Past exam of the advanced quantum field theory for particle physics course of the University of Oxford / 2020 / 3 / a

Goldstone's theorem states that in a relativistic quantum field theory, spontaneous breaking of an exact continuous global internal symmetry produces a massless Goldstone boson for each independent broken generator. More precisely, a conserved global current whose charge acts nontrivially on the vacuum expectation value has a massless pole in an appropriate current-field correlation function. The usual counting assumes a Lorentz invariant vacuum state and the standard locality and positivity hypotheses.

A local gauge symmetry is a gauge redundancy, so it does not by itself imply massless physical Goldstone bosons. In the Higgs mechanism, the corresponding scalar directions provide the longitudinal polarizations of massive gauge bosons.

For fundamental-doublet breaking of SU(2), assume $\lambda >0$ and $v>0$. The minima of the scalar potential satisfy ${\varphi }^{†}\varphi =v^2/2$, giving a three-sphere vacuum manifold. A chosen vacuum expectation value is ${\varphi }_0=(0,v{)}^T/\sqrt{2}$. No nonzero generator $a_a{\tau }^a$ annihilates ${\varphi }_0$: its squared norm is $v^2(a_1^2+a_2^2+a_3^2)/8$. At the group level, an $SU(2)$ matrix fixing the nonzero vector ${\varphi }_0$ must be the identity. Thus all three generators of the SU(2) gauge theory are broken in the usual perturbative Higgs mechanism description.

Make the radial and angular fields explicit:A local gauge transformation removes the three angular fields ${\pi }^a$. In unitary gauge, the Pauli matrices identity $\{{\tau }^a,{\tau }^b\}={\delta }^{ab}1/2$ givesThe mixed derivative term cancels between a term and its complex conjugate. The scalar potential becomesConsequently the complete physical Lagrangian density isReading the quadratic terms with their canonical factors $1/2$ givesAll physical scalar-vector and scalar self-interactions are displayed in the second line of $\mathcal{L}$. In particular the new cubic couplings proportional to $v$ are $h{A}^a{A}^a$ and $h^3$. Their Feynman vertices, together with the quartic ones, areThe three-vector and four-vector interactions are already contained in the non-Abelian gauge field strength term. There are no further physical fields or interactions in this specified model. Before fixing unitary gauge, the quadratic angular term is $\frac{1}{2}({\partial }_{\mu }{\pi }^a-m_A{A}_{\mu }^a{)}^2$. It exhibits three zero-curvature angular directions of the scalar potential, but also their mixing with the gauge fields. They supply the third polarization of each massive gauge boson, rather than massless physical particles.

The degrees of freedom count is $3\cdot 2+4=10$ before breaking and $3\cdot 3+1=10$ afterwards. If instead the symmetry were global, as in the decoupled $g=0$ scalar theory, the three angular fields would be physical massless Goldstone bosons. This is consistent with Goldstone's theorem. The language of a broken local symmetry refers to this gauge-fixed expansion; a gauge-dependent vacuum expectation value is not itself a gauge-invariant observable.