The Standard Model

1
Section VII

• The Standard Model

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The Standard Model

• Spin ½ Fermions
• 
  
• LEPTONS
  
• 
  
• QUARKS
  
• 
  
• PLUS antileptons and antiquarks.
• Spin 1 Bosons
  Mass (GeV/c2)
• Gluon g 0 STRONG
• Photon g 0 EM
• W and Z Bosons W?, Z0 91.2/80.3 WEAK
• The Standard Model also predicts the existence of
  a spin 0
• HIGGS BOSON
• which gives all particles their masses via its
  interactions.

Charge (units of e) -1 0 2/3 -1/3
3
Theoretical Framework

• Macroscopic
  Microscopic
• Slow Classical Mechanics
  Quantum Mechanics
• Fast Special Relativity
  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)

4
ASIDE Relativistic Kinematics

• Low energy nuclear reactions take place with
  typical kinetic energies of order 10 MeV ltlt
  nucleon rest energies ) non-relativistic formulae
  OK.
• In particle physics kinetic energy is of O (100
  GeV) gtgt rest mass energies so relativistic
  formulae usually essential.
• Always treat ?-decay relativisticly (me 0.5 MeV
  lt 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
• Definitions mass m velocity v ßc( ß)
  mom p
• Consequences

5
ASIDE cont (Decay example)

• Consider decay of charged pion at rest in the lab
  frame

(Simplify by taking m?0 )
6
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!123n where we observe
  particles 1 to n, we have

J/y
In the specific (and common) case of a two-body
decay, X!12, this reduces to the same formula
as on a preceding slide, i.e.
Typical invariant mass histogram
7
ASIDE cont Colliders and vs

• Consider the collision of two particles
• The invariant quantity
• ? is the angle between the momentum
  three-vectors
• is the energy in the zero momentum
  (centre-of-mass) frame
• It is the amount of energy available to the
  interaction e.g. in particle-antiparticle
  annihilation it is the maximum energy/mass of
  particle(s) that can be produced

8

• Fixed Target Collision
• For
• e.g. 450 GeV proton hitting a proton at rest
• Collider Experiment
• For then and
  ? ?
• e.g. 450 GeV proton colliding with a 450 GeV
  proton
• In a fixed target experiment most of the
  protons energy is wasted providing forward
  momentum to the final state particles rather than
  being available for conversion into interesting
  particles.

9
ASIDE over

• Back to the main topic of combining Quantum
  Mechanics with Special Relativity..

10
Problems with Non-Relativistic Schrodinger
Equation

• To describe the fundamental interactions of
  particles we need a theory of RELATIVISTIC
  QUANTUM MECHANICS.
• Schrodinger Equation for a free particle comes
  from classical
• or equivalently promoted to an
  operator equation
• using energy and momentum operators
• giving which has
  plane wave solutions
• Schrodinger Equation
• 1st Order in time derivative
• 2nd Order in space derivatives
• Schrodinger equation cannot be used to describe
  the physics of relativistic particles.

h 1 natural units
Not Lorentz Invariant!
11
Klein-Gordon Equation (relativistic)

• From Special Relativity
• From Quantum Mechanics
• Combine to give
• 
  KLEIN-GORDON
• 
  EQUATION
• Second order in both space and time derivatives
  and hence Lorentz invariant.
• Again plane wave solutions
  
• but this time requiring
  allowing
• Negative energy solutions required to form
  complete set of eigenstates
• ANTIMATTER

12
Antimatter

• Feynman-Stückelberg interpretation The negative
  energy solution is a negative energy particle
  travelling forwards in time. And then, since
• ? Interpret as a positive energy antiparticle
  travelling forwards in time.
• Then all solutions describe physical states with
  positive energy, going forward in time.
• Examples ee- annihilation
  pair creation
• 
  

All quantum numbers carried into vertex by e,
same as if viewed as outgoing e-.
13
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.
• 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.

14
Normalization

• Can show solutions of Schrodinger Equation
  (non-relativistic) satisfy
• and solutions of Klein-Gordon Equation
  (relativistic) satisfy
• Both are of the form
• and the divergence theorem tells us that
• this means that D is conserved, where
• If we interpret D as number of particles, and ?
  as number of particles per unit volume, then
• For plane wave of arbitrary normalization
  find that
• Non-relativistic particles per unit volume.
• Relativistic particles per unit volume.

15
Interaction via Particle Exchange

• Consider two particles, fixed at and ,
  which exchange a particle of mass m.
• Calculate shift in energy of state i due to this
  process (relative to non-interacting theory)
  using 2nd order perturbation theory
• Work in a box of volume L3, and normalize s.t.
• i.e. use
• 2E particles per box!

Sum over all possible states j with different
momenta.
16

• Consider ( transition from i to j by
  emission of m at )
• represents a free particle with
• Let g be the strength of the emission at .
  Then
• Similarly is the transition from j to
  i at and is
• Substituting these in, we see the shift in energy
  state is

Original 2 particles
17

• 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
• i.e. 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
  (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.

Lab. frame 1 particle per V/g
Rest frame 1 particle per V
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Beginning to put it together

• Different states j have different momenta .
  Therefore sum is actually an integral over all
  momenta
• And so energy change

19
Final throes

• The integral can be done by choosing the z-axis
  along .
• Then and
• so
• Write this integral as one half of the integral
  from -? to ?, which can be done by residues
  giving

Appendix D
20

• Can also exchange particle from 2 to 1
• Get the same result
• Total shift in energy due to particle exchange is
• ATTRACTIVE force between two particles which
  decreases exponentially with range r.

21
Yukawa Potential

• YUKAWA POTENTIAL
• Characteristic range 1/m
• (Compton wavelength of exchanged particle)
• For m?0,
  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.

Hideki Yukawa 1949 Nobel Prize
22
Scattering from the Yukawa Potential

• Consider elastic scattering (no energy transfer)
• Born Approximation
• Yukawa Potential
• The integral can be done by choosing the z-axis
  along , then
• and
• For elastic scattering,
  and exchanged massive particle is
  highly virtual

q2 is invariant VIRTUAL MASS
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Virtual Particles

• Forces arise due to the exchange of unobservable
  VIRTUAL particles.
• The mass of the virtual particle, q2, is given
  by
• 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
• If q2 m2, then the particle is real and can be
  observed.

h c1 natural units
24

• For virtual particle exchange, expect a
  contribution to the matrix element of
• where
• 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.
• For m ? 0 e.g. single g exchange
• q2? 0, very low energy transfer EM scattering

COUPLING CONSTANT STRENGTH OF INTERACTION PHYSIC
AL (On-shell) mass VIRTUAL (Off-shell)
mass PROPAGATOR
25
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.
• The sum of all time orderings are represented by
  FEYNMAN DIAGRAMS.

Space

FEYNMAN DIAGRAM
26

• 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.
• But each vertex gives a factor of g, so if g is
  small (i.e. the perturbation is small) only need
  the first few.
• Example QED

Total amplitude
Fermis Golden Rule
Total rate
27
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)
• and their interaction point (vertex) with a
  .
• Each vertex gives a factor of the coupling
  constant, g.

Spin ½ Quarks and Leptons
Spin 1 g, W? and Z0
Spin 1 g
28

• External Lines (visible particles)
• Internal lines (propagators)

Spin ½ Particle Incoming Outgoing
Spin ½ Antiparticle Incoming Outgoing
Spin 1 Particle Incoming Outgoing
Spin ½ Particle (antiparticle)
Spin 1 g, W? and Z0
Spin 1 g
Each propagator gives a factor of
29

• Examples

WEAK
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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
  dont 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.

31
Section VIII

• QED

32
QED

• QUANTUM ELECTRODYNAMICS is the gauge theory of
  electromagnetic interactions.
• Consider a non-relativistic charged particle in
  an EM field
• given in term of vector and scalar
  potentials

Maxwells Equations
Change in state of e- requires change in field
? Interaction via virtual g emission
33

• Schrodinger equation
• is invariant under the gauge transformation
• ? 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

Appendix E
34
The Electromagnetic Vertex

• All electromagnetic interactions can be described
  by the photon propagator and the EM vertex
• 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
• i.e. but not
  or
• QED vertex always conserves PARITY

STANDARD MODEL ELECTROMAGNETIC VERTEX
g
antiparticles
35
Pure QED Processes

• Compton Scattering (ge-?ge-)
• Bremsstrahlung (e-?e-g)
• Pair Production (g? ee-)

e-
g
e-
g
e-
The processes e-?e-g and g? ee-cannot occur
for real e?, g due to energy, momentum
conservation.
Ze
36

• ee- Annihilation
• p0 Decay
• J/????-
• The coupling strength determines order of
  magnitude of the matrix element. For particles
  interacting/decaying via EM interaction typical
  values for cross-sections/lifetimes
• sem 10-2 mb
• tem 10-20 s

37
Scattering in QED

• Examples Calculate the spin-less
  cross-sections for the two processes
• (1)
  (2)
• Fermis Golden rule and Born Approximation (see
  page 48)
• For both processes write the SAME matrix element
• is the strength of the
  interaction.
• measures the probability that the
  photon carries 4-momentum
• 
  i.e. smaller probability for higher mass.

Electron-positron annihilation
Electron-proton scattering
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(1) Spin-less e-p Scattering

• q2 is the four-momentum transfer
• Neglecting electron mass i.e. and
• Therefore for ELASTIC scattering

e?
e?
RUTHERFORD SCATTERING
39
Discovery of Quarks

• Virtual g carries 4-momentum
• Large q ? Large , small
• Large , large
• 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
lltlt size of proton q2 gt 1 (GeV)2
Elastic scattering from quarks in proton
40
(2) Spin-less ee- Annihilation

• Same formula, but different 4-momentum transfer
• Assuming we are in the centre-of-mass system
• Integrating gives total cross-section

41

• This is not quite correct, because we have
  neglected spin. The actual cross-section (using
  the Dirac equation) is
• Example Cross-section at GeV
• (i.e. 11 GeV electrons colliding with
• 11 GeV positrons)

42
The Drell-Yan Process

• Can also annihilate as in the Drell-Yan
  process.
• Example

See example sheet (Question 25)
43
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
  Dirac Equation
• However, higher order terms introduce an
  anomalous magnetic
• moment i.e. g not quite 2.

? ? ? ?
O(a)
O(a4) 12672 diagrams
O(1)
44

• O(a3)
• Agreement at the level of 1 in 108.
• QED provides a remarkable precise description of
  the electromagnetic interaction !

Experiment Theory
45
Higher Orders

• So far only considered lowest order term in the
  perturbation series. Higher order terms also
  contribute
• Second order suppressed by a2 relative to first
  order. Provided a is small, i.e. perturbation is
  small, lowest order dominates.

Lowest Order
? ? ? ?
Second Order
Third Order
? ? ? ?
46
Running of ?

• specifies the strength of the
  interaction between an electron and a photon.
• BUT a is NOT a constant.
• 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.
• Consider a free electron.
• The same effect can happen due to quantum
• fluctuations that lead to a cloud of virtual ee-
  pairs
• The vacuum acts like a dielectric medium
• The virtual ee- 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. a.

47

• Measure a(q2) from etc
• a increases with increasing q2
• (i.e. closer to the bare charge)
• At q20 a1/137
• At q2(100 GeV)2 a1/128

48
Summary

• QED is the physics of the photon charged
  particle vertex
• 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 a 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.