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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. xEt)/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 22(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
-              i1  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 22(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. xEt)/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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