Transcript Elementary Particle Physics
Elementary Particle Physics David Milstead
[email protected]
A4:1021 tel: 5537 8663/0768727608
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Format
19 lecture sessions 2
räkneövningnar
Homepage http://www.physto.se/~milstead/fk7003/course.html
Course book Particle Physics (Martin and Shaw,3 rd edition, Wiley) ● Earlier editions can be used – handouts to be provided where appropriate.
Supplementary books which may be useful but which are not essential Introduction to Elementary Particles (Griffiths, Wiley) Subatomic Physics (Henley and Garcia, World Scientific) Particles and Nuclei (Povh, Rith, Scholz and Zetsche, Springer) Quarks and Leptons (Halzen and Martin, Wiley) Assessment
2 x inlämningsuppgifter tenta
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Lecture outline
10 11 12 13 14 15 3 4 5 6 7 8 9 1 2 16 17 18 19
Lecture Topic
Antiparticles, Klein-Gordon and Dirac equations, Feynman diagrams, em and weak forces Units, fundamental particles and forces, Charged leptons and neutrino oscillations Quarks and hadrons, multiplets, resonances Räkneövning 1 Symmetries: Noether’s theorem, C, P and T Symmetries: C, P, CP violation, CPT Hadrons: isospin and symmetries Hadrons: bound states, quarkonia Quantum chromodynamics: asymptotic freedom, jets, elastic lepton-nucleon scattering Räkneövning 2 Relativistic kinematics: four-vectors, cross section Deep-inelastic lepton-nucleon scattering: quark parton model, structure functions, scaling violations, parton density functions Weak interaction: charged and neutral currents, Caibbo theory Standard Model: renormalisation, Electroweak unification, Higgs Beyond the Standard Model: hierarchy problems, dark matter, supersymmetry, grand unified theories Accelerators – synchrotron, cylcotron + LHC Detectors: calorimeter, tracking, LHC detectors, particle interactions in matter Revision lecture 1 Revision lecture 2
Martin and Shaw (2 nd edition)
1 2 2,5 4 10 5 6 7 Appendix B 7 8 9 11 3 3
Martin and Shaw (3 rd edition)
1 2 3 5 10 6 6 7 Appendix B 7 8 9 11 4 4
Extra info
Handout Handout Handouts Handouts Handouts Handouts FK7003 3
FK7003 particle physics research Particle physics is frontier research of fundamental importance.
4
The aim of this course
●
Survey the elementary constituents in nature
● Identification and classification of the fundamental particles Theory of the forces which govern them over short distances
Experimental techniques
Accelerator Particle detectors FK7003 5
Lecture 1 Basic concepts
Particles and antiparticles Klein-Gordon and Dirac equations Feynman diagrams Electromagnetic force Weak force FK7003 6
Going beyond the Schrödinger equation
E
2 2 4 Collider experiments typically involve energies of several hundred GeV.
Eg a proton (mass 1 GeV) with 50 GeV energy
pc
- relativity effects can't be ignored.
Dirac, Klein-Gordon equati ons and quantum field theory necessary.
small Classical mechanics Relativistic mechanics Quantum mechanics (Schrödingers equation) Quantum field theory (Dirac, Klein Gordon equations, QED, weak, QCD) fast FK7003 7
Implications of introducing special relativity
What is its energy ? Special relativity gives us a choice:
E
E
2 4 (1.2)
E
2 2 4 (1.1) 4 (1.3) Surely the negative energy solution is unphysical and daft.
Can't we just ignore it ? No - from quantum mechanics, every observable must have a complete set of eigenstates. The negative energy states are needed to form that complete set. They must mean something.... FK7003 8
Negative energy states
Positive energy solution:
px
Ne i
(1.4)
E
x
E
p t
(1.5) (moves to the right) 2 2
x
Negative energy solution:
x
E
p t
E
p
(1.7)
Ne i
(1.6)
E
E
0 Negative energy state moving forwards in time is equivalent to a positive energy state moving backwards in time.
E
,
E
0
x E
0
t t
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What does a particle moving backwards in time look like ?
What are the implications of moving backwards in time ? Lorentz force on particle (charge -
q B
time at a certain point in space and time and
t
(1.
8) Force on a particle with charge
m
q dr
B
(1.9)
dt
2
dt
q
moving backwards in time:
dt
m dt
2
q dr
dt
B
(1.10) easily rearranged to (1.9)
m dt
2
q dr dt
B
The equation of motion of charge in time.
q
moving backwards in time in a magnetic field is the same as the equation of motion of a particle with charge
-q
moving backwards FK7003 10
Antiparticles
Special relativity permits negative energy solutions and quantum mechanics demands we find a use for them.
(1) The wave function of a particle with negative energy moving forwards in time is the same as the wave function of a particle with positive energy moving backwards in time. Ok, the negative energy solutions must be used but we can convert them to positive energy states if we reverse the direction of time when considering their interactions. (2) A particle with charge with charge
–q q
moving backwards in time looks like a particle moving forwards in time.
General argument that a particle with negative energy and charge
q
behaves like a particle with positive energy and charge
-q
. We expect, for a given particle, to see the ”same particle” but with opposite charge:
antiparticles.
Antiparticles can be considered to be particles moving backwards in time - Feynman and Stueckelberg.
Hole theory
(not covered) provides an alternative, though more old fashioned way of thinking about antiparticles.
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Electron and the positron
B F
1.5
( )
B
(to left)
r
p eB
1897
e -
discovered by J.J. Thompson 1932 Anderson measured the track of a cosmic ray particle in a magnetic field.
Same mass as an electron but positive charge The positron (
e +
) - anti-particle of the electron Nobel prize 1936 Every particle has an antiparticle.
Some particles, eg photon, are their own antiparticles. Special rules for writing particles and antiparticles, eg antiproton p, given in next lecture.
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Klein-Gordon equation
Start with Schrödinger equation: Free particle:
i
t
Ne
2 2
m
2 (1.11) (1.12)
E
i
t
,
p
-
i
(1.13) This is the quantum analogue to the non-relativistic conservati on of energy:
E
1 2
mv
2 (1.14) Try to build a relativistic wave equation (Oskar Klein, Walter Gordon 1927) Based on
E
2 2 2 (1.1) 2 2
t
2 2
c
2 2 4 (1.15) The Klein Gordon Equation Two plane wave solutions:
Ne
(1.16); Energy
E
i
t
*
N e i
p
E
(1.17) Energy=
E
2 2 2 2 Two possible solutions for a free particle. One has posi tive energy and the other "negative energy" eigen values. Schrödinger's equation only had one energy solution.
Special relativity demands antiparticles.
Klein-Gordon equation describes spin-0 bosons.
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The Dirac Equation
Dirac (1928): Relativistic equation for spin 1 2 particles (fermions).
Look for an equation based on form: i
t
(1.18) Hamiltonian
H
i c i
3 1
i
x i
mc
2
c
mc
2 (1.19) cients constrained by need to satisfy the Klein-Gordon equation (1.15)
i
2 1, 2 1,
i
i
0 and
i j
j i
are not numbers - represent as matrices 4-component vector : Plane wave solutions: 1 2 4 3 (1.21)
i
(1.22)
i
j
) (1.20) are spinors Four solutions: Two positive energy energies
E
E
corresponding to two spin st ates of spin 1 2 particles Two negative energy
E
E
E
solutions corresponding to two spin states of spin 1 2 particles FK7003 14
Implications of the Dirac Equation
Dirac equation implies that for every spin 1 2 particle there is (a) a corresponding spin 1 2 antiparticle (b) two spin states Spin and antiparticles arises as a consequence of treating quantum mecha nics relativistically Intrinsic magnetic moment of elementary particle:
e g
2
mc S
(1.23) Prediction of Dirac equation for electrons:
g
2 Experiment
g
2.002..
Precision experimental result:
g
2 11 2
g
2 10 12 2 10 12 Quantum electrodynamics is "the best theory we've got!" FK7003 15
-
How particles interact – exchange forces Electromagnetic force
Particles carrying
charge
mass=
0
, spin=
1
(boson) interact via the exchange of
photons
( g )
-
photon
+ + Repulsion
Easy to visualise but beware this is a useful but limited "visual toy model" for the quantum world.
Attraction +
A photon is emitted - we don't know its momentum FK7003 0 and we don't know where it is 2
p
The "quantum path" between the start and end points is not like a classical path.
The reaction can take place as per the diagram.
16
Electromagnetic processes
Two possible interpretations (1) A particle moves forward in time, emits two photons at
x t
2 2 and moves back in time with negative energy to point where it scatters off a photon and 1 moves for ward in time. There is only one particle moving through space and time.
(2) At point to point
x t
1
x t
1 1 an antiparticle-particle pair is produced. The antiparticle moves forward where it annihila tes with another particle producing to two photons.
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Feynman diagrams
Important mathematical tool for calculating rates of processes - Feynman rules.
Qualitative treatment here but more detailed treatment later in the course.
Represent any process by contributing diagrams.
One possible diagram for
e
e
e
e
Strategy: (1) Build Feynman diagrams for electromagnetic processes (2) Consider energy-momentum conservation/violation (3) Consider how they can be used for simple rate estimates. (4) Show Feynman diagram formalism for other fundamental forces.
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(1) Electromagnetic processes
Convention - time flows to the right The lines do not represent trajectories of a particle.
Arrow for antiparticle goes "backward in time".
Lines should not be taken as "trajectories" of particles Interac tions occur at a vertex.
vertex g
e
e
A basic process.
Rule of thumb: a vertex carries a factor
em
associated with the probability of that interaction taking place.
Probability 1 4 0
e
2
c
1 137 (1.24) Fine structure constant
s t
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(1) Basic electromagnetic diagrams
Consider all electromagnetic processes built up from basic processes: to The basic processes are never seen since they violate energy conservation (next slide) They can be combined to make observa ble processes:
e
e
e
e
s
vertex
e
and FK7003
t
( ) vacuum
e
g ( )
e
vacuum ((g) and (h) become clear soon) 20
(2) Is energy conservation violated ?
e
e
e
e
(annihilation) Electron and positron in centre-of-mass frame:
E
1
e
Annihilate to form photon
E
g
E
1
e
E
1
e
,
p
g
p
1
e
E
1
e
E
1
e
E
g 0 (according
E
1
e
p
,
p
1
e
1
e
to conservation of momentum)
p
1
e
0 (1.25) (1.24)
E
g 2
E
g 2
p c
2 0!!
2
m c
4
p
1 g
E
1
e
0 ,
m
1 g
E
1
e
0 0 (1.26)
E
1
e
,
p
1
e
virtual particle ( g )
E
g ,
p
g
t
Nevertheless, the process happens. Two qualitative ways to interpret this (a) Energy-momentum conservation is violated for the short interaction time
E
1
e
,
p
1
e
E
2
e
,
p
2
e
real particle
E
2
e
,
p
2
e
Uncertainty principle: violation can happen over time
E
(1.27) (b) The mass of the photon is not zero (goes off mass-shell) for a short time
t
Mass chang e
m
E
(1.28) as permitted by the uncertainty principle
t c
2
c
2
m
(1.29) Internal lines correspond to virtual particles (we can never see them) External lines correspond to real particles which can be observed and always carry the expected mass.
s
Important: however we think about our diagrams, we
never
measure energy and momentum violation. The energy and momenta of the real particles always add up :
E
1
e
E
1
e
E
2
e
E
2
e
p
1
e
p
1
e
p
2
e
p
2
e
(1.30)
t
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(3) Using Feynman diagrams
Even with a qualitative treatment it is possible to see how the Feynman diagram picture agrees with observed reactions.
Aim: compare rates of
e
e
Start
e
e
and and consider possib
e
e
le contributing diagrams. Use the simplest possible diagrams (leading order) - in this case diagrams with two vertices. Life gets much easier if you don't think about things going back in time. Instead, take your diagram, think of the lines as "rubber" and see how they can be bent in such a way as to change the time order of the vertices. Two possibilities here : (a)
e
emits a photon and goes on to annihilate with a
e
leading to a photon.
(b)
e
emits a photon and goes on to annihilate with a
e
leading to a photon .
Usually only one such diagram is shown and the others implied.
Diagrams with two vertices : probability of process occuring 2 Negative energy solutions –antiparticles.
QM insists we use them!
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(3) Using Feynman diagrams
e
e
Three vertices probability
3
R
e
e
3 2 2 (1.31) Observed
R
10 3 Qualitative Feynman diagram picture gives suppression with (very) rough accuracy.
Full QED calculation gives correct rates.
+ 5 other contributions FK7003 23
Question
For the interaction
e
e
draw three Feynman diagrams which would be suppressed wrt to those we studied earlier.
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(3) Using Feynman diagrams
2 Two electrons are observed to repell each other:
e
e
e
e
Many different indistinguishable processes, eg one-photon, two-photon exchange, can contribute to the scattering Coupling is weak 1 137 1 higher order processes contribute less and less to the 4 calculation and can be safely be neglected in any approximate solution.
e e e e e -
6
e -
+ + + ….
FK7003
e e e e e e -
25
Question
For the interaction
e
e
draw all six possible time ordered Feynman diagrams for the leading order (3 vertices) processes FK7003 26
Go beyond EM force
Understanding forces
X
.
p
0,
E
m c A
2 (1.32) After vertex: Particle A:
p
Parti
X p
p A
,
E
p A
,
E
E A E X
2 2
p c
2
A
4 1/ 2 2
m c X
4 1/ 2 (1.33) (1.34) Energy difference between initial and final state:
E X
E A
m c A
2 2
pc
(
p
) (1.35)
M c X
2 (
p
0) (1.36) 0 (apparent energ y conservation violation!) Energy violation can only persist time:
t
E
minimum energy violation corresponds to longest time
t
max
M c X
2 Max speed
v
(1.37)
c
Range of force
R
M c X
(1.38) Electrom agnetic range
R
g due to massless photon.
FK7003
E X
,
p A
M c A
2 , 0
E A
,
p A
27
The weak force
e e e W -
n
e e +
(
decay)
(neutrinos – next lecture)
Use same formalism as for electromagnetic force Very brief overview: Exchange of 3 spin-1 particles:
Z
0 (mass= 91.2
(mass= 80.4
range
R
M c W
18 m (tiny - proton "radius" 1 Define coupling constant analagous to fine structure constant
W
g W
2 4
c
(1.39)
e
2 4 0
c
(1.24)
g W
analagous to electric charge (very few talk of "weak charge")
W
g W
2 4
c
1 240 (1.41) 1 137 , the weak force is only weak due to masses .
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The fundamental forces
Different exchange particles mediate the forces: strong electromagnetic weak Interaction Strong Electromagnetic Relative strength 1 1/137 Weak Gravitational 10 10 -9 -38 Range Exchange Short ( fm) Long
(1/r 2 )
Gluon Photon ( Short 10 -3 fm) Long
(1/r 2 ) W + W ,Z
Graviton ?
Mass (GeV)
0 0
Charge
0 0 80.4,80.4, 91.2
0 +e,-e,0 0
Spin
1 1 1 2
No quantum field theory yet for gravity FK7003 29
Summary
Antiparticles and spin states are predicted when when relativity and quantum mechanics meet up!
Antiparticles correspond to negative energy states moving backwards in time.
Feynman diagram formalism developed and used for (very basic) rate estimation Generic approach for all forces Weak force is weak because of the mass of the exchanged particles. FK7003 30