Part III, Chapter 6

More on Perturbation Theory

Connected diagrams, vacuum bubbles, and systematic QFT calculations

🔗Course Connections

📚Prerequisites

Feynman Diagrams

Diagrammatic expansion and Wick's theorem

Dyson series and perturbative expansion

▶️

Video Lecture

Lecture 12: More on Perturbation Theory and Feynman Diagrams - MIT 8.323

Advanced perturbation theory techniques and the linked cluster theorem (MIT QFT Course)

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

6.1 Connected vs Disconnected Diagrams

A Feynman diagram is connected if you can reach any part from any other part by following lines. Otherwise, it's disconnected.

💡Physical Interpretation

Connected diagrams: All particles actually interact with each other. This is the "interesting" physics!

Disconnected diagrams: Independent processes happening simultaneously. Example: Two particles scatter in one region while two others scatter far away.

For scattering amplitudes, we only care about connected diagrams - disconnected ones factor out and cancel in properly normalized S-matrix elements!

The generating functional decomposes as:

$$Z[J] = e^{iW[J]} \quad \Rightarrow \quad W[J] = -i\ln Z[J]$$

W[J] generates only connected diagrams! This is the linked cluster theorem.

6.2 Vacuum Bubbles

Vacuum bubbles are diagrams with no external lines - they're disconnected from the rest of the diagram and represent vacuum-to-vacuum fluctuations.

$$Z[0] = \langle 0|0\rangle = \int \mathcal{D}\phi \, e^{iS[\phi]} = e^{i\Gamma_{\text{vac}}}$$

where Γ_vac is the sum of all vacuum bubble diagrams. Key fact: Vacuum bubbles cancelin normalized correlation functions!

$$\langle 0|T\{\phi(x_1)\cdots\phi(x_n)\}|0\rangle = \frac{1}{Z[0]}\frac{\delta^n Z[J]}{\delta J(x_1)\cdots\delta J(x_n)}\Bigg|_{J=0}$$

The Z[0] in the denominator exactly cancels all vacuum bubble contributions! This is crucial - we don't have to compute them.

6.3 Linked Cluster Theorem

The linked cluster theorem states:

Linked Cluster Theorem

For properly normalized S-matrix elements and correlation functions:

Only connected diagrams contribute
Vacuum bubbles cancel in the normalization
Disconnected diagrams factor into products of connected pieces

This dramatically simplifies calculations - we only compute connected, non-vacuum diagrams!

Mathematically:

$$\langle f|S - \mathbb{1}|i\rangle = \text{(sum of connected, amputated diagrams)}$$

6.4 One-Particle Irreducible (1PI) Diagrams

A diagram is one-particle irreducible (1PI) if it cannot be split into two pieces by cutting a single internal line.

1PI Diagrams

• Cannot be split by cutting one line
• "Strongly connected"
• Build more complex diagrams
• Generate effective vertices

Non-1PI Diagrams

• Can be split by cutting one line
• Factorize into simpler pieces
• Chain of 1PI diagrams
• Sum up to full propagator

The effective action Γ[φ] generates 1PI diagrams:

$$\Gamma[\phi] = W[J] - \int d^4x \, J(x)\phi(x)$$

where φ(x) = δW/δJ is the classical field. Γ[φ] is the quantum effective action- the Lagrangian with all quantum corrections!

6.5 Complete Example: φ⁴ Scattering

Let's compute 2 → 2 scattering in φ⁴ theory at leading order:

Setup

• Initial state: |i⟩ = |p₁, p₂⟩ (two particles with momenta p₁, p₂)
• Final state: |f⟩ = |p₃, p₄⟩ (two particles with momenta p₃, p₄)
• Interaction: ℒ_int = -(λ/4!)φ⁴
• Goal: Compute scattering amplitude M_fi

Tree Level (Order λ)

At lowest order, there's one diagram: a single 4-point vertex connecting the external lines.

$$i\mathcal{M} = -i\lambda \times (2\pi)^4\delta^4(p_1 + p_2 - p_3 - p_4)$$

The delta function enforces energy-momentum conservation. The amplitude is:

$$\boxed{\mathcal{M} = -\lambda}$$

That's it! At tree level, the scattering amplitude is just the coupling constant.

One-Loop Level (Order λ²)

At next order, we have diagrams with 2 vertices and one internal loop. These give quantum corrections to the tree-level result:

$$\mathcal{M}_{\text{1-loop}} = -\lambda + \delta\mathcal{M} + O(\lambda^3)$$

where δM contains loop integrals (which are UV divergent and need renormalization!).

6.6 From Amplitudes to Cross Sections

The differential cross section for 2 → 2 scattering is:

$$\boxed{\frac{d\sigma}{d\Omega} = \frac{1}{64\pi^2 s}|\mathcal{M}|^2}$$

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

For our φ⁴ example at tree level:

$$\frac{d\sigma}{d\Omega} = \frac{\lambda^2}{64\pi^2 s}$$

This is a measurable quantity! We can compare with experiment to determine λ.

6.7 Systematic Calculation Procedure

Step-by-Step QFT Calculation

Identify process: Specify initial and final states
Draw diagrams: All connected, amputated Feynman diagrams at desired order
Apply Feynman rules: Convert each diagram to a mathematical expression
Compute loop integrals: Use regularization if divergent
Sum all diagrams: Add contributions from all diagrams at that order
Renormalize: Absorb infinities into redefined parameters
Calculate observables: Cross sections, decay rates, etc.
Compare with experiment!

🎯 Key Takeaways: Part III Complete!

Connected diagrams: Only these contribute to S-matrix (linked cluster theorem)
Vacuum bubbles: Cancel in normalized correlation functions
1PI diagrams: Building blocks for effective action and full propagator
φ⁴ scattering: M = -λ at tree level (order λ)
Cross sections: |M|² gives measurable quantities
Systematic procedure: Diagrams → Feynman rules → Loop integrals → Observables
Path integrals = Canonical quantization (different formulations, same physics!)
Next: Apply these techniques to real theories (QED, Standard Model)!

📖 What's Next in the QFT Course?

You've mastered the path integral formulation! The next parts of the course will build on this foundation:

• Part IV: Interacting theories - φ⁴, Yukawa, QED processes
• Part V: Gauge theories - Yang-Mills, QCD, Electroweak theory
• Part VI: Renormalization - taming the infinities
• Part VII: Advanced topics - anomalies, instantons, SUSY
• Part VIII: The Standard Model - the crown jewel of QFT!