Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 305 (The Standard Model)

The range of a force is controlled by the lightest state that can carry it. Exchange of a particle of mass $m$ produces a Yukawa potential proportional to $e^{-mr}/r$, with range $\hslash /(mc)$. The electromagnetic interaction is long range because its carrier, the photon, is massless. The weak interaction is short range because the W bosons and Z boson are massive. Although the gluons of the strong interaction are massless in the Lagrangian, Quantum chromodynamics confines color and has a mass gap; only color-singlet hadrons propagate over macroscopic distances. The residual nuclear force is consequently controlled at long distance by massive pion exchange.

Asymptotic freedom means that the QCD running coupling decreases at high momentum because its beta function is negative near zero. Quarks and gluons therefore interact weakly in sufficiently hard processes, justifying perturbation theory and explaining approximate parton behavior. Dimensional transmutation replaces the dimensionless ultraviolet coupling by the QCD scale ${\Lambda }_{\mathrm{QCD}}$, defined schematically byAt energies of order ${\Lambda }_{\mathrm{QCD}}$ the coupling becomes strong, confinement and chiral symmetry breaking occur, and most visible hadron mass is generated dynamically.

The near equality $m_u\simeq m_d$ makes the QCD Lagrangian approximately invariant under the isospin group $SU(2)$ acting on $(u,d)$. Treating $m_u,m_d,m_s$ as small compared with the hadronic scale enlarges this to approximate $SU(3)$ flavor symmetry, whose hadron multiplets form the eightfold way. The observed multiplet pattern motivated the quark model, while the need for a hidden three-valued quantum number and a consistent strong dynamics led to color and QCD.

QCD explains both symmetries because its gluon coupling is flavor blind; only the quark-mass matrix breaks flavor symmetry. There is no useful larger light-flavor symmetry because $m_c,m_b,m_t\gg {\Lambda }_{\mathrm{QCD}}$. Their mass terms break $SU(N_f)$ too strongly for hadrons containing them to join approximately degenerate extensions of the light-quark multiplets.

With the Standard Model field content, every gauge-invariant, Lorentz-invariant operator of dimension at most four automatically preserves baryon number and total lepton number. They are therefore accidental symmetries, rather than symmetries imposed in constructing the renormalizable Lagrangian. This explains perturbative conservation in ordinary reactions. They are not exact principles: the electroweak chiral anomaly violates $B+L$, the dimension-five Weinberg operator violates lepton number, and dimension-six proton-decay operators can violate baryon number. The combination $B-L$ is anomaly free when a right-handed neutrino is included in each generation.

After electroweak symmetry breaking, the Higgs field has vacuum expectation value $⟨H⟩=(0,v/\sqrt{2}{)}^T$. A Yukawa interaction $-y_f{\bar{f}}_{L}H{f}_R+h.c.$ then becomes a fermion mass with $m_f=y_{f}v/\sqrt{2}$. The Higgs gauge-covariant kinetic term $|D_{\mu }H{|}^2$ giveswhile the radial fluctuation is the massive Higgs boson. The photon and all eight gluons correspond to unbroken gauge generators and remain massless. In the minimal renormalizable Standard Model, which has no right-handed neutrinos, the neutrinos also remain massless; adding right-handed neutrinos permits Higgs-generated Dirac masses.

The unique dimension-five operator built solely from Standard Model fields is the Weinberg operatorwhich gives a Majorana mass term $m_{\nu }\sim c_5{v}^2/\Lambda$ after the Higgs condenses. A dimension-six example that mediates proton decay iswith Lorentz and weak indices contracted appropriately; $u^c{u}^c{d}^c{e}^c/{\Lambda }^2$ is another. Their dimensions explain why neutrino masses are tiny and proton decay is rare: both probe a high scale, while the stronger proton-lifetime bound places especially severe constraints on baryon-number violation.

In neutrino oscillation, a neutrino produced with definite flavor is a coherent superposition $|{\nu }_{\alpha }⟩={\sum }_i{U}_{\alpha i}^{\ast }|{\nu }_i⟩$ of mass eigenstates. Propagation gives the components different phases, producing a baseline- and energy-dependent probability to detect another flavor. Oscillation requires nonzero mass-squared differences and nontrivial mixing, so its observation proves that at least two neutrinos are massive and that individual flavor lepton numbers are not conserved. It measures mass differences and mixing parameters, though not the absolute neutrino-mass scale.

First, electroweak gauge invariance places left-handed quarks in $SU(2{)}_L$ doublets, pairing one charge-$2/3$ flavor with one charge-$-1/3$ flavor in every generation. Second, cancellation of Standard Model gauge anomalies ties each colored quark doublet and its singlets to a lepton doublet and its singlet; removing either member destroys the hypercharge-anomaly cancellations. Equivalently, the weak charged current and anomaly-free chiral spectrum organize quarks into complete up-down generations, so the number of quark flavors is even.

The Wigner classification labels a massive one-particle state by $|p;j,m_s⟩$, where $p^2=m^2>0$, $p^0>0$, $j$ is the spin representation of the $SO(3)$ little group, and $m_s=-j,\dots ,j$. A finite-spin massless state is $|p;\lambda ⟩$, where $p^2=0$, $p^0>0$, and the helicity $\lambda$ labels a one-dimensional representation of the rotational part of the $ISO(2)$ little group. A parity-invariant theory pairs nonzero helicities $\lambda$ and $-\lambda$.

The physical polarization counts areA scalar obeys the Klein-Gordon equation. A spinor obeys the Dirac equation, and a massless irreducible spinor additionally has a fixed chirality. A massive vector obeys the Proca equation, whose divergence gives ${\partial }_{\mu }{A}^{\mu }=0$ and leaves three polarizations. A massless vector instead has the gauge redundancy $A_{\mu }\sim A_{\mu }+{\partial }_{\mu }\alpha$, leaving two transverse polarizations. A massive symmetric tensor obeys the Fierz-Pauli equations: symmetry, transversality, and tracelessness leave five components. A massless tensor has linearized diffeomorphism redundancy $h_{\mu \nu }\sim h_{\mu \nu }+{\partial }_{\mu }{\xi }_{\nu }+{\partial }_{\nu }{\xi }_{\mu }$, leaving helicities $\pm 2$. A redundancy identifies field configurations representing the same physical state; gauge invariance is invariance under that identification.

Write an amplitude with an external massless vector of momentum $k$ asChanging the polarization representative by ${ϵ}_{\mu }\mapsto {ϵ}_{\mu }+c,k_{\mu }$ is a gauge transformation. The physical amplitude must be unchanged, so the Ward identity isThus a longitudinal pure-gauge polarization decouples from the S-matrix. This on-shell statement is the amplitude counterpart of the off-shell Ward-Takahashi identity.

The leading Soft photon theorem for a photon of momentum $q$ iswhere ${\eta }_i=+1$ for outgoing and $-1$ for incoming particles. Replacing $ϵ$ by $q$ and applying the Ward identity gives ${\sum }_i{\eta }_i{Q}_i=0$, which is electric charge conservation.

The analogous Weinberg soft theorem for helicity two forces every particle to couple to a massless spin-two field with the same strength; its Ward identity reduces to momentum conservation and yields the equivalence principle. For helicity strictly greater than two, the corresponding polynomial constraints have no nontrivial solution for a generic interacting S-matrix under the usual assumptions of Lorentz invariance, locality, and finitely many particle species. This is the soft-theorem obstruction to ordinary long-range interactions mediated by massless higher-spin particles.

The longitudinal polarization of a massive vector behaves as ${ϵ}_L^{\mu }(p)\sim p^{\mu }/m$ at high energy. Generic self-interactions therefore make amplitudes grow as powers of $E/m$, eventually violating partial-wave unitarity and destroying perturbative predictivity. Simply adding a Proca mass also spoils the gauge cancellations that control ultraviolet behavior.

The Standard Model obtains the $W$ and $Z$ masses through the Higgs mechanism inside a renormalizable gauge theory. Goldstone bosons supply the longitudinal polarizations, and relations among gauge and Higgs couplings enforce cancellations of the growing terms. In particular, Higgs exchange cancels the remaining $E^2$ growth in longitudinal vector-boson scattering.

For an infinitesimal spacetime-dependent parameter, the action variation has the Noether current formIn canonical quantization, apply the corresponding transformation inside an equal-time correlation function. Integrating the resulting Ward-Takahashi identity over a thin time interval around an insertion givesComparison with $\delta {\phi }_m=i{\alpha }^a{T}_{mn}^a{\phi }_n$ yieldsin the conventions of the question. Current conservation makes $Q^a$ time independent when the surface flux vanishes, so the same charge generates the symmetry at every time.

A symmetry generated by $Q^a$ is spontaneously broken when the vacuum is not invariant:A local order parameter is obtained by finding a field ${\phi }_m$ for whichwith the sign depending on the commutator convention. This criterion is useful in infinite volume because it can remain finite even when the spatially integrated charge itself is not a well-defined normalizable operator.

For an unbroken internal symmetry, $[H,Q^a]=0$ and $Q^a|0⟩=0$. If $H|n⟩=E_n|n⟩$, then $Q^a|n⟩$ is either zero or another state of the same energy, so particles fill degenerate symmetry multiplets.

In a spontaneously broken phase, applying the finite symmetry changes the vacuum and moves to a distinct infinite-volume superselection sector. The charge is not a well-defined finite-norm operator within one such sector, so the usual argument that $Q^a|n⟩$ is a physical partner state can fail; particle masses need not occur in multiplets of the broken group. Formally, however,Thus the broken directions are degenerate with the vacuum. In a local relativistic theory these zero-energy, long-wavelength excitations become the Goldstone bosons.

Goldstone's theorem states that every spontaneously broken generator of a continuous global internal symmetry gives a massless scalar excitation in a Lorentz-invariant quantum field theory, subject to the standard locality and positivity assumptions.

Choose a local field $\phi$ with $⟨0|[Q^a,\phi (0)]|0⟩\ne 0$ and write $Q^a=\int d^{3}x,J_0^a$. Translation invariance givesLorentz covariance forces the contribution of a spin-zero intermediate state to have ${\rho }_{\mu }(p)=p_{\mu }\rho (p^2)$. Current conservation then implies $p^2\rho (p^2)=0$. The nonzero equal-time integral demanded by the order parameter rules out $\rho =0$, so the spectral density must contain support at $p^2=0$. Hence there is a massless one-particle pole withwhich is the required Goldstone boson. Independent broken generators give independent massless modes in the ordinary relativistic case.

Up to a vacuum-energy constant and the topological Abelian $F\tilde{F}$ term, the most general renormalizable Abelian Higgs model isIt has three local dynamical parameters, $g,m^2,\lambda$. Stability requires $\lambda >0$, and spontaneous symmetry breaking occurs for $m^2<0$. Writingand choosing unitary gauge removes $\pi$. The physical masses areThe interactions include $g^{2}vh{A}_{\mu }{A}^{\mu }$, $g^2{h}^2{A}_{\mu }{A}^{\mu }/2$, $-\lambda v{h}^3$, and $-\lambda h^4/4$. The original massless vector's two polarizations and the complex scalar's two real components become a massive vector with three polarizations and one massive scalar. This rearrangement, in which the gauge field absorbs the would-be Goldstone mode and acquires mass without explicitly breaking gauge invariance, is the Higgs mechanism.

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.