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Physics of Neutrino Oscillation

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Physics of Neutrino Oscillation PHYSICS OF NEUTRINO OSCILLATION SPANDAN MONDAL UNDERGRADUATE PROGRAMME, INDIAN INSTITUTE OF SCIENCE, BANGALORE Project Instructor: PROF. SOUROV ROY INDIAN ASSOCIATION FOR THE CULTIVATION OF SCIENCE, KOLKATA Contents 1. Introduction ......................................................................................................................................................... 1 2. Particle Physics and Quantum Mechanics: The Basics ........................................................................................ 1 2.1. Elementary Particles ..................................................................................................................................... 1 2.2. Natural Units (ℏ = 푐 = 1) ........................................................................................................................... 2 2.3. Dirac’s bra-ket Notation in QM .................................................................................................................... 3 Ket Space ......................................................................................................................................................... 3 Bra Space ......................................................................................................................................................... 3 Operators ......................................................................................................................................................... 3 The Associative Axiom ..................................................................................................................................... 4 Matrix Representation..................................................................................................................................... 4 2.4. Rotations in Quantum Mechanics ................................................................................................................ 5 2.5. The Klein-Gordon Equation .......................................................................................................................... 6 2.6. The Dirac Equation ....................................................................................................................................... 6 2.7. Bilinear covariants and Weak Interactions ................................................................................................... 8 3. Neutrino Flavours, Mass Eigenstates and Mixing................................................................................................ 8 3.1. Introduction .................................................................................................................................................. 8 3.2. Leptonic Mixing ............................................................................................................................................ 9 4. Neutrino Oscillation in Vacuum ......................................................................................................................... 10 4.1. Detecting a flavour change ......................................................................................................................... 10 4.2. Finding the oscillation probability .............................................................................................................. 10 4.3. Analysis and Discussions ............................................................................................................................. 14 5. Neutrino Oscillation in Matter ........................................................................................................................... 16 5.1. Factors affecting Neutrinos in Matter ........................................................................................................ 16 5.2. Finding the Vacuum Hamiltonian ............................................................................................................... 17 5.3. Finding the Hamiltonian in Matter ............................................................................................................. 18 5.4. Finding the Oscillation Probability in Matter ............................................................................................. 19 5.5. Discussions .................................................................................................................................................. 19 6. Properties of Neutrinos ..................................................................................................................................... 21 6.1. The Mass-squared Splittings ....................................................................................................................... 21 6.2. The Flavour and Mass States and Mixing ................................................................................................... 21 7. Conclusions ........................................................................................................................................................ 22 8. References ......................................................................................................................................................... 23 [i] 1. Introduction In recent years, there have been several breakthroughs in neutrino physics and it has emerged as one of the most active fields of research. The Standard Model of particle physics describes neutrinos as massless, chargeless elementary particles that come in three different flavours [1]. However, recent experiments indicate that neutrinos not only have mass, but also have multiple mass eigenstates that are not identical to the flavour states, thereby indicating mixing. As an evidence of mixing, neutrinos have been observed to change from one flavour to another during their propagation – a phenomenon called neutrino oscillation. Neutrinos can be produced from four different sources, and accordingly they are termed as solar, atmospheric, reactor and accelerator neutrinos. In the past few decades, several experiments have confirmed the event of flavour change in each of these neutrinos. Besides, the values of the parameters affecting the probabilities of neutrino oscillation have been experimentally determined in most of the cases. In this project, we have studied the reasons and derived the probabilities of neutrino flavour change, both in vacuum and in matter. We have also studied the parameters affecting this probability. We have discussed the special case of two-neutrino oscillations. Lastly, we have discussed some basic properties of neutrinos that are reflected in the previous derivations and highlighted a few relevant open problems. To begin with, we have also studied the relevant topics in introductory High Energy Physics and Quantum Mechanics to familiarize with the notations, units and methodologies that would be required for the subsequent project work. 2. Particle Physics and Quantum Mechanics: The Basics 2.1. Elementary Particles During the 5th century B.C., Greek philosophers Leucippus and Democritus considered that if one keeps on cutting matter into smaller and smaller bits one will eventually end up with a fundamental bit which cannot be further subdivided. They called this smallest bit an atom. This is probably man’s first attempt at finding out what the world is made of. It goes without saying that 25 centuries worth of research has considerably refined the concept of atom, and today we have a finer understanding of what the world is made of. We know today that there is no single fundamental particle, but a number of them. Our current understanding of elementary particles can be summarized in Table 2.1 [1]. Table 2.1: Elementary Particles Baryon Lepton Name Spin Charge Q Number B Number L 1 1 2 up (u), charm (c), top (t) 0 + 2 3 3 Quarks 1 1 1 down (d), strange (s), bottom (b) 0 − 2 3 3 1 electron (푒−), muon (휇−), tau (휏−) 0 1 -1 2 Leptons 1 휈 , 휈 , 휈 (neutrino) 0 1 0 푒 휇 휏 2 훾 (photon) 1 0 0 0 ± Guage Bosons 푊 , 푍 (weak bosons) 1 0 0 ±1, 0 푔푖 (gluons) 1 0 0 0 1 Spin is in the units of ℏ. Charges are defined such that an electron has -1 units of charge. Each quark and lepton has a corresponding antiparticle with its B, L and Q having reversed signs. All the particles which undergo strong interactions are called hadrons. Electrons and neutrinos do not undergo such interactions and are called leptons. While the electron is charged and can therefore undergo electromagnetic interactions, the neutrino participates exclusively in weak interactions. Both quarks and leptons can be represented in terms of generation, wherein each generation forms a weak isospin doublet. The representation for leptons has been shown in Table 2.2. Table 2.2: Generations of Leptons First Generation Second Generation Third Generation 휈 휈휇 휈 푄 = 0 ( 푒 ) ( ) ( 휏 ) 푒− 휇− 휏− 푄 = −1 2.2. Natural Units (ℏ = 푐 = 1) The two fundamental constants, viz., the Planck’s constant, ℎ, and the velocity of light in vacuum, 푐, appear repeatedly in expressions in relativistic quantum mechanics and high energy physics. It is convenient to use a system of units in which the use of these two constants can be avoided, thereby hugely simplifying such expressions. This can be achieved by choosing units ℏ = 푐 = 1, where ℏ is the reduced Planck’s constant. ℎ ℏ = = 1.055 × 10−34 J sec 2휋 푐 = 2.998 × 108 m sec−1 In high energy physics, quantities are measured in units of GeV, which is convenient as rest mass-energy of a proton is of the order of 1 GeV. Keeping
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