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Particle Physics 2
𝒈
𝝂𝝁
𝝎
𝝓
𝑲+
𝒆
𝑩
𝑾±
𝑱/𝝍
𝝁
𝒒
𝒁𝟎
𝝌
𝝂𝝉
𝜸
𝑯𝟎
𝝂𝒆 𝝉
Prof. Glenn Patrick
U23525, Particle Physics, Year 3
University of Portsmouth, 2013 - 2014
1
Last Lecture
Particle Physics 1
Course Outline
Preliminaries - Assessment
Preliminaries - Books
Preliminaries - Course Material
Particle Physics, Cosmology & Particle Astrophysics
Natural Units
Rationalised Heaviside-Lorentz EM Units
Special Relativity and Lorentz Invariance
Mandelstam Variables (s, t and u)
Crossing Symmetry and s, t & u Channels
Spin and Spin Statistics Theorem – Fermions and Bosons
Addition of Angular Momentum
Non-Relativistic Quantum Mechanics (Schrödinger Equation)
Relativistic Quantum Mechanics (Klein-Gordon Equation)
Feynman-Stückelberg Interpretation of Negative Energy States
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Today’s Plan
Particle Physics 2
Dirac Equation
Dirac Interpretation of Negative States
Stückelberg & Feynman Interpretations
Discovery: Positron & e+e- Pair Production
Decays – Lifetimes
Scattering – Cross-sections
Feynman Diagrams
Quantum Electrodynamics (QED)
Higher Order Diagrams
LEP 𝑒 + 𝑒 − → 𝛾𝛾 𝛾 example
Running Coupling Constant (𝛼)
QED Renormalisation
Electron/Positron Annihilation & precision
LEP electroweak example
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Correction - Assessment
40 hours of lectures across two teaching blocks
plus 8 hours of tutorial classes.
The main aim is to improve your understanding of
fundamental physics.
However, we cannot forget the small matter of your degree….
Not 3 hours as I said last week.
1 Final written examination (2 hours)
– 80%
2 Coursework questions and problems
– 20%




Main thing is that you enjoy the course.
We will try and focus on understanding the underlying concepts.
Extra material/maths shown mainly to aid understanding.
Guidance will be given over essential knowledge needed for exam.
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Corrections - Timetable
 There is NO lecture this afternoon at 15:00 – 17:00. The
timetable was wrong.
 There are also 2 extra weeks added in March to the end of
teaching block 2 (weeks 34 and 35).
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Moodle Site
Slides - pptx
Slides - pdf
6
2013 Nobel Prize – Particle Physics
The Nobel Prize in Physics 2013 was awarded jointly to
François Englert and Peter W. Higgs
"for the theoretical discovery of a mechanism that
contributes to our understanding of the origin of mass
of subatomic particles, and which recently was
confirmed through the discovery of the predicted
fundamental particle, by the ATLAS and CMS
experiments at CERN's Large Hadron Collider”.
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Dirac Equation
The Klein Gordon equation was believed to be sick, but today
we now understand it was telling us something about
antiparticles.
To try to solve the problem of –ve energy solutions, Dirac:
 Wanted an equation first order in 𝜕 𝜕𝑡 .
 Started by assuming a Hamiltonian, which is local and
linear in p and of the form:
H D  (  p  m)  1 P1   2 P2   3 P3  m
To also make the equation relativistically covariant, it has to be linear in 𝛻:

i
 (i    m)
t
Solutions of this equation were also required to be solutions of the
Klein Gordon equation.
Only true if: 𝜶𝒊 𝟐 = 𝟏, 𝜷𝟐 = 𝟏, 𝜶𝒊 𝜷 + 𝜷𝜶𝒊 = 𝟎, 𝜶𝒊 𝜶𝒋 + 𝜶𝒋 𝜶𝒊 = 𝟎 (𝒊 ≠ 𝒋)
Coefficients do not commute so they cannot be numbers – require 4 matrices.
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Dirac Equation
In two dimensions, a natural set of matrices would be the Pauli spin matrices
(from atomic physics for electron spin):
0 1
0  i
1 0 
  2  
  3  

 1  
1 0
i 0 
 0  1
but there is no suitable fourth
anti-commuting matrix for β.
Convenient choice is to instead use the Dirac-Pauli representation:
0

0
1  
0

1

0
0
1
0
0
1
0
0
0 0
0 0 0  i
1




0
0 0 i 0    0 0
2  
 3 1 0

0

i
0
0
0




 1 1



i
0
0
0

0


1 0

0  1
0 0

0 0 
1 0 0 0 


0 1 0 0 
 
1 0 1 0 


 1  1 0  1


2 x 2 identity matrix
These can be abbreviated as:
(each element is a 2 x 2 matrix)
0  
1 0 
   

  
 0 
 0  1
Block matrix (2 x 2)
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Dirac Equation
Now that 𝛼 and 𝛽 are matrices, the equation
makes no sense unless the wave function Ψ is
itself a matrix with four rows and one column.
This is the Dirac Spinor:
(a four component
wavefunction)

i
 (i    m)
t
 1 (r , t ) 


 2 (r , t ) 
 (r , t )  
3 (r , t ) 


  (r , t ) 
 4

Plane wave solutions take the form:
 (r , t )  u ( p )ei ( p.r et ) 
where 𝑢(𝑝) is also a four-component spinor satisfying the eigenvalue equation:
H p u ( p )  (  p  m)u ( p )  Eu ( p )
There are four solutions: two with positive energy E = +Ep corresponding to
the two possible spin states of a spin 1 2 particle and two corresponding
negative energy solutions with E = - Ep.
Electron spin is therefore included in a natural way (rather than the previous
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ad-hoc attempts). All solutions also have positive probability density.
Dirac Interpretation
Dirac interpreted the negative energy solutions by:
 Postulating the existence of a “sea” of negative energy states, which are
almost always filled – each with two electrons (spin “up” and spin “down”):
 When an electron is added to the vacuum, it is confined to the positive
energy region since all negative energy states are occupied (Pauli exclusion
principle).
 When energy is supplied to promote a negative energy electron to a positive
energy level, an electron-hole pair is created. The hole is seen as a charge
+e and E > 0 state.
A false start when Dirac
identified negative
energy electrons as
protons.
Eventually, persuaded by
arguments of Weyl &
Oppenheimer that +ve particle had
to have the same mass as
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the electron.
Dirac Interpretation
Photon with 𝐸 > 2𝑚𝑒
excites electron from
–ve energy state.
𝜸
Leaves hole in vacuum
corresponding to a state
with more energy (less
negative energy) and a
positive charge wrt the
vacuum.
Sea
Problems with this picture.
 Bosons have no exclusion principle, so this picture does not work for them.
 It implies the Universe has infinite negative energy!
Nonetheless, in 1931 Dirac postulated the existence of the positron as the
electron’s antiparticle.
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It was discovered 1 year later.
Positron - First Antiparticle Discovered!
Positron = the anti-electron
6 mm lead plate
Discovered in 1932
by Carl Anderson
+
e
photographing
cosmic ray tracks in
a cloud chamber.
+
e
e-
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𝒆+ 𝒆− Pair Production
In 1933, Blackett & Occhialini observed pair production in a
triggered cloud chamber and confirmed that Anderson’s particle
was indeed Dirac’s positron.
𝜸 + 𝒏𝒖𝒄𝒍𝒆𝒖𝒔 → 𝒆+ + 𝒆− + 𝒏𝒖𝒄𝒍𝒆𝒖𝒔
𝒆−
𝜸
𝒆+
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Stückelberg Interpretation
As early as 1941, Stückelberg cultivated the view that positrons may be
understood as electrons running backward in time.
E.C.G. Stueckelberg, Helvetica Physica Acta, 14 (1941), 588-594
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Feynman Interpretation
“The problem of the behavior of positrons and
electrons in given external potentials, neglecting their
mutual interaction, is analyzed by replacing the theory
of holes by a reinterpretation of the solutions of the
Dirac equation…..”.
July
1962
…In this solution, the “negative energy states” appear
in a form which may be pictured (as by Stückelberg)
in space-time as waves traveling away from the
external potential backwards in time. Experimentally,
such a wave corresponds to a positron approaching
the potential and annihilating the electron….
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Feynman Interpretation
Waves can proceed
backward in time
Virtual pair production.
Positron goes forward
in time (4 to 3) to be
annihilated and electron
backwards (3 to 4)
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One electron in the entire Universe?
It was Feynman’s PhD supervisor, John Wheeler, who
suggested that there may be only a single electron in the
universe, propagating through space and time!
This obviously has a few problems! For
example:
• You would expect equal number of
electrons and positrons and yet we
observe extremely few positrons.
• How do we account for the fact that
electrons can be created/destroyed in
weak interactions.
• This poor single electron would have
had to traverse huge distances and be
very ancient.
World-line of
single electron
Nonetheless, Feynman kept the idea that
positrons could simply be represented as
electrons going from the future to the past18.
Reminder: this is Applied Physics!
This quantum theory is all
very well, but what can we
physically measure?
Quantum
Theorist
Applied
Physicist
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Decays - Lifetimes
In the case of particle decays, the most interesting physical quantity is the
lifetime of the particle.
 Measured in the rest frame of the particle. We can define the decay rate, Γ,
as the probability per unit time the particle will decay.
Similar to your Year 2
nuclear physics
𝑑𝑁 = −Γ𝑁𝑑𝑡
Mean Lifetime (natural units) τ =
1
Γ
𝑁 𝑡 = 𝑁(0)𝑒 −Γ𝑡
ℏ
𝐹𝑜𝑟 𝑠𝑒𝑐𝑜𝑛𝑑𝑠, 𝑢𝑠𝑒 𝜏 =
Γ
𝑛
 Most particles decay by several different routes. The total
decay rate is then the sum of the individual decay rates:
 The lifetime of the particle is then:
1
𝜏=
Γ𝑡𝑜𝑡
Γ𝑡𝑜𝑡 =
Γ𝑖
𝑖=1
 The different final states are known as decay modes.
Γ𝑖
 The branching ratio for i’th decay mode is: 𝐵𝑟𝑎𝑛𝑐ℎ𝑖𝑛𝑔 𝑅𝑎𝑡𝑖𝑜 =
Γ𝑡𝑜𝑡
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𝟎
𝑲
𝑺
Meson Example
Particle Data Group: http://pdg.lbl.gov/
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Scattering - Cross-Section
na = no. of beam particles
va = velocity of beam particles
nb = no. target particles/area
Incident flux F=nava
dN = no. scattered particles
in solid angle dΩ
dΩ  sin .d.d
dN (d)  na va nb d (d)  Fnb d (d)  Ld (d)
Differential d
1 dN
cross-section d  L d
Total
cross-section
Luminosity L = flux x no. targets (cm-2s-1)
d
 
d
d
Measured in barns. 1 b = 10-24 cm-2
Event rate N  L
Cross-section quantifies
rate of reaction. Depends
on underlying physics.22
Feynman Diagrams
“Like the silicon chip of more recent
years, the Feynman diagram was
bringing computation to the masses”.
Julian Schwinger
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Feynman Diagrams – The Problem
𝒆+ 𝒆− → 𝝁+ 𝝁−
Consider the calculation of the
cross-section of one of the simplest
QED processes.
In the Centre of Mass frame:
𝒑′ = 𝒑, 𝒌′ = 𝒌
𝐸
𝒑 = 𝒑′ = 𝒌 = 𝒌′ = 𝐶𝑀 2
where 𝑝, 𝑝′ , 𝑘, 𝑘 ′ are momenta
𝑒−
𝒌
𝒑
𝜇−
𝜃
𝒑′
𝜇+
𝑒+
𝒌′ ℳ = QM amplitude for
process.
d
1
2
Differential Cross - section


2 2
d 64 ECM
Bad News! Even for this simple process the exact expression is not known.
Best that can be done is to obtain a formal expression for ℳ as a perturbation
series in the strength of the EM interaction & evaluate the first few terms.
Feynman invented a beautiful way to organise, visualise and thereby calculate
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the perturbation series.
Quantum Field Theory - Basics
• In Quantum Field Theory, the scattering and decay of particles
is described in terms of transition amplitudes.
•
For a transition process |𝑖 ⟶ |𝑓 , the transition amplitude is
written as 𝑓 𝑆 𝑖 , where S is an operator known as the
Scattering Matrix or S-Matrix.
•
Exact calculations of 𝑓 𝑆 𝑖 are not possible, but it is possible
to use perturbation theory which allow approximate
calculations of the transition amplitude.
• A bit like a binomial/Taylor series expansion:
1
−
2
𝑥
1
3
−2 −2
e.g. (1 + 𝑥) = 1 − +
𝑥 2 + ⋯ and then keeping the
2
2
first 2 terms in the expansion (for small values of x).
• Feynman diagrams are a technique to solve quantum field
theory by calculating the amplitude for a state with specified
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incoming and outgoing particles.
Feynman Diagrams - Basics
Annihilation Diagram
External legs
represent amplitudes
of initial & final state
particles.
boson (wavy line)
Final
Initial vertex
vertex
State Internal lines (propagators) represent
State
amplitude of exchanged particle.
time 
Exchange Diagram
fermion (solid line)
CONVENTION
fermion
gluons
anti-fermion
H bosons
Virtual
Particle
photons,
W, Z bosons
•
•
•
•
Time runs from left to right.
Particles point in +ve time direction.
Anti-particles point in –ve time direction.
Charge, energy, momentum, angular momentum, baryon no. & lepton no.
26
conserved at interaction vertices. Quark flavour for strong & EM interactions.
Feynman Diagrams - Calculation
 Perturbation theory. Expand and keep the most important terms for
calculations.
 Associate each vertex with the square root of the appropriate coupling
constant, i.e. √𝛼.
 When the amplitude is squared to yield a cross-section there will be a
factor 𝛼 𝑛 , where n is the number of vertices (known as the “order” of the
diagram).
Second order
Lowest order
For QED:
e2

4

HL
1
137
Add the amplitudes from all possible diagrams to get the total amplitude,
M, for a process  transition probability.
2
2
Transition Rate 
M  (phase space)

Contains the fundamental physics
Fermi’s Golden Rule
“Just” kinematics
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Phase Space
Introduced by Willard Gibbs in 1901. Defines the space in which
all possible states of a system are represented, with each possible
state of the system corresponding to one unique point in the
phase space.
pp   0 0 0 s  2050 MeV
Example: 3 Body Decay
Phase space illustrated by this 2-D plot
of a three-body decay (a so-called
“Dalitz plot”).
The contour line shows the boundary
of what is kinematically possible - i.e.
the edge of phase space.
Contour showing limit
of kinematically available
phase space.
Crystal Barrel Experiment 28
Quantum Electrodynamics (QED)
Early relativistic
quantum theories
had to be
rethought after the
discovery of the
Lamb Shift in
hydrogen.
Sin-Itiro Tomonaga Julian Schwinger
Detailed results disagreed with the Dirac Equation.
Due to self-energy of the electron.
Led to concept of renormalisation (Bethe).
Willis Lamb,
Robert Retherford,
1947
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The Most Accurate Theory
Anomalous magnetic moment of electron (g-2).
Dirac theory predicts g=2 exactly, but
this is modified by quantum loops and ae  ( g  2)
2
the difference is defined as ………….
𝒂𝒆 𝐞𝐱𝐩 = 𝟏 𝟏𝟓𝟗 𝟔𝟓𝟐 𝟏𝟖𝟎. 𝟕𝟑 𝟎. 𝟐𝟖 × 𝟏𝟎−𝟏𝟐
𝒂𝒆 𝒕𝒉𝒆 = 𝟏 𝟏𝟓𝟗 𝟔𝟓𝟐 𝟏𝟖𝟏. 𝟕𝟖 𝟔 𝟒 𝟑 𝟕𝟕 × 𝟏𝟎−𝟏𝟐
𝒂𝒆 𝒆𝒙𝒑 − 𝒂𝒆 𝒕𝒉𝒆 = −𝟏. 𝟎𝟔 𝟎. 𝟖𝟐 × 𝟏𝟎−𝟏𝟐
T. Aoyami et al, Tenth-Order QED Contribution to
the Electron g-2 and an Improved Value of the Fine
Structure Constant, ArXiv:1205.5368
[0.24ppb]
[0.67ppb]
Calculated from
12,672 Feynman diagrams!
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QED Vertex Factor
Coupling constant, 𝑔, specifies the
strength between the 3 particles at
each vertex.
This is a measure of the probability
of spin 1 2 fermion emitting or
absorbing a photon.
𝒆−
𝜶
𝒆−
𝜸
Coupling strength 𝑔𝑄𝐸𝐷 = 𝑒
Because 𝛼 =
𝜸
𝑒2
, often we simply
4𝜋
use 𝛼 in place of 𝑔𝑄𝐸𝐷
Taking the 2 vertices, we get a total
factor of 𝛼 𝛼 = 𝛼
𝜶
𝒆−
𝒆−
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Basic QED Processes and Vertices (8)
Electron Bremsstrahlung


𝒆−
Photon Absorption

𝒆− 
e  e 
e   e
𝒆 − 𝒆−
Positron Bremsstrahlung
e   e    𝒆+
Photon Absorption
𝒆+ e     e 
𝒆−
𝒆+ 𝒆+
+ −
𝒆 𝒆 Annihilation
e  e  
𝒆
Vacuum
Production
𝒆−
−
Pair Production
𝒆
𝒆+
𝒆+
Vacuum
Extinction
e   e     vacuum
vacuum  e   e   
time
  e  e
−
𝒆+
𝒆+
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Charged Leptons
As well as electrons and positrons interacting with the EM field (i.e. the photon),
QED also includes the interactions of the other charged leptons –
the muon (𝜇 ± ) and the tau lepton (𝜏 ± ).
The Charged Leptons
𝟏
𝑱 = , 𝑪𝒉𝒂𝒓𝒈𝒆 = ±𝒆
𝟐
𝑀𝑎𝑠𝑠 𝑒 = 0.511 𝑀𝑒𝑉
𝑀𝑎𝑠𝑠 𝜇 = 105.66 𝑀𝑒𝑉
𝑀𝑎𝑠𝑠 𝜏 = 1776.82 𝑀𝑒𝑉
𝜏 > 4.6 × 1026 𝑦𝑟
𝜏 = 2.2 × 10−6 𝑠
𝜏 = 290.6 × 10−15 𝑠
All properties shared except for their very different masses and lifetimes.
Lepton Universality
Example
Electromagnetic properties of muons and tau leptons Dirac magnetic moment
are identical with those of electrons provided the
𝑒
𝝁= S
mass difference is taken into account.
𝑚ℓ
Both the 𝜇 and 𝜏 decay by the weak interaction and
we will return to this in a later lecture.
where 𝑚ℓ is the mass of
the lepton.
33
Also Charged Leptons and Quarks
Same interaction strength for all charged leptons – QED only cares about charge.
𝒆−
𝒆−
𝜶
𝝁−
𝝁− 𝝉−
𝜶
𝜸
𝝉−
𝜶
𝜸
𝜸
Coupling less for quarks due to fractional charge.
u
𝟐
𝜶
𝟑
𝜸
u
d
𝟏
𝜶
𝟑
𝜸
d
34
QED Propagator Factor
Propagator factor tells us about the contribution to the amplitude from an
intermediate (or virtual) particle travelling through space and time.
𝒆−
𝜶
Virtual Particles
Do not have mass of a physical particle.
q 2  m X2  E X2  p X2
Known as “off –mass shell”
(e.g. not zero for photon)
𝒆−
Propagator
1
𝑞2
𝜸
𝜶
𝒆−
𝒆−
• Virtual photons have propagators proportional to 1 𝑞 2 , where
𝑞 is the four-momentum transfer between the vertices (or the
four-momentum of the exchanged particle).
• Heavy bosons with mass 𝑚 have propagators 1 𝑞 2 − 𝑚2 .
35
Feynman Factors: Summary
 Each part of a Feynman diagram has factors associated with it.
Multiply them all together to get matrix element 𝓜.
 Initial and final state particles use wavefunction currents:
 Spin-0 bosons are plane waves.
 Spin-1/2 fermions have Dirac spinors.
 Spin-1 bosons have polarisation vectors 𝓔𝝁 .
 Vertices have dimensionless coupling constants.
 In the electromagnetic case, 𝜶 = 𝒆.
 In the strong interaction, 𝜶𝒔 = 𝒈𝒔 (more later).
 Vertices have propagators, 𝒒𝝁 , which is the momentum
transferred by boson.
 Virtual photon propagator is 𝟏 𝒒𝟐
 Virtual W/Z boson propagator is 𝟏 𝒒𝟐 − 𝑴𝟐 𝑾 or
𝟏 𝒒𝟐 − 𝑴𝟐 𝒁
 Virtual fermion propagator is 𝜸𝝁 𝒒𝝁 + 𝒎)/(𝒒𝟐 − 𝒎𝟐
36
Feynman Rules for QED
External Lines
Spin
𝟏
𝟐
Spin 𝟏
Incoming particle
𝑢(𝑝)
Outgoing particle
𝑢(𝑝)
Incoming antiparticle
𝑣(𝑝)
Outgoing antiparticle
𝑣(𝑝)
Incoming photon
𝜀 𝜇 (𝑝)
Outgoing photon
𝜀 𝜇 (𝑝)∗
Internal Lines (propagators)
Spin 𝟏
Spin 𝟏
Photon
𝟐
Vertex Factors
Spin 𝟏 𝟐
Matrix Element
Fermion
Taken from
Thomson,
page 124
−𝑖𝑔𝜇𝜈
𝜇
𝑞2
−𝑖 𝛾 𝜇 𝑞𝜇 + 𝑚
𝑞 2 − 𝑚2
Fermion (charge − 𝑒 )
𝜈
−𝑖𝑒𝛾 𝜇
∞
ℳ𝑛
Product of all factors −𝑖ℳ =
𝑛=1
37
Example: 𝒆− 𝝉− → 𝒆− 𝝉− scattering
𝒆−
𝑝1
𝝁
𝝉−
𝑝3
𝑞 = 𝑝1 − 𝑝3
𝜸
𝑝2
𝒆−
𝝂
𝑝4
𝝉−
𝑢 𝑝3 𝑖𝑒𝛾 𝜇 𝑢 𝑝1
electron
current
−𝑖𝑔𝜇𝜈
𝑞2
propagator
𝑢 𝑝4 𝑖𝑒𝛾 𝜈 𝑢 𝑝2
tau
current
In lowest order, the amplitude by applying the Feynman rules to the above
diagram is therefore:
𝜇
−𝑖ℳ = 𝑢 𝑝3 𝑖𝑒𝛾 𝑢 𝑝1
−𝑖𝑔𝜇𝜈
𝜈
𝑢
𝑝
𝑖𝑒𝛾
𝑢 𝑝2
4
2
𝑞
38
Higher Order Diagrams
Tree level Processes (or Born Diagrams) = diagrams that contain no loops
Higher Order Corrections (or Radiative Corrections) = loop diagrams
𝒆
+
Lowest order diagram
𝓜 ∝ 𝒆𝟐 ∝ 𝜶
𝜸
𝜶
Order = number of vertices in
each diagram
𝝁+
Any diagram of order 𝑛 gives a
contribution of 𝛼 𝑛
𝜶
𝒆−
𝝁−
Second order diagrams: 𝓜 ∝ 𝒆𝟒 ∝ 𝜶𝟐
𝒆+
𝝁+𝒆+
𝜶
𝜶
𝒆−
𝜶
𝜶
+
𝝁−
𝜸
𝒆−
Total amplitude: 𝓜𝒇𝒊 = 𝜶𝑴𝑳𝑶 + 𝜶𝟐
+..
𝜶
𝜶
𝜶
𝒋 𝑴𝒋
𝝁+
𝜶
𝝁−
+ …
Jargon ↠ Leading Order (LO) + Next-to-Leading-Order (NLO)
39
LEP 𝒆+ 𝒆− → 𝜸𝜸(𝜸) Example
Born level
diagrams
+
Excellent agreement
with QED
Radiative
corrections
Different for each
experiment due
to phase space
40
Alpha – Fine Structure Constant
o Need to take care with the Fine Structure Constant, 𝜶, and its role as a
coupling constant measuring the strength of the EM interaction.
o 𝛼 was introduced by A. Sommerfield in 1916 to explain the fine structure of
the energy levels of the hydrogen atom. In particle physics, it is not really a
constant.
o This is because in QED an electron can emit virtual photons, which form
virtual e+/e- pairs which “screen” the electron.
𝒆−
Vacuum polarisation
loops. Correction:
𝒆−
 (0)
 (Q ) 
1   (Q 2 ) 𝒆−
2
OPAL, CERN-EP/98-108
+
𝒆
𝒆
−
𝒆+
𝒆+
Alpha not really a constant!
1
137.035999074(44)
1
At Q2≈mW2, 𝛼~
41
128
At Q2=0,
𝛼=
QED Renormalisation
The strength of the coupling between a photon and an electron is determined
by the coupling at the QED vertex, which until now we have assumed constant
with value 𝑒.
• The value 𝛼 ≈ 1 137 is obtained from measurements of the static Coulomb
potential in atomic physics.
• This is not the same as the strength of the coupling in Feynman diagrams,
which can be written as 𝑒0 (and called the bare electron charge).
• The experimentally measured value of 𝑒 is the effective strength of the
interaction which results from summing over all relevant higher order
diagrams.
• There is an infinite set of higher-order corrections, including the ones below…
Corrections to electron four vector current
Correction to
propagator
(a)
Lowest order
(b)
(c)
(d)
𝒪 𝑒 2 corrections to QED vertex
(e)
42
QED Renormalisation
There are no restrictions on the momentum, 𝑘, of the virtual particles and the
self-energy terms include integrals of the form 𝑑𝑘 𝑘, which is logarithmically
divergent.
Techniques to deal with this are beyond scope of this course, but basically…..
1. Use cut-off procedures in integrals. It turns out that this enables the
calculations to be separated into two parts: a finite term and one which blows
up.
2. Amazingly, all the divergent terms then appear as additions to the bare
parameters such as mass (𝑚0 ), charge (𝑒0 ) and coupling constant (𝛼0 ).
i.e. 𝑚𝑝ℎ𝑦𝑠𝑖𝑐𝑎𝑙 = 𝑚0 + 𝛿𝑚
3. The strategy is then to absorb the infinities into renormalisable masses and
coupling constants. i.e. it means that we use the physical values as
determined by experiment and NOT the values (𝑚0 , 𝑒0 , 𝛼0 ) that appeared in
the original Feynman rules that we wrote down.
As already discussed, one
consequence is that the coupling
constant depends logarithmically on
energy scale ⟹⟹⟹⟹⟹⟹⟹⟹
𝛼 𝑞2
𝛼(𝜇2 )
=
𝑞2
1
2
1 − 𝛼(𝜇 ) ln 2
3𝜋
𝜇
43
CLIC after LHC at CERN?
CLIC= Compact Linear (e+e-) Collider
44
Electron Positron Annihilation
Electron-positron colliders have been central to the
development and understanding of the Standard Model.
WHY?
• It is easier to accelerate protons to very high energies than
leptons, but the detailed collision process of protons cannot be
well controlled or selected.
• Electron positron colliders offer a well-defined initial state.
• The collision energy 𝑠 is known and it is tuneable (e.g. for
scanning thresholds of particle production).
• Polarisation of electrons/positrons is possible.
• In proton collisions, the rate of unwanted collision processes is
very high, whereas the point-like nature of leptons results in low
backgrounds.
• Scattering of point-like particles can be calculated to very
45
high precision in theory.
LEP Electroweak Measurements Example
Cross-sections of electroweak Standard
Model (SM) processes. Dots with error
bars show the measurements, while
curves show theoretical predictions
based on SM.
Precision results on fundamental
properties of W boson & EW theory
𝑚𝑊 = 80.376 ± 0.033 𝐺𝑒𝑉
Γ𝑊 = 2.195 ± 0.083 𝐺𝑒𝑉
𝐵 𝑊 → ℎ𝑎𝑑𝑟𝑜𝑛𝑠 = 67.41 ± 0.27%
𝑔 𝑍 1 = 0.984+0.018 −0.020
𝜅𝛾 = 0.982 ± 0.042
𝜆𝛾 = −0.022 ± 0.019
Electroweak measurements in electronpositron collisions at W-boson-pair energies at
LEP, Physics Reports, 2013, in press
46
End
CONTACT
Professor Glenn Patrick
email: [email protected]
email: [email protected]