Renormalization Group

• Write the naive scaling law of an n-point vertex function, for the scalar massless theory, in momentum space. Why this behaviour is not correct in the quantum field theory?
• Write the correct scaling law in differential form.
• Write the relation between the unrenormalized and the renormalized vertex functions and derive the Callan-Symanzik equation.
• Compare the solution of the Callan-Symanzik equation and the behaviour expected on the basis of a naive scaling.
• What is the anomalous dimension of a field? What is the running coupling constant?
• What are the fixed points and how are they classified? How do they caracterize the asymptotic behaviour of a vertex function? Which fixed point can we study by using perturbation theory?
• Derive the beta function of QED in the one-loop approximation, starting from the expression of the photon self-energy in dimensional regularization.
• Explain the difference between QCD and QED in the determination of the one-loop beta function. Which among the quantum field theories you know are asymptotically free?
• Why we say that the typical scale in QCD is about 1 GeV? Would you be able to predict the mass scale of light hadrons by the knowledge of the strong coupling constant measured at high energies?