= Goldstone directions in the scalar mass matrix
{c}
{title2=$M^2(it^a\phi_0)=0$}
For a smooth invariant <scalar potential>, differentiating $\partial_I V\,(it^a\phi)^I=0$ at a <classical vacuum> gives $M^2(it^a\phi_0)=0$. 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.
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