Goldstone Boson - A Simple Example

A Simple Example

Consider a complex scalar field φ, with the constraint that φ*φ = v², a constant. One way to impose a constraint of this sort is by including a potential interaction term in its Lagrangian density,

and taking the limit as λ → ∞ (this is called the "Abelian nonlinear σ-model". It corresponds to the Goldstone sombrero potential where the tip and the sides shoot to infinity, preserving the location of the minimum at its base).

The constraint, and the action, below, are invariant under a U(1) phase transformation, δφ=iεφ. The field can be redefined to give a real scalar field (i.e., a spin-zero particle) θ without any constraint by

where θ is the Nambu–Goldstone boson (actually vθ is), and the U(1) symmetry transformation effects a shift on θ, namely

but does not preserve the ground state |0⟩, (i.e. the above infinitesimal transformation does not annihilate it—the hallmark of invariance), as evident in the charge of the current below.

The vacuum is degenerate and noninvariant under the action of the spontaneously broken symmetry.

The corresponding Lagrangian density is given by

and thus

Note that the constant term m²v² in the Lagrangian density has no physical significance, and the other term in it is simply the kinetic term for a massless scalar.

The symmetry-induced conserved U(1) current is

The charge, Q, resulting from this current shifts θ and the ground state to a new, degenerate, ground state. Thus, a vacuum with ⟨θ⟩=0 will shift to a different vacuum with ⟨θ⟩=−ε. The current connects the original vacuum with the Nambu–Goldstone state, ⟨0|J₀(0)|θ⟩ ≠ 0.

In general, in a theory with several scalar fields, φ_j, the Nambu–Goldstone mode φ_g is massless, and parameterises the curve of possible (degenerate) vacuum states. Its hallmark under the broken symmetry transformation is nonvanishing vacuum expectation ⟨δφ_g⟩, an order parameter, for vanishing ⟨φ_g⟩=0, at some ground state |0⟩ chosen at the minimum of the potential, ⟨∂V /∂φ_i⟩=0.

This nonvanishing vacuum expectation of the transformation increment, ⟨δφ_g⟩, specifies the relevant (Goldstone) null eigenvector of the mass matrix,

and hence the corresponding zero-mass eigenvalue.