Section VII
The Standard Model
The Standard Model
Spin ½ Fermions
LEPTONS
QUARKS
 e   μ   τ  
     
 νe   ν μ   ντ 
Charge (units of e)
-1
0
u  c t 
 d   s  b
     
2/3
-1/3

PLUS antileptons and antiquarks.
Spin 1 Bosons
Gluon
Photon
W and Z Bosons
g
g
W , Z 0
Mass (GeV/c2)
0
0
91.2/80.3
STRONG
EM
WEAK
The Standard Model also predicts the existence of a spin 0
HIGGS BOSON
which gives all particles their masses via its interactions.
201
Theoretical Framework
Slow
Macroscopic
Classical Mechanics
Fast
Special Relativity
Microscopic
Quantum Mechanics
Quantum Field Theory
The Standard Model is a collection of related GAUGE THEORIES
which are QUANTUM FIELD THEORIES that satisfy LOCAL GAUGE
INVARIANCE.
ELECTROMAGNETISM: QUANTUM ELECTRODYNAMICS (QED)
1948 Feynman, Schwinger, Tomonaga (1965 Nobel Prize)
ELECTROMAGNETISM: ELECTROWEAK UNIFICATION
+WEAK
1968 Glashow, Weinberg, Salam (1979 Nobel Prize)
STRONG:
QUANTUM CHROMODYNAMICS (QCD)
1974 Politzer, Wilczek, Gross (2004 Nobel Prize)
202
ASIDE : Relativistic Kinematics
Low energy nuclear reactions take place with typical kinetic energies of
order 10 MeV << nucleon rest energies ) non-relativistic formulae OK.
In particle physics kinetic energy is of O (100 GeV) >> rest mass
energies so relativistic formulae usually essential.
 Always treat -decay relativisticly (me ~ 0.5 MeV < 1.3 MeV ~ mn-mp).
WE WILL ASSUME UNDERSTANDING OF FOUR VECTORS and
FOUR MOMENTA from the Part II Relativity Electrodynamics and Light
course ... in natural units:
E
E

γm
 Definitions:
 
1
E
   px 
mass = m

γ

p      
velocity= v = βc(= β)
p
py
2



1

β
mom = p
p 
p  mγ β
 z
 Consequences:

2
p  p p  E  p  E 2  p x2  p y2  p z2  invariant  m 2
2
2
Kinetic Energy  T  E  m  ( γ  1)m
γE
m
203
ASIDE cont: (Decay example)
Consider decay of charged pion at rest in the lab frame
μ
Conservation of Energy
and Momentum says:
π
Appropriate choice
of axes gives, and noting
pion is at rest (pπ=0) gives:
νμ
π  μ  νμ
 E   E   E 
     
p  p  p 
     
 m2  0 2   m2  p 2   m2  p 2 

  

 
 
 

 
 

0
p
p



 
 

0
0
0

 
 







0
0
0

 
 

So have two simultaneous equations in two unknowns
Once you know p you know E etc via E 
(Simplify by taking mν=0 )
 Eμ 
mπ2  mμ2
2mπ
2
2

140 MeV   106 MeV 

2  140 MeV
m2  p 2
p
&
p .
etc ….
 110 MeV
pμ  p  30 MeV
204
ASIDE cont : Relevance of Invariant mass
 In Particle Physics, typically many particles are observed as a
result of a collision. And often many of these are the result of
decays of heavier particles. A common technique to identify
these is to form the “invariant mass” of groups of particles.
 e.g. if a particle X!1+2+3…n where we observe particles 1 to n,
2
we have:
2
2
n
n
  E1   E2 
 
 

M         ...    Ei     p i 
 p
  i 1   i 1 
p2 
1





2
X
In the specific (and common) case of a twobody decay, X!1+2, this reduces to the
same formula as on a preceding slide, i.e.

M X2  m12  m22  2 E1E2  p1 p 2 cos 
J/y

Typical invariant mass histogram:
205
ASIDE cont : Colliders and √s
Consider the collision of two particles:

p1  E1, p1


p2  E2 , p 2

s  ( p1  p2 ) 2
s  ( E1  E2 ) 2  ( p1  p 2 ) 2
The invariant quantity
 E12  | p1 |2  E22  | p 2 |2 2 E1 E2  2 p1. p 2

 m12  m22  2 E1 E2  p1 p 2 cos 


is the angle between the momentum three-vectors
s is the energy in the zero momentum (centre-of-mass) frame;
It is the amount of energy available to the interaction e.g. in particleantiparticle annihilation it is the maximum energy/mass of particle(s)
that can be produced
206
Fixed Target Collision

p1  E1, p1
s  m  m  2E1m2
2
1
2
2

p2 m2 ,0
s  2E1m2  s  2E1m2
For E1  m1 , m2
e.g. 450 GeV proton hitting a proton at rest:
s  2 Ep mp  2  450 1  30 GeV
Collider Experiment

s  m12  m22  2 E1E2  p1 p 2 cos 

 
p1  E, p

p2  E,  p

For E1  m1 , m2 then p  E and  =


s  2 E 2  E 2 cos θ  4 E 2  s  2 E
e.g. 450 GeV proton colliding with a 450 GeV proton:
s  2  450  900 GeV
In a fixed target experiment most of the proton’s energy is wasted
providing forward momentum to the final state particles rather than
being available for conversion into interesting particles.
207
ASIDE over
Back to the main topic of
combining Quantum Mechanics
with Special Relativity…..
208
Problems with Non-Relativistic Schrodinger Equation
To describe the fundamental interactions of particles we need a theory
of RELATIVISTIC QUANTUM MECHANICS.
2
Schrodinger Equation for a free particle comes from classical E  21 mv
2
ˆ
p
p2
or equivalently E 
promoted to an operator equation Eˆ ψ 
ψ
2m
2m

ħ =1
using energy and momentum operators: Ê  i , p̂  i
natural units
t
ψ
1 2

 ψ which has plane wave solutions:
giving i

t
2m
 i(Et- pr )

ψ(r ,t)  Ne
Schrodinger Equation:
 1st Order in time derivative
Not Lorentz Invariant!
nd
 2 Order in space derivatives
Schrodinger equation cannot be used to describe the physics of
relativistic particles.
209
Klein-Gordon Equation (relativistic)
E 2  p 2  m2
From Special Relativity:
From Quantum Mechanics:
Combine to give:

ˆ
E i
t
, pˆ  i


 2ψ
 2   2ψ  m 2ψ
t
 2ψ
2
2



m
ψ
2
t
KLEIN-GORDON
EQUATION
Second order in both space and time derivatives and hence Lorentz
invariant.

 i(Et- pr )
Again plane wave solutions

ψ(r ,t)  Ne
but this time requiring E  p  m
2
2
2
allowing E  
p  m2
2
Negative energy solutions required to form complete set of
eigenstates
ANTIMATTER
210
Antimatter
Feynman-Stückelberg interpretation: The negative energy solution is a
negative energy particle travelling forwards in time. And then, since:
e iEt  e i(  E)(  t)
 Interpret as a positive energy antiparticle travelling forwards in time.
Then all solutions describe physical states with positive energy, going
forward in time.
Examples: e+e annihilation
e
g
pair creation
g
e
T
T
e+
All quantum numbers carried
into vertex by e, same as if
viewed as outgoing e.
e+
211
More particle/anti-particle examples
In first diagram, the interpretation easy as the first photon emitted
has less energy than the electron it was emitted from. No need for
“anti-particles” or negative energy states.
e
g

e
g
e
T
e
g
g
e
T
In the second diagram, the emitted photon has more energy than the
electron that emitted it. So can either view the top vertex as
“emission of a negative energy electron travelling backwards in
time” or “absorption of a positive energy positron travelling
forwards in time”.
212
Normalization
Can show solutions of Schrodinger Equation (non-relativistic) satisfy:
@ ¹
¡ i ¹
¹ = 0
(
ÃÃ)
+
r
:(
(
Ãr
Ã
¡
Ãr
Ã))
@t
2m
and solutions of Klein-Gordon Equation (relativistic) satisfy:
@ù
@ ¹ @Ã
¹ Ã ¡ Ãr Ã)
¹ = 0
(
Ã
¡
Ã
)
+
r
:(
Ãr
@t
@t
@t
@½
Both are of the form:
+ r :j = 0
@t
and the divergence theorem tells us that
this means that D is conserved, where: D
Z
´
½dV
If we interpret D as number of particles, and  as number of particles
per unit volume, then:
For plane wave of arbitrary normalization à = N ei (E t ¡ p :x ) find that:
Non-relativistic: ½= jN j 2 particles per unit volume.
½= 2EjN j 2 particles per unit volume.
Relativistic:
213
Interaction via Particle Exchange


Consider two particles, fixed at r1 and r2, which exchange a particle
of mass m.
Space
1
state i
state j
state i

q  E, p 
μ
E  E j  Ei
2
Time
Calculate shift in energy of state i due to this process (relative to noninteracting theory) using 2nd order perturbation theory:
ΔEi  
j i
iH j jHi
Ei  E j
Sum over all possible states j
with different momenta.
Work in a box of volume L3, and normalize s.t. 1 1  2 2  3 3  1
i.e. use
à = L ¡ 3=2 ei (E t ¡ p :x ) :
(N = L ¡ 3=2 ) ½= 2E=L 3 ) 2E particles per box!
214
Consider j H i ( transition from i to j by emission of m at
ψ i  ψ1ψ 2 Original 2 particles
ψ j  ψ1ψ 2ψ 3
ψ 3 represents a free particle with

r1 )
 
3 / 2  i  Et  pr 
ψ3  L
e

q  E, p 
μ

Let g be the “strength” of the emission at r1 . Then:
g -3/2 i  Et  p r1 
g
3 * 
3  
j H i  1 1 2 2 3 H r 1 0   d r ψ3 r  2 E δ r  r1  
L e
2E

i
H
j
Similarly
is the transition from j to i at r2 and is
g -3/2 -i  Et  p r2 

g
3 3  
i H j  1 1 2 2 0 H r 2 3   d r δ r  r2  2 E ψ3 r  
L e
2E
Substituting these in, we see the shift in energy state is
ΔE
1 2
i
  
2 -3 ip(r2  r1 )
  
2 -3 ip(r2  r1 )
g L e
g L e


2
 2E
j  i 2E Ei  E j 
j i
E  E
j
 Ei 
215
Normalization of source strength “g” for relativistic situations
Previously normalized wave-functions to 1 particle in a box of side L.
In relativity, the box will be Lorentz contracted by a factor g
v
Rest frame
1 particle per V
Lab. frame
1 particle per V/g
 v2 
γ   1  2 
 c 

1
2

E
m
i.e. γ  E m particles per volume V. (Proportional to 2E particles per unit
volume as we already saw a few slides back)
Need to adjust “g” with energy (next year you will see “g” is “lorentz
invariant” matrix element for interaction)
Conventional choice:
g
g effective 
2E
(square root as g always occurs squared)
Source “appears” lorentz contracted to particles of high energy, and
effective coupling must decrease to compensate. The presence of the
E is important. The presence of the 2 is just convention. The absence
of the m (in gamma) is just a convention to keep g dimensionless here.
216
Beginning to put it together

Different states j have different momenta p . Therefore sum is
actually an integral over all momenta:
3
 L 
3

d

 2   p
 
i j
 L 
 
 2 
3
3
 1  2
ρ(p)    p dΩ
 2π 
2
p
 dp d
And so energy change:
ΔE
1 2
i
 L
 
 2 
3
  
2 -3 ip(r2  r1 )
g L e
  2E 2
2
g

3
22 


2 ip  r
p 2 dp d
p e
dp d
2
2
p m
  
r  r2  r1
E 2  p 2  m2
217
Final throes

The integral can be done by choosing the z-axis along p
 
Then p  r  prcos and dΩ  d d(cos )  2π d(cos )
so
ΔEi12
g2

3
22 

2
g

2
22 
2
g

2
22π 
.
  
r  r2  r1
p 2 eiprcos
dp 2d(cos )
2
2
p m



0
iprcos
cos  1
e

p
dp

2
2 
p  m  ipr  cos  1
2
 
ip  r
p
e e
p 2  m2
ipr
2
 
- ip  r
dp
Write this integral as one half of the integral from - to +, which can
be done by residues giving
Appendix D
ΔE
12
i
g 2 e  mr

8π r
218
Can also exchange particle from 2 to 1:
Space
1
state i
state j
state i
2
Get the same result:
2
 mr
g
e
ΔEi21  
8π r
Time
Total shift in energy due to particle exchange is
g 2 e  mr
ΔEi  
4π r
ATTRACTIVE force between two particles which decreases
exponentially with range r.
219
Yukawa Potential
YUKAWA POTENTIAL
g 2 e  mr
V(r)  
4π r
 Characteristic range = 1/m
(Compton wavelength of exchanged particle)
 For m0,
g2
V(r)  
4π r
Hideki Yukawa
1949 Nobel Prize
infinite range
Yukawa potential with m = 139 MeV/c2 gives good description of long
range interaction between two nucleons and was the basis for the
prediction of the existence of the pion.
220
Scattering from the Yukawa Potential
Consider elastic scattering (no energy transfer)
Born Approximation

M fi   e V(r)d r
 
ip  r
2
3
 mr
g e
V(r)  
4π r
g 2 e  mr ip r 3 
g2
M fi  
e d r  2

4π
r
p  m2
Yukawa Potential

pi

pf

p

q μ  E, p 
2
q E  p
2
2
q2 is invariant
“VIRTUAL MASS”

The integral can be done by choosing the z-axis along r , then
 

p  r  prcos and d 3 r  2πr 2 dr d(cos )

2
2
For elastic scattering, q  0, p , q   p and exchanged massive
μ
particle is highly “virtual”
g2
M fi  2
q  m2
221
Virtual Particles
Forces arise due to the exchange of unobservable VIRTUAL particles.
 The mass of the virtual particle, q2, is given by
2
q2  E 2  p
and is not the physical mass m, i.e. it is OFF MASS-SHELL.
 The mass of a virtual particle can by +ve, -ve or imaginary.
 A virtual particle which is off-mass shell by amount Dm can only exist
for time and range

1

1
t~

, range 

2
Δmc
Δm
Δmc Δm
ħ =c=1 natural units
 If q2 = m2, then the particle is real and can be observed.
222
For virtual particle exchange, expect a contribution to the matrix
element of
2
M fi 
where
g
g
q 2  m2
COUPLING CONSTANT
g2
STRENGTH OF INTERACTION
m2
PHYSICAL (On-shell) mass
q2
VIRTUAL (Off-shell) mass
1
2
2 PROPAGATOR
q m
Qualitatively: the propagator is inversely proportional to how far the
particle is off-shell. The further off-shell, the smaller the probability of
producing such a virtual state.
g2
 For m  0 ; e.g. single g exchange M fi  2
q
 q2 0, very low energy transfer EM scattering
223
Feynman Diagrams
Results of calculations based on a single process in Time-Ordered
Perturbation Theory (sometimes called old-fashioned, OFPT) depend on
the reference frame.
The sum of all time orderings is not frame dependent and provides the
basis for our relativistic theory of Quantum Mechanics.
+
Time
=
Space
Space
The sum of all time orderings are represented by FEYNMAN DIAGRAMS.
Time
FEYNMAN
DIAGRAM
224
Feynman diagrams represent a term in the perturbation theory
expansion of the matrix element for an interaction.
Normally, a matrix element contains an infinite number of Feynman
diagrams.
Total amplitude
Total rate
M fi  M 1  M 2  M 3  
Γ fi  2π M 1  M 2  M 3   ρE  Fermi’s Golden
2
Rule
But each vertex gives a factor of g, so if g is small (i.e. the perturbation
is small) only need the first few.
g2
g4
e2
1
~
Example: QEDg  e  4πα  0.30 , α 
4π 137
g6
225
Anatomy of Feynman Diagrams
Feynman devised a pictorial method for evaluating matrix elements for
the interactions between fundamental particles in a few simple rules.
We shall use Feynman diagrams extensively throughout this course.
Represent particles (and antiparticles):
Spin ½
Quarks and
Leptons
Spin 1
g, W and Z0
g
and their interaction point (vertex) with a “ “.
Each vertex gives a factor of the coupling constant, g.
226
External Lines (visible particles)
Particle
Incoming
Outgoing
Antiparticle
Incoming
Outgoing
Particle
Incoming
Outgoing
Spin ½
Spin 1
Internal lines (propagators)
Spin ½
Particle
(antiparticle)
Spin 1
g, W and Z0
g
Each propagator gives
a factor of
1
q 2  m2
227
Examples:
ELECTROMAGNETIC
e
1
q2
g
p
q
e
g
g
STRONG
q
g
g2
M~ 2
q
gs
gs
1
q2
q
g s2
M~ 2
q
q
p
Quark-antiquark annihilation
Electron-proton scattering
WEAK
u
n d
d
gWVCKM
W
gW
Neutron decay
u
d p
u
e
gW2 VCKM
M~ 2
q  M W2
νe
228
Summary
 Relativistic mechanics necessary for most calculations involving
Particle Physics and the Standard Model
 Anti-particles arise from requiring wave equation be relativistically
invariant
 Calculating Matrix Elements from Perturbation Theory from first
principles is a pain – so we don’t usually use it!
 Need to do time-ordered sums of (on mass shell) particles whose
production and decay does not conserve energy and momentum !
 Feynman Diagrams / Rules manage to represent the maths of
Perturbation Theory (to arbitrary order, if couplings are small
enough) in a very simple way – so we use them to calculate matrix
elements!
 Approx size of matrix element may be estimated from the simplest
valid Feynman Diagram for given process.
 Full matrix element requires infinite number of diagrams.
 Now only need one exchanged particle, but it is now off mass shell,
however production/decay now conserves energy and momentum.
229
Section VIII
QED
QED
QUANTUM ELECTRODYNAMICS is the gauge theory of
electromagnetic interactions.
Consider a non-relativistic charged particle in an EM field:

  
F  q E  v  B 

 
E, B given in term of vector and scalar potentials A,φ

  


A
Maxwell’s Equations
B    A ; E  φ 
t


1 
H
p  qA
2m

e
e
g

2
 qφ
Change in state of e requires change in field
 Interaction via virtual g emission
231

Schrodinger equation
2  qφψ(r,t)  i ψ(rt ,t)

Appendix E
is invariant under the gauge transformation

 1 
 2m  p  qA

iqα (r ,t)
ψ  ψ  e
ψ

 
α
A  A  α ; φ  φ 
t
 LOCAL GAUGE INVARIANCE
LOCAL GAUGE INVARIANCE requires a physical GAUGE FIELD
(photon) and completely specifies the form of the interaction between
the particle and field.
 Photons (all gauge bosons) are intrinsically massless
(though gauge bosons of the Weak Force evade this requirement by “symmetry breaking”)
 Charge is conserved – the charge q which interacts with the field
must not change in space or time.
DEMAND that
QED be a GAUGE THEORY
232
The Electromagnetic Vertex
All electromagnetic interactions can be described by the photon
propagator and the EM vertex:


e , μ ,τ
q

e , μ  , τ 
q
STANDARD MODEL
g  Qe
g
e2

4
ELECTROMAGNETIC
VERTEX
+antiparticles
 The coupling constant, g, is proportional to the fermion charge.
 Energy, momentum, angular momentum and charge always
conserved.
 QED vertex NEVER changes particle type or flavour


e

e
γ but not e   qγ or e   μ  γ
i.e.
 QED vertex always conserves PARITY
233
Pure QED Processes
Compton Scattering (gege)
g
g
g

e
e

e
e
e
Bremsstrahlung (eeg)
g
e
Nucleus

e
Ze
e
g2
M~ 2
q
M e
2
σ  M  e4
2
σ  4π  α 2
2
g
M  Ze 3
2
M  Z 2 e6
3 2 3
σ  4π  Z α
Pair Production (g ee)
3
e
M

Ze
g
e2

4
The processes eeg
and g eecannot occur
for real e, g due to
energy, momentum
conservation.
M  Z 2 e6
3
σ  4π  Z 2 α 3
2
Nucleus
e
234
e
e+e
Annihilation
e e  qq
0 Decay
  gg
g
g
M  Qu2 e 2
2
M  Qu4 e 4
2
σ  4π  Qu4  2
g
u
J/Y
M  Qq2 e 4
2
σ  4π  Qq2  2
q
u
0

c
c
M  Qq e 2
2
e
0
J/y
q
g


M  Qc e 2
2
M  Qc2 e 4
2
σ  4π  Qc2  2
The coupling strength determines “order of magnitude” of the matrix element.
For particles interacting/decaying via EM interaction: typical values for crosssections/lifetimes
sem ~ 10-2 mb
tem ~ 10-20 s
235
Scattering in QED
Examples: Calculate the “spin-less” cross-sections for the two processes:


e
e

e
(1)
(2)
e

g
p
g
1
q2
e
e
e
p
Electron-proton scattering
1
q2
e

Electron-positron annihilation
Fermi’s Golden rule and Born Approximation (see page 48):
dσ
E2

M
2
dΩ 2π 
2
For both processes write the SAME matrix element
e 2 4 πα
M 2  2
q
q
 e  4 πα is the strength of the interaction.
 1
measures the probability that the photon carries 4-momentum
2

2
q
q μ  E, p ; q 2  E 2  p i.e. smaller probability for higher mass.
2
236
(1) “Spin-less” e-p Scattering
e
e

1
q2
g
p
e
e
p
e 2 4 πα
M 2  2
q
q
2
dσ
E2
E 2 4 πα 
4α 2 E 2
2

M 

2
2
4
dΩ 2π 
q4
2π  q
q2 is the four-momentum transfer:

 2
2
2
μ

q

q
q

E

E

p

p
μ
f
i
f
i
 e
2 2
 
pf

2
2


E

E

2E
E

p

p

2
p
p
f
i
f i
f
i
f  pi
p
i

 
e
 2me2  2E f Ei  2 p f pi cos

2
p
Neglecting electron mass: i.e. me  0 and
f  Ef

 

q 2  2E f Ei (1  cos )
 4E f Ei sin 2 
2
Therefore for ELASTIC scattering Ei  E f
dσ
α2
 2 4
dΩ 4E sin 
2
RUTHERFORD
SCATTERING
237
Discovery of Quarks

q μ  E, p 

p
Large q  Large , small λ
Large E , large ω
Virtual g carries 4-momentum

p
λ
E  ω
High q wave-function oscillates rapidly in space and time  probes short
distances and short time.
Rutherford Scattering
q2
small
E = 8 GeV
Excited states
q2
increases
Expected Rutherford
scattering
q2 large
E
l<< size of proton
q2
>1
(GeV)2

Elastic scattering from
quarks in proton
238
(2) “Spin-less” e+e- Annihilation
e

e
e
e

g

1
q2

e 2 4 πα
M 2  2
q
q
2
dσ
E2
E 2 4 πα 
4α 2 E 2
2

M 

2
2
4
dΩ 2π 
q4
2π  q
Same formula, but different 4-momentum transfer:

 2
2
q 2  q μ q μ  Ee  Ee-    pe  pe- 
Assuming we are in the centre-of-mass system
dσ 4α 2 E 2 4α 2 E 2 α 2



4
4
dΩ
q
16E
s
Ee  Ee-  E


pe-   pe 
2
q 2  q μ q μ  2E   s
Integrating gives total cross-section:
4 πα 2
σ
s
239
This is not quite correct, because we have neglected spin. The actual
cross-section (using the Dirac equation) is

dσ α 2

1  cos 2 
dΩ 4s

σ (nb )
ee  μ  μ 
4 πα 2
σ(e e  μ μ ) 
3s
 


Example: Cross-section at s  22 GeV
(i.e. 11 GeV electrons colliding with
11 GeV positrons)
4 πα 2
4π
1
s GeV 
 
 
σ(e e  μ μ ) 

3s
137 2 3  22 2
 4.6  10 7 GeV  2
2
 4.6  10 7  0.197  fm2
 1.8  10 8 fm2
c  0.197 GeVfm
 0.18 nb
240
The Drell-Yan Process
Can also annihilate q q as in the Drell-Yan process.
Example: π - p  μ  μ   hadrons
d
u
u
u
See example sheet (Question 25)

g
e
Qu e
u
d

u


σ π - p  μ  μ   hadrons  Qu2 e 4  Qu2 α 2
241
Experimental Tests of QED
QED is an extremely successful theory tested to very high precision.
Example:
 Magnetic moments of e,  :
 For a point-like spin ½ particle:
e 

μg
s
2m
g  2 Dirac Equation
However, higher order terms introduce an anomalous magnetic
moment i.e. g not quite 2.

B
α
O(1)
v
v
α
O()
   
O(4)
12672 diagrams
242
O(3)
ge  2
 11596521.869  0.04110 10 Experiment
2
ge  2
Theory
 11596521.3  0.3 10 10
2
 Agreement at the level of 1 in 108.
 QED provides a remarkable precise description of the
electromagnetic interaction !
243
Higher Orders
So far only considered lowest order term in the perturbation series.
Higher order terms also contribute
e
Lowest
Order
Second
Order
Third
Order
g
e

e
e

e

g
e
g

g
e
e
   

g
e

4

e
e
2
+
e
e
2
1

M  α 4 ~ 137


 e

e
e
e
e
e
1

M  e 4  α 2 ~ 137
2
+
e

1

M  α 6 ~ 137
2
6
  
e2

4
Second order suppressed by 2 relative to first order. Provided  is
small, i.e. perturbation is small, lowest order dominates.
244

2
Running of 
e
4 specifies the strength of the interaction between an electron
and a photon.
 BUT  is NOT a constant.


 
 Q






-
Consider an electric charge in a dielectric medium.
Charge Q appears screened by a halo of +ve charges.
Only see full value of charge Q at small distance.

e
Consider a free electron.
The same effect can happen due to quantum
g
fluctuations that lead to a cloud of virtual e+e- pairs
e-
e-
g
e+
e+
g
e+
g
eg
e-
 The vacuum acts like a dielectric medium
 The virtual ee pairs are polarised
 At large distances the bare electron charge is screened.
 At shorter distances, screening effect reduced and see a larger
effective charge i.e. .
245
Measure (q2) from e  e   μ  μ  etc
e
g
e
e
e

1
q2

  increases with increasing q2
(i.e. closer to the bare charge)
 At q2=0:
=1/137
 At q2=(100 GeV)2: =1/128
246
Summary
 QED is the physics of the photon + “charged particle” vertex:
e , μ  , τ 
q
e , μ  , τ 
q
g  Qe
g
e2

4
 Every Feynman vertex has:





An arrow going in (lepton or quark)
An arrow going out (lepton or quark)
And a photon
Does not change the type of lepton or quark “passing through”
Conserves charge, energy and momentum.
 The coupling α “runs”, and is proportional to the electric charge of
the lepton or quark
 QED has been tested at the level of 1 part in 108.
247