Scattering in QED

With the QED Feynman rules, we can calculate cross sections for real processes. The paradigmatic examples — e⁺e⁻ annihilation into muons, Compton scattering, Mott scattering — illustrate the complete chain: amplitude from diagrams → spin-average using completeness relations → gamma-matrix traces → differential cross section. This trace technology is the workhorse of all QED and QCD calculations.

Key Concepts

e⁺e⁻ → μ⁺μ⁻: one photon exchange, amplitude ℳ = (−ie)²ū(k)γμv(k̄) (−igμν/q²) v̄(p̄)γνu(p)
Spin average: |M̄|² = ¼ Σ_{spins} |ℳ|² via spin-sum ΣuūΣvv̄ = (p̸₁+m₁)(p̸₂−m₂)
Key traces: tr[γμγν] = 4gμν, tr[γμγνγργσ] = 4(gμνgρσ−gμρgνσ+gμσgνρ)
High-energy result: |M̄|² = 8e⁴(t²+u²)/s² for e⁺e⁻→μ⁺μ⁻ at s≫m²
Compton scattering: two diagrams (s and u channel), Klein-Nishina cross section at low energy → Thomson limit
Optical theorem: Im ℳ(A→A) = 2E·p·σ_tot — unitarity of S-matrix

Key Equations

Spin-averaged amplitude

\overline{|\mathcal{M}|^2}=\frac{1}{4}\mathrm{tr}[(\not\!k+m)\Gamma(\not\!p+m)\bar\Gamma]

e⁺e⁻→μ⁺μ⁻ at high energy

\overline{|\mathcal{M}|^2}\xrightarrow{s\gg m^2}2e^4(1+\cos^2\theta)

Total cross section

\sigma(e^+e^-\to\mu^+\mu^-)=\frac{4\pi\alpha^2}{3s}

Thomson limit

\frac{d\sigma}{d\Omega}\bigg|_{E\to0}=\frac{\alpha^2}{m_e^2}\cos^2\theta=r_e^2\cos^2\theta

Worked Example

Example Problem

Problem

At √s = 10 GeV, compute σ(e⁺e⁻→μ⁺μ⁻) = 4πα²/3s with α = 1/137. Express in nanobarns (1 GeV⁻² = 0.3894 nb).

Solution

s = 100 GeV². σ = 4π/(3×137²×100) GeV⁻² = 4π/5629800 = 2.228×10⁻⁶ GeV⁻². In nanobarns: 2.228×10⁻⁶ × 0.3894×10⁶ = 0.868 nb. The standard unit is the point cross section σ_pt = 4πα²/3s ≈ 0.87 nb at 10 GeV.

Key Takeaways

QED cross sections follow the chain: Feynman amplitude → |ℳ|² → spin average → trace over γ matrices → integrate phase space
Trace identities for γ matrices reduce spin-summed amplitudes to Lorentz-invariant dot products of external momenta
The e⁺e⁻→μ⁺μ⁻ total cross section σ = 4πα²/3s falls as 1/s — measured to high precision at LEP and lower-energy colliders
The R-ratio R = σ(e⁺e⁻→hadrons)/σ(e⁺e⁻→μ⁺μ⁻) = Σ_f e²_f × N_c counts quark flavors and colors — key evidence for QCD