Linear Sigma Model

Explore Spontaneous Symmetry Breaking, Goldstone Theorem, and Higgs Mechanism

Symmetry Status O(n) 对称
Vacuum Expectation v 0.0
Higgs Mass m_σ 1.0
Goldstone Bosons 0

Model Parameters

Visualization Options

Symmetry Phase

Current Formula

V(Φ) = (λ/4)(Φ² - v²)²

Mexican Hat Potential

When v > 0, the potential forms a 'Mexican hat' shape. Any point on the circumference is a ground state, and this continuous degeneracy leads to the appearance of Goldstone bosons.

Field Configuration Space

The field Φ takes values in an n-dimensional internal space. O(n) symmetry ensures that the potential depends only on |Φ|² = ΦᵀΦ. The vacuum expectation value ⟨Φ⟩ indicates the direction of spontaneous symmetry breaking.

Particle Mass Spectrum

What is the Linear Sigma Model?

The Linear Sigma Model is the simplest toy model for understanding spontaneous symmetry breaking and the Higgs mechanism. It consists of n real scalar fields with O(n) global symmetry. By adjusting parameters, one can transition from the symmetric phase to the broken phase, observing the generation of Goldstone bosons and the Higgs mode mass.

Key Concepts

O(n) Symmetry

n-dimensional real scalar fields are invariant under orthogonal transformations. The O(n) group has n(n-1)/2 generators, corresponding to n(n-1)/2 conserved charges.

Spontaneous Symmetry Breaking

When v ≠ 0, the vacuum state does not preserve the original symmetry. The system chooses a specific vacuum direction, leading to O(n) → O(n-1) breaking.

Goldstone Theorem

Each broken continuous symmetry produces a massless Goldstone boson. There are n-1 Goldstone modes oscillating on the vacuum manifold.

Higgs Mode

The radial σ field acquires mass m_σ = √(2λ)v, the only massive excitation of the system, corresponding to the Higgs particle.

Lagrangian

$$\mathcal{L} = \frac{1}{2}(\partial_\mu\Phi)^T(\partial^\mu\Phi) - \frac{\lambda}{4}(\Phi^T\Phi - v^2)^2$$

Kinetic term: 1/2(∂_μΦ)^T(∂^μΦ) describes the dynamics of n uncoupled real scalar fields

Potential term: V(Φ) = λ/4(Φ^TΦ - v²)² has O(n) symmetry, forming a Mexican hat shape when v > 0

Symmetry Breaking Mechanism

O(n)
Symmetric Phase (v = 0)
  • n degenerate fields with mass m = √(λ)v
  • Vacuum at origin, unique and symmetric
  • Full O(n) symmetry preserved
→
O(n-1)
Broken Phase (v > 0)
  • 1 massive Higgs field with m_σ = √(2λ)v
  • n-1 massless Goldstone bosons
  • Vacuum on circle/sphere with |Φ| = v

Field Decomposition

In the broken phase, decompose the field into radial Higgs mode and transverse Goldstone modes:

$$\Phi = \begin{pmatrix} \pi_1 \\ \pi_2 \\ \vdots \\ \pi_{n-1} \\ v + \sigma \end{pmatrix}$$

σ Field (Higgs Mode)

Oscillates radially, restoring equilibrium radius, mass m_σ = √(2λ)v

π Fields (Goldstone Modes)

Oscillates tangentially on vacuum manifold, no restoring force, m_π = 0

Coupling to Gauge Fields

When the linear sigma model couples to gauge fields, Goldstone bosons become the longitudinal polarization of gauge bosons. This is the Higgs mechanism:

$$\mathcal{L}_{\text{gauge}} = \frac{1}{2}(D_\mu\Phi)^T(D^\mu\Phi) - V(\Phi) - \frac{1}{4}F_{\mu\nu}^a F^{\mu\nu}_a$$

where D_μ = ∂_μ + g A_μ^a T^a is the covariant derivative. In unitary gauge, π fields are 'eaten' and A_μ^a acquire mass m_A = gv.

History and Applications

1960
Yoichiro Nambu proposes spontaneous symmetry breaking to explain the near-masslessness of pions
1961
Goldstone theorem: Spontaneous breaking of continuous symmetry necessarily produces massless bosons
1964
Higgs, Englert/Brout, and Guralnik/Hagen/Kibble independently propose the Higgs mechanism
1967-68
Weinberg-Salam model: Applies Higgs mechanism to electroweak unification
2012
LHC discovers the Higgs boson, confirming the Standard Model's Higgs mechanism

Physical Applications

  • Particle Physics: Higgs sector of the Standard Model, explaining the origin of W/Z boson and fermion masses
  • Condensed Matter: Ginzburg-Landau theory of superfluidity and superconductivity, ferromagnetic phase transitions
  • Cosmology: Early universe symmetry breaking phase transitions, may produce cosmological defects (domain walls, cosmic strings)
  • Nuclear Physics: Pions as approximate Goldstone bosons in chiral perturbation theory