Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 305 / 4

Write every fermion as a left-handed Weyl field. One generation isThe perturbative anomaly coefficients vanish as follows:The common Dynkin-index factors have been suppressed in the mixed non-Abelian lines. An $SU(2{)}^3$ perturbative anomaly vanishes because the doublet is pseudoreal, while mixed anomalies containing one non-Abelian generator vanish because that generator is traceless. There are $3+1=4$ left-handed $SU(2)$ doublets after color multiplicity, so the nonperturbative Witten SU(2) anomaly also cancels. The gauge-singlet right-handed neutrino changes none of these sums.

An uncancelled gauge anomaly would make the quantum effective action vary along a gauge orbit. The resulting failure of the Ward-Takahashi identities and BRST symmetry prevents unphysical polarizations from decoupling, so the theory loses unitarity or renormalizability and cannot define a consistent gauge theory. An uncancelled mixed gauge-gravitational anomaly would similarly conflict with simultaneous gauge-current and stress-energy conservation. A global anomaly such as the Witten SU(2) anomaly would make the fermion determinant change sign under a large gauge transformation, so even the path integral would be ill defined.

The classical baryon current is anomalous under $SU(2{)}_L$:An electroweak instanton or electroweak sphaleron changes the topological charge and producesfor three generations. At zero temperature, instanton-induced baryon violation is exponentially tiny and irrelevant to proton decay. Above the electroweak scale, thermal sphaleron transitions are rapid; they can erase a pre-existing $B+L$ asymmetry or convert a $B-L$ asymmetry into the observed baryon asymmetry, making the anomaly central to baryogenesis.

Biunitary transformations diagonalize the up- and down-type Yukawa matrices:The neutral Higgs couplings are then diagonal, but the charged weak current becomesA $3\times 3$ unitary matrix has nine real parameters. Rephasing the six quark mass eigenfields removes five phases because their common phase is baryon number. The Cabibbo-Kobayashi-Maskawa matrix therefore has four physical parameters: three mixing angles and one CP-violating phase.

The Standard Model has two independent sources of CP violation. Weak CP violation comes from the irreducible phase of the Cabibbo-Kobayashi-Maskawa matrix; its basis-independent measure is the Jarlskog invariant. Strong CP violation is governed bywhich multiplies the Yang-Mills theta term $G_{\mu \nu }^a{\tilde{G}}^{a\mu \nu }$. Quark chiral rephasings shift ${\theta }_{\mathrm{QCD}}$ and the mass-matrix phase oppositely, leaving $\bar{\theta }$ invariant. The CKM phase and $\bar{\theta }$ are otherwise independent parameters: observed weak CP violation does not explain why neutron-electric-dipole bounds require $|\bar{\theta }|\ll 1$. This unexplained smallness is the Strong CP problem.