Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 305 / 2

A quantum anomaly occurs when a symmetry of the classical action cannot be preserved by the regulator or functional measure of the quantum theory. An anomaly in a global symmetry is physically allowed: it changes a classical current conservation law and can mediate observable processes. A gauge anomaly is fatal unless it cancels, because gauge symmetry is the redundancy that removes negative-norm and longitudinal unphysical states; its loss spoils the gauge Ward identities and unitarity.

For a massless Dirac fermion,The local gauge symmetry gives the vector current $J^{\mu }=e\bar{\psi }{\gamma }^{\mu }\psi$. The global chiral transformation $\psi \mapsto e^{i\beta {\gamma }^5}\psi$ gives the classical axial currentUsing the massless Dirac equation and $\{{\gamma }^5,{\gamma }^{\mu }\}=0$ gives ${\partial }_{\mu }{J}_{\mathrm{ax}}^{\mu }=0$ classically.

Quantum mechanically, a gauge-invariant regulator for the fermion measure is not invariant under the axial rotation. In the Fujikawa form, its infinitesimal Jacobian containsThe first nonzero term in the heat kernel expansion is quadratic in $F_{\mu \nu }$. Usingand the Gaussian momentum integral gives the chiral anomalyThe overall sign follows the gamma-matrix, charge, and Levi-Civita conventions stated in the question. The same coefficient is obtained from the one-loop axial-vector-vector triangle diagram.

Write every fermion as a left-handed Weyl field. One Standard Model generation then haswith an optional neutral ${\nu }_R^c:(1,1{)}_0$. The purely colored anomaly vanishes becauseThe mixed non-Abelian-hypercharge coefficients areThe cubic hypercharge anomaly also cancels:as does the mixed gauge-gravitational anomaly,There is no perturbative cubic $SU(2)$ anomaly because its representations are pseudoreal. There are four left-handed weak doublets after counting the three colors of $Q_L$, so the global Witten SU(2) anomaly also vanishes. This proves Standard Model anomaly cancellation generation by generation.

The equality of proton and positron charges follows from the same constraints. Let the Higgs hypercharge be $Y_H=1/2$. Gauge-invariant Yukawa interactions implywhile cancellation of $[SU(2{)}_L{]}^{2}U(1{)}_Y$ gives $3Y_Q+Y_L=0$. Since electric charge is $Q=T^3+Y$,Thereforeshowing how anomaly cancellation, together with Yukawa gauge invariance and the normalization fixed by the neutral Higgs vacuum, enforces this instance of charge quantization.