Transcript Elementary Particle Physics

Elementary Particle Physics
David Milstead
[email protected]
A4:1021
tel: 5537 8663/0768727608
FK7003
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Format
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19 lecture sessions
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2 räkneövningnar
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Homepage http://www.physto.se/~milstead/fk7003/course.html
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Course book

Particle Physics (Martin and Shaw,3rd edition, Wiley)
●
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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)

●
Earlier editions can be used – handouts to be provided where appropriate.
Quarks and Leptons (Halzen and Martin, Wiley)
Assessment

2 x inlämningsuppgifter

tenta
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Lecture outline
Lecture
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
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 particle physics research
Particle physics is frontier
research of fundamental
importance.
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The aim of this course
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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
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Lecture 1
Basic concepts
Particles and antiparticles
Klein-Gordon and Dirac equations
Feynman diagrams
Electromagnetic force
Weak force
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Going beyond the Schrödinger equation
E 2  p 2c 2  m2c 4
Collider experiments typically involve energies of several hundred GeV.
Eg a proton (mass  1 GeV) with 50 GeV energy
 E  pc - relativity effects can't be ignored.
 Dirac, Klein-Gordon equations and quantum field theory necessary.
small
Classical
mechanics
Quantum
mechanics
(Schrödingers
equation)
fast
Relativistic
mechanics
Quantum field
theory
(Dirac, KleinGordon equations,
QED, weak, QCD)
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Implications of introducing special relativity
Consider a particle of charge q, mass m with momentum p moving along the x-axis
What is its energy ?
Special relativity gives us a choice: E   p 2c 2  m 2c 4
E 
p 2c 2  m2c 4
(1.2)
E   p 2 c 2  m 2 c 4
(1.1)
(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....
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Negative energy states
Plane wave corresponding to momentum p along x-direction.
Positive energy solution:   x, t   Ne
px  E t
 x
 px  E t 
i



(1.4) E  
 pc 
2
  mc 2 
2
x
E
t (1.5) (moves to the right)
p
Negative energy solution:   x, t   Ne
 px  E t 
i



E  0
(1.6) E   E  0
E
E
 x   t    t  (1.7)
p
p
t
E , E  0
 Negative energy state moving forwards in time is equivalent to a
positive energy state moving backwards in time.
x
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) in a B-field travelling forwards in
time at a certain point in space and time r and t
d 2r
dr
F  r , t   qv  B (1.8)  F  r , t   m 2  q  B (1.9)
dt
dt
Force on a particle with charge  q moving backwards in time: dt  -dt.
d 2r
dr
d 2r
dr
F r ,t   m 2  q
 B (1.10) easily rearranged to (1.9) F  r , t   m 2  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.
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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.
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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
B  1.5T (out of page)


F  q v  B (to left)
p
r
eB
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:   r , t   Ne
i  p  r  Et 
(1.11)
2
  r , t 

i

 2  r , t  (1.12) E  i
, p  -i  (1.13)
t
2m
t
This is the quantum analogue to the non-relativistic conservation of energy:
1
E  mv 2 (1.14)
2
 Try to build a relativistic wave equation (Oskar Klein, Walter Gordon 1927)
Based on E 2   pc    mc 2  (1.1)
2
2

2
 2  r , t 
  2 c 2 2  r , t   m 2 c 4 (1.15) The Klein Gordon Equation
2
t
Two plane wave solutions:
  r , t   Ne
i  p  r  E t 
 r ,t   N e
2
  mc 2 
2
i   p  r  E  t 
*
i
 p c 
(1.16); Energy E 

  E
t
(1.17)
 Energy=  E  -
 p c 
2
  mc 2 
2
Two possible solutions for a free particle. One has positive 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):
1
particles (fermions).
2
Look for an equation based on form:
Relativistic equation for spin
  r , t 
 H  r , t   r , t  (1.18)
t
3

Hamiltonian H  i c   i
  mc 2  c  pˆ   mc 2 (1.19)
xi
i 1
i
 , co-efficients constrained by need to satisfy the Klein-Gordon equation (1.15)
  i2  1,  2  1,
 i    i  0 and  i j   j i  0 (i  j ) (1.20)
  ,  are not numbers - represent as matrices
 1  r , t  


 2  r , t  
  4-component vector :   r , t   
 (1.21)

r
,
t


 3

  r , t  
 4

Plane wave solutions:   r , t   u  p  e
i
p
r  Et 
(1.22)
u  p  ,  r , t  are spinors
Four solutions:
1
particles
2
1
solutions corresponding to two spin states of spin
particles
2
Two positive energy energies  E  E  corresponding to two spin states of spin
Two negative energy  E  E   E 
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Implications of the Dirac Equation
Dirac equation implies that for every spin
(a) a corresponding spin
1
particle there is
2
1
antiparticle
2
(b) two spin states
Spin and antiparticles arises as a consequence of treating quantum mechanics relativistically
Intrinsic magnetic moment of elementary particle:   g
e
S (1.23)
2mc
Prediction of Dirac equation for electrons: g  2
Experiment g  2.002..
g 2
Precision experimental result:
 1159652180.7  0.3 1012
2
g 2
Dirac prediction + quantum corrections for e  :
 1159652153.5  28   1012
2
Quantum electrodynamics is "the best theory we've got!"
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How particles interact – exchange forces
Electromagnetic force
Particles carrying charge interact via the exchange of photons (g)
mass=0, spin=1 (boson)
-
-
+
photon
-
-
-
+
-
-
-
+
Attraction
Repulsion
Easy to visualise but beware this is a
useful but limited "visual toy model"
for the quantum world.
A photon is emitted - we don't know its momentum
p  0 and we don't know where it is x 
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.
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Electromagnetic processes
Two possible interpretations
(1) A particle moves forward in time, emits two photons at ( x2 , t2 ) and moves
back in time with negative energy to point ( x1 , t1 ) where it scatters off a photon and
moves forward in time. There is only one particle moving through space and time.
(2) At point ( x1 , t1 ) an antiparticle-particle pair is produced. The antiparticle moves forward
to point ( x1 , t1 ) where it annihila tes with another particle producing to two photons.
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Feynman diagrams
One possible diagram for
Important mathematical tool for calculating
rates of processes - Feynman rules.
Qualitative treatment here but more detailed
e  e  e  e
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.
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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".
vertex
Lines should not be taken as "trajectories" of particles
Interactions occur at a 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
e
2
4 0 c

s
1
(1.24) Fine structure constant
137
t
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(1) Basic electromagnetic diagrams
vertex
Consider all electromagnetic processes
built up from basic processes: (a) to (h)
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
t
(e) and ( f )
( g ) vacuum  e  e  g
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(h) e  e  g  vacuum
((g) and (h) become clear soon)
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(2) Is energy conservation violated ?
e   e   e   e  (annihilation)
virtual particle (g)
E , p 
Electron and positron in centre-of-mass frame: E1e  E1e , p1e   p1e (1.24)
Annihilate to form photon  Eg  E1e  E1e , pg  p1e  p1e  0 (1.25)
E  E1e  E1e  Eg  0 (according to conservation of momentum)
g
E
1e

, p1e
g
2 e
Eg2  pg2 c 2  mg2 c 4  p1g  0 , m1g  0
 Eg  0!!  E  E1e  E1e  0 (1.26)

E1e , p1e

, p2 e

real particle
t
Nevertheless, the process happens. Two qualitative ways to interpret this
(a) Energy-momentum conservation is violated for the short interaction time
E

E
2 e
, p2 e

Uncertainty principle: violation can happen over time t 
(1.27)
E
(b) The mass of the photon is not zero (goes off mass-shell) for a short time t
E
c2
(1.28)
as
permitted
by
the
uncertainty
principle

t

(1.29)
c2
m
Internal lines correspond to virtual particles (we can never see them)
Mass change m 
s
External lines correspond to real particles which can be observed and
always carry the expected mass.
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:
 E1e  E1e  E2 e  E2 e
t
p1e  p1e  p2 e  p2 e (1.30)
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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  g  g and e   e   g  g  g
Start e  e  g  g and consider possible 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
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2
Negative energy
solutions –antiparticles.
QM insists we use them!
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(3) Using Feynman diagrams
e  e  g  g  g
Three vertices  probability
R
Rate  e   e   g  g  g 
Rate  e  e  g  g 


3

3
 2


   0.7  102 (1.31)
Observed R  103
 Qualitative Feynman diagram picture gives
suppression with (very) rough accuracy.
 Full QED calculation gives correct rates.
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
+ 5 other contributions
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Question
For the interaction e  e  g  g draw three Feynman diagrams which would be
suppressed wrt to those we studied earlier.
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(3) Using Feynman diagrams
e-
2
e-
e-
Two electrons are observed to repell each other:
e-
e  e  e  e
+
Many different indistinguishable processes,
eg one-photon, two-photon exchange,
can contribute to the scattering

e4
e-
1
 1
137
 higher order processes contribute less and less to the
calculation and can be safely be neglected in any approximate
Coupling is weak  
solution.
e-
e+
e-
e-
6
ee+
….
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Question
For the interaction e  e  g  g  g draw all six possible time ordered Feynman diagrams for
the leading order (3 vertices) processes
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Understanding forces
Go beyond EM force
Generic force between particles A and B via exchange of particle X .
Inititally particle A at rest: p  0, E  mAc 2 (1.32)
After vertex:
Particle A: p  p A , E  E A   p 2c 2  mA2 c 4 
1/2
(1.33)
Particle X : p   p A , E  E X   p 2c 2  mX2 c 4 
1/2
(1.34)
Energy difference between initial and final state:
E  E X  E A  mAc 2  2 pc ( p  ) (1.35)
 EX ,  p A 
 M X c 2 ( p  0) (1.36)
E  0 (apparent energy conservation violation!)
Energy violation can only persist time: t 
E
 minimum energy violation corresponds to longest time
 tmax 
M X c2
M
A
c2 , 0
 EA , p A 
(1.37)
Max speed v  c
 Range of force R 
MXc
(1.38)
 Electromagnetic range Rg   due to massless photon.
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The weak force
e-
e-
e-
W-
(decay)
e+
ne
(neutrinos – next lecture)
Use same formalism as for electromagnetic force
Very brief overview:
Exchange of 3 spin-1 particles: Z 0 (mass=91.2 GeV/c 2 ) ,W  , W  (mass=80.4 GeV/c 2 )
 range RW , Z 
MW c
 2  1018 m (tiny - proton "radius"  1 0 15m)
Define coupling constant analagous to fine structure constant
gW2
W 
4 c
(1.39)

e2
4 0 c
(1.24)
gW analagous to electric charge (very few talk of "weak charge")
gW2
1
W 

4 c 240
1


(1.41)   
, the weak force is only weak due to W , Z masses  .
137


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The fundamental forces
Different exchange particles mediate the forces:
electromagnetic
strong
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
2
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No quantum field
theory yet for gravity
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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.
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