Spontaneous Symmetry Breaking

A Lagrangian can be symmetric while the ground state is not. When the vacuum spontaneously selects a direction in field space, the symmetry is broken without any explicit breaking term. Goldstone's theorem then guarantees one massless scalar — a Nambu-Goldstone boson — per broken symmetry generator. This underlies the pseudo-Goldstone pions of QCD and (with a gauge-theory twist) the mass generation of W and Z.

Key Concepts

Order parameter ⟨φ⟩ = v ≠ 0 selects a vacuum that breaks the symmetry of the Lagrangian
Mexican hat potential: V = −μ²|φ|² + λ|φ|⁴, minimum at |φ| = v = √(μ²/2λ)
Goldstone's theorem: one massless scalar per broken generator (dim G − dim H)
Massive mode (Higgs-like): oscillation in radial direction, mH = √(2μ²)
Massless Goldstone: oscillation along flat angular direction, no restoring force
Pseudo-Goldstone bosons: explicit breaking makes them light but not massless (e.g., pions, mπ ≪ mρ)

Key Equations

Mexican hat potential

V(\phi)=-\mu^2|\phi|^2+\lambda|\phi|^4,\quad\langle\phi\rangle=v=\sqrt{\frac{\mu^2}{2\lambda}}

Field decomposition

\phi(x)=\frac{1}{\sqrt{2}}\bigl(v+h(x)\bigr)e^{i\pi(x)/v},\quad m_h=\sqrt{2\lambda}\,v,\;m_\pi=0

Goldstone counting

N_{\rm GB}=\dim G-\dim H_{\rm unbroken}

Pion EFT

\mathcal{L}=\frac{f_\pi^2}{4}\mathrm{tr}(\partial_\mu U^\dagger\partial^\mu U)+\cdots,\quad U=e^{2i\boldsymbol{\pi}\cdot\boldsymbol{T}/f_\pi}

Worked Example

Example Problem

Problem

For V = −μ²|φ|²+λ|φ|⁴ with μ = 100 GeV, λ = 0.5, find v and mh.

Solution

v = √(μ²/2λ) = √(10000/1) = 100 GeV. The Higgs mass comes from V″ at the minimum: V″(v) = −2μ²+12λv² = −20000+60000 = 40000, so mh = √40000 = 200 GeV = 2μ.

Key Takeaways

SSB: the Lagrangian has symmetry G but the vacuum ⟨φ⟩=v breaks G → H; physics observes the broken symmetry
Goldstone's theorem: N_GB = dim G − dim H massless scalars must appear in the spectrum for each broken continuous symmetry generator
Radial (Higgs-like) mode is massive (mh = √(2λ)v); angular (Goldstone) mode is exactly massless if the symmetry is exact
Pions are pseudo-Goldstone bosons of approximate chiral symmetry of QCD, explaining why mπ ≪ mρ ≈ 770 MeV