Quarks, Leptons, Baryons, Mesons, Conservation Laws & Feynman Diagrams

1

Four Fundamental Interactions & Exchange Bosons

1. Strong Nuclear Force: range ~0.5 to 3 fm (repulsive below 0.5 fm to prevent nuclear collapse, attractive from 0.5 to 3 fm to overcome electrostatic proton repulsion); acts on hadrons/quarks only; exchange particle = gluon (between quarks) or pion π (between nucleons). 2. Electromagnetic Force: infinite range; acts on charged particles; exchange particle = virtual photon (γ). 3. Weak Nuclear Force: range ~10⁻¹⁸ m; acts on both quarks and leptons (responsible for β⁻/β⁺ decay, electron capture, and strange particle decay); exchange particles = W⁺ and W⁻ bosons (which have mass, charge, and short range). 4. Gravity: infinite range; acts on all mass; negligible at subatomic scales.

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2

Hadrons (Baryons vs Mesons) & Quark Compositions

Hadrons feel the strong nuclear force and are made of quarks: Up (u: Q = +2/3, B = +1/3, S = 0), Down (d: Q = -1/3, B = +1/3, S = 0), Strange (s: Q = -1/3, B = +1/3, S = -1); antiquarks (ū, d̄, s̄) have opposite signs for Q, B, and S. 1. Baryons (qqq, B = +1; antibaryons q̄q̄q̄, B = -1): Proton p = uud (the only stable free baryon; free neutrons decay into protons with half-life ~10 min); Neutron n = udd. 2. Mesons (qq̄, B = 0, unstable): Pions (no strange quarks): π⁺ = ud̄, π⁻ = dū, π⁰ = uū or dd̄. Kaons (contain a strange or antistrange quark, decay into pions via the weak force): K⁺ = us̄ (S = +1), K⁻ = sū (S = -1), K⁰ = ds̄ (S = +1).

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3

Leptons & Lepton Number Conservation (Lₑ and L_μ)

Leptons are fundamental particles that do not experience the strong nuclear force. Electron family (Lₑ = +1 for leptons, Lₑ = -1 for antileptons): electron (e⁻), electron neutrino (νₑ); positron (e⁺, Lₑ = -1), electron antineutrino (ν̄ₑ, Lₑ = -1). Muon family (L_μ = +1 for leptons, L_μ = -1 for antileptons): heavy negative muon (μ⁻, decays into an electron via μ⁻ → e⁻ + ν̄ₑ + ν_μ), muon neutrino (ν_μ). Electron lepton number (Lₑ) and muon lepton number (L_μ) are conserved separately in every interaction.

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4

Strangeness (S) & Associated Production vs Weak Decay

Strange particles (such as Kaons K and Sigma Σ baryons) contain at least one strange (s, S = -1) or antistrange (s̄, S = +1) quark. Strange particles are always created in pairs via the strong interaction (associated production, e.g. π⁻ + p → K⁰ + Λ⁰), where Strangeness is strictly conserved (ΔS = 0). However, strange particles decay slowly (long lifetimes ~10⁻¹⁰ s) via the weak interaction (e.g. K⁺ → μ⁺ + ν_μ or K⁺ → π⁺ + π⁰), which is the only interaction in which strangeness can change by ΔS = ±1.

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5

Quark-Level Equations & Feynman Diagrams for Weak Interactions

1. Beta-Minus (β⁻) Decay: n → p + e⁻ + ν̄ₑ (quark level: d → u + e⁻ + ν̄ₑ via a W⁻ boson). 2. Beta-Plus (β⁺) Decay: p → n + e⁺ + νₑ (quark level: u → d + e⁺ + νₑ via a W⁺ boson). 3. Electron Capture: an inner-shell atomic electron is captured by a proton in a proton-rich nucleus: p + e⁻ → n + νₑ (W⁺ boson goes from p→n vertex to e⁻→νₑ vertex). 4. Electron-Proton Collision: p + e⁻ → n + νₑ (identical overall equation to electron capture, but in the Feynman diagram the W⁻ boson goes from the e⁻→νₑ vertex to the p→n vertex).

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6

Pair Production & Matter-Antimatter Annihilation

Every particle has a corresponding antiparticle with the exact same rest mass (and rest energy E₀ = mc²) but opposite charge and opposite baryon/lepton/strangeness quantum numbers. 1. Annihilation: when a particle and its antiparticle (e.g. e⁻ and e⁺) meet, their entire mass is converted into two gamma-ray photons travelling in opposite directions (to conserve momentum): E_total = 2E₀ + KE = 2hf_min, so each photon has minimum energy hf_min = E₀. 2. Pair Production: a single high-energy photon passing near a nucleus converts into a particle-antiparticle pair (e.g. γ → e⁻ + e⁺), requiring minimum photon energy hf_min = 2E₀ (for e⁻/e⁺, 2 × 0.511 MeV = 1.022 MeV).

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1. Q1

   A negative kaon (K⁻) has a charge of -1e and a strangeness of -1. Deduce the quark composition of the K⁻ meson, state the two particles formed when a K⁻ decays into a muon and one other particle, and use conservation laws to explain why this decay must occur via the weak interaction. (5 marks)

   Check Mark Scheme

   Click “Check Mark Scheme” to compare your answer, 1. Since K⁻ is a meson, it consists of one quark and one antiquark (qq̄); a strangeness of S = -1 means it contains a strange quark (s, charge -1/3e) [1 mark]. 2. To give a total charge of -1e, the antiquark must have charge -2/3e (and S = 0), which is an anti-up quark (ū); therefore K⁻ = sū [1 mark]. 3. Decay equation into a muon: K⁻ → μ⁻ + ν̄_μ (a negative muon and a muon antineutrino) [1 mark]. 4. Conservation check: Charge Q (-1 = -1 + 0), Baryon number B (0 = 0 + 0), and Muon Lepton number L_μ (0 = +1 + (-1)) are all conserved [1 mark]. 5. However, Strangeness changes from S = -1 on the left to S = 0 on the right (ΔS = +1), and because a hadron (quarks) decays into leptons and strangeness is not conserved, the decay must occur via the weak nuclear interaction [1 mark].

2. Q2

   Compare the Feynman diagrams and physical processes of electron capture and an electron-proton collision, both of which have the overall equation p + e⁻ → n + νₑ. (4 marks)

   Check Mark Scheme

   Click “Check Mark Scheme” to compare your answer, 1. In both processes, a proton (uud) and an electron (e⁻) enter at the bottom of the Feynman diagram and a neutron (udd) and an electron neutrino (νₑ) leave at the top [1 mark]. 2. In electron capture, an unstable proton-rich nucleus spontaneously pulls in one of the atom's own inner-shell (K-shell) orbital electrons [1 mark]. 3. Because the proton initiates the decay in electron capture, a positively charged W⁺ exchange boson travels from the proton-neutron vertex (left) to the electron-neutrino vertex (right) [1 mark]. 4. In an electron-proton collision, a high-energy external beam of electrons is fired into a proton target, so a negatively charged W⁻ exchange boson travels from the electron-neutrino vertex to the proton-neutron vertex [1 mark].

3. Q3

   A high-energy gamma photon of wavelength 8.20 × 10⁻¹³ m passes close to a heavy nucleus and undergoes pair production to create an electron and a positron (rest energy of an electron = 0.511 MeV). Calculate the energy of the photon in Joules and MeV, and calculate the total kinetic energy shared by the electron and positron in MeV. (h = 6.63 × 10⁻³⁴ J s, c = 3.00 × 10⁸ m s⁻¹, 1 eV = 1.60 × 10⁻¹⁹ J). (4 marks)

   Check Mark Scheme

   Click “Check Mark Scheme” to compare your answer, 1. Calculate photon energy in Joules: E = hc ÷ λ = (6.63 × 10⁻³⁴ × 3.00 × 10⁸) ÷ (8.20 × 10⁻¹³) = 2.426 × 10⁻¹³ J [1 mark]. 2. Convert photon energy to MeV (÷ 1.60 × 10⁻¹³ J MeV⁻¹): E = (2.426 × 10⁻¹³) ÷ (1.60 × 10⁻¹³) = 1.52 MeV (1.516 MeV) [1 mark]. 3. Minimum rest energy required to create both the electron and the positron: 2E₀ = 2 × 0.511 MeV = 1.022 MeV [1 mark]. 4. By conservation of energy, total kinetic energy of the pair = E_photon - 2E₀ = 1.516 - 1.022 = 0.494 MeV [1 mark].

4. Q4

   Determine whether the following proposed strong interaction can occur by testing four conservation laws: p + p → p + n + π⁺ + π⁰. (4 marks)

   Check Mark Scheme

   Click “Check Mark Scheme” to compare your answer, 1. Conservation of Charge (Q): Left-hand side = (+1) + (+1) = +2; Right-hand side = (+1) + 0 + (+1) + 0 = +2 (Conserved) [1 mark]. 2. Conservation of Baryon Number (B): Left-hand side = (+1) + (+1) = +2; Right-hand side = (+1) + (+1) + 0 + 0 = +2 (Conserved, since pions are mesons with B = 0) [1 mark]. 3. Conservation of Lepton Number (L) and Strangeness (S): Both sides have L = 0 and S = 0 (Conserved) [1 mark]. 4. Conclusion: Provided the colliding protons have sufficient kinetic energy to supply the rest mass energy of the created pions (and neutron-proton mass difference), the interaction IS allowed [1 mark].

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