Theory :

(1) Scattering theory

• definition of in/out states

• Kallen-Lehmann spectral decomposition

• LSZ reduction formulas

• optical theorem

(2) Spinors and symmetries 

• Clifford algebra and spinorial representations of the Lorentz group

• Dirac, Weyl and Majorana spinors

• P, C, T discrete symmetries 

(3) QFT beyond tree-level: 

• regularization and renormalization

• renormalization of QED

• Ward identities

• anomalous magnetic moment of the electron

• beta function and running coupling

Applications :

(1) Bhabha scattering (e+e- → e+e-)

(2) Parity violation and neutron polarization in beta decay (n → p e- 𝜈̅e)

(3) Muon decay (𝜇- → e- 𝜈̅e 𝜈𝜇) 

(4) Pion decay (𝜋- → 𝜇- 𝜈̅𝜇) 

(5) Pion form factor determination from e-𝜋+ → e-𝜋+ and e+e- → 𝜋+𝜋- scatterings

(6) Anomalous electron magnetic dipole moment (g-2)

Summer session

• Written exam: June 18 in Aula Amaldi

Oral exam:

• Written exam: July 9 in Aula Amaldi

Oral  exam: 

Fall session

• Written exam: September 10 in Aula Amaldi

Oral exam: 

Sessione straordinaria 

[riservata alle categorie di studenti previste dal regolamento di ateneo (elenco)]

• Written exam: November 13 in Aula 4 "Giustina Baroni"

Oral exam:

The exam consists of a written and an oral session.

The written session typically consists in the calculation of a cross section or a decay width. Books, notes and calculators are not allowed during the written exam, this formula sheet with all the needed formulas will be distributed to students before starting the exam. The results of the written exam will be published on classroom a few days after, students admitted to the oral session will be able to choose a date from an excel file that will be published together with the results. 

The oral session covers the programme of the course and is done at the blackboard. In particular, students should be able to give a logical derivation of the results.

The evaluation criteria include:

- the correctness of the arguments reported in the written and oral sessions

- the ability in solving the written session

- the clarity and the mathematical rigour in the oral session

The final score is the result of a global assessment of the written and oral sessions.

1. 2/03/2025 [3h]: [Contino] Introduction to the course. Particles as a new phenomenon: the particle revolution from 1890's to 1960's (very brief summary, see slides). Scattering theory: assumptions on the theory, vacuum and 1-particle states.

2. 4/03/2025 [2h]: [Contino] Wave packets and multi-particle states; asymptotic states as eigenstates of smeared energy and momentum operators, asymptotic condition. Description of a scattering experiment: Born rule for scattering probability, Heisenberg picture and Schroedinger picture; scattering operator and the S matrix. [PS], [W]

3. 9/03/2025 [3h]: [Contino] In and out states, asymptotic completeness and decomposition of asymptotic states. Cross section: classical definition in the rest frame of the target, formula in a generic frame, differential cross section; definition in a quantum theory,  formula in terms of n-body phase space and Lorentz-invariant scattering amplitude. [T] sec. 3-c and 3-d, [LL] sec. 2.12, [M] sec. 6.3, [PS]

4. 11/03/2025 [2h]: [Vittorio] Kinematics: decay rates and cross sections in the general case of an arbitrary number of final-state particles. Classification of interesting cases: (i) 2-body decays, (ii) 2-to-2 scatterings and (iii) 3-body decays. General discussion on case (i): definition of rest frame of the decaying particle; computation of the energies and of the 3-momenta of the final-state particles; simplied expressions for the 2-body phase space and for the differential decay width. General discussion on case (ii): definition of center-of-mass frame; Mandestal variables; analysis of the ranges of values that s and t can assume (physical region); simplified expression for the differential cross section. Discussion on the Lorentz-invariance of the flux factor. Introduction to case (iii). [PS]

5. 16/03/2025 [3h]: [Contino] From particles to fields: free vs interacting fields, Kallen-Lehmann spectral decomposition, interpolating fields. [W], [PS]

6. 18/03/2025 [2h]: [Vittorio] General discussion on 3-body decays: analysis of the ranges of values that s and t can assume (physical region); simplified expression for the 3-body phase space and for the differential decay width. Dalitz plots and their connection with dynamics. Introduction to Bhabha scattering (e+e- → e+e-). Discussion on s-channel and t-channel Feynman diagrams and their connection with interference terms. [PS] [Review 49 of the PDG on Kinematics]

7. 23/03/2025 [3h]: [Vittorio] Bhabha scattering (e+e- → e+e-). Computation of the squared amplitude, derivation of the final expression of the differential cross section. Recap of Dirac algebra: traces of an arbitrary number of gamma matrices and contraction identities. Phenomenology: comparison of QED predictions for the differential cross section with the experimental data by TASSO Collaboration. [Chapters 4 & 5 of PS] [Basically we have solved Problem 5.2 of PS]

8. 25/03/2025 [2h]: [Contino] Finite dimensional representations of the Lorentz group: SO(3) vs SU(2), SO(3,1) vs SL(2,C); algebra of SO(3,1) and its finite-dimensional irreps. Chiral fields (1/2,0) and (0,1/2): transformation rules under Lorentz, local Lagrangian and Dirac mass term; Dirac theory in 2-component and 4-component notation, equations of motion, quantized field. [W], [PS], [S]

9. 30/03/2025 [3h]: [Contino] Clifford algebras and spinorial representations of the Lorentz group. Massless Weyl fermions in 2-component notation: quantization and physical interpretation, helicity. Massive Majorana fermions in 2-component notation: quantization and physical interpretation.  [S] and article by Pal.

10. 1/04/2025 [2h]: [Contino] Massive Majorana fermions in 4-component notation: Majorana representation of gamma matrices, reality condition, Majorana mass term, symmetries of the Majorana Lagrangian. Symmetries of the S matrix: the case of rotations, translations, parity and internal symmetries. 

11. 8/04/2025 [2h]: [Contino] Rotational and translational invariance as implication of the existence of a representation of the Poincarè group on the Hilbert space.  Symmetries of the S matrix: passive point of view in terms of a transformed reference frame. Parity: action on the generators of the Poincarè group, unitarity of the parity operator, action on one-particle states, intrinsic parity and its possible values. [W]

12. 13/04/2025 [3h]: [Contino] Transformation of fields under parity (scalar field, Dirac field). Invariance of the QED Lagrangian and of the free Majorana Lagrangian under parity; parity-violating interactions. Physical consequences of parity invariance of the S-matrix: condition on scattering amplitude; [W] example: decay of para-positronium to two photons. [P] sec. 3.4

13. 15/04/2025 [2h]: [Vittorio] Beta decay (n → p e- 𝜈̅e) - part I. Historical introduction. Computation of kinematics, definition of non-relativistic scheme. Dynamics: first proposal of Lagrangian by Fermi (1934). Digression on parity violation in weak decays and its experimental validation through Wu's experiment: V-A structure of the Langrangian as the proper way to describe available data. Computation of the S-matrix element, evaluation of products of (nucleonic) spinors in the non-relativistic limit. Expressions for the amplitudes in the V and A cases. Comment on the absence of interference terms in the case of decay of an unpolarized neutron. [Chapter 10 of BD] [Chapter 15 of MB] [Chapter 6 of CG]

14. 20/04/2025 [3h]: [Vittorio] Beta decay (n → p e- 𝜈̅e) - part II. Computation of the squared matrix element |M|^2 = |M_V|^2 + |M_A|^2. Derivation of the differential decay width and of the total decay width. Fermi-Kurie plot, its relation with the mass of the (anti)neutrino. Spin 4-vector formalism: basic relations, definition of the projectors, structure (without derivation) of the completeness relation for the u spinor. How to infer the numerical value of the ratio CA/CV and its sign from data: i) e- - 𝜈̅e angular correlation, and ii) angular correlation between the momentum of the electron and the spin of the polarized neutron. Numerical determination of CV, and thus of the Fermi constant GF, from the measurement of the neutron lifetime. Introduction to muon decay (𝜇- → e- 𝜈̅e 𝜈𝜇): form of the weak interaction Lagrangian in this case, discussion on the helicities of neutrinos and antineutrinos. [Chapter 10 of BD] [Chapter 15 of MB] [Chapter 6 of CG]

15. 22/04/2025 [2h]: [Vittorio] Muon decay (𝜇- → e- 𝜈̅e 𝜈𝜇) - part I. Dynamics: step-by-step computation of the squared amplitude |M|^2 starting from the expression of the weak interaction Lagrangian. Method #1 for kinematics: computation of the total decay width by explicitly using Mandelstam variables. Method #2 for kinematics: computation of phase space without exploiting Mandelstam variables. [Chapter 10 of BD] [Chapter 15 of MB]

16. 27/04/2025 [3h]: [Contino] Physical consequences of parity invariance: intrinsic parity of  the charged pion, theta-tau puzzle and parity violation in weak decays. Time reversal: action on the generators of the Poincarè group, anti-unitarity of the T operator, action on one-particle states, Kramers' degeneracy; action on fields, transformation of fermion bilinears,  invariance of the QED Lagrangian under time reversal; invariance of the S-matrix under time reversal. [W]

17. 29/04/2025 [2h]: [Contino] Charge conjugation C: action on one-particle states; action on fields (4-component and 2-component notations); invariance of the QED Lagrangian, C violation in theories with Weyl fermions; selection rules from C invariance of the S-matrix; positronium decays. [W], [PS]

18. 4/05/2025 [3h]: [Vittorio] Muon decay (𝜇- → e- 𝜈̅e 𝜈𝜇) - part II. Computation of the differential decay width without exploiting Mandelstam variables. Digression on the effective Lagrangians. Definition of branching ratios, comments on the lepton flavour universality of weak interactions. Pion decay (𝜋- → 𝜇- 𝜈̅𝜇) - part I. Interaction Lagrangian, computation of the amplitude and of its absolute value squared. Discussion on the suppression of the branching ratio of 𝜋- → e- 𝜈̅e w.r.t. the  branching ratio of 𝜋- → 𝜇- 𝜈̅𝜇 due to the difference in the electron and muon masses. Interpretation of this suppression in terms of conservation of total angular momentum and its connection to the helicity of the anti-neutrino. Computation of the squared amplitude |M|^2 through the spin 4-vector formalism in the case of a polarized muon.  [Chapter 10 of BD] [Chapter 15 of MB]

19. 6/05/2025 [2h]: [Vittorio] Pion decay (𝜋- → 𝜇- 𝜈̅𝜇) - part II. Case of a polarized muon: how to write down the spin 4-vector in the case of a polarization vector (anti-)parallel to the direction of motion. Vanishing of the total squared amplitude |M|^2  when the polarization vector of the muon is chosen to be anti-parallel to its 3-momentum vector. Introduction to e-𝜋+ → e-𝜋+. Case #1: pion as a point-like object. Derivation of the structure of the interaction Lagrangian and of the hadronic 4-current J𝜇, discussion on the form of the matrix element of J𝜇 between one-pion states. Case #2: pion as a composite object. Derivation of the expression of the matrix element of J𝜇 between one-pion states, appearance of the electromagnetic form factor of the pion Fπ(q2). Derivation of the charge conservation condition Fπ(q2 = 0) = 1, discussion on its physical meaning. [Chapter 10 of BD]

20. 11/05/2025 [3h]: [Contino] Implication of invariance under C: Furry's theorem. CP: rules of transformation, condition for the invariance of the Lagrangian (example of Fermi's theory). CPT: rules of transformation, conditions for the invariance of the Lagrangian (CPT theorem), implication of CPT invariance on the physical spectrum. [W], [MB] sec. 12.6 (no 12.6.1)  [BD2] sec. 15.14.   LSZ reduction formula: sketch of proof using wave packets - part I. [PS]

21. 13/05/2025 [2h]: [Contino] LSZ reduction formula: sketch of proof using wave packets - part II. Crossing relation. Optical theorem. [PS]

22. 18/05/2025 [3h]: [Contino] Lorentz invariance of Green functions and the S-matrix. Computing Green functions and the S-matrix in perturbation theory: interaction picture (review); master formula for the perturbative calculation of Green functions; Wick's theorem and diagrammatic approach (review). [PS]

23. 20/05/2025 [2h]: [Vittorio] e-𝜋+ → e-𝜋+ scattering: computation of the squared amplitude |M|^2 and of the differential cross section. Comments on how to extract information on the electromagnetic form factor of the pion from experimental data on e-𝜋+ → e-𝜋+ scattering. Parenthesis on kinematics: the Mandelstam variable t is always negative for elastic 2-to-2 scatterings. e-e+ → 𝜋-𝜋+ scattering: application of crossing symmetry to derive the form of the hadronic matrix element, computation of the squared amplitude |M|^2 and of the total cross section. Definition of spacelike and timelike regions for the form factor. Extras: comments on the importance of a parametrization of the form factor to determine its shape in the whole kinematical region. Discussion on the Vector Meson Dominance (VMD) ansatz and its phenomenological interpretation. Derivation of the dispersion relation for the the electromagnetic form factor of the pion through analiticity and unitarity (=first principles of QFT!). [Chapter 10 of BD]

24. 25/05/2025 [3h]: [Contino] Computing Green functions and the S-matrix in perturbation theory: exponentiation and cancellation of vacuum diagrams; S-matrix elements in terms of (fully connected) amputated Green functions; resummed propagator, conditions for pole and residue. Introduction to radiative corrections: locality and counterterms, renormalization through a redefinition of Lagrangian parameters. [PS] Dimensional regularization: axiomatic definition and properties (quick intro). [Co] chap. 4

25. 27/05/2025 [2h]: [Contino] Power counting divergences: superficial degree of divergence, primitively divergent diagrams; classification of theories as renormalizable, super renormalizable and non-renormalizable; dimensionality of counterterms. General strategies for renormalization, renormalization conditions and renormalization scale. Primitively divergent diagrams in QED. [PS], [W] chap. 12

26. 3/06/2025 [2h]: [Contino] Electron self-energy: decomposition in terms of Passarino-Veltman integrals B0 and B1, calculation of B0 in dimensional regularization; resummed propagator,  physical electron mass and on-shell renormalization condition. [Ry] sec. 9.5, [PS] sec. 7.1, [W] sec. 11.4

27. 4/06/2025 [3h]: [Contino] Electron self-energy: residue of the resummed propagator at the physical pole, infrared divergence and its physical interpretation; counterterm Lagrangian, renormalization in terms of renormalized Lagrangian parameters and fields, minimal subtraction schemes (MS and MSbar) , derivation of Z2 and Zm in the MSbar scheme, physical electron mass in terms of the renormalized mass and coupling. Photon-electron vertex: off-shell computation of the divergent part in dimensional regularization, renormalization through the vertex counterterm, derivation of Z1 in the MSbar scheme. [Ry] secs. 9.5, 9.6

28. 8/06/2025 [3h]: [Contino] Ward identities: general form of WI identities with one insertion, derivation of the WI for the QED vertex and its physical implications; general form of WI identities with more insertions, WI for the two-point current Green function. [W] sec. 10.4.  Photon self-energy: overview of the calculation at 1-loop in dimensional regularization, check of Ward identity, resummed propagator and pole residue. Running coupling: beta-function and RG equation, 1-loop solution,  Landau pole; q^2-dependent coupling and its explicit form in the high-energy limit, vacuum polarization, QED as an effective theory. [Ry], [PS], [W]

29. 9/06/2025 [2h]: [Vittorio] Magnetic moment of the electron: quick recap of the O(⍺0) contribution, extension to higher-orders. Derivation of the general structure of the 4-current and definition of the form factors F1(q2), F2(q2) (derivation as well of the Gordon identity). Quick recap on Feynman parameters. Part I of the computation of the vertex correction: separation of divergent and convergent parts, comparison of this structure with what obtained before by symmetry arguments [PS Chapter 6] [CMB Chapters 11 & 12]

30. 10/06/2025 [2h]: [Vittorio] Part II of the computation of the vertex correction: evaluation of F2(0) as the O(⍺) correction to the Landé-factor g. Final result of this detailed computation: F2(0) = ⍺ / (2𝜋). Extras: current state-of-the-art of the g-2 of the muon. Discussion of different types of contributions: QED, EW, HVP, HLbL. Several comments on the impact of our knowledge of the electromagnetic form factor of the pion on the precise estimate of the HVP contribution to the g-2 of the muon. Discussion on possible effects of New Physics: exotic contributions to the g-2, how to derive constraints on the parameter space of a specific New Physics model. Importance of the complementarity of this information with bounds coming from other decay channels / physical observables [PS Chapter 6] [CMB Chapters 11 & 12]

Books (main references):

• [PS] M. Peskin, D. Schroeder, An Introduction to Quantum Field Theory, Perseus Books

• [W] S. Weinberg, The Quantum Theory of Fields, vol.1, Cambridge Univ. Press

More books:

• [BD] J.D. Bjorken, S.D. Drell, Relativistic Quantum Mechanics, McGraw-Hill 

• [BD2] J.D. Bjorken, S.D. Drell, Relativistic Quantum Fields, McGraw-Hill 

• [Ry] L. H. Ryder, Quantum Field Theory, Cambridge (2nd edition)

• [S] M. Schwartz, Quantum Field Theory and the Standard Model, Cambridge

• [T] J. R. Taylor, Scattering Theory, Dover

• [LL] L. D. Landau, E. M. Lifshitz, Course on Theoretical Physics vol. II: The classical theory of fields, Pergamon Press, Oxford.

• [M] M. Maggiore, A modern introduction to quantum field theory, Oxford

• [MB] L. Maiani, O. Benhar, Relativistic Quantum Mechanics: An Introduction to Relativistic Quantum Fields, CRC Press

• [CG] R. Cahn, G. Goldhaber, The Experimental Foundations of Particle Physics 2nd Edition, Cambridge University Press

• [P] D. H. Perkins, Introduction to High Energy Physics, Cambridge (4th edition)

• [Co] J. Collins, Renormalization, Cambdridge

• [CMB] N. Cabibbo, L. Maiani, O. Benhar, An Introduction to Gauge Theories, CRC Press (1st edition)

Articles:

On Dirac, Weyl and Majorana fermions:

• P. B. Pal, Am. J. Phys. 79 (2011) 485, arXiv:1006.1718.

Lecture notes:

• On the derivation of a simplified expression for the 3-body phase space [file ]

• On Dirac, Weyl and Majorana fermions [ file]

• On the symmetries of the S-matrix [file]

• On Ward identities [file]

Further material:

• Slides on the Particle Revolution from 1890's to 1960's.

• Slides on some examples of Dalitz plots

• Review 49 of the PDG on Kinematics

• Slides on the comparison between QED predictions for Bhabha scattering and the experimental data by TASSO Collaboration

• Slides on the electromagnetic form factor of the pion 

• Slides on the state-of-the-art of theoretical computations and measurements of the g-2 of the muon

Registrations :

• Video recording (and audio) concerning the first lesson on the g-2 of the electron (Tuesday 9th, June)