Quantum Field Theory Course

Total: 440+ pages covering classical field theory through the Standard Model

Part IV, Chapter 4

Elementary QED Processes I

Computing real physics: e⁺e⁻ annihilation, Compton scattering, pair production

🔗Course Connections

📚Prerequisites

QFT Part IV
QED
Feynman rules for QED
→
QFT Part IV
Cross Sections
Computing observables from amplitudes
→
▶️

Video Lecture

Lecture 24: Elementary Processes in QED (I) - MIT 8.323

Classic QED processes at tree level (MIT QFT Course)

💡 Tip: Watch at 1.25x or 1.5x speed for efficient learning. Use YouTube's subtitle feature if available.

4.1 e⁺e⁻ → μ⁺μ⁻ Annihilation

Electron-positron pairs annihilate into muon pairs via virtual photon exchange. At tree level, one Feynman diagram (s-channel):

$$\mathcal{M} = \frac{-ie^2}{s}[\bar{v}(p_2)\gamma^\mu u(p_1)][\bar{u}(p_3)\gamma_\mu v(p_4)]$$

where s = (p₁ + p₂)² is the center-of-mass energy squared.

After spin averaging/summing:

$$\overline{|\mathcal{M}|^2} = \frac{2e^4}{s^2}(t^2 + u^2)$$

Differential cross section in CM frame:

$$\frac{d\sigma}{d\Omega} = \frac{\alpha^2}{4s}(1 + \cos^2\theta)$$

4.2 Compton Scattering (γe⁻ → γe⁻)

Photon scattering off electron. Two diagrams at tree level (s and u channels):

$$|\mathcal{M}|^2 \propto \frac{s}{u} + \frac{u}{s} + 4m^2\left(\frac{1}{s} - \frac{1}{u}\right) - 2m^4\left(\frac{1}{s} - \frac{1}{u}\right)^2$$

In low-energy limit (Thomson scattering):

$$\frac{d\sigma}{d\Omega} \approx \frac{\alpha^2}{2m^2}(1 + \cos^2\theta) = r_e^2(1 + \cos^2\theta)$$

where re = α/m ≈ 2.8 × 10-15m is the classical electron radius.

4.3 Pair Production (γγ → e⁺e⁻)

Two photons create electron-positron pair. Minimum energy: 2√s ≥ 2me.

At threshold (√s = 2me):

$$\sigma \approx \frac{\pi\alpha^2}{m^2}$$

4.4 Bhabha Scattering (e⁺e⁻ → e⁺e⁻)

Both s-channel (annihilation) and t-channel (scattering) diagrams contribute:

$$\mathcal{M} = \mathcal{M}_s + \mathcal{M}_t$$

Interference between channels gives characteristic angular distribution!

🎯 Key Takeaways

  • e⁺e⁻ → μ⁺μ⁻: dσ/dΩ ∝ α²(1 + cos²θ)/s
  • Compton scattering: Two diagrams (s and u channels)
  • Spin averaging: Sum over final, average over initial spins
  • Trace techniques: Tr(γ-matrices) for spin sums
  • Pair production: Threshold at 2me
  • Next: More complex QED processes!