/nuclear-force-shell-overview) of
the nuclear force.

$$
% caption: Confinement in action. Adding energy to separate a quark creates a new
% quark-antiquark pair; the fragments rebind into hadrons, so no free quark
% emerges.
\begin{tikzpicture}[>=stealth, font=\footnotesize, scale=1.0,
  qk/.style={circle, draw=acc, fill=acc!12, minimum size=7mm, font=\scriptsize}]
  \definecolor{acc}{HTML}{4A6FA5}
  % before: quark pair with flux tube
  \node[qk] (a1) at (0,0) {$q$};
  \node[qk] (a2) at (2.2,0) {$q$};
  \draw[acc, very thick] (a1) -- (a2);
  \node[font=\scriptsize] at (1.1,-0.9) {energy added};
  % arrow
  \draw[->, acc, very thick] (3.0,0) -- (4.0,0);
  % after: new pair created in the middle
  \node[qk] (b1) at (4.8,0) {$q$};
  \node[qk] (b2) at (6.2,0) {$\bar{q}$};
  \node[qk] (b3) at (7.2,0) {$q$};
  \node[qk] (b4) at (8.6,0) {$\bar{q}$};
  \draw[acc, very thick] (b1)--(b2) (b3)--(b4);
  \node[font=\scriptsize] at (5.5,-0.9) {new hadron};
  \node[font=\scriptsize] at (7.9,-0.9) {new hadron};
\end{tikzpicture}
$$

## The electroweak theory

The electroweak theory treats the electromagnetic and weak interactions as two
faces of one interaction. Above $\sim 100$ GeV it is mediated by four massless
bosons of equal footing: a triplet $W^+$, $W^0$, $W^-$ and a singlet $B^0$. At
ordinary energy the symmetry is **spontaneously broken**: the observed photon is
one combination of $W^0$ and $B^0$, and the $Z^0$ is the orthogonal combination.

> **Definition (Spontaneous symmetry breaking).** The Hamiltonian retains a
> symmetry that its lowest-energy state does not. A ferromagnet is invariant
> under rotation, yet each magnetic domain picks a definite spin direction; the
> ground state breaks the rotational symmetry the interaction respects.

The symmetry-breaking agent is the **Higgs field**, requiring a new spin-0 boson,
the **Higgs boson**. Particles acquire mass by interacting with the Higgs field:
the photon, which does not couple to it, stays massless, while the $W^\pm$ and
$Z^0$ reach $\sim 80$–$91\ \text{GeV}/c^2$. The Higgs boson's rest energy is of
order $1$ TeV, produced in high-energy proton collisions and a primary target of
the Large Hadron Collider.

$$
% caption: A symmetry-breaking potential. The peak is symmetric but unstable; the
% state rolls to one point in the trough, a lowest-energy configuration that no
% longer shows the full symmetry.
\begin{tikzpicture}[>=stealth, font=\footnotesize, scale=1.0]
  \definecolor{acc}{HTML}{4A6FA5}
  % Mexican-hat cross section: two wells with central bump
  \draw[acc, very thick] plot[domain=-2.6:2.6, samples=90] (\x, {0.34*(\x*\x-1.7)*(\x*\x-1.7)*0.34 + 0.2});
  \draw[->, black] (-3.0,0) -- (3.0,0) node[right, font=\scriptsize] {field value};
  \draw[->, black] (0,-0.2) -- (0,2.6) node[above, font=\scriptsize] {energy};
  % ball at broken minimum
  \fill[acc] (1.85,0.2) circle (3pt);
  \node[acc, below, font=\scriptsize] at (1.85,0.15) {broken};
  % symmetric top
  \node[black, font=\scriptsize, above] at (0,1.4) {symmetric (unstable)};
\end{tikzpicture}
$$

## The model assembled

The Standard Model's fundamental particles are the leptons and quarks, each in
three generations, plus the force carriers.

- **Matter particles.** Six leptons and six quarks, all spin-$\tfrac{1}{2}$
  fermions obeying the exclusion principle.
- **Force carriers.** The photon, the $W^\pm$ and $Z^0$, and eight gluons — all
  integer-spin bosons. The graviton lies outside the model.
- **The Higgs.** A spin-0 boson whose field gives the other particles mass.

$$
% caption: The Standard Model particles: three generations of quarks and leptons,
% the gauge bosons that carry the three included interactions, and the Higgs.
\begin{tikzpicture}[>=stealth, font=\footnotesize, scale=1.0,
  cell/.style={draw, minimum size=9mm, font=\scriptsize, align=center},
  bos/.style={draw=acc, fill=acc!10, minimum size=9mm, font=\scriptsize, align=center}]
  \definecolor{acc}{HTML}{4A6FA5}
  % header
  \node[font=\scriptsize] at (0.7,3.2) {gen 1};
  \node[font=\scriptsize] at (1.9,3.2) {gen 2};
  \node[font=\scriptsize] at (3.1,3.2) {gen 3};
  % quarks up-type row
  \node[cell] at (0.7,2.4) {$u$};
  \node[cell] at (1.9,2.4) {$c$};
  \node[cell] at (3.1,2.4) {$t$};
  % quarks down-type row
  \node[cell] at (0.7,1.4) {$d$};
  \node[cell] at (1.9,1.4) {$s$};
  \node[cell] at (3.1,1.4) {$b$};
  \node[left, font=\scriptsize] at (0.1,1.9) {quarks};
  % charged leptons
  \node[cell] at (0.7,0.4) {$e$};
  \node[cell] at (1.9,0.4) {mu};
  \node[cell] at (3.1,0.4) {tau};
  % neutrinos
  \node[cell] at (0.7,-0.6) {nu e};
  \node[cell] at (1.9,-0.6) {nu mu};
  \node[cell] at (3.1,-0.6) {nu tau};
  \node[left, font=\scriptsize] at (0.1,-0.1) {leptons};
  % bosons column
  \node[bos] at (4.7,2.4) {photon};
  \node[bos] at (4.7,1.4) {gluon};
  \node[bos] at (4.7,0.4) {$W$};
  \node[bos] at (4.7,-0.6) {$Z$};
  \node[bos] at (6.0,0.9) {Higgs};
  \node[font=\scriptsize, acc] at (4.7,3.2) {carriers};
\end{tikzpicture}
$$

Every force is one of four basic interactions, and a particle feels an
interaction only if it carries the corresponding charge: electric charge (quarks
and charged leptons), weak charge (all quarks and leptons), color charge (quarks
and gluons), and mass (all particles). Hadrons are the color-neutral bound states;
leptons exist only as isolated particles. The strong interaction acts in two
ways: the fundamental color interaction between quarks, mediated by gluons, and
the residual strong interaction between color-neutral nucleons, mediated by meson
exchange, analogous to the residual electromagnetic force that binds neutral
atoms into molecules.

> **Example (Reading quark content).** For $\bar{u}d$: baryon number
> $0$, so a meson; no strange quark, so strangeness $0$; charge
> $-\tfrac{2}{3} - \tfrac{1}{3} = -1$. This is the $\pi^-$. For $dds$: baryon
> number $1$; strangeness $-1$; charge $-\tfrac{1}{3} - \tfrac{1}{3} - \tfrac{1}{3}
> = -1$. This is the $\Sigma^-$.

Decays are tracked at the quark level by following flavor changes, each mediated
by the weak $W$ boson when strangeness changes.

> **Example (Quark-level decay of the omega).** The $\Omega^-$ ($sss$) decays via
> $\Omega^- \to \Lambda^0 + K^-$. In quarks,
>
> $$
> sss \to uds + s\bar{u},
> $$
>
> where one $s$ becomes a $d$ (emitting a $W^-$) and a $u\bar{u}$ pair is created
> from the color field. The daughter $\Lambda^0$ ($uds$) then decays
> $uds \to uud + \bar{u}d$, again with an $s \to d$ transition, and the $K^-$
> ($s\bar{u}$) annihilates to a $W^-$ that decays leptonically. Every flavor
> change is a weak process, so the $\Omega^-$ lifetime is the long
> $\sim 10^{-10}$ s characteristic of the weak interaction.

The Standard Model leaves questions open: why four interactions, why their
strengths differ so widely, and what lies at higher energy. These drive the
[beyond-Standard-Model](/particle-physics/beyond-standard-model/beyond-standard-model)
program.[^tl-sm]

[^tl-sm]: Tipler & Llewellyn, §12-4 — the eightfold way and SU(3), the quark model, color and confinement, quantum chromodynamics with asymptotic freedom, and electroweak unification through spontaneous symmetry breaking and the Higgs field.