Past exam of the particle physics course of the University of Oxford / 2015 / 8

The Goldhaber experiment inferred neutrino helicity from the circular polarization of a nuclear gamma ray. A spin-zero metastable europium-152 nucleus captures an inner-shell electron, producing a spin-one excited samarium-152 daughter and an electron neutrino; the daughter promptly emits a roughly $960\mathrm{keV}$ gamma ray to its spin-zero ground state. For a nearly stationary initial atom, the daughter recoils opposite the neutrino. Selecting photons emitted along that recoil relates the daughter spin projection, the photon helicity and the neutrino helicity.

The selection used resonant absorption and scattering in samarium. Nuclear recoil normally shifts an emitted gamma ray below the absorption resonance; the daughter recoil from electron capture supplies a compensating forward Doppler shift. Resonant scattering therefore preferentially selects forward photons, fixing their direction relative to the otherwise invisible neutrino. A samarium scatterer and scintillation counting identify this resonant component. Transmission through magnetized iron, whose Compton scattering depends on the relative electron and photon spins, then analyzes the circular polarization. Reversing the magnetization reverses the analyzing asymmetry and helps control instrumental offsets.

The selected photon and neutrino normalized helicities have the same sign: their momenta are opposite, while the daughter angular-momentum accounting gives opposite spin projections along the recoil axis. The observed negative photon helicity therefore establishes predominantly left-helicity neutrinos, consistent with the V-A interaction. It measures a helicity correlation, not a tiny neutrino mass or a literal observation of the neutral particle's trajectory.^[ref]

Nuclear fusion in the Sun produces predominantly electron neutrinos, through the pp chain and subsidiary reactions. The historical solar-neutrino deficit could have reflected either a source-model error or flavour transformation. In a two-flavour approximation, mixing gives the vacuum survival law derived in a. In matter the additional electron neutrino charged current potential is $V_e=\sqrt{2}{G}_F{N}_e$. Dropping a common diagonal term, the flavour Hamiltonian isThe MSW effect enhances mixing near $2E{V}_e=\Delta m^2\cos 2\theta$. An initially electron-like high-density state follows its instantaneous mass eigenstate approximately adiabatically as solar density falls. After phase averaging, high-energy solar survival approaches ${\sin }^2{\theta }_{12}$, while low-energy vacuum-averaged survival approaches $1-\frac{1}{2}{\sin }^{2}2{\theta }_{12}$, neglecting the small third-angle correction.

The Sudbury Neutrino Observatory used about a kilotonne of heavy water viewed by photomultipliers. Its three channels separate flavour from total active flux:The charged current channel measures electron flavour, the neutral current channel measures all active flavours nearly equally, and elastic scattering is more sensitive to electron flavour but also detects the others. Cherenkov radiation gives the charged-particle energy and direction. Neutrons were detected through capture gamma rays, with later salt loading and dedicated counters improving the neutral-current separation. The different angular patterns, light isotropy and energy spectra permit a combined channel fit, with radioactive backgrounds and calibrations controlled.

The measured electron flux was lower than the total active flux, while the latter agreed with solar fusion expectations. This directly established a non-electron component and removed a simple overall source-flux error as the explanation. ^[ref] Super-Kamiokande supplied high-statistics directional elastic-scattering data; radiochemical chlorine/gallium experiments supplied complementary thresholds, and reactor antineutrino disappearance independently confirmed the solar mass-squared difference. Together these support large-angle mixing at $\Delta m_{21}^2\sim 7.5\times 10^{-5}{\mathrm{eV}}^2$ rather than the atmospheric splitting in c.

The bottom quark has charge $-1/3$ and left-handed weak isospin $-1/2$, requiring a charge-$+2/3$ partner in a complete Standard Model weak doublet. Completing the third quark generation also maintains the quark/lepton cancellation of gauge anomalies. A three-generation CKM matrix permits an irreducible CP violation phase. Flavour constraints and loop corrections, including their strong top-mass dependence in electroweak precision observables, provided indirect evidence before direct production. These arguments identify the required partner and its interactions; they do not by themselves fix its mass.

The Tevatron collided protons and antiprotons at $\sqrt{s}=1.8\mathrm{TeV}$. Quantum chromodynamics produces $t\bar{t}$, mainly through $q\bar{q}$ annihilation there. Each top quark decays predominantly to $Wb$ before hadronization, giving $W^{+}{W}^{-}b\bar{b}$. The dilepton channel has two isolated oppositely charged leptons, missing transverse momentum from neutrinos, and bottom jets. The single-lepton channel has an isolated electron or muon, missing transverse momentum, and four leading jets, two consistent with bottom decay. The all-hadronic channel has more background. Secondary-vertex and soft-lepton tags identify bottom hadrons.

CDF and D0 reported independent excesses in 1995, with event kinematics and reconstructed masses consistent with a common very heavy quark. Fits assign jets to the two decay chains and compare with expected $W+$jets, multijet and instrumental backgrounds. Agreement of counts, bottom content and reconstruction across channels is stronger evidence than an excess in one event class alone. ^{[ref] [ref]}

The CKM matrix appears in the charged current of b. For a tree semileptonic transition, a schematic extraction is $\Gamma =G_F^2|V_{ij}{|}^2\times \text{phase space}\times \text{hadronic factor}\times (1+\text{corrections})$. Measure an absolute rate or differential spectrum, calculate or independently determine the hadronic factor, and solve for the modulus. The process is not a direct count of quark flavours in a detector: form factors, decay constants, nuclear structure and radiation can limit the result.

Rate moduli alone do not determine the complex phase. CP-asymmetric decay distributions and interference in neutral-meson systems, or tree-level $B\to DK$ interference, constrain the phase and the CKM unitarity triangle. Unitarity supplies consistency checks and indirect determinations, but a fit that imposes it must be distinguished from a direct measurement of an element.

A general-purpose collider detector surrounds the interaction region with complementary layers. The layout makes the measured signals useful together: tracks establish charged trajectories, calorimeters sum deposited energy, and outer detectors identify penetrating muons.

Near-hermetic coverage permits a missing transverse momentum estimate from the calibrated vector sum, signaling neutrinos or other invisible particles but also sensitive to detector failures and escaping jets. At a hadron collider the initial parton longitudinal momenta are unknown, so missing longitudinal energy cannot be inferred in the same way; an $e^{+}{e}^{-}$ collider supplies stronger initial-state constraints. Every subsystem needs alignment, calibration and independent control samples for reliable combined reconstruction.