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Electroweak interactions

A.Y. 2020/2021

Learning objectives

The objective of the course is an introduction to the quantum field theory and to the standard model of particle physics. Special emphasis is on the development of weak interactions theory and to the unification with quantum electrodynamics

Expected learning outcomes

At the end of the course the student:

1.Will know the Klein-Gordon and the Dirac equations as the relativistic generalization of the Schrödinger equation;

2.Will have understood the difficulties of the interpretation of the relativistic equations and will understand the necessity of a quantum field theory

3.Will have learnt to perform cross sections and decay widths calculations for tree level processes

4.Will have understood the connections between the structure of the weak current and the consequent phenomenology as, for example, angular distributions, energy distributions, polarization effects;

5.Will have studied the most important experiments useful for the undertanding of chaged current weak interactions and parity violation;

6.Wiil have understood neutral currents and the unification of electroweak interactions and the spontaneous symmetry breaking mechanism.

1.Will know the Klein-Gordon and the Dirac equations as the relativistic generalization of the Schrödinger equation;

2.Will have understood the difficulties of the interpretation of the relativistic equations and will understand the necessity of a quantum field theory

3.Will have learnt to perform cross sections and decay widths calculations for tree level processes

4.Will have understood the connections between the structure of the weak current and the consequent phenomenology as, for example, angular distributions, energy distributions, polarization effects;

5.Will have studied the most important experiments useful for the undertanding of chaged current weak interactions and parity violation;

6.Wiil have understood neutral currents and the unification of electroweak interactions and the spontaneous symmetry breaking mechanism.

**Lesson period:** First semester
(In case of multiple editions, please check the period, as it may vary)

**Assessment methods:** Esame

**Assessment result:** voto verbalizzato in trentesimi

Course syllabus and organization

### Single session

Responsible

Lesson period

First semester

During COVID-19 emergency, the program of the course will be the same as during the no-emergency condition. Lectures will be delivered remotely and synchronously according to the official schedule of the Physics Course and using the virtual classrooms system of the Physics Department implemented through the Zoom platform.

Assessment method will still consist of an oral exam, still lasting from 45 to 60 minutes, and will take place in one of the virtual classrooms of the Physics Department. The criteria that will be used to assess the preparation of the student will the same as those used in the non-emergency mode.

Assessment method will still consist of an oral exam, still lasting from 45 to 60 minutes, and will take place in one of the virtual classrooms of the Physics Department. The criteria that will be used to assess the preparation of the student will the same as those used in the non-emergency mode.

**Course syllabus**

Relativistic quantum mechanics.

Klein-Gordon's equation. Plane wave solutions of Klein-Gordon's equation.

Negative energy solutions. Probability current. Negative probability.

Relativistic invariance of Klein-Gordon's equation.

Klein's paradox. Quantum mechanics and relativity.

Dirac's equation. Properties of alpha and beta matrices. Pauli-Dirac's representation.

Plane wave solutions of Dirac's equation. Spinors. Negative energy solutions.

Covariant form of Dirac's equation: gamma matrices.

Properties of gamma matrices.

Probability current. Spinor adjont e conserved probability current.

Normalization of spinors.

Negative energy solutions interpretation: Dirac sea model: holes.

Generators of four-vectors Lorentz transformations.

Generators of spinors Lorentz transformations.

Relativistic invariance of Dirac's equation.

Orbital angular momentum non conservation in Dirac's equation.

Intrinsic angular momentum (spin) non conservation.

Conservation of total angular momentum.

Spin relativistic description: helicity.

Spin relativistic description using particle's rest frame. Four vector polarization s.

Completeness relations for spinors.

Electromagnetic interaction trough minimal substitution.

Low energy limit of Dirac's equation.

Coulomb scattering of spin 0 particle. Cross section.

Coulomb scattering of spin ½ particle. Gamma matrices trace techniques.

Trace of gamma matrices products.

Scattering of polarized particles. Extension of gamma matrices trace techniques: spin projectors.

Gamma5 matrix.

Polarization effects in Coulomb scattering. Helicity conservation in the high energy limit.

Introduction to quantum field theory

Fourier analysis of the Klein-Gordon field

The classical electromagnetic field. Coulomb gauge. Electromagnetic waves.

Fourier analysis of the electromagnetic field. Field decomposition in oscillators.

Blackbody radiation. Classical model of blackbody radiation. Ultraviolet catastrophe.

Plank Hypothesis. Quantum distribution of blackbody radiation.

Introduction to quantum field theory.

Quantum harmonic oscillator. Algebraic solution. Creation and annihilation operators.

Number operator and Hamiltonian. Eigenstates of number and of Hamiltonian operators.

Quantization of a scalar real field. Occupation number Fock space.

States normalization. Symmetry of the states and commutation rules.

Number operators and Hamiltonian.

Field operators. Lagrangian for classical field.

Canonical quantization of the real Klein-Gordon field. Calculation of the Hamiltonian.

Noether's theorem. Space-Time invariances and energy-momentum tensor.

Complex Klein-Gordon scalar field. Quantization of the complex Klein-Gordon scalar field.

Global gauge invariance. Hamiltonian and charge operator. Particles and anti-particles.

Quantization of the Dirac field. Hamiltonian and commutation rules.

Global gauge invariance and conserved current. Particles and anti-particles.

Local gauge invariance and Quantum Electrodynamics (QED).

Gupta-Bleuler electromagnetic field quantization.

Interacting fields. Quantum oscillator with anharmonic term.

Scattering and S-matrix formalism. Perturbative expansion and Dyson series.

Observables: cross sections and decay widths.

Coulomb scattering in quantum field theory.

Electromagnetic interaction and fermions scattering. Photon propagator.

Feynman rules.

Two bodies final state kinematics. Two bodies final state phase space.

Photon propagator and gauge invariance.

Weak Interactions

Fermi's theory of neutron beta decay. Current-Current interaction.

Three bodies decay kinematics. Phase space for three bodies final state.

Non relativistic approximation. Nuclear spin variation.

Fermi's theory generalization.

Scalar, vector, axial vector, tensor and pseudo-scalar couplings.

Decay width calculation. Electron energy distribution. Experimental measurements.

Electron-neutrino angular correlations. Allen experiment.

Beta decay lifetime.

Lorentz group. Structure of the Lorentz group.

Spinors inversion. Lorentz transformation of bilinear covariants.

Parity violation. Theta-tau paradox. Dalitz plot. Experiments through a mirror.

Parity and charge symmetry violation in beta decay. CP invariance of beta decay.

Introduction of parity violating terms in the beta decay Hamiltonian.

Electron polarization in beta decay. Electron polarization measurement.

Bargman-Telegdi-Michel formula and electron spin rotation. Frauenfelder experiment.

Beta decay Hamiltonian. Chiral projections.

Chiral projections and zero mass particles.

Chiral projections and anti-particles. Left-handed chiral currents.

Universality of weak interaction. Mu lepton decay.

Weak hadronic current. Pi meson decay.

Electron polarization in pi meson decay

Tau lepton decay. Decay fraction for the pi meson neutrino final state.

Polarized tau lepton decay.

Tau lepton polarization measurement in positron electron collisions at LEP.

Hadrons weak decays.

Isotopic properties of hadronic current: isospin formalism.

Hadronic current matrix elements and weak form factors. Vector hadronic current.

Hadronic electromagnetic current and electromagnetic form factors.

Gordon's electromagnetic current decomposition.

Electromagnetic form factors of proton, neutron and pi meson.

Electromagnetic current and Isospin. Gell-Mann's CVC hypothesis.

CVC hypothesis test: weak beta decay of pi meson.

Weak decays with variation of strangeness.

Selection rules DeltaS = ±1 e DeltaS = DeltaQ.

Cabibbo's hypothesis. Cabibbo's angle.

Neutrino fermion interactions. Neutrino electron and anti-neutrino electron scattering.

Neutrinos quark scattering.

Fermi interaction difficulties and intermediate vector boson W (IVB).

W boson propagator.

Neutrino cross-sections in the IVB model. Difficulties of IVB model.

Divergence of the longitudinally polarized W boson pair cross section at high energies.

Neutrino- nucleon deep inelastic scattering.

Kinematic variables. x and y variables. Bjorken's scaling.

Nucleon parton model.

Neutrino nucleon cross-sections calculation in the parton model. Experimental measurements.

Quark distributions in nucleons measurement.

Electroweak interactions and Standard Model

Discovery of neutral currents. Neutral weak current interactions parametrization.

Vector and axial vector couplings.

Neutral current mediated neutrino quark cross sections.

Quarks vector and axial vector coupling constants measurement.

Neutrino electron scattering and lepton's coupling constants measurement.

Standard Model symmetries. Weak isospin and weak hypercharge.

The SU(2)_L x U(1)_Y group. Weak isospin and weak hypercharge currents.

Local gauge invariance and interaction vertices.

Spontaneous symmetry breaking. Charged currents. Neutral electromagnetic current. Weak neutral current.

Higgs model: complex scalar field, local gauge invariance and electromagnetic field.

Interaction terms and photon mass.

Higgs mechanism in the standard model. Higgs field and spontaneous symmetry breaking.

Masses of the vector bosons Z and W. Coupling constants.

Radiative corrections. Introduction to renormalization.

Electromagnetic propagator renormalization. Lamb's shift.

Klein-Gordon's equation. Plane wave solutions of Klein-Gordon's equation.

Negative energy solutions. Probability current. Negative probability.

Relativistic invariance of Klein-Gordon's equation.

Klein's paradox. Quantum mechanics and relativity.

Dirac's equation. Properties of alpha and beta matrices. Pauli-Dirac's representation.

Plane wave solutions of Dirac's equation. Spinors. Negative energy solutions.

Covariant form of Dirac's equation: gamma matrices.

Properties of gamma matrices.

Probability current. Spinor adjont e conserved probability current.

Normalization of spinors.

Negative energy solutions interpretation: Dirac sea model: holes.

Generators of four-vectors Lorentz transformations.

Generators of spinors Lorentz transformations.

Relativistic invariance of Dirac's equation.

Orbital angular momentum non conservation in Dirac's equation.

Intrinsic angular momentum (spin) non conservation.

Conservation of total angular momentum.

Spin relativistic description: helicity.

Spin relativistic description using particle's rest frame. Four vector polarization s.

Completeness relations for spinors.

Electromagnetic interaction trough minimal substitution.

Low energy limit of Dirac's equation.

Coulomb scattering of spin 0 particle. Cross section.

Coulomb scattering of spin ½ particle. Gamma matrices trace techniques.

Trace of gamma matrices products.

Scattering of polarized particles. Extension of gamma matrices trace techniques: spin projectors.

Gamma5 matrix.

Polarization effects in Coulomb scattering. Helicity conservation in the high energy limit.

Introduction to quantum field theory

Fourier analysis of the Klein-Gordon field

The classical electromagnetic field. Coulomb gauge. Electromagnetic waves.

Fourier analysis of the electromagnetic field. Field decomposition in oscillators.

Blackbody radiation. Classical model of blackbody radiation. Ultraviolet catastrophe.

Plank Hypothesis. Quantum distribution of blackbody radiation.

Introduction to quantum field theory.

Quantum harmonic oscillator. Algebraic solution. Creation and annihilation operators.

Number operator and Hamiltonian. Eigenstates of number and of Hamiltonian operators.

Quantization of a scalar real field. Occupation number Fock space.

States normalization. Symmetry of the states and commutation rules.

Number operators and Hamiltonian.

Field operators. Lagrangian for classical field.

Canonical quantization of the real Klein-Gordon field. Calculation of the Hamiltonian.

Noether's theorem. Space-Time invariances and energy-momentum tensor.

Complex Klein-Gordon scalar field. Quantization of the complex Klein-Gordon scalar field.

Global gauge invariance. Hamiltonian and charge operator. Particles and anti-particles.

Quantization of the Dirac field. Hamiltonian and commutation rules.

Global gauge invariance and conserved current. Particles and anti-particles.

Local gauge invariance and Quantum Electrodynamics (QED).

Gupta-Bleuler electromagnetic field quantization.

Interacting fields. Quantum oscillator with anharmonic term.

Scattering and S-matrix formalism. Perturbative expansion and Dyson series.

Observables: cross sections and decay widths.

Coulomb scattering in quantum field theory.

Electromagnetic interaction and fermions scattering. Photon propagator.

Feynman rules.

Two bodies final state kinematics. Two bodies final state phase space.

Photon propagator and gauge invariance.

Weak Interactions

Fermi's theory of neutron beta decay. Current-Current interaction.

Three bodies decay kinematics. Phase space for three bodies final state.

Non relativistic approximation. Nuclear spin variation.

Fermi's theory generalization.

Scalar, vector, axial vector, tensor and pseudo-scalar couplings.

Decay width calculation. Electron energy distribution. Experimental measurements.

Electron-neutrino angular correlations. Allen experiment.

Beta decay lifetime.

Lorentz group. Structure of the Lorentz group.

Spinors inversion. Lorentz transformation of bilinear covariants.

Parity violation. Theta-tau paradox. Dalitz plot. Experiments through a mirror.

Parity and charge symmetry violation in beta decay. CP invariance of beta decay.

Introduction of parity violating terms in the beta decay Hamiltonian.

Electron polarization in beta decay. Electron polarization measurement.

Bargman-Telegdi-Michel formula and electron spin rotation. Frauenfelder experiment.

Beta decay Hamiltonian. Chiral projections.

Chiral projections and zero mass particles.

Chiral projections and anti-particles. Left-handed chiral currents.

Universality of weak interaction. Mu lepton decay.

Weak hadronic current. Pi meson decay.

Electron polarization in pi meson decay

Tau lepton decay. Decay fraction for the pi meson neutrino final state.

Polarized tau lepton decay.

Tau lepton polarization measurement in positron electron collisions at LEP.

Hadrons weak decays.

Isotopic properties of hadronic current: isospin formalism.

Hadronic current matrix elements and weak form factors. Vector hadronic current.

Hadronic electromagnetic current and electromagnetic form factors.

Gordon's electromagnetic current decomposition.

Electromagnetic form factors of proton, neutron and pi meson.

Electromagnetic current and Isospin. Gell-Mann's CVC hypothesis.

CVC hypothesis test: weak beta decay of pi meson.

Weak decays with variation of strangeness.

Selection rules DeltaS = ±1 e DeltaS = DeltaQ.

Cabibbo's hypothesis. Cabibbo's angle.

Neutrino fermion interactions. Neutrino electron and anti-neutrino electron scattering.

Neutrinos quark scattering.

Fermi interaction difficulties and intermediate vector boson W (IVB).

W boson propagator.

Neutrino cross-sections in the IVB model. Difficulties of IVB model.

Divergence of the longitudinally polarized W boson pair cross section at high energies.

Neutrino- nucleon deep inelastic scattering.

Kinematic variables. x and y variables. Bjorken's scaling.

Nucleon parton model.

Neutrino nucleon cross-sections calculation in the parton model. Experimental measurements.

Quark distributions in nucleons measurement.

Electroweak interactions and Standard Model

Discovery of neutral currents. Neutral weak current interactions parametrization.

Vector and axial vector couplings.

Neutral current mediated neutrino quark cross sections.

Quarks vector and axial vector coupling constants measurement.

Neutrino electron scattering and lepton's coupling constants measurement.

Standard Model symmetries. Weak isospin and weak hypercharge.

The SU(2)_L x U(1)_Y group. Weak isospin and weak hypercharge currents.

Local gauge invariance and interaction vertices.

Spontaneous symmetry breaking. Charged currents. Neutral electromagnetic current. Weak neutral current.

Higgs model: complex scalar field, local gauge invariance and electromagnetic field.

Interaction terms and photon mass.

Higgs mechanism in the standard model. Higgs field and spontaneous symmetry breaking.

Masses of the vector bosons Z and W. Coupling constants.

Radiative corrections. Introduction to renormalization.

Electromagnetic propagator renormalization. Lamb's shift.

**Prerequisites for admission**

1. Non relativistic quantum mechanics

2. Relativistic kinematics, Lorentz transformations

3. Lagrangian and Hamiltonian formulation of mechanics

4. Basics concepts of classical electrodynamics

2. Relativistic kinematics, Lorentz transformations

3. Lagrangian and Hamiltonian formulation of mechanics

4. Basics concepts of classical electrodynamics

**Teaching methods**

Classroom lectures (42 hours)

**Teaching Resources**

Aitchison , Hey - Gauge Theories in Particle Physics (vol.I) - IOP

Horejsi J. - Fundamentals od Electroweak Theory - Charles University, Prague 2002

Halzen, Martin - Quarks and Leptons - Wiley

Slides (pdf) used for the lectures and registrations of lectures (mp4) are available at

http://www0.mi.infn.it/~ragusa/2019-2020/elettrodeboli/

http://www0.mi.infn.it/~ragusa/2019-2020/elettrodeboli/registrazioni

Horejsi J. - Fundamentals od Electroweak Theory - Charles University, Prague 2002

Halzen, Martin - Quarks and Leptons - Wiley

Slides (pdf) used for the lectures and registrations of lectures (mp4) are available at

http://www0.mi.infn.it/~ragusa/2019-2020/elettrodeboli/

http://www0.mi.infn.it/~ragusa/2019-2020/elettrodeboli/registrazioni

**Assessment methods and Criteria**

Assessment methods: oral exam.

The oral exam (lasting from 45 to 60 minutes) aims to evaluate the body of knowledge achieved by the student, his/her ability to critically discuss fundamental concepts of quantum field theory and its use to build the electroweak interactions theory guided by the relevant experimental results.

The oral exam (lasting from 45 to 60 minutes) aims to evaluate the body of knowledge achieved by the student, his/her ability to critically discuss fundamental concepts of quantum field theory and its use to build the electroweak interactions theory guided by the relevant experimental results.

FIS/04 - NUCLEAR AND SUBNUCLEAR PHYSICS - University credits: 6

Lessons: 42 hours

Professor:
Ragusa Francesco

Professor(s)

Reception:

please contact francesco.ragusa@unimi.it to arrange an appointment

Via Celoria 16 - Edificio LITA, IV floor/ Zoom videoconference platform