Goldstone theorem

In a relativistic quantum field theory, each spontaneously broken generator of a continuous internal symmetry produces a massless scalar excitation. For $G\to H$, the standard setting therefore has $\dim G-\dim H$ Goldstone bosons.

For a smooth invariant scalar potential, differentiating ${\partial }_{I}V(i{t}^a\varphi {)}^I=0$ at a classical vacuum gives $M^2(i{t}^a{\varphi }_0)=0$. The symmetry tangent space has dimension $\dim G-\dim H_0$, where $H_0$ is the full stabilizer subgroup. With a positive nonsingular scalar kinetic term, these are massless directions of the quadratic theory. Further zero eigenvalues are possible unless the Hessian matrix is positive on the normal complement.

The Goldstone theorem guarantees zero directions tangent to the vacuum orbit, not the absence of other zero eigenvalues. For $V(\phi )=\kappa ({\phi }^T\phi -v^2{)}^4$, $\kappa >0$, the minima form the same sphere as in ordinary $O(n)$ breaking, but the entire Hessian matrix vanishes there. Thus all n scalar quadratic masses vanish, although only n-1 are symmetry-protected Goldstone bosons. Exact Goldstone counting requires nonzero curvature transverse to the orbit.

For $V=m^2{\varphi }^2/2+\lambda ({\varphi }^2{)}^2/4$, $m^2<0$, the vacuum manifold is a sphere of radius $v=\sqrt{-m^2/\lambda }$. At a chosen vacuum the Hessian matrix is $2\lambda v^{2}n{n}^T$, where $n$ is its unit direction. The radial scalar has mass squared $2\lambda v^2$; the $n-1$ angular modes are physical Goldstone bosons when the symmetry is global.

A pseudo-Goldstone boson is a light mode produced when a spontaneously broken continuous symmetry is also weakly broken explicitly. Its mass vanishes with the explicit symmetry-breaking parameters.