Section IV
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.
68
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)
69
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.
70
Klein-Gordon Equation (relativistic)
From Special Relativity:
From Quantum Mechanics:
E 2  p 2  m2

ˆ
E i
t
, pˆ  i


 2ψ
 2   2ψ  m 2ψ
t
 2ψ
2
2



m
ψ
2
t
Combine to give:
KLEIN-GORDON
EQUATION
Second order in both space and time derivatives and hence Lorentz
invariant.


 i(Et- pr )
Plane wave solutions ψ(r ,t)  Ne
provided that E 2  p 2  m2
allowing
E   p  m2
2
Negative energy solutions required to form complete set of
eigenstates
ANTIMATTER
71
Antimatter
Feynman-Stückelberg interpretation: The negative energy solution is a
negative energy particle travelling backwards 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+
72
More particle/anti-particle examples
In first diagram, the interpretation easy as the first photon emitted
has less energy than the photon 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”.
73
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:
74
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!
75
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
E

E

2
E
j i
j i
i
j
E  E
j
 Ei 
76
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.77
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
78
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
79
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.
80
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.
81
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
82
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.
83
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
84
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
85
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
86
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.
87
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
88
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
89
Section V
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
91

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
92
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
93
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
94
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/
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
95
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 55):
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
96
(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
97
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
98
(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
99
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
100
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 1 (Question 11)

g
e
Qu e
u
d

u


σ π - p  μ  μ   hadrons  Qu2 e 4  Qu2 α 2
101
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
102
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 !
103
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.
104

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. .
105
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
106