Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 305 / 4

Before diagonalizing quark masses, the charged weak current couples fields in the same weak doublet:The Higgs Yukawa matrices produce mass matrices $M_u$ and $M_d$. Biunitary transformations diagonalize them, with $u_L^{\prime }=U_{uL}{u}_L$ and $d_L^{\prime }=U_{dL}{d}_L$. The charged current therefore contains the Cabibbo-Kobayashi-Maskawa matrixNeutral gauge currents remain flavor diagonal because the same unitary matrix occurs on both sides, and neutral Higgs couplings are diagonal together with each mass matrix.

For $N$ generations, rephasing the quark fields leaves $N(N-1)/2$ mixing angles and $(N-1)(N-2)/2$ irreducible phases, a total of $(N-1{)}^2$ parameters. Three generations are the minimum needed for a physical phase; the observed $3\times 3$ matrix has

CP conjugates the charged-current coupling and replaces $V_{\mathrm{CKM}}$ by $V_{\mathrm{CKM}}^{\ast }$. CP is conserved only if quark-field rephasings can make the entire matrix real. For three generations an irreducible phase prevents this. Equivalently, the rephasing-invariant Jarlskog invariantis nonzero for distinct rows and columns. Since $J$ changes sign under complex conjugation, $J\ne 0$ proves CP violation in electroweak interactions.

The QCD Yang-Mills theta term isThe chromoelectric field is a spatial vector and the chromomagnetic field is a pseudovector, so their scalar product is odd under parity. It is also odd under time-reversal symmetry because time reversal reverses ${\mathbf{B}}^A$ but not ${\mathbf{E}}^A$. Charge conjugation reverses both color fields in the gauge-invariant trace and leaves the product even. The term therefore violates P, T, and CP, while it preserves CPT because the two sign reversals from P and T cancel.

An anomalous chiral redefinition moves phase between the bare QCD angle and the quark mass matrix, so the measurable parameter isThe neutron electric dipole moment requires $|\bar{\theta }|$ to be extremely small, although no generic Standard Model symmetry enforces this. This is the Strong CP problem.

In Peccei–Quinn theory, an anomalous spontaneously broken global symmetry replaces $\bar{\theta }$ by a dynamical axion field. Nonperturbative QCD generates an axion potential whose minimum lies at the CP-conserving value, dynamically relaxing the effective angle to zero.

The $SU(2{)}_L$ electroweak theta angle can be shifted by an anomalous global rephasing generated by baryon plus lepton number. Because the renormalizable Standard Model has no operator that explicitly breaks this global phase in a way that fixes it, one may choose the rephasing to set ${\theta }_{\mathrm{EW}}=0$ without changing any other physical parameter. The weak theta angle is therefore unobservable in the renormalizable Standard Model; it could enter a physical invariant only after adding suitable explicit violation of baryon plus lepton number.