Past exam of the advanced quantum field theory for particle physics course of the University of Oxford / 2016 / 3 / b / Solution

For $\lambda >0$, the scalar potential is minimized at ${\varphi }^{†}\varphi =v^2/2$. With the specified upper-component vacuum expectation value ${\varphi }_0=(v,0{)}^T/\sqrt{2}$ and hypercharge $Y=-I/2$, the unbroken generator isThe other three independent generator directions act nontrivially on ${\varphi }_0$. Thus the connected unbroken gauge subgroup is $U(1{)}_{\mathrm{em}}$, while three gauge directions are broken. The normalization here is $Q=T^3+Y$, not the alternative convention in which hypercharge is doubled.

Apply the covariant derivative directly to the vacuum:In the mostly-plus convention the canonical scalar kinetic term is $-(D_{\mu }\varphi {)}^{†}{D}^{\mu }\varphi$. Its quadratic vector terms areDefine $s_W=\sin {\theta }_W=g_1/\sqrt{g_1^2+g_2^2}$ and $c_W=\cos {\theta }_W=g_2/\sqrt{g_1^2+g_2^2}$. Diagonalization givesThe neutral combination $Z$ lies along the nonzero eigenvector $(g_2,-g_1)$; the orthogonal combination $(g_1,g_2)$ is the massless photon. Comparing the mass terms with $-m_W^2{W}^{+}\cdot W^{-}-m_Z^2{Z}^2/2$ yieldsFinally insert the inverse neutral rotation into the coupling:Both coefficients in the photon term agree, givingThe upper vacuum component is electrically neutral and the lower scalar component has charge $-e$, consistently with the source's negative hypercharge. This doublet is the conjugate convention to the commonly displayed positive-hypercharge doublet; it does not change the vector masses or the electromagnetic coupling. The four real scalar components supply three would-be Goldstone modes and one physical radial Higgs mode, preserving the total physical degrees of freedom.