Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 305 (The Standard Model)

The little group of a massive four-momentum is $SO(3)$, so a massive spin-one one-particle state has $2s+1=3$ polarizations. For $p^{\mu }=(E,0,0,p)$ and metric signature $(+---)$, one convenient basis isAll three obey $p_{\mu }{ϵ}^{\mu }=0$ and ${ϵ}^{\ast }\cdot ϵ=-1$. Their completeness relation isFor a massless momentum, the finite-helicity representations of the $ISO(2)$ little group carry only the two helicity states $\lambda =\pm 1$. Gauge equivalence removes the timelike and longitudinal polarizations; correspondingly, ${ϵ}_L$ has no finite $M_X\to 0$ limit.

The quadratic Proca action isIts momentum-space kinetic operator isInverting it on transverse and longitudinal projectors gives the Proca propagator

For static sources, the spatial Fourier transform of a massive scalar propagator isThus massive-vector exchange likewise produces a Yukawa potential, $V(r)\propto e^{-M_{X}r}/r$, with range $M_X^{-1}$ instead of the infinite range of the Coulomb potential.

Quantum chromodynamics supplies the requested massless-vector counterexample. Its gluons are massless, but confinement and the QCD mass gap prevent a long-range color force between color-singlet asymptotic states. The short range is generated by strong dynamics rather than by a vector-boson mass.

A gauge boson couples to the conserved current of a charged fermion throughwith the appropriate chiral projector when the gauge representation is chiral. At momentum transfer $|q^2|\ll M_X^2$, its propagator can be expanded aswhere the longitudinal term drops for a conserved current. The operation of integrating out a field, applied to $X_{\mu }$, gives the local four-fermion interactionup to the normalization used for $J^{\mu }$. Hence the effective Fermi coupling scales as $G\sim g_X^2/M_X^2$. For Standard Model charged currents, the conventional normalization is

At energy $E\gg M_X$, the longitudinal polarization behaves as ${ϵ}_L^{\mu }\sim p^{\mu }/M_X$. A Proca theory with arbitrary interactions can therefore produce longitudinal-vector scattering amplitudes growing like powers of $E/M_X$, eventually violating partial-wave unitarity and invalidating perturbation theory.

In a gauge-theory completion, the Higgs mechanism identifies the longitudinal mode with a would-be Goldstone boson. Gauge relations fix the vector, Goldstone, and Higgs couplings. Higgs-exchange diagrams then cancel the leading energy growth of the gauge diagrams, as in longitudinal $W$-boson scattering, leaving amplitudes compatible with perturbative unitarity up to the scale where the full renormalizable theory itself becomes strongly coupled.

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.

The Goldstone theorem states that every spontaneously broken generator of a continuous internal global symmetry in a Lorentz-invariant quantum field theory produces a massless scalar particle.

For the classical proof, let $V(\varphi )$ be the invariant scalar potential and let $v_i$ be a vacuum. Infinitesimal invariance givesDifferentiate with respect to ${\varphi }_k$ and evaluate at the stationary point $v$, where ${\partial }_{i}V(v)=0$. The scalar mass matrix then obeysFor every broken generator, $(T^{a}v{)}_i\ne 0$, so $T^{a}v$ is a zero eigenvector of the hessian matrix. It is a massless fluctuation tangent to the vacuum manifold.

For the quantum proof, spontaneous breaking means that some local field hasWrite $Q_a=\int d^{3}x{j}_a^0(x)$ and insert a complete set of momentum eigenstates into the current-field correlation function. Current conservation and Lorentz covariance imply that a scalar intermediate state couples asThe nonzero equal-time commutator requires a pole at $p^2=0$; otherwise the conserved-current spectral integral vanishes at zero momentum. Thus a massless Goldstone boson exists for every independent broken direction.

For $U(x)\in SU(3)$ and $A_{\mu }=A_{\mu }^a{T}^a$, choose $D_{\mu }={\partial }_{\mu }-ig{A}_{\mu }$. The Yang-Mills gauge transformation iswhich ensures $D_{\mu }^{\prime }(U\Phi )=U{D}_{\mu }\Phi$. Sinceconjugating this commutator immediately givesThus the gauge field strength is gauge covariant.

For one complex scalar in the fundamental representation, the most general power-counting-renormalizable gauge-invariant Lagrangian isapart from a constant and the four-dimensional Yang-Mills theta term. Stability requires $\lambda >0$. A cubic candidate ${ϵ}_{ijk}{\Phi }_i{\Phi }_j{\Phi }_k$ vanishes because the scalar components commute.

For ${\mu }^2>0$, the minima satisfy ${\Phi }^{†}\Phi =v^2/2$ with $v^2={\mu }^2/\lambda$. A global $SU(3)$ rotation can chooseThe transformations acting as $SU(2)$ on the first two entries leave this vector fixed, so the symmetry-breaking pattern isThere are $8-3=5$ broken generators. If the symmetry were global, the Goldstone theorem would therefore give five massless scalar modes, exactly the five tangent directions of the vacuum manifold $SU(3)/SU(2)$.

Parameterize the scalar locally aswhere the five $X^{\alpha }$ are broken generators. A gauge transformation removes all ${\pi }^{\alpha }$ in unitary gauge. Their derivative terms combine with the corresponding gauge fields in $(D_{\mu }\Phi {)}^{†}{D}^{\mu }\Phi$, supplying the longitudinal polarizations of five massive vectors. This is the Higgs mechanism.

With $T^a={\lambda }^a/2$, the gauge bosons $A_{\mu }^{1,2,3}$ of the unbroken $SU(2)$ remain massless. The four bosons $A_{\mu }^{4,5,6,7}$ haveand $A_{\mu }^8$ hasThe five eaten Goldstone modes complete their third polarizations. The sixth real component of the complex triplet remains as a radial Higgs boson withThis realizes the degree-of-freedom count summarized by Fundamental-Higgs breaking of SU(3) to SU(2).

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.