---
title: Spontaneous Symmetry Breaking
module: Electroweak Unification and the Higgs
moduleNumber: 9
lessonNumber: 2
order: 902
summary: >
  A symmetry of the Lagrangian need not be a symmetry of the ground state. When the
  lowest-energy configuration sits away from the symmetric point, the symmetry is
  spontaneously broken and the vacuum is one of a degenerate family. Breaking a
  continuous global symmetry produces one massless scalar — a Goldstone boson — for
  every broken generator, the flat direction along the vacuum manifold. The
  Mexican-hat potential and the ferromagnet below its Curie point are the working
  pictures.
topics: [Electroweak Unification and the Higgs]
sources:
  - book: Griffiths
    ref: "Ch. 11 §11.1–11.5 (spontaneous symmetry breaking, the Goldstone boson)"
  - book: Thomson
    ref: "Ch. 17 §17.1–17.3 (spontaneous symmetry breaking and Goldstone's theorem)"
  - book: Halzen & Martin
    ref: "Ch. 14 (spontaneous breaking of a global symmetry; the Higgs mechanism)"
  - book: Tong
    ref: "The Standard Model (Cambridge Part III), §2.1–2.2 (spontaneous symmetry breaking, Goldstone's theorem)"
draft: false
---

The [electroweak gauge theory](/particle-physics/electroweak-higgs/electroweak-su2-u1)
forbids explicit masses for the gauge bosons and, through their chiral assignments,
for the fermions: a mass term $M^2 W_\mu W^\mu$ or $m\bar\psi\psi$ is not invariant
under $SU(2)_L \times U(1)_Y$. Yet the $W$ and $Z$ weigh in at eighty and ninety
GeV. The resolution is that the symmetry is not violated but **spontaneously
broken**: the Lagrangian keeps its full symmetry while the ground state does not.
This lesson develops spontaneous breaking for global symmetries — the setting where
it produces massless Goldstone bosons — and holds the gauge case for the
[next lesson](/particle-physics/electroweak-higgs/higgs-mechanism), where the
Goldstone modes are converted into gauge-boson masses.

## Symmetry of the law, asymmetry of the state

A symmetry of a theory is a transformation that leaves the Lagrangian invariant. A
symmetry of a state is a transformation that leaves that state unchanged. The two
usually coincide, but they need not. When the lowest-energy state fails to share a
symmetry of the Lagrangian, that symmetry is **spontaneously broken**.

> **Definition (Spontaneous symmetry breaking).** A symmetry is spontaneously
> broken when the Lagrangian is invariant under a transformation $G$ but the ground
> state (vacuum) is not. The transformation then maps the vacuum to a distinct,
> degenerate vacuum of the same energy, so the set of ground states forms a
> nontrivial orbit under $G$.

The elementary example is a real scalar field $\phi$ with the potential

$$
V(\phi) = -\tfrac12\mu^2\phi^2 + \tfrac14\lambda\phi^4,
\qquad \lambda > 0,
$$

which is invariant under the discrete reflection $\phi \to -\phi$. The sign of the
quadratic term decides everything. For $\mu^2 < 0$ the term $-\tfrac12\mu^2\phi^2$
is a positive parabola, the minimum sits at $\phi = 0$, and the vacuum shares the
reflection symmetry. For $\mu^2 > 0$ the point $\phi = 0$ becomes a local maximum
and the minima move to

$$
\phi = \pm v, \qquad v = \sqrt{\frac{\mu^2}{\lambda}}.
$$

There are now two degenerate ground states, and neither is invariant under
$\phi \to -\phi$ — the reflection exchanges them. The field must choose one, say
$\phi = +v$, and the symmetry is spontaneously broken. Expanding $\phi = v + \eta$
about the chosen vacuum, the potential becomes $V = \mu^2\eta^2 + \ldots$, so the
fluctuation $\eta$ is a scalar of mass $m_\eta^2 = 2\mu^2 = 2\lambda v^2$. A broken
discrete symmetry leaves a massive particle and no massless one.

$$
% caption: The double-well potential. For a positive mass-squared parameter the
% symmetric point at the origin is a maximum and the two minima sit at plus and
% minus v. The field settles into one well, breaking the reflection symmetry; small
% oscillations about that well are the massive scalar.
\begin{tikzpicture}[>=stealth, font=\footnotesize, scale=1.0]
  \definecolor{acc}{HTML}{4A6FA5}
  \draw[->, black] (-3.0,0) -- (3.0,0) node[right, font=\scriptsize] {scalar};
  \draw[->, black] (0,-0.4) -- (0,2.7) node[above, font=\scriptsize] {$V$};
  % V = 0.25*(phi^2 - 1.7)^2 shifted; minima near +-1.3
  \draw[very thick] plot[domain=-2.55:2.55, samples=100]
    (\x, {0.28*(\x*\x - 1.7)*(\x*\x - 1.7) + 0.15});
  % minima markers at phi = +- sqrt(1.7) ~ 1.30, V=0.15
  \fill[acc] (1.304,0.15) circle (2.4pt);
  \fill[black] (-1.304,0.15) circle (2.4pt);
  \node[acc, below, font=\scriptsize] at (1.304,0.1) {chosen vacuum $+v$};
  \node[black, below, font=\scriptsize] at (-1.304,0.1) {other minimum};
  \node[black, font=\scriptsize, above] at (0,1.2) {symmetric point (unstable)};
\end{tikzpicture}
$$

## A broken continuous symmetry

Replace the reflection with a continuous symmetry. Take a complex scalar
$\phi = (\phi_1 + i\phi_2)/\sqrt2$ with the potential

$$
V(\phi) = -\mu^2\,\phi^\ast\phi + \lambda\,(\phi^\ast\phi)^2,
\qquad \mu^2, \lambda > 0,
$$

invariant under the global $U(1)$ phase rotation $\phi \to e^{i\alpha}\phi$. In
terms of the two real components this is the **Mexican-hat** (or wine-bottle)
potential: a central bump surrounded by a circular trough. The minima form a whole
circle,

$$
\phi^\ast\phi = \frac{\mu^2}{2\lambda} \equiv \frac{v^2}{2},
\qquad
|\phi| = \frac{v}{\sqrt2},
$$

parametrized by the phase. Every point on the circle is a legitimate vacuum, all
degenerate, and the phase rotation slides one into another. Choosing a vacuum — say
the real direction, $\langle\phi\rangle = v/\sqrt2$ — breaks the $U(1)$.

$$
% caption: The Mexican-hat potential drawn as a cross-section through the two
% degenerate real directions. The rim is a circle of minima; the center is a local
% maximum. A radial displacement climbs the potential and costs energy; a tangential
% displacement stays on the rim and costs none.
\begin{tikzpicture}[>=stealth, font=\footnotesize, scale=1.0]
  \definecolor{acc}{HTML}{4A6FA5}
  \draw[->, black] (-3.4,0) -- (3.4,0) node[right, font=\scriptsize] {radial direction};
  \draw[->, black] (0,-0.4) -- (0,2.7) node[above, font=\scriptsize] {$V$};
  \draw[very thick] plot[domain=-2.9:2.9, samples=110]
    (\x, {0.22*(\x*\x - 1.9)*(\x*\x - 1.9) + 0.15});
  % vacuum points on rim
  \fill[acc] (1.378,0.15) circle (2.4pt);
  \fill[acc] (-1.378,0.15) circle (2.4pt);
  % radial arrow (uphill) and tangential note
  \draw[black, ->, thick] (1.378,0.15) -- (2.15,0.62);
  \node[black, right, font=\scriptsize] at (2.1,0.72) {radial: costs energy};
  \node[acc, below, font=\scriptsize] at (1.378,0.1) {rim of minima};
\end{tikzpicture}
$$

Now expand about the chosen vacuum in a radial and an angular coordinate,
$\phi = \tfrac{1}{\sqrt2}(v + h)\,e^{i\theta/v}$. Substituting into the potential,

$$
V = \lambda v^2\, h^2 + \ldots,
$$

with no $\theta$ dependence at all. Two facts stand out.

- **The radial mode $h$ is massive**, $m_h^2 = 2\lambda v^2$. Displacing the field
  radially climbs the wall of the hat, which costs energy in proportion to the
  square of the displacement.
- **The angular mode $\theta$ is massless.** It slides along the flat bottom of the
  trough, from one degenerate vacuum to another, so no restoring force and no mass
  term appears. This is a **Goldstone boson**.

The massless mode is the direct signature of the broken continuous symmetry: the
flat direction of the potential traces the orbit of the symmetry through the
vacuum, and motion along it costs nothing.

$$
% caption: Radial and angular excitations about a point on the rim. The radial
% direction (h) climbs the potential wall and is the massive Higgs-like mode; the
% angular direction (theta) runs along the circle of degenerate minima and is the
% massless Goldstone mode. The two are orthogonal at the chosen vacuum.
\begin{tikzpicture}[>=stealth, font=\footnotesize, scale=1.0]
  \definecolor{acc}{HTML}{4A6FA5}
  % circle of minima (top-down view)
  \draw[very thick] (0,0) circle (1.7);
  \fill (0,0) circle (2pt);
  \node[font=\scriptsize] at (0,-0.35) {maximum};
  % chosen vacuum point
  \fill (1.7,0) circle (2.6pt);
  \node[below right, font=\scriptsize] at (1.75,-0.1) {vacuum};
  % radial arrow (massive)
  \draw[black, ->, very thick] (1.7,0) -- (2.75,0);
  \node[black, above, font=\scriptsize] at (2.35,0.12) {radial (massive)};
  % tangential arrow (Goldstone)
  \draw[acc, ->, very thick] (1.7,0.28) arc (9:45:1.72);
  \node[acc, above right, font=\scriptsize] at (1.5,1.25) {Goldstone mode};
  \node[black, font=\scriptsize, below] at (0,-1.95) {circle of degenerate minima};
\end{tikzpicture}
$$

## Goldstone's theorem

The pattern generalizes. Let a theory have a continuous global symmetry group $G$,
and let the vacuum be invariant only under a subgroup $H \subset G$. The generators
of $G$ split into those that fix the vacuum (the generators of $H$, unbroken) and
those that move it (the broken generators). Each broken generator produces one flat
direction in field space and one massless scalar.

> **Theorem (Goldstone).** If a global continuous symmetry $G$ is spontaneously
> broken to a subgroup $H$, the spectrum contains exactly
> $\dim G - \dim H = \dim(G/H)$ massless scalar bosons, one for each broken
> generator. These are the Goldstone bosons, and they parametrize the vacuum
> manifold $G/H$.

> **Proof.** Let the vacuum be $\phi_0$ with $V(\phi_0)$ a minimum, so the mass
> matrix is $M^2_{ij} = \partial_i\partial_j V(\phi_0)$, positive semi-definite. A
> generator $T^a$ acts on the fields by $\delta\phi = i\alpha\,(T^a\phi)$, and
> invariance of $V$ means $\partial_i V\,(T^a\phi)_i = 0$ identically. Differentiate
> this with respect to $\phi_j$ and evaluate at $\phi_0$: the term
> $\partial_j\partial_i V\,(T^a\phi_0)_i + \partial_i V\,(T^a)_{ij}$ reduces, since
> $\partial_i V(\phi_0) = 0$, to
>
> $$
> M^2_{ij}\,(T^a\phi_0)_j = 0.
> $$
>
> If $T^a$ is unbroken, $T^a\phi_0 = 0$ and the statement is empty. If $T^a$ is
> broken, $T^a\phi_0 \ne 0$ is a nonzero eigenvector of $M^2$ with eigenvalue zero —
> a massless mode. Distinct broken generators give linearly independent null
> vectors, so the number of massless bosons equals the number of broken generators,
> $\dim(G/H)$.

For the broken $U(1)$ above, $\dim G = 1$ and $\dim H = 0$, giving one Goldstone
boson — the angular mode $\theta$, confirming the explicit calculation. For a broken
$O(N) \to O(N-1)$, the count is $\tfrac12 N(N-1) - \tfrac12(N-1)(N-2) = N-1$, the
number of coordinates on the sphere $S^{N-1}$ of vacua.

## Global versus gauge symmetry

Whether the broken symmetry is global or gauged changes the physics entirely, and
the distinction is the hinge of the whole electroweak construction.

- **Global symmetry broken.** Goldstone's theorem applies: physical massless scalar
  particles appear in the spectrum. In particle physics genuinely massless scalars
  are not observed (the pion is light but not massless — it is a **pseudo-Goldstone**
  boson of the approximate chiral symmetry of QCD), so an exactly broken global
  symmetry would be a problem.
- **Gauge symmetry broken.** The would-be Goldstone bosons are not physical. A local
  symmetry means the phase $\theta(x)$ can be rotated away point by point by a gauge
  transformation, so the massless mode is a gauge artifact rather than a particle.
  Instead of appearing as a massless scalar, it is absorbed into the gauge field,
  which acquires a mass and a longitudinal polarization. This is the
  [Higgs mechanism](/particle-physics/electroweak-higgs/higgs-mechanism), and it is
  why the electroweak theory has no massless Goldstone scalars in its spectrum.

The bookkeeping is exact: each broken gauge generator removes one Goldstone scalar
and adds one longitudinal polarization to a gauge boson, turning a massless
spin-1 field (two polarizations) into a massive one (three). Degrees of freedom are
conserved; the count is the subject of the next lesson.

$$
% caption: The two fates of a Goldstone mode. When the broken symmetry is global,
% the massless scalar is a physical particle. When the broken symmetry is gauged,
% the same mode is unphysical: it is absorbed by the gauge boson, which gains mass
% and a third (longitudinal) polarization.
\begin{tikzpicture}[>=stealth, font=\footnotesize, scale=1.0,
  bx/.style={draw=black, minimum width=30mm, minimum height=8mm, font=\scriptsize, align=center},
  bxa/.style={draw=acc, fill=acc!10, minimum width=30mm, minimum height=8mm, font=\scriptsize, align=center}]
  \definecolor{acc}{HTML}{4A6FA5}
  \node[bx] (start) at (0,2.2) {broken continuous symmetry};
  \node[bx] (glob) at (-2.6,0.4) {global: physical\\ massless scalar};
  \node[bxa] (gauge) at (2.6,0.4) {gauge: scalar eaten,\\ boson gains mass};
  \draw[->, black] (start) -- (glob);
  \draw[->, acc] (start) -- (gauge);
  \node[black, below, font=\scriptsize] at (-2.6,-0.35) {Goldstone's theorem};
  \node[acc, below, font=\scriptsize] at (2.6,-0.35) {Higgs mechanism};
\end{tikzpicture}
$$

## The ferromagnet as a picture

Spontaneous symmetry breaking is not exotic; it governs ordinary phase transitions.
A ferromagnet is the cleanest analogy. The interaction between spins — the
Heisenberg Hamiltonian $H = -J\sum_{\langle ij\rangle}\vec S_i\cdot\vec S_j$ — is
rotationally invariant: it has no preferred direction. Above the Curie temperature
$T_c$ the spins point randomly and the average magnetization vanishes, respecting
the rotational symmetry. Below $T_c$ the spins align, and the sample acquires a net
magnetization $\vec M$ pointing in some direction. That direction is arbitrary — the
physics does not prefer one — but the state must choose, and the choice breaks the
rotational symmetry $O(3) \to O(2)$ (rotations about $\vec M$ still leave the state
invariant).

The correspondence is exact term by term:

| Ferromagnet | Field theory |
| --- | --- |
| rotational symmetry of $H$ | symmetry $G$ of the Lagrangian |
| magnetization direction $\vec M$ | vacuum $\langle\phi\rangle$ |
| residual rotations about $\vec M$ | unbroken subgroup $H$ |
| spin waves (magnons) | Goldstone bosons |
| temperature $T$ vs $T_c$ | sign of $\mu^2$ |

The magnons — long-wavelength twists of the magnetization that cost vanishing
energy as the wavelength grows — are the Goldstone modes of the broken $O(3)$. The
broken-generator count, $\dim O(3) - \dim O(2) = 3 - 1 = 2$, matches the two
transverse spin-wave polarizations. Raising the temperature through $T_c$ flips the
sign of the effective $\mu^2$ and restores the symmetry, the thermal analogue of
tuning the potential from a single well to a Mexican hat.

$$
% caption: A ferromagnet across its Curie temperature. Above T_c the spins are
% disordered and the average magnetization is zero, so the rotational symmetry is
% intact. Below T_c the spins align along one arbitrary direction, giving a net
% magnetization that breaks the symmetry; long-wavelength twists of the alignment
% are the Goldstone magnons.
\begin{tikzpicture}[>=stealth, font=\footnotesize, scale=1.0]
  \definecolor{acc}{HTML}{4A6FA5}
  % disordered (left)
  \foreach \x/\y/\a in {0/0/20, 0.6/0/200, 0/0.6/110, 0.6/0.6/300,
    0/1.2/70, 0.6/1.2/250, 1.2/0/150, 1.2/0.6/30, 1.2/1.2/210}
    {\draw[black, ->] (\x,\y) -- ++({0.42*cos(\a)},{0.42*sin(\a)});}
  \node[black, below, font=\scriptsize] at (0.6,-0.55) {above $T_c$: disordered};
  % ordered (right)
  \begin{scope}[xshift=4.2cm]
    \foreach \x/\y in {0/0, 0.6/0, 0/0.6, 0.6/0.6, 0/1.2, 0.6/1.2,
      1.2/0, 1.2/0.6, 1.2/1.2}
      {\draw[->, thick] (\x,\y) -- ++(0,0.42);}
    \node[below, font=\scriptsize] at (0.6,-0.55) {below $T_c$: aligned};
    \draw[acc, ->, very thick] (2.0,0.0) -- (2.0,1.2);
    \node[acc, right, font=\scriptsize] at (2.1,0.6) {net M};
  \end{scope}
\end{tikzpicture}
$$

## Summary

A symmetry of the Lagrangian is spontaneously broken when the ground state does not
share it, leaving a degenerate family of vacua related by the symmetry. Breaking a
discrete symmetry (the double well) leaves a massive scalar and nothing else.
Breaking a continuous global symmetry (the Mexican hat) leaves a massive radial mode
and one massless Goldstone boson per broken generator — Goldstone's theorem, with
$\dim(G/H)$ massless scalars filling the flat directions of the vacuum manifold. The
ferromagnet below its Curie point realizes the same structure, with magnons for
Goldstone bosons. The decisive fork is global versus local: a broken gauge symmetry
does not yield physical Goldstone particles but feeds them to the gauge bosons as
mass and longitudinal polarization. That conversion, applied to
$SU(2)_L \times U(1)_Y$, is the
[Higgs mechanism](/particle-physics/electroweak-higgs/higgs-mechanism).[^tong-ssb][^gr-ssb]

[^tong-ssb]: The double-well and Mexican-hat potentials, the radial-versus-angular mode analysis, and Goldstone's theorem with the count $\dim G - \dim H$ are developed in Tong, _The Standard Model_ (Cambridge Part III), §2.1–2.2, [damtp.cam.ac.uk/user/tong/standardmodel.html](http://www.damtp.cam.ac.uk/user/tong/standardmodel.html); the classical and quantum versions of the theorem trace to Goldstone, Salam, and Weinberg (1962).

[^gr-ssb]: Griffiths, §11.1–11.5, and Thomson, §17.1–17.3, give the same construction at textbook level, including the global-versus-gauge distinction; Halzen & Martin, Ch. 14, present the abelian Goldstone model as the warm-up to the Higgs mechanism. The ferromagnet analogy and the $O(3) \to O(2)$ magnon count follow Thomson, §17.1.