Elementary Particle Physics
Post on 11-Jan-2016
53 Views
Preview:
DESCRIPTION
Transcript
1FK7003
Elementary Particle Physics
David Milstead
milstead@physto.seA4:1021
tel: 5537 8663/0768727608
2FK7003
Format● 19 lecture sessions● 2 räkneövningnar● Homepage http://www.physto.se/~milstead/fk7003/course.html● Course book
Particle Physics (Martin and Shaw,3rd 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
3FK7003
Lecture outlineLecture Topic Martin and
Shaw (2nd edition)
Martin and Shaw (3rd edition)
Extra info
1 Antiparticles, Klein-Gordon and Dirac equations, Feynman diagrams, em and weak forces
1 1 Handout
2 Units, fundamental particles and forces, Charged leptons and neutrino oscillations
2 2 Handout
3 Quarks and hadrons, multiplets, resonances 2,5 3
4 Räkneövning 1
5 Symmetries: Noether’s theorem, C, P and T 4 5
6 Symmetries: C, P, CP violation, CPT 10 10
7 Hadrons: isospin and symmetries 5 6
8 Hadrons: bound states, quarkonia 6 6
9 Quantum chromodynamics: asymptotic freedom, jets, elastic lepton-nucleon scattering
7 7
10 Räkneövning 2
11 Relativistic kinematics: four-vectors, cross section Appendix B Appendix B Handouts
12 Deep-inelastic lepton-nucleon scattering: quark parton model, structure functions, scaling violations, parton density functions
7 7 Handouts
13 Weak interaction: charged and neutral currents, Caibbo theory 8 8
14 Standard Model: renormalisation, Electroweak unification, Higgs 9 9
15 Beyond the Standard Model: hierarchy problems, dark matter, supersymmetry, grand unified theories
11 11
16 Accelerators – synchrotron, cylcotron + LHC 3 4 Handouts
17 Detectors: calorimeter, tracking, LHC detectors, particle interactions in matter
3 4 Handouts
18 Revision lecture 1
19 Revision lecture 2
4FK7003
Particle physics is frontier research of fundamental importance.
particle physics research
5FK7003
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
6FK7003
Lecture 1 Basic concepts
Particles and antiparticles
Klein-Gordon and Dirac equations
Feynman diagrams
Electromagnetic force
Weak force
7FK7003
Going beyond the Schrödinger equation
Classical mechanics
Quantum mechanics (Schrödingers equation)
Relativistic mechanics
Quantum field theory (Dirac, Klein-Gordon equations,QED, weak, QCD)
2 2 2 2 4
Collider experiments typically involve energies of several hundred GeV.
Eg a proton (mass 1 GeV) with 50 GeV energy
- relativity effects can't be ignored.
Dirac, Klein-Gordon equati
E p c m c
E pc
ons and quantum field theory necessary.
small
fast
8FK7003
Implications of introducing special relativity
2 2 2 4
2 2 2 4 2 2 2 4
Consider a particle of charge , mass with momentum moving along the -axis
What is its energy ?
Special relativity gives us a choice: (1.1)
(1.2)
q m p x
E p c m c
E p c m c E p c m c
(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....
9FK7003
Negative energy states
22 2,
Plane wave corresponding to momentum along -direction.
Positive energy solution: (1.4)
(1.5) (moves to the right)
Negative energy solution:
px E ti
p x
x t Ne E pc mc
Epx E t x t
p
x
, 0 (1.6)
(1.7)
Negative energy state moving forwards in time is equivalent to a
positive energy state moving backwards in time.
px E ti
t Ne E E
EEx t t
p p
t
x
t
x
0E
, 0E E
10FK7003
What does a particle moving backwards in time look like ?
-
,
What are the implications of moving backwards in time ?
Lorentz force on particle (charge ) in a -field travelling forwards in
time at a certain point in space and time and
(1.
q B
r t
F r t qv B
2
2
2 2
2 2
,
- .
, ,
8) (1.9)
Force on a particle with charge moving backwards in time:
(1.10) easily rearranged to (1.9)
d r drF r t m q B
dt dtq dt dt
d r dr d r drF r t m q B F r t m q B
dt dt dt dt
The equation of motion of charge 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 in time.
11FK7003
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 q moving backwards in time looks like a particle with charge –q 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.
12FK7003
Electron and the positron
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.
1.5 ( )
(to left)
B T out of page
F q v B
pr
eB
13FK7003
Klein-Gordon equation
2
2
,
,, -
2
Start with Schrödinger equation:
Free particle: (1.11)
(1.12) , (1.13)
This is the quantum analogue to the non-relativistic conservati
i p r Et
r t Ne
r ti r t E i p i
t m t
2
222 2
22 2 2 2 2 4
2
1
2
,,
on of energy:
(1.14)
Try to build a relativistic wave equation (Oskar Klein, Walter Gordon 1927)
Based on (1.1)
(1.15) The Klein Gordon Equation
Two
E mv
E pc mc
r tc r t m c
t
22 2
*
22 2
,
,
plane wave solutions:
(1.16); Energy
(1.17)
Energy= -
Two possible solutions for a free particle. One has posi
i p r E t
i p r E t
r t Ne E p c mc
r t N e
i E E p c mct
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.
14FK7003
The Dirac Equation
3
2 2
1
1
2
,, ,
ˆ
Dirac (1928):
Relativistic equation for spin particles (fermions).
Look for an equation based on form:
i (1.18)
Hamiltonian (1.19)
, co-effi
ii i
r tH r t r t
t
H i c mc c p mcx
2 21, 1, 0 0 ( )
,
cients constrained by need to satisfy the Klein-Gordon equation (1.15)
and (1.20)
are not numbers - represent as matrices
4-component vector :
i i i i j j i i j
1
2
3
4
,
,,
,
,
,
, ,
(1.21)
Plane wave solutions: (1.22)
are spinors
Four solutions:
Two positive energy energies corresponding to two spin st
p r Eti
r t
r tr t
r t
r t
r t u p e
u p r t
E E
1
21
2
ates of spin particles
Two negative energy solutions corresponding to two spin states of spin particlesE E E
15FK7003
Implications of the Dirac Equation
1
21
2
Dirac equation implies that for every spin particle there is
(a) a corresponding spin antiparticle
(b) two spin states
Spin and antiparticles arises as a consequence of treating quantum mecha
22
2.002..
211
2
nics relativistically
Intrinsic magnetic moment of elementary particle: (1.23)
Prediction of Dirac equation for electrons:
Experiment
Precision experimental result:
eg S
mcg
g
g
12
12
59652180.7 0.3 10
21159652153.5 28 10
2Dirac prediction + quantum corrections for :
Quantum electrodynamics is "the best theory we've got!"
ge
16FK7003
How particles interact – exchange forces Electromagnetic force
02
A photon is emitted - we don't know its momentum
and we don't know where it is
The "quantum path" between the start and end points
is not like a classical path.
The reaction can take place as
p xp
per the diagram.
- -
--
+
- -
+
+-
-
-
RepulsionAttraction
Easy to visualise but beware this is a
useful but limited "visual toy model"
for the quantum world.
photon
Particles carrying charge interact via the exchange of photons () mass=0, spin=1 (boson)
17FK7003
Electromagnetic processes
2 2
1 1
( , )
( , )
Two possible interpretations
(1) A particle moves forward in time, emits two photons at and moves
back in time with negative energy to point where it scatters off a photon and
moves for
x t
x t
1 1
1 1
( , )
( , )
ward in time. There is only one particle moving through space and time.
(2) At point an antiparticle-particle pair is produced. The antiparticle moves forward
to point where it annihila
x t
x t tes with another particle producing to two photons.
18FK7003
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.
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.
One possible diagram for
e e e e
19FK7003
(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
2
0
1 1
4 137
tions occur at a vertex.
Rule of thumb: a vertex carries a factor associated
with the probability of that interaction taking place.
Probability
(1.24) Fine structure constant
em
e
c
A basic process.
e e
t
s
vertex
20FK7003
( ) ( )
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
a h
( ) ( )
ble processes:
and
e e e e
e f
(1) Basic electromagnetic diagrams
t
s
( ) vacuum g e e ( ) vacuumh e e
((g) and (h) become clear soon)
vertex
21FK7003
(2) Is energy conservation violated ?
1 1 1 1
1 1 1 1
1 1
0
0
(annihilation)
Electron and positron in centre-of-mass frame: , (1.24)
Annihilate to form photon , (1.25)
(according
e e e e
e e e e
e e
e e e e
E E p p
E E E p p p
E E E E
2 2 2 2 41 1
1 1
0 0
0!! 0
to conservation of momentum)
,
(1.26)
Nevertheless, the process happens. Two qualitative ways to interpret this
(a) Energy-momentum conservation
e e
E p c m c p m
E E E E
is violated for the short interaction time
Uncertainty principle: violation can happen over time (1.27)
(b) The mass of the photon is not zero (goes off mass-shell) for a short time
Mass chang
tE
t
2
2e (1.28) as permitted by the uncertainty principle (1.29)
Internal lines correspond to virtual particles (we can never see them)
External lines correspond to real particles which can
E cm t
c m
be observed and
always carry the expected mass.
Important: however we think about our diagrams, we measure energy and momentum
violation. The energy and momenta of the real particles always add up
never
1 1 2 2 1 1 2 2
:
(1.30)e e e e e e e e
E E E E p p p p
2 2,
e eE p
t
s
treal particle
virtual particle ()
1 1,
e eE p
1 1,
e eE p
,E p
2 2,
e eE p
22FK7003
(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 and
Start and consider possib
e e e e
e e
2
le contributing diagrams. Use the simplest possible
diagrams (leading order) - in this case diagrams with two vertices.
Diagrams with two vertices : probability of process occuring
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) emits a photon and goes on to annihilate with
a leading to a photon.
(b) emits a photon and goes on to annihilate with
a leading to a photon
e
e
e
e
.
Usually only one such diagram is shown and the others
implied.Negative energy solutions –antiparticles.QM insists we use them!
23FK7003
(3) Using Feynman diagrams
3
3
2
2
3
0.7 10
10
Three vertices probability
(1.31)
Observed
Qualitative Feynman diagram picture gives
suppression with (very) rough accuracy.
e e
Rate e eR
Rate e e
R
Full QED calculation gives correct rates.+ 5 other contributions
24FK7003
QuestionFor the interaction draw three Feynman diagrams which would be
suppressed wrt to those we studied earlier.
e e
25FK7003
(3) Using Feynman diagrams
6
4
e-
e-
e-
e-
e-
e-
e-
e-
e-
e-
e-
e-
11
137
Two electrons are observed to repell each other:
Many different indistinguishable processes,
eg one-photon, two-photon exchange,
can contribute to the scattering
Coupling is weak
e e e e
higher order processes contribute less and less to the
calculation and can be safely be neglected in any approximate
solution.
+
2
+
….+
26FK7003
QuestionFor the interaction draw all six possible time ordered Feynman diagrams for
the leading order (3 vertices) processes
e e
27FK7003
Understanding forces
,A AE p
,X AE p
2 ,0AM c
2
1/22 2 2 4
.
0,
,
Go beyond EM force
Generic force between particles and via exchange of particle
Inititally particle at rest: (1.32)
After vertex:
Particle A: (1.33)
Parti
A
A A A
A B X
A p E m c
p p E E p c m c
1/22 2 2 4
2
2
,
2 ( )
( 0)
0
cle : (1.34)
Energy difference between initial and final state:
(1.35)
(1.36)
(apparent energ
A X X
X A A
X
X p p E E p c m c
E E E m c pc p
M c p
E
max 2
y conservation violation!)
Energy violation can only persist time:
minimum energy violation corresponds to longest time
(1.37)
Max speed
Range of force (1.38)
Electrom
X
X
tE
tM c
v c
RM c
agnetic range due to massless photon.
R
28FK7003
e- e-e-
W-
ee+
0 2 2
18,
91.2 , 80.4
2 10
Use same formalism as for electromagnetic force
Very brief overview:
Exchange of 3 spin-1 particles: (mass= GeV/c ) , (mass= GeV/c )
range m (tiny - proton "radius" 1W ZW
Z W W
RM c
15
2 2
0
2
4 4
1
4
0 m)
Define coupling constant analagous to fine structure constant
(1.39) (1.24)
analagous to electric charge (very few talk of "weak charge")
WW
W
WW
g e
c c
g
g
c
1
, , .240 137
(1.41) the weak force is only weak due to massesW Z
The weak force
(neutrinos – next lecture)(decay)
29FK7003
The fundamental forcesDifferent exchange particles mediate the forces:
strongelectromagnetic
weak
Interaction Relative strength
Range Exchange Mass (GeV)
Charge Spin
Strong 1 Short ( fm)
Gluon 0 0 1
Electromagnetic 1/137 Long (1/r2) Photon 0 0 1
Weak 10-9 Short ( 10-3 fm)
W+ W-,Z 80.4,80.4, 91.2
+e,-e,0 1
Gravitational 10-38 Long (1/r2) Graviton ? 0 0 2No quantum field theory yet for gravity
30FK7003
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.
top related