Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 305 / 3 / b / iv

Parameterize the scalar locally aswhere the five $X^{\alpha }$ are broken generators. A gauge transformation removes all ${\pi }^{\alpha }$ in unitary gauge. Their derivative terms combine with the corresponding gauge fields in $(D_{\mu }\Phi {)}^{†}{D}^{\mu }\Phi$, supplying the longitudinal polarizations of five massive vectors. This is the Higgs mechanism.

With $T^a={\lambda }^a/2$, the gauge bosons $A_{\mu }^{1,2,3}$ of the unbroken $SU(2)$ remain massless. The four bosons $A_{\mu }^{4,5,6,7}$ haveand $A_{\mu }^8$ hasThe five eaten Goldstone modes complete their third polarizations. The sixth real component of the complex triplet remains as a radial Higgs boson withThis realizes the degree-of-freedom count summarized by Fundamental-Higgs breaking of SU(3) to SU(2).