= Accidental massless radial modes beyond Goldstone modes
= Accidental massless radial mode beyond Goldstone modes
{synonym}
The <Goldstone theorem> guarantees zero directions tangent to the vacuum orbit, not the absence of other zero eigenvalues. For $V(\varphi)=\kappa(\varphi^T\varphi-v^2)^4$, $\kappa>0$, 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.
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