Modern Physics · Upper Division

Particle Physics & The Standard Model

The Standard Model of particle physics describes all known matter and three of the four fundamental forces in terms of quantum fields. It has been tested to extraordinary precision and remains the most successful scientific theory ever constructed.

PrerequisitesQuantum mechanics (Ch. 20) \cdot Special relativity (Ch. 19) \cdot Spin & angular momentum (Ch. SP)
Learning Goals
  • Classify all Standard Model particles by spin, charge, and color and identify the force carrier for each interaction.
  • ExplainasymptoticfreedomandconfinementinQCDusingtherunningcouplingconstantαs(Explain asymptotic freedom and confinement in QCD using the running coupling constant \alpha_s(Q2Q^{2}.
  • Apply quark model addition rules to verify charges and strangeness of baryons and mesons.
  • Use the Weinberg angle to derive the W and Z boson masses from electroweak unification.
  • Draw leading-order Feynman diagrams for QED and weak processes and extract cross sections from the amplitude M.

PP.1 Elementary Particles

Definition PP.1 — The Standard Model Particles
Quarks (spin-½, fractional charge): up(+2/3e),down(−1/3e),charm,strange,top,bottom.Quarkscombineintohadrons:baryonsup(+2/3e), down(-1/3e), charm, strange, top, bottom. Quarks combine into hadrons: baryons(3 quarks, e.g. proton uud) and mesons (quark-antiquark, e.g. pion u-dbar).Leptons (spin-½, integer charge): electron(e−),muon(μ−),tau(τ−),andtheirneutrinos(\nue,νμ,ντ).Eachhasanantipartelectron (e^{-}), muon (\mu^{-}), tau (\tau^{-}), and their neutrinos (\nue, \nu\mu, \nu\tau). Each has an antipartcle.Force carriers (spin-1 bosons): photonγ(EM),W±,Z0(weak),8gluons(strongphoton \gamma (EM), W\pm, Z^{0} (weak), 8 gluons (strongHiggs boson (spin-0): discovered at LHC 2012. Mass 125 GeV. Responsible for electroweak symmetry breaking and gives fermions and W/Z their masses.

PP.2 The Three Forces

Electromagnetism (QED): mediated by massless photon. Coupling strength α = e²/(4πε₀ℏc) = 1/137.036. Tested to 12 decimal places (electron g-factor). Range: infinite (massless mediator).

Weak force: mediated by W± (80.4 GeV) and Z⁰ (91.2 GeV) — massive because of spontaneous symmetry breaking (Higgs mechanism). Responsible for β decay, neutrino interactions, flavor-changing processes. Parity-violating: W only couples to left-handed fermions. Range: 10⁻¹⁸ m.

Strong force (QCD): mediated by gluons. Quarks carry "color charge" (red, green, blue). Two unique features distinguish QCD:

αs(Q2)=12π/[(33−2nf)ln(Q2/ΛQ2CD)](runningcoupling)\alpha_s(Q^{2}) = 12\pi / [(33 - 2n_f) ln(Q^{2}/\Lambda^{2}_QCD)] \qquad (running coupling)(PP.1)

Asymptotic freedom: α_s → 0 at high Q² (short distances) — quarks behave like free particles inside the proton at high energies. 2004 Nobel Prize.

Confinement: α_s → large at low Q² — color flux tubes form between quarks with energy density ~1 GeV/fm. If you try to separate a quark, the string breaks and produces a new quark-antiquark pair. Isolated quarks cannot exist.

PP.3 Symmetries and Conservation Laws

The Standard Model is built on gauge symmetry: SU(3) × SU(2) × U(1). Each factor corresponds to a force: SU(3) is QCD (8 gluons = 3²−1), SU(2)×U(1) gives electroweak (W±, Z⁰, γ = 4 bosons = 3 + 1, mixed by Weinberg angle θ_W).

Conservation laws from symmetry (Noether):Gauge invariance → charge conservation. Lorentz invariance → energy-momentum conservation. Baryon number B, lepton numbers L_e, L_μ, L_τ conserved (at low energy).

Discrete symmetries: C (charge conjugation), P (parity), T (time reversal). QED and QCD conserve all three. The weak interaction violates P maximally (Wu experiment 1956). CP violation (1964, K meson) explains matter-antimatter asymmetry — without it, the Big Bang would have produced equal matter and antimatter.

Example PP.1 — The Quark Model: Mass and Charge

Theproton(uud),neutron(udd),andπ+meson(udˉ).Verifycharges.EstimateprotonmassThe proton (uud), neutron (udd), and \pi^{+} meson (ud̄). Verify charges. Estimate proton mass.

Proton charge:uud:Q=+2/3+2/3−1/3=+1✓uud: Q = +2/3 + 2/3 - 1/3 = +1 ✓
Neutron charge:udd:Q=+2/3−1/3−1/3=0✓udd: Q = +2/3 - 1/3 - 1/3 = 0 ✓
Pion charge:π+=udˉ:Q=+2/3+1/3=+1✓(dˉhascharge+1/3\pi^{+} = ud̄: Q = +2/3 + 1/3 = +1 ✓ (d̄ has charge +1/3
Proton mass:mu≈2.2MeV,md≈4.7MeV.Sumofquarks:2×2.2+4.7=9.1MeV.Butprotonmass=938m_{u} \approx 2.2 MeV, m_{d} \approx 4.7 MeV. Sum of quarks: 2\times2.2 + 4.7 = 9.1 MeV. But proton mass = 938.3 MeV! The remaining ~929 MeV comes from gluon fields and quark kinetic energy — "dynamical mass generation" from QCD. 99% of the proton mass (and your mass) is energy, not quark rest mass.

PP.4 Feynman Diagrams and Cross Sections

Feynman diagrams are pictorial representations of perturbation theory expansions in the coupling constant. Each diagram corresponds to a term in the S-matrix amplitude. The rules are: external lines (particles in/out), internal lines (propagators = virtual particles), vertices (coupling constant), loops (higher-order corrections).

The cross section is the quantum mechanical transition rate normalized to incoming flux:

σ=(1/flux)×∑∣M∣2×(phasespace)[m2]\sigma = (1/flux) \times \sum |M|^{2} \times (phase space) \qquad [m^{2}](PP.2)

where M is the Feynman amplitude. For e⁺e⁻ → μ⁺μ⁻ at center-of-mass energy √s ≫ m_μ: σ = 4πα²/(3s) — the classic QED result, confirmed to percent level at LEP. The ratio R = σ(hadrons)/σ(μ⁺μ⁻) counts quark colors and flavors, proving quarks come in 3 colors.

Definition PP.2 — Common Traps
  • Conservation laws control reactions: charge, energy-momentum, baryon number, and lepton number must be checked.
  • Virtual particles are calculation terms: they are internal lines, not directly observed particles.
  • Cross section is not literal size: it is an effective interaction probability.
  • Running couplings depend on scale: interaction strength changes with momentum transfer.
Exercises — PP.1–PP.4 Particle Physics
1.Identifythequarkcontentof:(a)∑+(Q=+1,S=−1,B=1),(b)K0(Q=0,S=−1,B=0),(c)Ω−(Identify the quark content of: (a) \sum^{+} (Q=+1, S=-1, B=1), (b) K^{0} (Q=0, S=-1, B=0), (c) \Omega^{-} (Q=−1,S=−3,B=1).WhywastheΩ−discoverysignificantQ=-1, S=-3, B=1). Why was the \Omega^{-} discovery significant?
Straightforward
2.
Whydoesπ+→e++\nuehaveamuchsmallerratethanπ+→μ++νμ,eventhoughtheelectronWhy does \pi^{+} \to e^{+} + \nue have a much smaller rate than \pi^{+} \to \mu^{+} + \nu\mu, even though the electron is lighter? Calculate the ratio.
Intermediate
3.
UsetheWeinbergangle(sin2θW=0.231)topredicttheWandZmassesfromtheZmass.HoUse the Weinberg angle (sin^{2}\theta_W = 0.231) to predict the W and Z masses from the Z mass. How does electroweak unification work?
GeV
Intermediate
4.ExplaintheHiggsmechanism:howdoesspontaneoussymmetrybreakingofSU(2)\timesU(1)givemasExplain the Higgs mechanism: how does spontaneous symmetry breaking of SU(2)\timesU(1) give masses to the W and Z bosons while leaving the photon massless?
Challenging
Key Takeaways
  • Matter: 6 quarks (fractional charge, color) + 6 leptons (integer charge). Antiparticles for each.
  • Forces:γ(EM),W±/Z0(weak),8gluons(strong).HiggsgivesW/Z/fermionmassesForces: \gamma (EM), W\pm/Z^{0} (weak), 8 gluons (strong). Higgs gives W/Z/fermion masses.
  • QCD: asymptotic freedom (free at high E), confinement (confined at low E). Color flux tubes.
  • GaugesymmetrySU(3)\timesSU(2)\timesU(1).Electroweakunificationat 100GeVGauge symmetry SU(3)\timesSU(2)\timesU(1). Electroweak unification at ~100 GeV.
  • CP violation: weak interaction. Explains matter-antimatter asymmetry.
  • Feynmandiagrams→S−matrixamplitudes.Crosssectionσ∝∣M∣2.Rratioproves3colorsFeynman diagrams \to S-matrix amplitudes. Cross section \sigma \propto |M|^{2}. R ratio proves 3 colors.