Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 305 (The Standard Model)

Use unitary gauge and write $\varphi =v+h/\sqrt{2}$. The quadratic Lagrangian containsThus the radial Higgs excitation is a spin-zero particle with $m_h^2=2\lambda v^2$, while the gauge boson is a massive spin-one particle with $m_A^2=2e^2{v}^2$. The phase of $\varphi$ is the Goldstone boson eaten by the Higgs mechanism to supply the longitudinal polarization of $A_{\mu }$.

Varying $A_{\nu }$ gives the Maxwell equationsOn the vacuum manifold, choose unitary gauge so that $\varphi =v$. Then $\mathbf{J}=-2e^2{v}^2\mathbf{A}$ and the London equation isFor a static configuration, $\nabla \times \mathbf{B}=\mathbf{J}$ and $\nabla \cdot \mathbf{B}=0$, soThe magnetic field therefore decays exponentially inside the condensate. This is the Meissner effect, with penetration depth equal to the inverse gauge-boson mass.

Writing $\Phi =({\varphi }_1,{\varphi }_2{)}^T$, the scalar sector has a global $SU(2)$ flavour symmetry in addition to the gauged common $U(1)$ phase. A vacuum can be chosen as ${\Phi }_0=(0,v{)}^T$. A diagonal combination of the gauge phase and the $SU(2)$ rotation generated by ${\sigma }_3$ leaves it invariant.

The radial scalar has $m_h^2=2\lambda v^2$, the gauge boson has $m_A^2=2e^2{v}^2$, and two real scalars remain massless. The four real scalar directions consist of one radial mode and three angular modes; one angular mode is eaten, leaving the two Goldstone bosons expected from the two broken physical global generators. The vacuum set before quotienting is $S^3$, and the manifold of gauge-inequivalent vacua is the complex projective line

For $q=(u,d{)}^T$, massless two-flavour Quantum chromodynamics hasClassically its continuous global symmetry is $SU(2{)}_L\times SU(2{)}_R\times U(1{)}_V\times U(1{)}_A$, up to finite central quotients. The Adler-Bell-Jackiw anomaly destroys the continuous axial $U(1{)}_A$ in the quantum theory, leaving $SU(2{)}_L\times SU(2{)}_R\times U(1{)}_V$, together with an anomaly-preserved discrete axial subgroup.

The condensate transforms as $⟨{\bar{q}}_L{q}_R⟩\mapsto L⟨{\bar{q}}_L{q}_R⟩R^{†}$, so its value proportional to the identity produces chiral symmetry breakingThe vacuum manifold is the coset $(SU(2{)}_L\times SU(2{)}_R)/SU(2{)}_V\cong SU(2)$ and has three Goldstone modes. A field $U(x)\in SU(2)$ transforms as $U\mapsto LU{R}^{†}$. Lorentz invariance and chiral symmetry leave, at two-derivative order, one invariant metric on this coset, soSince the Lagrangian density has dimension four and $U$ is dimensionless, the pion decay constant $f_{\pi }$ has mass dimension one.

For $U=e^{2i\pi /f_{\pi }}$, the Maurer-Cartan form expands asSubstituting, using cyclicity of the trace, and collecting even powers givesOdd terms cancel because the sigma-model metric is invariant under $U\leftrightarrow U^{†}$.

Electromagnetism gauges the vector flavour transformation $U\mapsto e^{ie\alpha Q}U{e}^{-ie\alpha Q}$, so covariance requires $D_{\mu }U={\partial }_{\mu }U-ie{A}_{\mu }[Q,U]$. The identity part of $Q=I/6+{\sigma }_3/2$ drops out of the commutator. Writinggives $[Q,\pi {]}_{12}={\pi }_{12}$ and $[Q,\pi {]}_{21}=-{\pi }_{21}$. Hence ${\pi }^{+}$ has electric charge $+e$, ${\pi }^{-}$ has charge $-e$, and ${\pi }^0$ is neutral.

A gauge anomaly makes gauge redundancy inconsistent and must cancel. A chiral anomaly is the quantum nonconservation of a classically conserved axial current and may be a physical effect. A 't Hooft anomaly is an obstruction to gauging a global symmetry; it is preserved by renormalization-group flow and must be reproduced by the infrared theory, through massless degrees of freedom, symmetry breaking, or topological order.

The axial $U(1)$ is anomalous, while vector fermion number survives. In a left-handed convention, put ${\tilde{\psi }}_R=({\psi }_R{)}^c$. The continuous quantum symmetry and charges areup to finite quotients and the surviving discrete axial symmetry.

Normalize a fundamental cubic anomaly to $A({\mathbf{N}}_{\mathbf{f}})=1$ and its Dynkin index to $I({\mathbf{N}}_{\mathbf{f}})=1$. The two gauge colours giveThe left and charge-conjugated right fermions have opposite vector charges and equal multiplicities, soIf the theory confined with unbroken chiral symmetry, gauge-invariant operators contain an even number of $SU(2)$ fundamentals and are bosonic, so no massless composite fermions could match the nonzero chiral anomalies. 't Hooft anomaly matching therefore forces chiral symmetry breaking, with its infrared Goldstone theory carrying the anomaly.

There is no perturbative $SU(2)$ gauge anomaly because both the fundamental and adjoint representations are real or pseudoreal and have vanishing cubic gauge index. The Witten SU(2) anomaly also vanishes: the $N_f$ Dirac fermions give $2N_f$ fundamental Weyl doublets, an even number, while the adjoint Weyl fermion is in an integer-isospin real representation.

In left-handed variables, both ${\psi }_L$ and ${\tilde{\psi }}_R$ have $U(1{)}_X$ charge $-1$. Cancellation of the mixed $SU(2{)}_{\mathrm{gauge}}^{2}U(1{)}_X$ anomaly requireswhere $I(2)=1$ and $I(\mathrm{adj})=4$. Therefore

The adjoint fermion is a flavour singlet, soThere are $4N_f$ fundamental Weyl components of charge $-1$ and three adjoint components of charge $q=N_f/2$. Hence

For $N_f=3$, $q=3/2$. Under $SU(3{)}_L\times SU(3{)}_R\times U(1{)}_X$, the proposed left-handed composites transform asFor example, the charge of $\chi ={\bar{\psi }}_R{\lambda }_L{\psi }_L$ is $-1+3/2-1=-1/2$, and the antisymmetric product of two $SU(3)$ fundamentals is an antifundamental.

For $SU(3{)}_L^3$, the three right-flavour copies of $\chi$ contribute $3$, while $\eta$ contributes $-1$, giving $2$. For $SU(3{)}_L^{2}U(1{)}_X$, these fields contribute $3(-1/2)+(-1/2)=-2$. All fifteen components of $\chi$, $\eta$, and $\tilde{\eta }$ have charge $-1/2$, soThe ultraviolet formulas at $N_f=3$ give $2$, $-2$, $-15/8$, and $-15/2$ respectively. Every 't Hooft anomaly therefore matches.

Under $SU(3{)}_c\times SU(2{)}_L\times U(1{)}_Y$, one generation consists ofThese are the fundamental fermions of the Standard Model augmented by a right-handed neutrino.

A Dirac mass term directly pairs a left-handed weak doublet with a right-handed weak singlet and is therefore not electroweak gauge invariant. Introduce the Higgs doublet $H:(1,2{)}_{1/2}$ and $\tilde{H}=i{\sigma }_2{H}^{\ast }:(1,2{)}_{-1/2}$. The gauge-invariant Yukawa terms areWhen electroweak symmetry breaking gives $⟨H⟩=(0,v_H/\sqrt{2}{)}^T$, these become Dirac mass matrices $M_f=Y_f{v}_H/\sqrt{2}$.

Two complex $N\times N$ quark Yukawa matrices contain $4N^2$ real parameters. Flavour-basis transformations $U(N{)}_Q\times U(N{)}_u\times U(N{)}_d$ remove $3N^2-1$, because the common baryon-number phase remains unbroken. Thus there arephysical parameters: $2N$ quark masses, $N(N-1)/2$ mixing angles, and $(N-1)(N-2)/2$ complex phases. A physical phase exists only for $N\ge 3$ and produces CP violation, equivalently time-reversal violation when CPT is conserved.

Biunitary diagonalization uses independent left rotations for the up and down Yukawa matrices. Their mismatch is the Cabibbo-Kobayashi-Maskawa matrix $V=U_{uL}^{†}{U}_{dL}$. For two generations it can be made real and isIn the mass basis,The off-diagonal entries permit charged-current flavour change, for example $W^{+}\to u\bar{s}$ or semileptonic kaon decay; both vanish when ${\theta }_c=0$.