The Renormalization Group

Motivations

• Some among the motivation of the renormalization group are the study of the asymptotic behaviour of Green functions and the summation of large logarithms occurring in perturbation theory. Describe such problems with a concrete example.
• Another motivation is the construction of an effective theory, valid at energy scales much lower than the cut-off associated with the underlying fundamental theory. How does this effective theory look like? (Consider the example of a scalar theory.)

Evolution equations

• Derscribe the naive scaling law of a proper Green function in a massless theory.
• Write the renormalization conditions characterized by an arbitrary euclidean scale M for a scalar theory.
• Write the relation between bare and renormalizable proper Green functions. Derive the homogeneous Callan-Symanzik equation.
• By making use of dimensional analisys and the CS equation derive an equation for the asymptotic behaviour of proper Green functions. Write the general solution and discuss its physical implications.
• How the previous results look like in dimensional regularization with the MS scheme?
• What is a fixed point of the renormalization group equations? What kinds of fixed points can we find in a quantum field theory with one coupling constant? What are the fixed points that we can study in perturbation theory?

Explicit examples

• Is the renormalizable scalar theory asymptotically free? What is the one-loop anomalous dimension of the scalar field in such a theory?
• Derive the general expression for the beta function in QED in the MS scheme of dimensional regularization. Do you remember the one-loop approximation to this result?
• Which among the Quantum Field Theories are asymptotically free?
• The beta function of QCD in the one-loop approximation has two terms with opposite signs. Can you trace back the origin of these two terms by making a list of the diagrams contributing to wave function renormalizations and to the quark-quark-gluon vertex?
• Up to which value of N_f (the number of quark copies) is QCD asymptotically free?
• Draw a plot displaying the dependence of the running coupling constants of QED and QCD on the energy E'=e^tE. Assume E=m_e in QED and E=m_p (proton mass) in QCD. In which regions of E' is the perturbative expansion reliable for the two theories and why?