PHYS3036 · Condensed Matter and Particle Physics
Interactions & Feynman Diagrams
This chapter of University of Sydney PHYS3036 is the grammar of particle interactions. Feynman diagrams are built from allowed Standard-Model vertices (each carrying a coupling constant) joined by internal propagators (virtual particles); the amplitude is a product of couplings and the rate ∝ (coupling)². Fermi's golden rule ties the rate to the squared matrix element and the density of final states, and branching fractions follow from competing squared couplings. The exam asks you to draw allowed diagrams and estimate relative rates.
What this chapter covers
- 01Decay vs scattering; Feynman diagrams as space-time pictures built from vertices and internal lines
- 02Vertices carry a coupling constant; internal lines are virtual particles (propagators); only allowed Standard-Model vertices may be used (EM, strong, weak, Higgs)
- 03Amplitude ∝ product of vertex couplings; probability/rate ∝ |amplitude|² ⇒ rate ∝ (coupling constant)²
- 04Fermi's golden rule: Γ = 2π |T_fi|² ρ(E_f) — rate ∝ |matrix element|² × density of final states; lifetime τ = ℏ/Γ
- 05Branching fractions ∝ squared couplings of the competing decay channels
- 06Higher-order diagrams are suppressed by extra powers of the coupling constant
- 07Resonances: the mass ↔ width relationship, Γ = ℏ/τ (a shorter-lived state is broader)
Relative rate from squared charges: electron scattering off proton quarks
- +1Identify the quarks: a proton is uud — two up quarks (charge +2/3 each) and one down quark (charge −1/3). [+1]
- +1Rate per quark ∝ (charge)². Up quark: (2/3)² = 4/9. Down quark: (−1/3)² = 1/9. (The sign of the charge does not matter once squared.) [+1]
- +1Total off the up quarks (there are two): 2 × 4/9 = 8/9. Total off the single down quark: 1 × 1/9 = 1/9. [+1]
- +1Ratio = (8/9)/(1/9) = 8. So the electron is eight times as likely to scatter off the up-quark content as off the down-quark content of the proton. [+1]
Key terms
- Feynman diagram
- A space-time picture of an interaction built from vertices (points where particles meet) and lines (external real particles, internal virtual ones); it encodes the amplitude for a process.
- Vertex
- A point in a Feynman diagram where particles interact; each carries a coupling constant, and only the allowed Standard-Model vertices (EM, strong, weak, Higgs) may be used.
- Propagator
- An internal line representing a virtual particle exchanged in a process; it need not satisfy E² = (pc)² + (mc²)² and mediates the interaction between vertices.
- Coupling constant
- The strength attached to a vertex; the amplitude is a product of couplings and the rate is proportional to the square of the coupling(s).
- Fermi's golden rule
- Γ = 2π |T_fi|² ρ(E_f): the transition rate is proportional to the squared matrix element times the density of final states; the lifetime is τ = ℏ/Γ.
- Branching fraction
- The share of decays going to a particular final state, proportional to that channel's squared coupling relative to all competing channels.
Interactions & Feynman Diagrams FAQ
Why is the rate proportional to the coupling squared?
Because a Feynman diagram gives an amplitude that is a product of the vertex couplings, and the physical probability (and hence rate) is the square of the amplitude. So a process with one vertex of coupling g has a rate ∝ g². This is why the electromagnetic, strong and weak interactions have such different rates: their couplings differ, and squaring amplifies the gap.
What rules constrain which Feynman diagrams I can draw?
Only the allowed Standard-Model vertices are permitted — specific EM, strong, weak and Higgs couplings — and every conserved quantity (charge, energy–momentum, baryon and lepton number, and the flavour rules of each force) must be respected at each vertex and overall. Internal lines are virtual and can be off-mass-shell, but the external legs are real particles. Drawing only allowed vertices is itself an examined skill.
What does Fermi's golden rule tell me?
That a transition rate has two ingredients: the squared matrix element |T_fi|² (how strongly the initial and final states are connected, set by the couplings) and the density of available final states ρ(E_f). Multiply them (with the 2π) to get Γ, and invert to get the lifetime τ = ℏ/Γ. It is the bridge between a diagram's coupling structure and a measurable rate or lifetime.
How is this examined in PHYS3036?
You are typically asked to draw the allowed Feynman diagram for a given process and to estimate relative rates or branching fractions from the couplings (rate ∝ coupling²). The exam provides constants like α = 1/137 and αs on the data sheet. Practise both the drawing and the squared-coupling estimates; confirm the paper's exact style on Canvas and the unit outline.
Exam move
Build two reflexes. First, draw Feynman diagrams using only allowed Standard-Model vertices, checking conservation laws at every vertex — practise decays and scatterings until the vertex set is second nature. Second, estimate relative rates with rate ∝ (coupling)²: for electromagnetic processes the coupling is the charge, so square charges and count particles; for weak processes bring in the CKM factors from the next chapter. Keep Fermi's golden rule and τ = ℏ/Γ handy for lifetime/width questions, and remember higher-order diagrams are suppressed by extra powers of the coupling. The exam supplies α and αs, so focus on setup and clean ratios rather than recall. When a diagram won't close, ask Sia to check your vertices and the squared-coupling estimate step by step.
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