Past exam of the theoretical physics major option course of the University of Oxford / 2015 / 5 / d / Solution

A real antisymmetric generator of the connected rotation group isChoose $D_{\mu }\mathbf{\varphi }=({\partial }_{\mu }-g{A}_{\mu }T)\mathbf{\varphi }$. Under $\mathbf{\varphi }\mapsto R(\alpha (x))\mathbf{\varphi }$ and $A_{\mu }\mapsto A_{\mu }+{\partial }_{\mu }\alpha /g$, the gauge covariant derivative transforms in the same way as the field. Include the canonical gauge kinetic term $-F_{\mu \nu }{F}^{\mu \nu }/4$ so the vector's mass normalization is specified.

In unitary gauge, $\mathbf{\varphi }=(v+h,0{)}^T$. The relevant Lagrangian density isReading the quadratic vector term as $m_A^2{A}^2/2$ givesThe angular scalar supplies the longitudinal polarization of the massive vector; it is not a separate physical massless particle after gauging. The physical scalar still has $m_h^2=8\lambda v^2$.

The interactions among the physical fields areThus the complete physical vertex list is $hhh$, $hhhh$, $AAh$, and $AAhh$. An Abelian gauge kinetic term generates no $AAA$ or $AAAA$ self-interaction. Gauge-dependent Goldstone vertices in another gauge do not add physical particle species to this list. These statements concern the broken phase $v\ne 0$; when $v=0$ the unitary-gauge broken-phase parametrization is unavailable.