Goldstone directions in the scalar mass matrix ($M^2(i{t}^a{\varphi }_0)=0$)

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.

For a complex scalar field with canonical kinetic term $|\partial \varphi {|}^2$ and potential $V=F(|\varphi {|}^2)$, take a broken-symmetry minimum $\varphi =v>0$, where $F^{\prime }(v^2)=0$. Writing $\varphi =v+(h+i\pi )/\sqrt{2}$ gives quadratic potential $F(v^2)+v^2{F}^{\prime \prime }(v^2)h^2$. Hence the canonical squared masses are $m_h^2=2v^2{F}^{\prime \prime }(v^2)$ and $m_{\pi }^2=0$. Coupling $D_{\mu }={\partial }_{\mu }+ie{A}_{\mu }$ gives gauge-boson squared mass $m_A^2=2e^2{v}^2$; the factor of two depends on defining $v$ as the complex field's vacuum value rather than its canonically normalized real component.

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.