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Fundamental principles of particle physics
G.Ross, CERN, July03
Fundamental principles of particle physics
G.Ross, CERN, July03
Outline

Introduction - Fundamental particles and interactions

Symmetries I - Relativity

Quantum field theory

Cross sections and decay rates - theory confronts experiment


Symmetries – Gauge symmetries and QED
Mainly fermions and the weak interactions
Fundamental principles of particle physics
G.Ross, CERN, July03
Fundamental Interactions
Strength
Strong
 s  4 c 1†
2
1
Electromagnetic  em  4e c 137
Weak
GF m 2p 105†
Gravitational
GN m 2p 1036
g s2
†
Short range
Fundamental principles of particle physics
G.Ross, CERN, July03
Fundamental Interactions
Strength
Strong
 s  4 c 1†
2
1
Electromagnetic  em  4e c 137
Weak
GF m 2p 105†
Gravitational
GN m 2p 1036
g s2
Fundamental Particles
Fundamental principles of particle physics
G.Ross, CERN, July03
Fundamental Interactions
Strength
Strong
 s  4 c 1†
2
1
Electromagnetic  em  4e c 137
Weak
GF m 2p 105†
Gravitational
GN m 2p 1036
g s2
Fundamental Particles
Fundamental principles of particle physics
G.Ross, CERN, July03
Fundamental Interactions
Fundamental Particles
Strength
Strong
 s  4 c 1†
2
1
Electromagnetic  em  4e c 137
Weak
GF m 2p 105†
Gravitational
GN m 2p 1036
g s2
Strength
Size
E  PE + KE = -
Energy
e2
4 r

p2
2 me
p.r 
Heisenberg's
Uncertainty principle
r
1
2
3
5
E-
-5
-10
E
4
-15
e2
4 r
 2 mr 2
2
-20
-25
E
r
0 
 em 
e2
4 c

me rc
Units
c  3.10 m/sec
8
 1034 kg m2 /sec
Natural Units
Length : L
Time : T
Energy : E
or Mass : m
}
Choose units such that :
c 1 L /T
 1 E.T
1 unit left : choose
E  1 GeV
( M .L2 / T )
( 109 electron volts = 1.6 10-10 J)
Natural Units
1  c  3.108 m/sec = 3.1023 fm/sec
1   10 kg m /sec = 10 J/sec
34
2
1.6 10-10 J
energy of 1 GeV
length of 1 GeV 1
time of 1 GeV 1
me
†
GeV fm
...typical of elementary particles
0.2 fm
0.7 1024 sec
 em 
1010 m 105 fm
3
1
5
1.8 10-27 kg
mass of 1 GeV
ratom
34
0.510 GeV
†
e2
4 c

me rc

1
me r
em 
e2
4
2
10
Fundamental Interactions and sizes
Strength
Strong
 s  4 c 1
2
1
Electromagnetic  em  4e c 137
Weak
GF m 2p 105
Gravitational
GN m 2p 1036
g s2
  m1r (dimensionless - same in any units)
ratom
me
rnucleus
mp
1010 m 105 fm
0.5103 GeV
1015 m 1 fm
1 GeV
em 
e2
4
 strong 
 102
g2
4
1
Elementary particles
Leptons :
e ,  ,



 e ,   , 
Quarks :
2/ 3
2/ 3
u ,c , t
2/ 3
d 1/ 3 , s 1/ 3 , b 1/ 3
Elementary particles
Leptons :
e ,  ,



 e ,   , 
Quarks :
u, c, t
u, c, t
u, c, t
Hadrons : 3 strong “colour” charges
d , s, b
d , s, b
d , s, b
Elementary forces
Exchange forces
e
Electromagnetism
Vem (r ) 


e1 e2 1
4 r

Nucleus
Experiments conducted in momentum space :
Vem ( q )
 ik .r 3
V
(
r
)
e
d r
 em

k
2
The strength
of the force
Exchange forces
e
Electromagnetism
Vem (r ) 

 (Q )
2
e1 e2 1
4 r

Q
1
R
Nucleus
Experiments conducted in momentum space :
Vem ( q )
 ik .r 3
V
(
r
)
e
d r
 em

k
2
Q 2  k 2
" virtual photon "
Exchange forces
q
Strong interactions

Vstrong (r ) 
g s2 1
4 r
g(Q2)

q
In momentum space :
Vs ( q )
 V ( r )e
s
 ik .r
3
d r
s
k
2
" virtual gluon "
Exchange forces
e
Weak force


Vweak (r ) 
g1 g2 1
4 r
e M Z r
Z (Q2)

q
In momentum space :
Vweak ( k )
 ik .r 3
V
(
r
)
e
d r
 weak
 weak
k
2
 M Z2
" virtual Z boson "
GF 
1
M Z2
Exchange forces
m1

Gravitational force
G (Q2)
Vgravity (r )  G
m1 m2
N
r

m2
Vgravity ( k )
V
gravity
( r )e
 ik .r
3
d r
GN
m1m2
k
2
" virtual graviton "
Exchange forces
m1

Gravitational force
G (Q2)
Vgravity (r )  G
m1 m2
N
r

m2
GN  6.6 1011 m3 / kg.sec 2 ....could provide fundamental scale :
mass  ( c / GN )1/ 2  1.2 1019 GeV
length  1033 cm
or maybe M Planck not fundamental !
Units
c  3.10 m/sec
8
 1034 kg m2 /sec
c 1 L /T
 1 E.T
( M .L2 / T )
Symmetries
Central to our description of the fundamental forces :
Relativity - translations and Lorentz transformations
Lie symmetries -
SU (3)  SU (2)  U (1)
Copernican principle : “Your system of co-ordinates and units is nothing special”
Physics independent of system choice
Special relativity
a   (ct , x, y, z )

Space time point

Space-time vector
not invariant under translations
(a  a)   a   a   (ct , x, y, z )
Invariant under translations …but not invariant under rotations or boosts

Einstein postulate : the real invariant distance is
 a

0 2
  a

1 2
  a

2 2
  a

3 2

3


, 0
g  a  a  a  a   a 
2
g   diag (1, 1, 1, 1)

Physics invariant under all transformations that leave all such distances invariant :
Translations and Lorentz transformations
Lorentz transformations :
3
x   x   x  x

 

 0
 



 
g x x  g x  x
 g     g
(Summation assumed)
Solutions :

0
1

 0 cos 
0
0

 0  sin 


3 rotations R
3 boosts B
 cosh 

 sinh 
 0

 0
0
0 

0 sin  
1
0 

0 cos  
Space reflection – parity P
1 0 0 0 


0

1
0
0


 0 0 1 0 


0
0
0

1



sinh 
cosh 
0
0
0
0
1
0
0

0
0

1 
Time reflection, time reversal T
 1

0
0

0
0
1
0
0
0
0
1
0
0

0
0

1 

The Lorentz transformations are a group
g1 g2  L
g1 , g 2  L
if
Rotations
R( )  e
 iJ . /
[ Ji , J j ]  i
 ijk
,
Angular momentum
operator
(c. f . J  r  p)
3

k 1
ijk
Jk
totally antisymmetric Levi-Civita symbol, 123  1
Representations
e.g. J  1/ 2 J i   i / 2
J
J z  i ( x y  y x )..
z 1/ 2
0
 
1
J  0, 12 ,1,...
0 1
0
i
1
0
 
 2  
 3  

1
0

i
0
0

1






1
Demonstration that
RZ ( )  e iJ z
 x   cos 
RZ ( )    
 y    sin 
sin   x   x ' 
    
cos   y   y ' 
For small  = ,
 x'  x  y
 

y
'
y


x
  



RZ ( ) (t , x, y, z )   (t , x   y, y   x, z )  (t , x, y, z )   ( y
x
)
x
y
 (1  i ( xp y  ypx )) (t , x, y, z )
i.e. RZ ( )  (1  i ( xp y  ypx ))  1  i J z
Hence
iJ z
RZ (  n ) (1  i J Z )n 

e
n
Derivation of the commutation relations of SU(2)
Rx ( ) Ry ( ) Rx ( ) Ry ( )
1 0

 0 1
 0 

0  1 0   1 0 0  1 0  



 
0
1
0
0
1


0
1
0







1   0 1  0  1   0 1 
 1  0 




1
0

  Rz ( ) (1  i J z )
 0

0
1


Rx ( ) Ry ( ) Rx ( ) Ry ( )
 (1  i J x )(1  i J y )(1  i J x )(1  i J y )   ( J x J y  J y J x )
Equating the two equations implies
[ J x , J y ]  iJ z
QED
Rotations Ji Boosts Ki
The Lorentz group
[J i , J j ]  i ijk J k
}
[J i , K j ]  i ijk K k
[K i , K j ]  i ijk J k
(M   i( x

x
 x

x 
)
Generate the group SO(3,1)
J i  12  ijk M jk
Ki  M 0i )
To construct representations a more convenient (non-Heritian) basis is
Ni  12 ( J i  iKi )
[N i , N j ]  i ijk N k
[N i† , N j† ]  i ijk N k†
[N i , N j† ]  0
}
SU (2)  SU (2) representation
(n, m)
Rotations Ji Boosts Ki
The Lorentz group
[J i , J j ]  i ijk J k
}
[J i , K j ]  i ijk K k
[K i , K j ]  i ijk J k
(M   i( x

x
 x

x 
)
Generate the group SO(3,1)
J i  12  ijk M jk
Ki  M 0i )
To construct representations a more convenient (non-Hermitian) basis is
Ni  12 ( J i  iKi )
[N i , N j ]  i ijk N k
Representations
(n, m)
J i  N i  N i†
J  nm
[N i† , N j† ]  i ijk N k†
(0, 0) scalar J=0
[N i , N ]  0
( 12 , 0), (0, 12 ) LH and RH spinors J= 12
†
j
( 12 , 12 ) vector J=1, etc
( 12 , 0) (0, 12 )
Weyl spinors
L
R
2-component spinors of SU(2)
Rotations and Boosts
 L( R)  SL( R)  L( R)
SL( R)  e
i σ2 .ω
SL( R)  e
 σ2 . ν
: Rotations
: Boosts
Dirac spinor
Can combine
 L, R
Lorentz transformations
where
Note :
0
0  
 I
to form a 4-component “Dirac” spinor
  ei ,
I 
 0
, i  

0
 σi
 L ( R )  12 (1  5 )
 L 
= 
 R 
       2i   ,     
σi 
I 0 
3
0
1
2
,  5  i     


o
0

I


Weyl basis
Relativistic quantum field theory
Fundamental division of physicist’s world :
speed
slow
A
c
large
fast
Classical
Newton
Classical
relativity
(QM amplitude  e
t
i
o
n
S
i( )
small
Classical
Quantum
mechanics
Quantum
Field
theory
(S )
c
)
The Klein Gordon equation (1926)
Scalar field (J=0) :
E p m
2
2
2
Energy eigenvalues
 (x)
2
  t 2  2  m2
E  (p2  m2 )1/ 2 ???
1927 Dirac tried to eliminate negative solutions by writing a relativistic
equation linear in E
1934 Pauli and Weisskopf revived KG equation with E<0 solutions as E>0
Solutions for particles of opposite charge (antiparticles). Unlike Dirac’s hole
theory this interpretation is applicable to bosons (integer spin) as well as to
fermions (half integer spin).
Physical interpretation of Quantum Mechanics
i t  21m  2  0
Schrödinger equation (S.E.)

t
i ( S .E.)  i (S .E.)
*
*
= 
 .j  0 continuity eq.
j   2im ( *   * )
2
“probability density”
“probability current”
  i( * t   t )
*
Klein Gordon equation
  Neip. x ,   2 E N
V
}
3

dV


d

 x  2E
Lorentz invariant
2
f p  e
j  i( *   * )
j   (  , j)
ip. x
1
2 p0V
Normalised free particle solutions
Field theory of


Scalar particle – satisfies KG equation
   ( 1c t , ),    ( 1c t , )
(    m )  0


2
Classical electrodynamics, motion of charge –e in EM potential
Is obtained by the substitution :

Quantum mechanics :
A   (A0 , A)
p   p   eA
i   i   eA
The Klein Gordon equation becomes:
(     m2 )  V where V  ie(  A  A   )  e2 A2
em  4e
2
The smallness of the EM coupling,
1
137 ,
Make a “perturbation” expansion of V in powers of
means that it is sensible to
 em
(     m2 )  V
Want to solve :
 ( x)   ( x)   d 4 x '  F ( x ' x)V ( x ') ( x ')
Solution :
(     m2 ) F ( x ' x)   4 ( x ' x)
where
Dirac Delta function
Feynman propagator
Simplest to solve for propagator in momentum space by taking Fourier transform
1
(2
2 2) 2
 ip .( x '  x )
e

(     m2 ) F ( x ' x)d 4 ( x ' x)  (221 )2  e ip.( x ' x ) 4 ( x ' x)d 4 ( x ' x)
2
( p 2  m 2 ) F ( p) 
 F ( p)  (2 )2
1
1
 p2  m2 i
,
11
2(22) 2
 F ( x)   (2 )4  d p e
1
4
ip. x
1
p2 m2 i
The Born series
 ( x)   ( x)   d x '  F ( x ' x)V ( x ') ( x ')
4
Since V(x) is small can solve this equation iteratively :
 ( x)   ( x)   d 4 x '  F ( x ' x)V ( x ') ( x ')
  d 4 x '  d 4 x '' F ( x ' x)V ( x ') F ( x '' x ')V ( x '') ( x '')
 ....
The Born series
 ( x)   ( x)   d x '  F ( x ' x)V ( x ') ( x ')
4
Since V(x) is small can solve this equation iteratively :
 ( x)   ( x)   d 4 x '  F ( x ' x)V ( x ') ( x ')
  d 4 x ''  d 4 x '''  F ( x '' x)V ( x '')  F ( x ''' x '')V ( x ''') ( x ''')
 ....
x
Interpretation :
V ( x ')
x'
x '''
x ''
 F ( x ' x)
But energy eigenvalues
E  (p2  m2 )1/ 2
???
Feynman – Stuckelberg interpretation

  ( E  0)
e
  ( E  0)
ei (  E )( t )
 iEt
Two different time orderings giving same observable event :
t1
t2
  
   
 
time
space
t1
t2
  
     
 
But energy eigenvalues
E  (p2  m2 )1/ 2
???
Feynman – Stuckelberg interpretation

  ( E  0)
e
  ( E  0)
ei (  E )( t )
 iEt
Two different time orderings giving same observable event :
t1
t2
  
   
 
time
space
t1
t2
  
     
 
t1
t2
  
   
 
t1
t2
  
     
 
time
space
 F ( x ' x)   (2 )4  d p e
4
1

ip.( x '  x )
1
2
p  m2 i
 i  (2 )3 2 p0 e
d3 p
ip0 t ' t ip.( x '  x )

 F ( x)  i  d p f ( x ') f ( x) (t ' t )  i  d p f ( x ') f p* ( x ) (t  t ')
3
p
where
f p  e
ip. x
1
2 p0V
*
p
3
p
are positive and negative solutions to free KG equation
Scattering in Quantum Mechanics

Prepare state at t  
| in (t  )   | i 

Time evolution (possibly scattering)
| in (t  )   S | in (t  ) 

Observe resulting system in state
| out (t  )   | f 
QM : probability amplitude :
  out (t  ) | in (t  )   out (t  ) | S |  in (t  ) 
  f | S | i  S fi
S fi   fi  iT fi
S matrix for Klein Gordon scattering
Relativistic probability density
3
*
d
x
f
S p '  , p    out (t  ) |  in (t  )   lim
p
'  i 0 ( x)
t  
 ( x)   ( x)   d 4 x '  F ( x ' x)V ( x ') ( x ')  ...


 F ( x)  i  d p f ( x ') f ( x) (t ' t )  i  d p f ( x ') f p* ( x) (t  t ')
3
*
p
p
3
p
S p '  , p    ( p  p )  i  d y f
3
'

4
*
p
( y ) V ( y ) f p ( y )  ...
Feynman rules
iT fi  i  d 4 y f p* ( y ) V ( y ) f p ( y )  i  d 4 y f p* ( y ) ie( A      A ) f p ( y )
 i  d y j  A
4
fi
f p  e


jfi  ie f p* ( y)   f p ( y)     f p* ( y)  f p ( y )
 e( p f  pi )  e
ip. x

i ( p f  pi ). x
k, 
ie( p  p ' )
p
p'
Feynman rule associated
with Feynman diagram
1
2 p0V
Fundamental experimental objects
(a1  b1b2 ...bn )
Decay width = 1/lifetime
 (a1a2  b1b2 ...bn )
Cross section
Units “barn”
1 barn  102 fm2
1 mb  101 fm2 " milli "
1 b  104 fm2 " micro "
1 nb  107 fm 2 " nano "
1 pb  1010 fm2 " pico "
1 fb  1013 fm 2 " fempto "
(Natural Units
1GeV 2  0.39mb)
(Dimension 1/T=M)
(Dimension L2=M-2)
Fundamental experimental objects
(a1  b1b2 ...bn )
Decay width = 1/lifetime
 (a1a2  b1b2 ...bn )
Cross section
(Dimension 1/T=M)
(Dimension L2=M-2)
Transition rate x Number of final states
Cross section =
Initial flux
Fundamental experimental objects
(a1  b1b2 ...bn )
Decay width = 1/lifetime
 (a1a2  b1b2 ...bn )
Cross section
(Dimension 1/T=M)
(Dimension L2=M-2)
Momenta of final state forms phase space
Transition rate x Number of final states
Cross section =
Initial flux
For a single particle the number of final states in volume V with momenta
in element
3
d p
is
Vd 3 p
(2 )3

n Vd 3 p
i 1 (2 )3
Fundamental experimental objects
(a1  b1b2 ...bn )
Decay width = 1/lifetime
 (a1a2  b1b2 ...bn )
Cross section

n Vd 3 p
i 1 (2 )3
Transition rate x Number of final states
Cross section =
Initial flux
a1
v a1
va1
V
( incident  1)
 V1
T fi    d 4 x  *f ( x)V ( x )i ( x )  ...
The transition rate
i , f  f p  e
e.g.
A
C
B
D
W fi 
Transition rate per unit volume
ip. x
1
2 p0V
T fi

N
V
e
ip . x
2
TV
 f ,i  e
T fi   N A NVB N2 C N D (2 ) 4  4 ( pC  pD  p A  pB )M fi
4
 1  1   1   1 
4  ( pC  pD  p A  pB ) M
2
W fi  (2 )
V4





2
E
2
E
2
E
2
E
 A  B   C   D 
ip. x
The cross section
Transition rate x Number of final states
Cross section =
Initial flux
V2
1
d 
M
4
v A 2 E A 2 EB V
2
d 3 pC d 3 pD 2
(2 ) 4 4
 ( pC  pD  p A  pB )
V
6
(2 )
2 EC 2 ED
2
M
d 
dQ
F
dQ  (2 )4  4 ( pC  pD  p A  pB )
F  v A 2 EA 2 EB
 4(( pA . pB )2  mA2 mB2 )1/ 2
3
3
d pC
d pD
(2 )3 2 EC (2 )3 2 ED
Lorentz
Invariant
Phase
space
The decay rate
1
2
d 
M dQ
2EA
dQ  (2 ) 4  4 ( p A  pB1 ...  pBn )
d 3 pB1
(2 ) 2 EB1
3
...
d 3 pBn
(2 )3 2 EBn
  
Compton scattering of a π meson
k ',  '
k, 
p
pk
Feynman rules
k ',  '
k, 
p
p'
pk
Klein Gordon
k ',  '
k, 
p'
p'
p
(     m2 )  V
V  ie(  A  A   )  e2 A2
k, 
k ',  '
k, 
ie( p  p ' )
ie 2
p
p'
p'
p
p
i
p 2  m2
External photon

Compton scattering of a π meson
k ',  '
k, 
p
pk
iM
k ',  '
k, 
p
p'
pk'
k ',  '
k, 
p'
p
p'
i
 '.(2 p ' k ')
fi
2
2
( p  k)  m
i
  .(2 p ' k )
 '.(2 p  k ')  2i . ']
2
2
( p  k ')  m
 (ie)2 [ .(2 p  k )
V2
1
d 
M
4
v A 2 E A 2 EB V
2
d 3 pC d 3 pD 2
(2 ) 4 4
 ( pC  pD  p A  pB )
V
6
(2 )
2 EC 2 ED
Compton scattering of a π meson
k ',  '
k, 
p
pk
M
fi
k ',  '
k, 
p
p'
pk
p
p'
p'
i
 '.(2 p ' k ')
2
2
( p  k)  m
i
  .(2 p ' k ')
 '.(2 p ' k ')  2i . '
2
2
( p  k)  m
  .(2 p  k )

( . ')
 d 

  2
2
 d  lab m 1  mk (1  cos  )
2
 total
k ',  '
k, 
d
8 2
|k 0  
.d  
d
3m2
2
8.10 GeV
2
2
2
 3.10 mb
( . p   '. p  0 gauge)
 total |k / m1
2 2
mk
Construction of a relativistic field theory
Lagrangian
L  T V
(Nonrelativistic mechanics)
t2
Action
S   L dt
t1


Classical path … minimises action
Quantum mechanics … sum over all paths with amplitude
Lagrangian invariant under all the symmetries of nature
-makes it easy to construct viable theories
 eiS /
Lagrangian formulation of the Klein Gordon equation
L   L d 3 x, L
Klein Gordon field
lagrangian density
 ( x)
L =    ( x)    ( x)  m2 ( x)† ( x)
†
}
}
T
V
L
L


0


(  )
(     m2 )  0
Manifestly Lorentz
invariant
Euler Lagrange equation
Klein Gordon equation
The Lagrangian and Feynman rules
Associate with the various terms in the Lagrangian a set of propagators
and vertex factors


The propagators determined by terms quadratic in the fields, using the Euler
Lagrange equations.
The remaining terms in the Lagrangian are associated with interaction vertices.
The Feynman vertex factor is just given by the coefficient of the corresponding
term in iL
e.g.  (   ieA ) ( x)  (   ieeA ) ( x)   ieA ( *   (  * ) )
†
k, 
iL  ie( pi  p f ) A *
}
p
p'
New symmetries
L =    ( x)    ( x)  m2 ( x)† ( x)
†
Is invariant under
 ( x)  ei  ( x)
…an Abelian (U(1)) gauge symmetry
A symmetry implies a conserved current and charge.
e.g.
Translation
Rotation
Momentum conservation
Angular momentum conservation
What conservation law does the U(1) invariance imply?
Noether current
L =    ( x)    ( x)  m2 ( x)† ( x)
†
Is invariant under
 ( x)  ei  ( x)
i
…an Abelian (U(1)) gauge symmetry
i 
L
L
0  L =
 
 (  )  (   † )

(  )
0 (Euler lagrange eqs.)
 L
 L  
 L

 i 
  
  i  
   (   † )


  (  )  
  (  ) 
 







 j  0,

ie  L
L
†
j  

 
†

2   (   )
 (  ) 
Noether current
The Klein Gordon current
L =    ( x)    ( x)  m2 ( x)† ( x)
†
Is invariant under
 ( x)  ei ( x)

 j  0,
…an Abelian (U(1)) gauge symmetry

ie  L
L
†
j  

 
†

2   (   )
 (  ) 
j  ie      
KG
*
*

This is of the form of the electromagnetic current we used for the KG field
Q   d 3 x j 0 is the associated conserved charge
Suppose we have two fields with different charges :
1,2 ( x)  ei Q 1,2 ( x)
1,2
L =   1 ( x)   1 ( x)  m21 ( x)† 1 ( x)
†
   2 ( x)   2 ( x)  m22 ( x)† 2 ( x)
†
..no cross terms possible (corresponding to charge conservation)
U(1) local gauge invariance and QED
 ( x)  ei ( x )Q ( x)
L =    ( x)    ( x)  m2 ( x)† ( x)
(Q  1)
†
not invariant due to derivatives
     ei ( x )  ei ( x)   iei ( x)  
To obtain invariant Lagrangian look for modified derivative transforming covariantly
D  ei ( x ) D
D     ieA
A  A  1e  
 ( x)  ei ( x ) ( x)
D  ei ( x ) D
A  A  1e  
L =  D ( x)  D  ( x)  m2 ( x)†  ( x)
†
Note :
   D     ieA
is invariant under local U(1)
is equivalent to
p   p   eA
universal coupling of electromagnetism follows from local gauge invariance
L =    ( x)    ( x)  m2 ( x)†  ( x)  j  A  O(e2 )
†
The electromagnetic Lagrangian
0

 E1
 E2

 E3
F    A   A
F  F ,
A  A  1e  
LEM   14 F F   j  A
M 2 A A
 E1
0
B3
 E2
 B3
0
 B2
B1
 E3 

B2 
 B1 

0 
Forbidden by gauge invariance
The Euler-Lagrange equations give Maxwell equations !
L
L



0

 
A
( A )
  F `  j

.E   ,
.B  0,
B
0
t
E
B 
j
t
E 
The photon propagator
1
   A
  F       A   (  A )  j

Gauge ambiguity
A  A  1e  
  A    A  1e  2
i.e. with suitable “gauge” choice of α (“ξ” gauge) want to solve
1
1
    A  (1  ) (  A )  ( g  2  (1  )   ) A  j


In momentum space
1
p p 
  2
1  
i 
g
p

(1

)
p
p


g

(1


)
 




p2 
p2 


(‘t Hooft Feynman gauge ξ=1)
The inclusion of fermions
Weyl spinors
( 12 , 0) (0, 12 )
L
R
2-component spinors of SU(2)
Rotations and Boosts
 L( R)  SL( R)  L( R)
SL( R)  e
i σ2 .ω
SL( R)  e
 σ2 . ν
: Rotations
: Boosts
Dirac spinor
Can combine
 L, R
Lorentz transformations
where
0
0  
 I
  ei ,
I 
 0
, i  

0
 σi
(Dirac gamma matrices,
Note :
to form a 4-component “Dirac” spinor
       2i   ,     
σi 
I 0 
3
0
1
2
,  5  i     


o
0

I


Weyl basis
  …new 4-vector)
 L ( R )  12 (1  5 ) ,
 L 
= 
 R 


,               2g 
The Dirac equation
  † 0  

New 4-vector
Fermions described by 4-cpt Dirac spinors

Lorentz invariant

The Lagrangian
L = i       m 
From Euler Lagrange equation obtain the Dirac equation
p
(i     m)  0
U(1) symmetry
  ei
i
( p  m)


j   e   
ie 
The Standard Model
SU (3)  SU (2)  U (1)
e.g. SU (2) local gauge invariance
i ( x ).
Q e
0 1
 0 i
1 0 
1  
 2  
 3  

1
0

i
0




 0 1

2
Q
L = i QD   Q  mQ Q
D      ig
i
2
Wi
u
Q 
d 
W ,W , Z
where
W ,i
1
 W ,i    i   ijk jW ,k
g
  i  j 
k 
  ,   i ijk

2
2
2



The strong interactions
QCD Quantum Chromodynamics
Strong coupling, α3
q
 
q
q
 
Symmetry :
Local conservation of
3 strong colour charges
a  e

iα ( x ).λ a
b
a
Gr  Gr  g13   r  f rst s Gt
q


q
(   ig3r Gr )  
Ga=1..8
Gauge boson
(J=1)
“Gluons”
QCD : a non-Abelian (SU(3))
local gauge field theory
Weak Interactions
Fermi theory of  decay
n  pe  e
u n  (1   5 )u p  u e   (1   5 )ue 
2 


GF
}
velocity
Left-handed
m=0
Right-handed
ue -  u
L
c
e+ R
Weak Interactions
Fermi theory of  decay
L
n  pe  e
u n  (1   5 )u p  u e   (1   5 )ue 
2 


GF
}
p
p
n
V-A
P, C
n
2
GF
g2

2
8
M
2
W
W+
e
e
e
e
g2
Weak Interactions
Symmetry :
SU(2) local gauge theory
 u   
   
 d L  e L
uR , d R , eR
e
a  e
iα ( x ).

a
b
a
Wr  Wr  g12   r  f rst sWt
Weak coupling, α2
u
Local conservation of
2 weak isospin charges
(   ig2rWr )  
d
Wa=1..3
Gauge boson
(J=1)
e
W , Z
Neutral currents
A non-Abelian (SU(2))
local gauge field theory
Massive vector propagator (W, Z bosons)

2
2



g
(


M
)



B

j



 
2
i
(

g

p
p
/
M
)

2
2
1
( g ( p  M )  p p ) 
p2  M 2
B    eip.x
Free particle solution
 (  1)  (0,1, i, 0) / 2
 (  0)  ( p , 0, 0, E ) / M
  
   g  
(  )* (  )
p p
M2
Helicity polarisation vectors
Propagation of unstable scalar particle
 iJ 2  F ( p )
 F ( p)
iJ
iJ
.
.
iJ
Particle decays
into final state n
 iJ  F ( p) D(n)
….
 F ( p)
No decay
iD (n)
Optical theorem – conservation of probability, time evolution is unitary
S fi   fi  iT fi
S † S  SS †  1
}
Im(Tkk )  12  Tnk
mtot
2
n
 J Im( F ( p)) 
2
1
2
 iJ 
2
F
( p) D  J  F ( p)
n
 F ( p) 
1
p 2  m 2  imtot
2
2

1
2
2
D(n) dQ
( F ( x)  e m tot t )
e
Z
e
q , l ,
q , l ,
12 ( Z  ee)( Z  hadrons)
 (e e  hadrons) 
( E 2  M Z2 ) 2  M Z2  2Z
 
e
e
Z
q , l ,
q , l ,
2
 (e e  hadrons)
n
 3 f 1 Q
 
 
f
 (e e    )
f
  (k )
 decay

e  ( q1 )
 e (q2 )
Fermi theory (‘40s)
GF
M
u (k )  (1   5 )u ( p) u (q )  (1   5 )v( p)
2
tot 
1
5
2
m
G
 F
192 3
The hard part!
 exp t 
1
 2.19703(4) 106 sec

GF  1.16637(1) 105 GeV 2
  (k )
 decay

  (k )

e  ( q1 )
W (Q)
e  ( q1 )
 e (q2 )
 e (q2 )
Q  Q
M W2

M  igW u (k ) (1   5 )u ( p)
In μ decay
g  
MW2
2
W
 gW2
M W2
0
gW u (q )  (1   5 )v( p)
Q2  O(m2 )  MW2
gW2
Q 2  M W2
M W2
Q Q
Q  M  i
2
m me
gW2
GF

M W2
2
Fundamental principles of particle physics

Introduction - Fundamental particles and interactions

Symmetries I - Relativity

Quantum field theory

Cross sections and decay rates - theory confronts experiment


Symmetries II– Gauge symmetries and QED
Mainly fermions and the weak interactions