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

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 ${\mathcal{L}}_{hVV}$.