Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 305 / 1

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.