Spontaneous Symmetry Breaking and Goldstone's Theorem

A symmetry of the Lagrangian need not be a symmetry of the vacuum. When the lowest-energy state fails to share the symmetry of the dynamics, the symmetry is spontaneously broken — and a sharp theorem follows: every broken continuous symmetry produces a massless particle, a Goldstone boson. This mechanism underlies superconductivity, pions, and — once combined with gauge symmetry — the Higgs mechanism.

Conventions: $\hslash =c=1$, ${\eta }_{\mu \nu }=\mathrm{diag}(+1,-1,-1,-1)$.

Symmetry of the dynamics vs. symmetry of the vacuum

A symmetry can be realized two ways:

• Wigner–Weyl (unbroken): the vacuum is invariant, $Q|0⟩=0$, and states fall into degenerate multiplets (the usual case — e.g. isospin).
• Nambu–Goldstone (spontaneously broken): the Lagrangian is symmetric but the vacuum is not, $Q|0⟩\ne 0$. The symmetry still acts, but it moves one vacuum to a different, degenerate vacuum.

"Spontaneous" means no term in the Lagrangian breaks the symmetry explicitly; the breaking is a property of which ground state the system chooses. The classic image is a ball at the top of a symmetric "Mexican hat" potential: the dynamics are rotationally symmetric, but the ball must roll down to some point on the circular trough, picking a direction and hiding the symmetry.

The order parameter and the Mexican hat

The prototype is a complex scalar with the $U(1)$-symmetric Lagrangian and potential

$$\mathcal{L}={\partial }_{\mu }{\varphi }^{\ast }{\partial }^{\mu }\varphi -V(\varphi ),V(\varphi )=-{\mu }^2|\varphi {|}^2+\lambda |\varphi {|}^4.$$

The crucial sign: with ${\mu }^2>0$, the "mass-squared" term is negative, so $\varphi =0$ is a local maximum. The minima form a circle in field space:

$$|\varphi {|}^2=\frac{{\mu }^2}{2\lambda }\equiv \frac{v^2}{2},⟨\varphi ⟩=\frac{v}{\sqrt{2}}{e}^{i\theta }.$$

The vacuum expectation value (vev) $v$ is the order parameter. The theory must pick one phase $\theta$; the $U(1)$ symmetry, which rotates $\theta$, is spontaneously broken because $⟨\varphi ⟩\ne 0$ is not invariant.

Excitations: radial and angular modes

Expand around the chosen vacuum ($\theta =0$) in terms of two real fields,

$$\varphi (x)=\frac{1}{\sqrt{2}}(v+\sigma (x))e^{i\pi (x)/v},$$

with $\sigma$ the radial fluctuation and $\pi$ the angular fluctuation. Substituting into $V$:

• The radial mode $\sigma$ climbs the steep wall of the hat: it is massive, $m_{\sigma }^2=2{\mu }^2=2\lambda v^2$. This is the "Higgs-like" mode.
• The angular mode $\pi$ moves along the flat trough — the potential does not change along it — so it is exactly massless: $m_{\pi }=0$. This is the Goldstone boson.

The masslessness is geometric: the flat direction of $V$ is precisely the direction the broken symmetry moves the vacuum.

Goldstone's theorem (Theorem)

Statement. For every spontaneously broken continuous global symmetry generator, there is a massless, spin-0 Goldstone boson:

$$\boxed{\#\text{Goldstone bosons}=\dim G-\dim H,}$$

where $G$ is the symmetry group of the Lagrangian and $H\subset G$ is the unbroken subgroup (the stabilizer of the vacuum).

Proof sketch. The broken Noether charge satisfies $Q|0⟩\ne 0$ but still commutes with $H$: the state $Q|0⟩=\int d^{3}x{j}^0|0⟩$ is a zero-energy, zero-momentum excitation (since $[H,Q]=0$ and $Q$ carries no momentum). A zero-energy excitation at zero momentum with a continuum above it is exactly a massless particle created by the broken current, $⟨0|j^{\mu }|\pi (p)⟩=i{f}_{\pi }{p}^{\mu }\ne 0$. The current interpolates the Goldstone boson. $■$

Examples

Pseudo-Goldstone bosons

If the symmetry is only approximate (explicitly broken by a small term), the would-be Goldstone boson gains a small mass proportional to the breaking — a pseudo-Goldstone boson. The pions are the paradigm: light but not massless, because the small quark masses break chiral symmetry explicitly. Their dynamics are described by chiral perturbation theory, an effective field theory.

The decisive twist: gauge symmetry

Goldstone's theorem assumes the broken symmetry is global. If instead the broken symmetry is local (gauged), the theorem is evaded: the would-be Goldstone boson does not appear as a physical massless particle — it is "eaten" by the gauge field, which becomes massive. This is the Higgs mechanism, and it is how the electroweak $W$ and $Z$ acquire mass without spoiling renormalizability.

Summary

• Spontaneous breaking: symmetric Lagrangian, non-symmetric vacuum ($⟨\varphi ⟩=v\ne 0$).
• The Mexican-hat potential gives a massive radial mode and a massless angular mode.
• Goldstone's theorem: one massless boson per broken global generator, $\#=\dim G-\dim H$.
• Explicit breaking ⇒ pseudo-Goldstones (pions); gauged breaking ⇒ Higgs mechanism.

Where this leads

• Gauged breaking and massive gauge bosons: the Higgs mechanism.
• Radiative (loop-induced) breaking: the effective potential.
• Pions as an EFT: effective field theory.

References

• Peskin & Schroeder, An Introduction to Quantum Field Theory, Ch. 11.1.
• Goldstone, Salam & Weinberg, Phys. Rev. 127, 965 (1962).
• Weinberg, The Quantum Theory of Fields, Vol. 2, Ch. 19.
• Srednicki, Quantum Field Theory, Ch. 32.