Past exam of the advanced quantum field theory for particle physics course of the University of Oxford / 2021 / 3 / a

Put $r^2={\Phi }^T\Phi$. The potential has its minimum atChoose the vacuum expectation value ${\Phi }_0=(0,0,v{)}^T$. Since the weak hypercharge is zero, $Q=T^3$ on this real scalar triplet, and $T^3{\Phi }_0=0$. Neither $T^1$ nor $T^2$ annihilates ${\Phi }_0$. The stabilizer subgroup is consequentlyIn particular, spontaneous symmetry breaking has not reduced the two neutral gauge generators to a single one: $Y$ annihilates every component and remains unbroken independently of $T^3$.

The scalar mass matrix isWith canonical kinetic normalization $\frac{1}{2}(\partial h{)}^2$, the radial Higgs boson therefore hasThe two tangential zero eigenvalues are Goldstone modes, absorbed by the two massive gauge bosons through the Higgs mechanism.

Let $g$ and $g^{\prime }$ be the two gauge couplings. In unitary gauge, $\Phi =(0,0,v+h{)}^T$ and the weak hypercharge term in the gauge covariant derivative vanishes. Using $(T^a{)}_{bc}=-i{ϵ}_{abc}$, the kinetic term becomesThus, in the basis $(W^1,W^2,W^3,B)$, the gauge-boson mass matrix and its eigenvalues areHere $W^{\pm }=(W^1\mp i{W}^2)/\sqrt{2}$. The two massless neutral gauge bosons may be rotated to any orthonormal basis; no neutral mass term selects a unique mixing angle.

Expanding the same kinetic term to first order in $h$ givesFunctional differentiation gives the Feynman rulesand, in the real basis, $h{W}_{\mu }^a{W}_{\nu }^b:2i{g}^{2}v{\delta }^{ab}{g}_{\mu \nu }$ for $a,b\in \{1,2\}$. All such three-point Feynman vertices involving $W^3$ or $B$ vanish. The factor two in the real basis comes from the two identical gauge boson fields; in the charged basis it is already explicit in ${\mathcal{L}}_{hVV}$.

This cannot be the sole symmetry-breaking sector of the Standard Model. Its gauge-boson mass matrix has two zero eigenvalues, so besides the electromagnetic photon it predicts an extra massless neutral gauge boson. Every linear combination of $W^3$ and $B$, including the combination normally called $Z$, is massless. The observed massive Z boson is a concrete contradiction. Choosing $g,g^{\prime },m,\lambda$ differently cannot change these two zero eigenvalues.

A real scalar triplet can occur as an additional sector, but a further vacuum expectation value, for example that of a Higgs doublet, is then necessary to obtain the observed neutral gauge boson mass pattern. This does not rescue the triplet-only model posed here.