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Particle Physics Lecture course with exercise sessions in 2008 Period 4 Course book; ”Particle Physics” by B.R.Martin and G.Shaw Second Edition 1997 Publisher Wiley Price ca 400 kr Available at Studentbokhandeln and Akademibokhandeln From authors preface “Our aim in writing this book is to provide a short introduction to particle physics, which emphasizes the foundations of the standard model in experimental data rather than its more formal and theoretical aspects. The book is intended for undergraduate students who have previously taken introductory courses in quantum mechanics and special relativity. No prior knowledge of particle physics is assumed.” ELEMENTARY PARTICLE PHYSIC Uppdated 2008-04-03 • F4 TEKNISK FYSIK COURSE 1SV041 ÅK 4 PERIOD 44 VT 2008 http://www.tsl.uu.se/~ekelof TEACHER Professor Tord Ekelöf Department of Physics and Astronomy • Uppsala University • Tord.Ekelof@physast.uu.se Tel. 018 4713847 • LITERATURE B.R.Martin & G.Shaw, Particle Physics, second edition, 1997; Wiley • COURSE START 3 April 2008 • COURSE PLAN http://www.teknat.uu.se/student/ • 1. Antiparticles and Feynman Diagrams (Ch 1.1, 1.2, 1.3) Thu 03/04 13.15 Å2004 • 2. Particle Exchange and Leptons (Ch 1.4, 1.5, 2.1) Mon 07/04 15.15 Polhem • 3. Strongly Interacting Particles (Ch. 2) Tue 08/04 15.15 Å2002 • 4. Exercises 1.1, 1.2, 1.3, 1.5, 2.1, 2.2 Wed 09/04 10.15 Å80101 • 5. Translational and Rotational Invariance and Parity (Ch 4.1, 4.2, 4.3) Thu 10/04 10.15 Å4001 • 6. Accelerators (Ch. 3.1) Thu 17/04 08.15 Å2005 • 7. Particle interactions with matter and Particle detectors, (Ch. 3.2, 3.3) Fri 18/04 13.15 Å80109 • 8. Detector systems and Experiments (Ch. 3.4) Fri 18/04 15.15 Å80109 • 9. Exercises 2.3, 2.4, 2.5 3.2, 3.3 ,3.4 Mon 21/04 10.15 Siegbahn 10 C conjugation, Positronium and Hadron States (Ch 4.4,4.5,4.6,5.1) Tue 22/04 08.15 Å2003 • 11. Exercises 3.5, 3.6, 4.2, 4.3, 4.4, 4.5 Wed 23/04 10.15 Å4003 • 12. Isospin, Resonances, Quark Diagrams and Exotics (Ch 5.2 5.3, 5.4) Thu 24/04 10.15 Å80115 • 13. Exercises 4.7, 5.1, 5.2, 5.3, 5.4, 5.5 Thu 24/04 13.15 Å80121 • 14. Charmonium, Bottonium and Hadrons (Ch 6.1, 6.2) Fri 25/04 08.15 Å80115 • 15. Colour and Quantum Chromodynamics (Ch 6.3, 7.1) Mon 28/04 10.15 Å4005 • 16. Exercises 6.1, 6.2, 6.3, 6.4, 6.5, 6.6 Mon 28/04 13.15 Å4003 • 17. Electron-Positron and Electron-Proton Scattering (Ch 7.2,7.3) Tue 29/04 15.15 Å2002 • Muon decay lab Tue 06/05 8.15 • 18. Inelastic Electron, Muon and Neutrino Scattering (Ch. 7.4, 7.5) Wed 07/05 08.15 Å2005 • 19. Exercises 7.1, 7.2, 7.3, 7.4, 7.5, 7.6 Thu 08/05 13.15 Hägg • 20. The W and Z Bosons and Charged Current Reactions (Ch. 8.1, 8.2) Mon 12/05 10.15 Å2004 • 21. The Third Generation (Ch. 8.3) Mon 12/05 10.15 Å2004 • 22. Exercises 8.2, 8.3, 8.4, 8.5, 8.8, 8.9 Thu 15/05 10.15 Å4007 • 23. Neutral Currents and Unified Theory (Ch. 9.1) Thu 15/05 13.15 Å2003 • 24. Gauge Invariance and the Higgs Boson (Ch. 9.2) Fri 16/05 13.15 Å2004 • 25. P-violation, C-violation and CP-conservation (Ch. 10.1) Tue 20/05 15.15 Å4004 • 26. Exercises 9.1, 9.2, 9.3, 9.4, 9.5, 9.6 Thu 22/05 13.15 Å2002 • 27. Neutral Kaons (Ch.10.2) Thu 22/05 15.15 Å2002 • 28. Neutrino Mixing and Grand Unification (Ch. 11.1, 11.2) Fri 23/05 13.15 Å2002 • 29. Supersymmetry and Dark Matter (Ch. 11.3, 11.4) Fri 23/05 15.15 Å2002 • 30. Recent Developments in Neutrino and Astroparticle Physics Tue 27/05 13.15 Å2003 • 31. Recent Developments in Collider Physics Wed 28/05 10.15 Å2001 • 32. Exercises 10.1, 10.2, 10.5, 11.1, 11.2, 11.5 Thu 29/05 10.15 Å4007 • Examination Fri 30/05 08.00 • For more information see www.tsl.uu.se/~ekelof • • The Elementary Particle Physics course comprises two exercises: • 1. Determination of the muon life-time • 06 May 08:15 to ca 17 in room Å – Instructor David Eriksson (David.Eriksson@tsl.uu.se) • In this exercise you get to calculate the life-time of the muon starting with the Feynman diagram for the muon decay. The goal is to achieve basic training in the application of Dirac spinors, gamma matrix trace calculations, and determination of the phase space for a given process. A written report is required. This exercise is compulsory for MN students and postgraduate students. Other students are most welcome to participate. 2. Studies of e+e- annihilations and Z decays in data from CERN's LEP collider • The aim of this exercise is to give you an idea of how measurements are done in modern particle physics. The form is that of a web tutorial, which you may find at the web address http://www.particle.kth.se/group_docs/particle/courses/lep/ The tutorial was originally developed for students at the KTH in Stockholm. You should click on “Instructions here” under “Laboratory Instructions, Tilläggskurs till 5A1400”. Under “Events and Questions” is written that “you should restrict yourself to events 1, 5 and 6 from the first ‘box’ and event 8 from the second ‘box’”. This is not so for the course in Uppsala. You are expected to treat all 11 events in the two boxes. You should work through the entire tutorial and write a report containing a short description of the LEP accelerator and of the OPAL experiment, a short description of how different particle types (muons, electrons, tauons, photons and hadrons) can be distinguished in the OPAL detector, and answers to all questions turning up during the tutorial. You are free to do this exercise whenever you like on your own computer. However, since a certain knowledge of the electro-weak interaction is required it is recommended to wait until this subject has been treated in the lecture course. This exercise is compulsory for all students. • DEAD LINE for BOTH reports: Tuesday 27/05. Organization of lectures and exercises • Student participation in the lectures For each lecture I will formulate in advance 6-10 questions to be answered during the lecture. During a lecture I will first present answers to some of these questions and will then ask a student, assigned beforehand to that specific lecture, to present answers to the remaining questions (marked with * in the list of questions) on the computer screen, the overhead projector or the black-board. • Student participation in the problem exercises During each problem exercise session there are 6 of the problems in the course book to be solved. There are two students assigned beforehand to each exercise. The first (a) student is asked to present the solution of the three first problems and the second (b) student is asked to present the solution of the three last problems on the black board. Students have individually assigned tasks. However, students grouped together are encouraged to collaborate and divide the tasks between them if they wish to do so. For more information see http://www.tsl.uu.se/~ekelof Questions for Lecture 1 Antiparticles and Feynman Diagrams (Ch 1.1, 1.2, 1.3) Thu 3 April 2008 13.15 Ång_2004 • *Which are the known elementary particles and fundamental forces in Nature? • What is the origin of the Klein-Gordon and Dirac equations and how is the existence of antiparticles inferred from them? • How are the positrons described in Dirac’s Hole Theory? • *How does a cloud chamber work? • *How was the positron discovered with the cloud chamber? • Which are the basic virtual Feynman interaction diagrams and how do they relate to Dirac’s Hole Theory description? • How are real particle interactions described by combining basic virtual Feynman interaction diagrams? • How is annihilation and pair production described with Feynman diagrams? Chapter 1 – Some basic concepts *Which are the known elementary particles and fundamental forces in Nature? • The Standard Model of Particle Physics • Relativistic Quantum Field Theory • Electromagnetic, Weak and Strong interactions – and Gravity • Constituent and exchange particles • Leptons, quarks and gauge bosons • The photon, the W+W-Z0 and the gluon • The Higgs boson What is the origin of the Klein-Gordon and Dirac equations and how is the existence of antiparticles inferred from them? Prediction of antiparticles by Dirac in 1931 based on the combination of special relativity and quantum mechanics.The de Broglie expression for free particle of momentum p is (please note typo: in all formulae Plancks constant ”h” should be ”ħ”=h/2π); i(p. xEt)/h (x,t) Ne - with frequency ν = E/h and wavelength λ= h/p. If E=p2/2m the corresponding wave equation is that of Schrödinger; h2 2 ih (x, t) (x,t) - t 2m Relativistically, however, - 2 E p 2c 2 m 2c 4 where m is the rest mass, and the corresponding wave equation is that of Klein-Gordon (first proposed by de Broglie); 2(x,t) h2 h2c 22(x,t) m2c 4(x,t) - t 2 which is quadratic i E. For every plane wave solution; i(p. x E p t) /h - (x,t) N e with momentum p and positive energy E; 1/2 E E p (p 2c2 m 2c4) mc2 - there is also a solution; ˜ ( x,t) * (x, t) N* e i( p.x E p t) /h , corresponding to momentum -p and negative energy: 1/2 E E p (p 2c2 m2c4) mc2. - At the time the KG-equation was thought to have unavoidable problems with the probability density function for position not being positive-definite. The mysterious negative energy is connected to the second order time derivative. Dirac postulated in 1928; (x, t) ˆ ih H(x, p) (x,t) , - t Where H is the Hamiltonian and ˆ p ih - is the momentum operator. Lorenz invariance requires the space derivatives d/dx to be of first order like the time derivative is. So which is this Hamiltonian? Dirac set up a general first order expression; 3 H ihc i 2 ˆ mc2 c .p mc - i1 x i in which the coefficients b and ai (i=1,2,3) are determined by requiring that solutions of the Dirac equation are also solutions of the Klein-Gordon equation. Acting on (x, t) ˆ ih H(x, p) (x,t) , with iħd /dt gives; t - 2 2 h2 ih ihc i mc t 2 i xi t ihc i mc2 ihc j mc2 , i x i j x j which coincides with the Klein-Gordon equation; 2(x,t) - h2 h2c 22(x,t) m2c 4(x,t) t 2 if, an only if, 2 - 2 1 i 1 - i i 0 - i j j i 0 (i j) These conditions cannot be satisfied by any set of ordinary numbers. The simplest assumption is that they are matrices, which must be Hermitian so that the Hamiltonian is Hermitian. The smallest matrices satisfying these requirements are of dimension 4x4. We thus arrive at the Dirac four-dimensional matrix equation; H ihc i 2 - ih mc t i xi where the Ψ are four-component wavefunctions called spinors; 1(x, t) (x, t) - (x, t ) 2 3 (x, t) (x, t) 4 Plane-wave solutions take the form; - i(p. xEt)/h (x,t) u p e , Where u(p) is a four-component spinor satisfying the eigenvalue equation; Hp u(p) (c .p mc 2 ) u(p) E u(p) This equation has four solutions, two with positive energy states corresponding to two possible spin states of a spin-one-half state and two corresponding to negative energy solutions First successes of Dirac theory: •calculation of relativistic corrections, including spin- orbit effects, in atomic spectroscopy •prediction of the magnetic moment of point-like spin- one-half particles; D q S / m - For the electron this was measured to be correct but for the proton and the neutron measurements (Frisch and Stern 1933) gave that; e e p 2.79 S, n 1.91 S, - mp mp How are the positrons described in Dirac’s Hole Theory? •What about the negative energy states? Dirac: we do not see the negative energy states because they are nearly always filled –> the invisible vacuum sea The positive energy states are nearly all unoccupied (Fig 1.1).This picture is valid for fermions which obey the Pauli exclusion principle. •When an electron is excited from the sea a hole is created in the sea and a positive energy state is filled (Fig 1.5).The positive energy electron may then fall back into the hole in the sea and annihilate. •The hole in the sea is identified with the antiparticle of the electron, i.e. the positron which has opposite charge to the electron. *How does a cloud chamber work? ¤The cloud chamber devised by C.R.T.Wilson Saturated vapour – ionising tracks – expansion – supersaturation – droplets – flashlight – photograph *How was the positron discovered with the cloud chamber? C.D.Andersson Positron track in Fig. 1.2 – -decrease in curvature above 5 mm lead plate->direction -amount of curvature (B=1.2 T) -> momentum 23 MeV/c -range -> cannot be slowly moving proton mass, can only be relativistic mass < 20 me -direction of curvature -> positive particle -> the positron -All charged fermions have antiparticles and so do also all charged bosons (RQFT). For neutral particles there is no general rule – there is an antineutrino but no antiphoton. Which are the basic Feyman interaction diagrams and how do they relate to Dirac’s Hole Theory? In the hole theory electromagnetic interactions are pictured as electrons jumping from one state to another with emission or absorption of a photon, e.g. like in Fig 1.3. (a) e e (b) e e Richard Feyman developed in the 1940’s another pictorial technique, see Fig 1.4. - The time flows to the right - Antiparticles have arrows pointing to the left The processes (c) e e (d) e e Are pictured in Fig 1.4 c and d Finally there are the processes in which a positive energy electron falls into a vacant level(hole) in the negative energy sea or in which an electron is excited from a negative energy state to a positive energy state, leaving a ’hole’ behind : (e) e e (f ) e e (g) vacuum e e (h) e e vacuum The corresponding hole theory graphs are shown in Fig 1.5 and The corresponding Feyman graphs in Fig 1.4 e-h Each of these processes have a probability of occurring proportional to the electromagnetic fine structure constant; 1 e2 1 - . 4 0 hc 137 How are real particle interactions described by combining virtual Feynman interaction diagrams? In the basic processes discussed so far energy conservation is violated. Take e.g. the reaction Fig 1.4 a in the rest frame of the electron; - e (E 0 , 0) e (E k ,k) (ck,k) Momentum conservation has been imposed. In free space, 2 2 2 2 4 1/ 2 - E 0 mc , E k (k c m c ) and E E k kc E 0 - Satisfies kc E 2kc for all finite k. (Set k<<mc and k>>mc) Energy conservation is thus violated by between kc and 2kc. According to the Heisenberg uncertainty relation this can occur but only during a very short time τ~ħ/ΔE. Such processes are said to be virtual. A real process, i.e. one that conserves energy, can be obtained by combining two basic, virtual processes and having the energy violation at one vertex compensated at the other vertex. See e.g. Fig 1.6 which shows single photon exchange in elastic electron-electron scattering. e e e e Can be view in two equivalent ways (time ordering) Each vertex carries a factor α. The probability for elastic single photon scattering is thus α2 and for elastic double- photon scattering α4 (see Fig 1.7). All multi photon scatterings are therefore negligible and the first order diagram is quite accurate. How is annihilation and pair production described with Feynman diagrams? The lowest order Feynmann diagram for electron- positron annihilation is shown in Fig 1.8. The next to lowest order is shown i Fig 1.9 (has 3!=6 different time orderings). The first diagram is proportional to α2 and the second to α3 i.e. Rate(e e 3) R O() . Rate (e e 2) Measurements using positronium (e+e- bound states) gives R=0.9x10-3. OK if comparing with α/2π=1.2x10-3. Fig 1.10 shows pair creation from a photon near an atomic nucleus of charge Z. The recoil is taken up by the nucleus. The rate is proportional to Z2α3 .

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