# Master 2 courses in Physics - Nuclei, Particles, Astroparticles and Cosmology track

The NPAC track trains students for research in nuclear physics, particle physics, astroparticles and cosmology. Its objective is to prepare students to begin an experimental or theoretical thesis in major research organisations.

This training provides the fundamentals of the physics of the infinitely small and the infinitely large. On the one hand, the physics of elementary particles, their fundamental interactions as well as their assembly into atomic nuclei and the properties of these nuclei are studied. On the other hand, students work on understanding the universe, its geometry, its dark matter and dark energy content. They are also made aware of the cosmic messengers that are the astroparticles, which can also provide answers to some of these still open questions.

## 2nd year Master courses (M2) taught in English

### Semester 1

**2nd year Master - 1st Semester - 6 ECTS - English Level: B2 (no test required)**

**Brief Description**

Preliminary course on Group Theory. Classical field theory: Lagrangian formalism, variational principle, Euler-Lagrange equations. Relativistic fields: free real scalar field, Klein-Gordon equation, Noether theorem: Complex scalar field and U(1) invariance. Quantization of the free scalar field. Free Dirac field: Spinors, Dirac equation. Quantization of the free Dirac field. U(1) gauge-invariance; Maxwell field. Scalar field coupled to a source: Klein-Gordon propagator. Interactions: Asymptotic states, scattering amplitudes, Smatrix, reduction formulae, correlation functions. Perturbation expansion; Feynman diagrams in scalar field theory. Quantum Electrodynamics: Fermion and Photon propagators, Feynman diagrams, calculation of tree level processes. Spontaneous symmetry breaking: Goldstone and Higgs in abelian theories. Non-abelian gauge theories. The standard model.

**Prerequisites**

- Lagrangian formulation of classical systems of point particles
- Non-relativistic quantum mechanics: Heisenberg and Schrödinger pictures time-independent and time-dependent perturbation theory. Quantization of the Harmonic Oscillator in terms of raising and lowering operators.
- Fourier transforms; Complex analysis (contour integration in the complex plane, Cauchy theorem).

**Contact**

Delphine Hardin (delphine.hardin@sorbonne-universite.fr)

Eli Ben Haim (eli.ben_haim@sorbonne-universite.fr)

**2nd year Master - 1st Semester - 6 ECTS - English Level: B2 (no test required)**

**Contact**

Delphine Hardin (delphine.hardin@sorbonne-universite.fr)

Eli Ben Haim (eli.ben_haim@sorbonne-universite.fr

**2nd year Master - 1st Semester - 3 ECTS - English Level: B2 (no test required)**

**Brief Description**

The detectors used in nuclear physics and accelerator-based particle physics, in the field of astroparticles or even in cosmology are of different natures but they must meet the same requirement: to enable the desired quantity to be evaluated with the best sensitivity and resolution. Modern experiments often employ numerous sub-detectors using different and complementary techniques to measure energies, directions or to identify particles.

The teaching unit "Detector Physics" gives a complete overview of these different techniques. Since the detection of particles is related to how they interact, the unit deals in detail with the interaction of neutral or charged particles in matter, focusing on application rather than theory. The basic concepts of how detectors work are discussed: triggering, dead time, quenching, etc. Finally, based on numerous examples, the different types of detectors used in current experiments are reviewed: gas detectors, solid-state detectors, scintillators, bolometers, etc.

This teaching unit complements the "'Laboratory Work”, carried out at the same time, where students implement some of the detectors discussed during the course.

**Prerequisites**

Some basic knowledge in particle physics, nuclear physics, semi-conductors and solid-state physics would help, but is not mandatory.

**Contact**

Delphine Hardin (delphine.hardin@sorbonne-universite.fr)

Eli Ben Haim (eli.ben_haim@sorbonne-universite.fr)

**2nd year Master - 1st Semester - 6 ECTS - English Level: B2 (no test required)**

**Brief Description**

Topics treated in the first part are: the saturation of nuclear forces, internal structure of the nucleus (shell model) and an introduction to pairing and quadrupole correlations. All the topics are illustrated through recent experiments.

The second part treats the following topics: many-body quantum mechanics, symmetries in nucleon-nucleon interaction, second quantization formalism, Wick theorem and its application to the nuclear Hamiltonian, mean-field and beyond mean-field methods for the description of atomic nuclei.

Introduction to nuclear astrophysics is given in the third part. Observed properties of stars, the chemical evolution of elements, and principles of steliar structure and evolution are first presented. This is followed by a description of the nucleosynthesis processes at work in the Universe from primordial to stellar synthesis of elements. Cross-section and thermonuclear reaction rates are then introduced, and an overview of key nuclear reactions is given, together with the different experimental approaches used in nuclear astrophysics.

**Prerequisites**

Quantum Mechanics.

**Contact**

Delphine Hardin (delphine.hardin@sorbonne-universite.fr)

Eli Ben Haim (eli.ben_haim@sorbonne-universite.fr)

**2nd year Master - 1st Semester - 6 ECTS - English Level: B2 (no test required)**

**Brief Description**

This series of lectures focuses on the experimental and theoretical foundations of the Standard Model of Particle Physics, which describes elementary particles and their interactions.

This course is a basic element in the training of a physicist at M2 level but also the starting point for the pursuit of a doctorate in the field of particle physics.

**Prerequisites**

Quantum Mechanics (notions on the composition of kinetic moments, time-dependent perturbations, identical particles); Special Relativity.

**Contact**

Eli Ben Haim (eli.ben_haim@sorbonne-universite.fr)

**2nd year Master - 1st Semester - 6 ECTS - English Level: B2 (no test required)**

**Brief Description**

This teaching unit is divided in two parts treated separately: Astroparticle Physics and Cosmology.

The lectures on Astroparticles deal with various aspects of high energy astrophysics. The list of topics treated during the course include the acceleration of cosmic rays at astrophysical shock waves, their transport in the turbulent interstellar magnetic field and their interactions with matter and radiation. The main radiative signatures of cosmic ray acceleration and transport are also discussed going from observations in the radio band to the very high energy gamma-ray domain.

The Cosmology lectures are organized into four main parts. We first give a formal description of the smooth expanding Universe: metric of Friedmann-Robertson-Walker, cosmological parameters, Friedmann equations and their solutions. Particular emphasis is given to the observables (redshift, cosmological distances and volumes) and to the classical cosmological probes relying on these observables (Supernovae and BAO). The second part of the course deals with the thermal history of the Universe (equilibrium distributions, Boltzmann equation, decoupling, relic densities and recombination). ln the third part, we give an introduction to the physics of the cosmic microwave background and to the linear theory of structure formation. Finally, the last two lectures are devoted to an introduction to primordial inflation. We discuss how this solves some of the problems of the hot big bang model and how it can produce the primordial curvature perturbations through the amplification of quantum fluctuations.

**Contact**

Eli Ben Haim (eli.ben_haim@sorbonne-universite.fr)

**2nd year Master - 1st Semester - 3 ECTS - English Level: B2 (no test required)**

**Brief Description**

ln this introduction to the physics of conventional accelerators, we discuss the modeling of the dynamics of charged particles in a particle accelerator. First, acceleration techniques are discussed and a simplified modeling of RF accelerator cavities is described. The dynamics in the longitudinal phase space is then presented. An introduction to collective effects, such as load space effects, is given.

ln a second step, the magnetic elements for manipulating a charged particle beam are described and modelled. Particle dynamics in the plane transverse to a reference trajectory is modelled ab-nihilo in order to grasp the numerous underlying hypotheses and to introduce the Snyder Current parameterisation and the notion of emittance. This model is applied to a few particular cases. The peculiarities of transverse dynamics in the important case of circular accelerators are discussed and the coupling between longitudinal and transverse effects is addressed by synchrotron radiation. Finally, the notion of luminosity of a particle collider is discussed and all the limitations of the analytical formulas often encountered are addressed.

**Prerequisites**

Electromagnetism (Maxwell-equations)

Special relativity (Lorentz transformations): Classical mechanics.

**Contact**

Eli Ben Haim (eli.ben_haim@sorbonne-universite.fr)

**2nd year Master - 1st Semester - 3 ECTS - English Level: B2 (no test required)**

**Brief Description**

This course begins with an overview of general relativity and its link to special relativity. We discuss the role of inertial and gravitational mass, the equivalence principle and introduce the idea of understanding gravity as a manifestation of the geometry and curvature of space-time. We continue with a review of the salient features of Special Relativity in standard Minkowski coordinates and in curvilinear coordinates. We discuss geodesic motion in curvilinear coordinates. This naturally introduces the concept of a metric. The next section introduces general covariance, vectors and tensors, the covariant derivative and the Riemann tensor. We discuss geodesic motion in an arbitrary spacetime together with the geodesic deviation equation. This is followed by the introduction of Einstein’s equations, and spherically symmetric space-time solutions are derived (Birkhoff theorem), leading on to a discussion of symmetries in GR and the Lie Derivative. We then derive the trajectories of particles around a massive body, and to tests of GR in the solar system. The next section is dedicated to Black Holes of different kinds and in different sets of coordinate systems. After a brief chapter on stars and gravitational collapse, we turn to gravitational waves (GWs). After linearizing the Einstein Equations, we obtain the wave equation and discuss the polarization and number of degrees of freedom carried by GWs, as well as their interactions with a GW detector. The last part focuses on the calculation of the GW signal from binary black holes and neutron stars. We also derive the quadrupole formula for the rate of energy loss. We compare the theoretical calculation to the data from LIGOVirgo. We introduce the cosmology that can be done with GWs, such as the measurement of the Hubble constant.

**Prerequisites**

Special relativity

**Contact**

Eli Ben Haim (eli.ben_haim@sorbonne-universite.fr)

### Semester 2

**2nd year Master - 2nd Semester - 3 ECTS - English Level: B2 (no test required)**

**Contact**

Eli Ben Haim (eli.ben_haim@sorbonne-universite.fr)

**2nd year Master - 2nd Semester - 3 ECTS - English Level: B2 (no test required)**

**Contact**

Eli Ben Haim (eli.ben_haim@sorbonne-universite.fr)