Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 305 / 4

Use unitary gauge to setThenDefineThe propagating fields are $W_{\mu }^{\pm },Z_{\mu },A_{\mu }$, and the Higgs boson $h$. Expandingand reading the quadratic terms givesThe factors $(v+h{)}^2$ in the kinetic term also give the corresponding Higgs-vector interactions.

Solved by gpt-5.6-sol high.

The lower Higgs component has $T^3=-1/2$ and hypercharge $Y=1/2$, so the generator annihilating the vacuum isThis is the unbroken electromagnetic $U(1{)}_Q$. Since $[Q,T^{\pm }]=\pm T^{\pm }$, the two W boson fields have charges $Q(W^{\pm })=\pm 1$. The Z boson, photon, and physical Higgs excitation are neutral.

Solved by gpt-5.6-sol high.

Under $SU(3{)}_c\times SU(2{)}_L\times U(1{)}_Y$, one Standard Model fermion generation including a right-handed neutrino is

Solved by gpt-5.6-sol high.

A bare Dirac bilinear couples left- and right-handed fields, but these fields transform in different electroweak representations, so ${\bar{f}}_L{f}_R$ is not a gauge singlet. Let $\tilde{H}=i{\sigma }^2{H}^{\ast }$. The four gauge-invariant Yukawa interaction terms areReplacing $H$ by its vacuum expectation value gives Dirac masses $m_f=y_{f}v/\sqrt{2}$. Applying $Q=T^3+Y$ givesfor both chiralities, so the generated mass terms preserve the unbroken electromagnetic gauge symmetry.

Solved by gpt-5.6-sol high.