Part IV, Chapter 6

Radiative Corrections & Renormalization

Beyond tree level: quantum loops, infinities, and how QED makes sense

🔗Course Connections

📚Prerequisites

QFT Part III

Feynman Diagrams

Loop diagrams and UV divergences

▶️

Video Lecture

Lecture 26: Quantum Fluctuations and Renormalization - MIT 8.323

Loop corrections and introduction to renormalization (MIT QFT Course)

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

6.1 Beyond Tree Level: Loop Diagrams

Tree-level calculations give leading-order predictions. But quantum corrections come from loop diagrams where virtual particles circulate!

💡Virtual Processes

Even in vacuum, quantum fields fluctuate. Virtual particle-antiparticle pairs pop in and out of existence constantly!

These fluctuations affect:

Electron mass: Self-energy corrections
Photon propagation: Vacuum polarization
Vertex: Higher-order interactions

6.2 The Problem: UV Divergences

Loop integrals diverge at high momentum (ultraviolet or UV divergences)!

Example - electron self-energy one-loop:

$$\Sigma(p) \sim \int \frac{d^4k}{(2\pi)^4} \frac{1}{k^2(p-k)^2} \sim \int_0^\Lambda \frac{k^3 dk}{k^4} \sim \ln\Lambda$$

This diverges logarithmically as cutoff Λ → ∞! The electron appears to have infinite self-energy from virtual photon clouds.

⚠️ The Crisis

Naive QED gives infinity for:

Electron mass correction: δm ~ ∞
Charge screening: δe ~ ∞
Vertex correction: δΓ ~ ∞

This looked like a fatal flaw in QFT! But renormalization saves the day...

6.3 The Solution: Renormalization

Key insight: We never measure "bare" parameters m₀, e₀. We measure renormalized(physical) mass m_phys and charge e_phys!

\begin{align*} m_{\text{phys}} &= m_0 + \delta m \\ e_{\text{phys}} &= e_0 + \delta e \end{align*}

If m₀ → -∞ and δm → +∞ in just the right way, m_phys can be finite! This is renormalization.

Renormalization Procedure

Regularize: Introduce cutoff Λ or use dimensional regularization
Identify divergences: Compute loop diagrams with regulator
Absorb into counterterms: Redefine m₀, e₀ to cancel infinities
Remove regulator: Take Λ → ∞, physical predictions remain finite!

6.4 Renormalized QED Predictions

After renormalization, QED makes finite, testable predictions:

🏆 QED Triumphs

Anomalous magnetic moment:
Theory: (g-2)/2 = 0.001 159 652 181 78 (77)
Experiment: (g-2)/2 = 0.001 159 652 180 73 (28)
Agreement to 10 significant figures!
Lamb shift:
1057 MHz energy difference in hydrogen 2S_1/2 - 2P_1/2
QED loop corrections explain it precisely!
Vacuum polarization:
Running coupling constant α(Q²) measured at different energies
Confirms QED loop predictions!

6.5 Running Coupling Constant

The effective charge depends on energy scale Q due to vacuum polarization:

$$\alpha(Q^2) = \frac{\alpha(0)}{1 - \frac{\alpha(0)}{3\pi}\ln(Q^2/m^2)}$$

At low energies: α(0) ≈ 1/137
At high energies (Z boson mass): α(M_Z²) ≈ 1/128

The coupling "runs" with energy - a purely quantum effect!

🎯 Key Takeaways

Loop diagrams: Virtual particles give quantum corrections
UV divergences: Naive loop integrals are infinite
Renormalization: Absorb infinities into parameter redefinitions
Physical predictions: Finite and extremely accurate!
Running coupling: α(Q²) depends on energy scale
QED success: Most precisely tested theory in physics
This is just the beginning - full renormalization theory in Part VI!

📖 What's Next?

You've completed Part IV! You can now:

• Compute tree-level QED processes
• Calculate cross sections and decay rates
• Understand where loop divergences come from
• Appreciate renormalization's role in QFT

Continue to: Part V (Gauge Theories) or Part VI (Renormalization Theory) for deeper understanding of the mathematical structure that makes QFT work!

Quantum Field Theory Course