Beyond the tree level approximation

Explain the properties of the one-loop contribution to the vacuum polarization.
How is the divergence removed?
What are the renormalization conditions?
Provide some example of renormalization schemes.
Discuss the relation between renormalized coupling constant "e" and the physical electric charge "e_phys". Is this relation universal or does it depend on the renormalization scheme?
Describe the physical consequences of the vacuum polarization correction. Why the energy levels of the hydrogen atom are shifted compared to Dirac's result? What does it mean running coupling constant?
What is the Landau pole?

Define the generating functional of the connected Green function.
What is a 1PI diagram?
Define the effective action. How can you represent graphically a connected Green function in terms of 1PI functions?
What is the expansion parameter of the loop expansion?
Why is the generating functional of 1PI functions termed effective action? What is the tree-level approximation to the effective action?

In classical electrodynamics the electromagnetic current is conserved. What is the counterpart of this property for QED? (I mean properties of Green functions where the operator associated to the electromagnetic current is inserted.)
Write the functional identity satisfied by the effective action in QED as a consequence of gauge invariance.
Derive the identities satisfied by 1PI functions with n photon legs.

Write the most general parametrization of the electron-photon-electron vertex with on-shell electrons. Discuss the consequences of the conservation of the electromagnetic current and of the existence of a conserved charge.
Discuss the properties of the 1-loop contribution to the form factors F(q²) and G(q²) from the vertex diagram: finiteness, gauge dependence, Schwinger term.
Is F(0)+G(0)=1 satisfied by the contribution of the vertex diagram alone?
Discuss the wave function renormalization of the Dirac field. How does it contribute to F and G?
Write the relation between bare and renormalized electric charge and explain what does it mean universality of the electric charge.
Derive the relation Z₁=Z₂ starting from the functional identity expressing the gauge invariance of the effective action.

Ricapitulate the 1-loop corrections to the elastic electron muon scattering when -q² is much smaller than the muon mass squared, in terms of the form factors F₁ and F₂. Which form factor is affected by the infrared divergence? Explain in simple terms the origin of this divergence.
Discuss the emission of a soft photon from an electron leg. How is the cross-section for the elastic scattering modified? Try to remember the result for -q² much larger than the electron mass squared.
How do the infrared divergences cancel in the physical process? How is defined the measured cross-section? What is the Sudakov form factor?
List all the renormalization conditions defining the on-shell scheme in QED.