Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 305 / 2

With $T^a={\tau }^a/2$, hypercharge $Y_H=1/2$, andthe most general local renormalizable bosonic Lagrangian isCanonical normalization leaves four parameters:Stability requires $\lambda >0$. If $m^2>0$, the minimum is $H=0$ and the electroweak symmetry is unbroken. If $m^2<0$,and the electroweak symmetry breaking pattern is

Write $H=2^{-1/2}{e}^{-i{\xi }^a{T}^a}(0,v+h{)}^{\mathsf{T}}$. Differentiating the exponential to first order in the fields, or using its Maurer-Cartan form exactly, shows that the angular fields enter $D_{\mu }H$ throughwith higher-order commutator terms dictated by the same group-valued combination. A local $SU(2{)}_L$ transformation can set ${\xi }^a=0$. In this unitary gauge,The three Goldstone modes have become the longitudinal polarizations of the three massive electroweak gauge bosons, which is the Higgs mechanism.

Defineand rotate the neutral fields byThe quadratic mass terms from $|D_{\mu }⟨H⟩{|}^2$ are diagonal in this basis and giveThe scalar mass isThe unbroken generator is $Q=T^3+Y$, and $A_{\mu }$ is its gauge field. Its exact masslessness and coupling $e=g\sin {\theta }_W=g^{\prime }\cos {\theta }_W$ identify it as the photon.

Let $\tilde{H}=i{\tau }^2{H}^{\ast }$. Gauge invariance permits the Yukawa interactionswhere $Y_u,Y_d$ are arbitrary complex $3$ by $3$ matrices in generation space. Their color and weak indices are contracted to singlets, and the hypercharges in each term sum to zero.

After symmetry breaking,The same terms couple the physical Higgs field proportionally to the quark mass matrices.

Use singular value decomposition to choose unitary matrices satisfyingThe neutral currents remain flavour diagonal because the same unitary matrix occurs on both sides of each bilinear. The charged current contains the mismatchand becomes

A general unitary $3$ by $3$ matrix has nine real parameters. Independent rephasings of the six quark fields remove five phases because one common baryon-number phase changes nothing. The Cabibbo-Kobayashi-Maskawa matrix therefore has four physical parameters: three mixing angles and one CP-violating phase.

The leptons consist ofwith charged-lepton Yukawa coupling $-{\overline{L}}_L{Y}_{e}H{e}_R+h.c.$

Adding gauge-singlet right-handed neutrinos ${\nu }_R\sim (1,1{)}_0$ permits the Dirac Yukawa couplingwhich gives $M_D=v{Y}_{\nu }/\sqrt{2}$. Because ${\nu }_R$ carries no Standard Model gauge charge, a Majorana mass term ${\nu }_R^{\mathsf{T}}C{M}_R{\nu }_R/2$ is also allowed. For $M_R\gg M_D$, the seesaw mechanism produces light neutrino masses of order $M_D{M}_R^{-1}{M}_D^{\mathsf{T}}$.