Past exam of the advanced quantum field theory for particle physics course of the University of Oxford / 2017 / 3 / b / i / Solution

Gauge invariance of the scalar potential implies the infinitesimal identityThe irrelevant common factor $i$ from the real representation's infinitesimal rotation has been divided out. Differentiate with respect to ${\varphi }_k$ and evaluate at the stationary vacuum $v$:The second term vanishes by stationarity, leavingWhenever $T^{a}v\ne 0$, this is a nonzero vector in the kernel of the scalar mass-squared matrix, so it is a zero-eigenvalue direction. It is tangent to the degenerate vacuum orbit. In a global theory it is a Goldstone direction; in the gauged theory the corresponding would-be Goldstone field supplies a longitudinal polarization of a massive vector.

For the specific potential, stationarity at a nonzero vacuum gives $v^2=m^2/(4\lambda )$, assuming $m^2>0$ and $\lambda >0$. Its Hessian isThe antisymmetry of the triplet generators gives $v_i(T^{a}v{)}_i=0$, verifying the claimed kernel directly.