Spontaneous Symmetry Breaking

The electroweak gauge theory 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, 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.

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

which is invariant under the discrete reflection $\varphi \to -\varphi$. The sign of the quadratic term decides everything. For ${\mu }^2<0$ the term $-\frac{1}{2}{\mu }^2{\varphi }^2$ is a positive parabola, the minimum sits at $\varphi =0$, and the vacuum shares the reflection symmetry. For ${\mu }^2>0$ the point $\varphi =0$ becomes a local maximum and the minima move to

There are now two degenerate ground states, and neither is invariant under $\varphi \to -\varphi$ — the reflection exchanges them. The field must choose one, say $\varphi =+v$, and the symmetry is spontaneously broken. Expanding $\varphi =v+\eta$ about the chosen vacuum, the potential becomes $V={\mu }^2{\eta }^2+\dots$, 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.

A broken continuous symmetry

Replace the reflection with a continuous symmetry. Take a complex scalar $\varphi =({\varphi }_1+i{\varphi }_2)/\sqrt{2}$ with the potential

invariant under the global $U(1)$ phase rotation $\varphi \to e^{i\alpha }\varphi$. 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,

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, $⟨\varphi ⟩=v/\sqrt{2}$ — breaks the $U(1)$.

Now expand about the chosen vacuum in a radial and an angular coordinate, $\varphi =\frac{1}{\sqrt{2}}(v+h)e^{i\theta /v}$. Substituting into the potential,

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.

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.

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 $\frac{1}{2}N(N-1)-\frac{1}{2}(N-1)(N-2)=N-1$, 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, 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.

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 $H=-J{\sum }_{⟨ij⟩}{\vec{S}}_i\cdot {\vec{S}}_j$ — 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:

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.

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.¹²