Feynman Diagrams

Feynman diagrams provide a pictorial and computational tool for calculating scattering amplitudes in perturbation theory. Each diagram corresponds to a mathematical expression involving propagators, vertices, and external lines, summed to give the S-matrix element.

Key Concepts

S-matrix: amplitude for scattering process
Propagator: D(k) = i/(k²-m²+iε) for scalar field
Vertex factor: coupling constant (e.g., -iλ for λφ⁴)
Loop diagrams contribute quantum corrections
Renormalization handles UV divergences in loop integrals

Key Equations

Scalar propagator

D(k) = \frac{i}{k^2 - m^2 + i\varepsilon}

QED photon propagator

D_{\mu\nu}(k) = \frac{-ig_{\mu\nu}}{k^2+i\varepsilon}

QED vertex

-ie\gamma^\mu

Cross section from amplitude

d\sigma = \frac{1}{4E_1 E_2 v_{rel}}|\mathcal{M}|^2 d\Phi

Worked Example

Example Problem

Problem

At lowest order in QED (tree level), the amplitude for electron-positron annihilation to two photons is ℳ ∝ e². Find the cross section scaling with energy.

Solution

|ℳ|² ∝ e⁴ = (4π α)². The cross section σ ∝ e⁴/s where s=(2E)² is the CM energy squared, giving σ ∝ α²/E² ∝ α²/s.

Practice

Exercises

7 problems

1 of 7

The lowest-order QED amplitude for e⁺e⁻→μ⁺μ⁻ goes as α. The cross section σ ∝ α²/s. At E_cm=100 GeV (s=10⁴ GeV²), using α=1/137, find σ in units of 4πα²/s.

Your answer

4πα²/s

2 of 7

The fine structure constant runs: α(M_Z) ≈ 1/128. Find α(M_Z) as a decimal.

Your answer

3 of 7

A loop diagram has a UV divergence regulated by cutoff Λ. The one-loop correction to a mass is δm² = λm²/(16π²) ln(Λ/m). For λ=0.1, Λ=10m, find δm²/m².

Unlock Exercise 3