Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 305 (The Standard Model)

The first operator is a Lorentz scalar formed from commuting translations. Since $[P_{\mu },P_{\nu }]=0$,and the vector transformation law of $P^{\mu }$ gives

The Pauli-Lubanski pseudovector commutes with every $P_{\mu }$, so $[W^2,P_{\mu }]=0$. Its Lorentz-vector commutator similarly givesbecause the two tensor-rotation terms cancel after relabelling the contracted index. Thuscommute with every generator of the Poincare algebra. They are its two algebraically independent Casimir operators in four spacetime dimensions.

For $P^2=m^2>0$, choose the future-pointing rest momentum $p^{\mu }=(m,0,0,0)$. Its little group is the rotation group $SO(3)$, or $SU(2)$ on the double cover. Its finite-dimensional unitary irreducible representations are labelled byand have dimension $2j+1$. These are the spin states of a massive particle.

For $P^2=0$, choose $p^{\mu }=(E,0,0,E)$. Its little group is the two-dimensional Euclidean groupNontrivial unitary action of the translation subgroup produces infinite-dimensional continuous-spin representations. For the finite-helicity particles observed in ordinary relativistic field theory, that subgroup acts trivially, leaving a one-dimensional representation of rotations about $\mathbf{p}$. Such representations are labelled by helicity.

The helicity of a massless state is the component of angular momentum along its momentum,When the translation part of the massless little group is trivial, the Pauli-Lubanski pseudovector obeysA rotation through angle $\theta$ about $\mathbf{P}$ acts by the phase $e^{-i\lambda \theta }$. On the double cover, a $4\pi$ rotation is the identity, soHence $\lambda =0,\pm \frac{1}{2},\pm 1,\pm \frac{3}{2},\dots$. A parity-invariant theory pairs the $\lambda$ and $-\lambda$ representations, although one chiral massless representation need not contain both.

For a null momentum $q^{\mu }$, impose transversalityThis removes one of four vector components. The remaining equivalenceremoves a second component, leaving two transverse polarizations. For momentum along the third axis they may be chosen aswhich carry helicity $\pm 1$.

The equivalence is a gauge redundancy: vectors that differ by a multiple of $q^{\mu }$ describe the same physical state rather than distinct measurable configurations. A Lorentz transformation of a chosen transverse representative can require a compensating gauge transformation. Therefore the amplitudemust be invariant under ${ϵ}_{\mu }\mapsto {ϵ}_{\mu }+\alpha q_{\mu }$. For arbitrary $\alpha$, this is exactly the Ward identity

For emission of a soft photon of momentum $q$ from external charged particles, the Soft photon theorem giveswhere ${\eta }_i=+1$ for outgoing and $-1$ for incoming particles. Contracting with $q_{\mu }$ and applying the Ward identity givesFor a nonzero underlying amplitude, total outgoing charge equals total incoming charge. Thus photon gauge redundancy enforces conservation of electric charge.

The analogous Soft graviton theorem for helicity two requires universal coupling to energy-momentum and yields conservation of total four-momentum, the seed of the Equivalence principle. For massless helicity greater than two, the corresponding soft consistency conditions demand conserved higher-rank momentum charges that generic interacting scattering cannot satisfy. Under the assumptions of a Lorentz-invariant local S-matrix with long-range interactions, such particles therefore have no nontrivial coupling; this is the soft-theorem obstruction to interacting massless higher-spin particles.

With $T^a={\tau }^a/2$, hypercharge $Y_H=1/2$, andthe most general local renormalizable bosonic Lagrangian isCanonical normalization leaves four parameters:Stability requires $\lambda >0$. If $m^2>0$, the minimum is $H=0$ and the electroweak symmetry is unbroken. If $m^2<0$,and the electroweak symmetry breaking pattern is

Write $H=2^{-1/2}{e}^{-i{\xi }^a{T}^a}(0,v+h{)}^{\mathsf{T}}$. Differentiating the exponential to first order in the fields, or using its Maurer-Cartan form exactly, shows that the angular fields enter $D_{\mu }H$ throughwith higher-order commutator terms dictated by the same group-valued combination. A local $SU(2{)}_L$ transformation can set ${\xi }^a=0$. In this unitary gauge,The three Goldstone modes have become the longitudinal polarizations of the three massive electroweak gauge bosons, which is the Higgs mechanism.

Defineand rotate the neutral fields byThe quadratic mass terms from $|D_{\mu }⟨H⟩{|}^2$ are diagonal in this basis and giveThe scalar mass isThe unbroken generator is $Q=T^3+Y$, and $A_{\mu }$ is its gauge field. Its exact masslessness and coupling $e=g\sin {\theta }_W=g^{\prime }\cos {\theta }_W$ identify it as the photon.

Let $\tilde{H}=i{\tau }^2{H}^{\ast }$. Gauge invariance permits the Yukawa interactionswhere $Y_u,Y_d$ are arbitrary complex $3$ by $3$ matrices in generation space. Their color and weak indices are contracted to singlets, and the hypercharges in each term sum to zero.

After symmetry breaking,The same terms couple the physical Higgs field proportionally to the quark mass matrices.

Use singular value decomposition to choose unitary matrices satisfyingThe neutral currents remain flavour diagonal because the same unitary matrix occurs on both sides of each bilinear. The charged current contains the mismatchand becomes

A general unitary $3$ by $3$ matrix has nine real parameters. Independent rephasings of the six quark fields remove five phases because one common baryon-number phase changes nothing. The Cabibbo-Kobayashi-Maskawa matrix therefore has four physical parameters: three mixing angles and one CP-violating phase.

The leptons consist ofwith charged-lepton Yukawa coupling $-{\overline{L}}_L{Y}_{e}H{e}_R+h.c.$

Adding gauge-singlet right-handed neutrinos ${\nu }_R\sim (1,1{)}_0$ permits the Dirac Yukawa couplingwhich gives $M_D=v{Y}_{\nu }/\sqrt{2}$. Because ${\nu }_R$ carries no Standard Model gauge charge, a Majorana mass term ${\nu }_R^{\mathsf{T}}C{M}_R{\nu }_R/2$ is also allowed. For $M_R\gg M_D$, the seesaw mechanism produces light neutrino masses of order $M_D{M}_R^{-1}{M}_D^{\mathsf{T}}$.

In four dimensions the action is dimensionless and $[{\partial }_{\mu }]=1$. The kinetic terms giveso $[\Phi ]=1$ and $[\Psi ]=3/2$.

Up to a constant and total derivatives, the most general renormalizable Lagrangian invariant underisHere $\lambda >0$ is required for stability at large field. The last line is the allowed charge-conserving scalar-fermion Yukawa interaction; its precise two-component expression depends on the chosen spinor notation.

Forwith $\lambda >0$, spontaneous symmetry breaking occurs when $m_{\Phi }^2<0$. ThenSince $\Phi$ has charge two, the transformations preserving a chosen nonzero vacuum satisfy $e^{2i\alpha }=1$. The unbroken subgroup is therefore ${\mathbb{Z}}_2$, generated by $\alpha =\pi$; it acts as $\Psi \mapsto -\Psi$.

With the convention that the fields have charges $q_{\Phi }=2$ and $q_{\Psi }=1$, the Noether current may be writtenup to an overall sign convention for the generator. The field equations, including the invariant Yukawa interaction, give ${\partial }_{\mu }{J}^{\mu }=0$. Subject to vanishing flux at spatial infinity, the Noether chargeis conserved and generates the global $U(1)$ transformations.

Promoting $\alpha$ to a spacetime-dependent parameter makes the scalar kinetic term vary by both terms linear and quadratic in the gauge field. Merely adding $e{J}^{\mu }{A}_{\mu }$ accounts for the linear term but is not invariant because the scalar current itself changes under a local transformation.

Withminimal coupling givesThe missing term is the seagull vertex $4e^2{A}^2|\Phi {|}^2$. This agrees exactly with replacing partial derivatives by gauge covariant derivatives.

The Dirac kinetic term is first order in derivatives:Its expansion is only linear in $A_{\mu }$, so the current coupling already completes the gauge-invariant fermion kinetic term and no $A^2\overline{\Psi }\Psi$ term arises.

The invariant interaction isbecause ${\Phi }^{\ast }$ has charge $-2$ while the fermion pair has charge $+2$. Once $\Phi$ has a vacuum expectation value, it generates a Majorana-type fermion mass and can split the two Majorana components of the original Dirac field.

The operator has dimensionso its coupling is classically marginal in four dimensions. By comparison, $\overline{\Psi }\Psi$ has dimension three and its coefficient is a relevant mass parameter.

Separating variables givesThereforeFor $b>0$, define the strong-coupling scaleThenIt decreases logarithmically toward zero in the ultraviolet and grows toward the infrared, becoming nonperturbative when $\mu$ approaches ${\Lambda }_{\mathrm{QCD}}$.

For $b>0$, the theory is asymptotically free: it is weakly coupled at high energies and strongly coupled in the infrared. For $b<0$, the coupling decreases toward the infrared but grows toward a finite ultraviolet Landau pole; such a theory is infrared free and requires an ultraviolet completion before that pole.

LetRunning each coupling down from the common value ${\alpha }_G$ givesSubtracting the $i=1,2$ equations determines $L$, and eliminating it from the $i=3,2$ difference giveswhich is the claimed gauge coupling unification relation.

Below the strange-quark threshold, the gauge-field and two-flavour quark terms arewhereThis is the low-energy Quantum chromodynamics plus electromagnetism Lagrangian for the light flavours.

Every renormalizable gauge-invariant term contains equal numbers of quark and antiquark fields. The common phase symmetryis therefore automatic rather than imposed in constructing the Lagrangian. Its conserved charge is baryon number, making $U(1{)}_B$ an accidental symmetry of the renormalizable low-energy theory.

For $m_u=m_d=0$ and electromagnetism neglected, the classical flavour symmetry isup to finite quotients. The chiral anomaly breaks $U(1{)}_A$ quantum mechanically, while $U(1{)}_V=U(1{)}_B$ remains. The quark condensatespontaneously breaksSmall quark masses and electromagnetism explicitly break parts of this approximate symmetry.

Introduce the linear sigma model fieldA suitable renormalizable potential iswith a small symmetry-breaking source proportional to the quark-mass matrix when desired. A vacuum $⟨\Sigma ⟩=vI$ is fixed precisely by $L=R$, so the unbroken group is the diagonal isospin $SU(2{)}_V$. The three broken axial generators produce the pion triplet ${\pi }^{\pm },{\pi }^0$, which become pseudo-Goldstone bosons when $m_u,m_d$ are restored.

At energies well below ${\Lambda }_{\mathrm{QCD}}$, collect the pion fields intoThe leading chiral perturbation theory Lagrangian iswhere $M=\mathrm{diag}(m_u,m_d)$ is treated as a symmetry-breaking spurion. The derivative and quark-mass expansion organizes the omitted terms.

Exact isospin symmetry would make the proton and neutron degenerate. Their observed mass difference is generated by explicit isospin breaking from $m_d-m_u$ together with electromagnetic effects. The approximate symmetry explains why the splitting is small compared with the nucleon mass, while chiral perturbation theory parametrizes it through symmetry-breaking operators and low-energy constants; symmetry alone does not predict its numerical value.

The strange quark is light enough compared with typical hadronic scales to permit an approximatechiral expansion. It adds kaons and the eta to the pseudo-Goldstone multiplet, although convergence is poorer because $m_s$ is substantially larger than $m_u,m_d$.

The charm quark is too heavy for the same light-flavour chiral expansion. At low energies it is integrated out; processes containing charm are instead treated using heavy-quark expansions or other effective theories. Thus the analysis extends approximately to $s$, but not to $c$ as another pseudo-Goldstone-producing light flavour.