Important Feynman Diagrams

Why This Matters

Feynman diagrams aren't just pretty pictures—they're the computational backbone of quantum field theory. When you're working through QFT, you're being tested on your ability to translate physical processes into these diagrams and then extract measurable quantities like scattering amplitudes and cross-sections. The diagrams encode everything: conservation laws, coupling strengths, virtual particle exchanges, and the perturbative structure of interactions. Master these, and you've got a visual language for the entire Standard Model.

Here's the key insight: every diagram represents a term in a perturbation expansion, and higher-order diagrams (those with loops) give quantum corrections that distinguish QFT from classical physics. Don't just memorize what each diagram looks like—understand what physical process it represents, what conservation laws constrain it, and whether it's a tree-level or loop-level contribution. That conceptual framework will carry you through any problem.

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Tree-Level Scattering Processes

These are the simplest Feynman diagrams with no internal loops. They represent the leading-order contributions to scattering amplitudes and are your starting point for any calculation. Tree-level diagrams give classical-like results; quantum corrections come from loops.

Electron-Positron Annihilation

• Two photons produced in the simplest case—single-photon production is forbidden by momentum conservation in the center-of-mass frame
• Vertex structure shows electron and positron lines meeting at a point where they convert entirely to electromagnetic radiation
• Conservation laws (energy, momentum, charge, lepton number) are all satisfied and easily verified from the diagram topology

Compton Scattering

• Photon scatters off electron with momentum transfer, demonstrating light's particle-like behavior in the quantum regime
• Two contributing diagrams at tree level—the photon can be absorbed then re-emitted, or vice versa (s-channel and u-channel)
• Klein-Nishina formula for the cross-section derives directly from summing these diagram contributions

Bhabha Scattering

• Electron-positron elastic scattering with two distinct diagram types: photon exchange (t-channel) and annihilation-creation (s-channel)
• Interference between channels produces the characteristic angular distribution used to test QED precision
• Standard candle process at $e^+e^-$ colliders for luminosity measurements due to its clean, calculable signature

Møller Scattering

• Electron-electron scattering mediated by virtual photon exchange between identical fermions
• Exchange symmetry requires including both direct and exchange diagrams, with a relative minus sign from Fermi statistics
• Foundational QED process that demonstrates how the Pauli exclusion principle manifests in scattering amplitudes

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Pair Creation and Annihilation

These diagrams show matter-energy conversion at the quantum level. They're related by crossing symmetry—rotate a diagram and you transform one process into another. This symmetry is a powerful calculational tool in QFT.

Pair Production

• High-energy photon converts to $e^+e^-$ pair when interacting with a nucleus or another photon (required for momentum conservation)
• Threshold energy of $E_\gamma \geq 2m_e c^2 \approx 1.022 \text{ MeV}$ follows directly from $E = mc^2$
• Crossing-related to annihilation—the same vertex appears, just with external lines reinterpreted as incoming vs. outgoing

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Propagators: The Internal Lines

Propagators describe how particles travel between interaction vertices. They're the mathematical objects assigned to internal lines in diagrams and encode mass, spin, and causal structure. Without propagators, you can't write down amplitudes.

Photon Propagator

• Massless spin-1 propagator takes the form $\frac{-i g_{\mu\nu}}{q^2 + i\epsilon}$ in Feynman gauge, where $q$ is the four-momentum
• $1/q^2$ behavior produces the familiar Coulomb potential in the non-relativistic limit
• Gauge dependence means different gauge choices give different propagator forms, but physical observables remain unchanged

Fermion Propagator

• Spin-1/2 propagator is $\frac{i(\not{p} + m)}{p^2 - m^2 + i\epsilon}$, incorporating both mass and the Dirac structure
• Pole at $p^2 = m^2$ corresponds to on-shell particles; off-shell (virtual) fermions can have any four-momentum
• Connects vertices in any diagram involving electrons, quarks, or other fermions—you'll use this constantly

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Loop Corrections and Renormalization

Loop diagrams contain closed paths of virtual particles and represent quantum corrections to tree-level results. They're where QFT gets interesting—and where infinities appear that require renormalization. These corrections give QED its famous precision.

Electron Self-Energy

• Electron emits and reabsorbs a virtual photon, creating a loop that modifies the electron's effective mass and wavefunction
• Divergent integral requires renormalization—the bare mass is adjusted to give the observed physical mass
• Mass renormalization is conceptually crucial: the "dressed" electron you observe differs from the "bare" electron in the Lagrangian

Vacuum Polarization

• Virtual $e^+e^-$ pair briefly appears in the photon propagator, screening the effective charge at large distances
• Running coupling constant $\alpha(q^2)$ increases at higher energies (shorter distances) due to this effect
• Modifies Coulomb potential at short range—experimentally verified in precision atomic physics (Lamb shift)

Vertex Correction

• Loop correction to the QED vertex where a virtual photon connects the incoming and outgoing electron lines
• Anomalous magnetic moment of the electron ($g-2$) arises from this correction—one of physics' most precise predictions
• Ward identity relates vertex corrections to self-energy, ensuring gauge invariance is preserved after renormalization

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Quick Reference Table

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Self-Check Questions

1. Which two diagrams contribute to Compton scattering at tree level, and how do they differ in their time-ordering of photon absorption and emission?

2. Explain why single-photon electron-positron annihilation is forbidden. Which conservation law would be violated?

3. Compare the photon and fermion propagators: how does the presence or absence of mass affect their mathematical structure?

4. If asked to calculate the anomalous magnetic moment of the electron, which loop diagram provides the leading correction, and why does it modify $g$ from its tree-level value of 2?

5. Vacuum polarization and electron self-energy are both one-loop corrections. Which propagator does each modify, and what physical quantity does each renormalize (mass, charge, or wavefunction)?