course ID





14 Weeks

Semester DD


Course details

1. Introduction to group-theory, [5]
• non-abelian Lie groups;
• representations of a group, the exponential map and the generators of the algebra;
• structure constants and the adjoint representation;

2. Non-abelian gauge theories, [2, 7]
• the geometric origin of the covariant derivative;
• gauge-invariant quantities built using the parallel transporter;
• the gauge field and its transformation rule;
• gauge field action starting from Wilson-loops;
• Haar measure;
• covariant quantization of a non-abelian gauge theory with the Faddeev-Popov method: the ghosts;

3. Chiral gauge theories
• left and right fermions with different quantum numbers;
• gauge-invariant connection of the left and right worlds, phi_LR;
• gauge-invariant quantities built with phi_LR;
• the action for the phi_LR field;

4. The Standard Model of Fundamental Interactions, [2, 8]
• matter fields in the SM: leptons and quarks;
• gauge symmetries of the SM: SU(3) × SU(2)L × U(1)Y ;
• the quantum numbers of the matter fields;
• phi_LR within the SM: the Higgs field;
• Majorana spinors as possible extensions of the SM in the neutrino sector;
• the Yukawa interactions within the SM;
• spontaneous symmetry breaking within the SM;
• the quark mass terms and the CKM matrix;
• CP violation within the SM;
• focus on symmetry breaking: the Goldstone theorem;

5. Introduction to hadronic physics, [1]
• the flavour quantum numbers of QCD;
• classification of hadrons in terms of quantum numbers;
• non-perturbative calculation of the mass of an hadron by starting from a two-point correlator;

6. The renormalization group, [2, 3]
• an example in lambda phi^4: physics must not depend from the observables used to fix the parameters of the theory;
• auxiliary renormalization schemes: at intermediate levels of the calculations there is some freedom to define finite quantities;
• renormalization group equations as conditions on the invariance of physics from the choice of the renormalization scale;
• the MS scheme;
• beta–functions in the MS scheme;
• analysis of UV divergences in QED by using Ward identities;
• explicit calculation of the QED beta–function;
• generalization to the case of QCD: asymptotic freedom;

Suggested books
[ 1] S. Weinberg, “The Quantum theory of fields. Vol. 1: Foundations,”
[ 2] S. Weinberg, “The quantum theory of fields. Vol. 2: Modern applications,”
[ 3] J. Zinn-Justin, “Quantum field theory and critical phenomena,” Int. Ser. Monogr. Phys. 113 (2002) 1.
[ 4] A. Duncan, “The conceptual framework of quantum field theory”
[ 5] H. Georgi, “Lie algebras in particle physics,” Front. Phys. 54 (1999) 1.
[ 6] C. Itzykson and J. B. Zuber, “Quantum Field Theory,” New York, Usa: Mcgraw-hill (1980) 705 P.(International Series In Pure and Applied Physics)
[ 7] L. H. Ryder, “Quantum Field Theory,”
[ 8] M. D. Schwartz, “Quantum Field Theory and the Standard Model,”
[ 9] S. Coleman, “Aspects of Symmetry : Selected Erice Lectures,” doi:10.1017/CBO9780511565045
[10] R. G. Newton, “Scattering Theory Of Waves And Particles,” New York, Usa: Springer ( 1982) 743p


The students will have an advanced knowledge of:
- the non-abelian field theories;
- the modern theoretical tools such as Ward Identities and Dyson-Schwinger equations;
- the renormalization group;
- the Standard Model of fundamental interactions;
The students will have also a good knowledge of hadronic physics and will be introduced to non-perturbative methods in quantum field theory.

At the end of the class the students should be able to:
- study in full details a quantum field theory;
- apply advanced theoretical tools such as Ward Identities and the renormalization group;
- be able to analyze in details a new quantum field theory;
- be able to understand the details of new researches in the field of theoretical particle physics;

At the end of the class the students should be able to:
- calculate cross-sections in the Standard Model at low orders of covariant perturbation theory;
- analyze a quantum field theory at the level of details needed to provide a solid theoretical interpretation of recent experimental measurements in particle physics;

At the end of the class the students should be able to:
- judge the complexity of a research project in theoretical particle physics;
- find all the bibliography needed to make a feasibility study of a new research project and to do a new theoretical calculation in particle physics;
- to judge the relevance and originality of a new result in theoretical particle physics;

At the end of the class the students should be able to communicate their knowledge in theoretical particle physics:
- clearly, at the required level of details, correctly;
- in such a way to be comprensible to collegues and expert researchers in theoretical and experimental particle physics;

At the end of the class the students should be able to understand by themselves the technical subleties and the phenomenological implications of new theories in the field of theoretical particle physics.