1. Scattering in QM
Consider a beam of particles scattering in potential V(r):
NOTE: natural units
The scattering rate is characterized by the interaction cross-section
Use Fermi’s Golden Rule to get the transition rate
where Mfi is the matrix element and r(Ef) is the density of final states.
Number of particles scattered per unit time
Incident flux

2. 1st Order Perturbation Theory using plane wave solutions of form
Require:
Wave-function normalization
Matrix element in perturbation theory
Expression for flux
Expression for density of states.
Normalization: Normalize wave-functions to one particle in a box of side L:

3. Matrix Element: This contains the physics of the interaction
where
Incident Flux: Consider a “target” of area A and a beam of particles travelling at velocity vi towards the target. Any incident particle within a volume viA will cross the target area every second.
Flux (F) = number of incident particles crossing unit area per second:
where n is the number density of incident particles = 1 per L3

4. Density of States: for a box of side L states are given by the periodic boundary conditions:
Each state occupies a volume in p space.
Number of states between in p and p+dp in solid angle dW
Density of states in energy
For relativistic scattering: (Ep)

5. Putting all the separate bits together:
Relativistic scattering vi= c = 1 and p  E

6. Breit-Wigner Cross-section
Some particle interactions proceed via an intermediate RESONANT state which then decays
Two stages: (Bohr Model)
Formation
Occurs when the collision energy s  the natural frequency of a resonant state.
Decay
The decay of the resonance Z is independent of the mode of formation and depends only on the properties of the Z.

7. The RESONANCE CROSS-SECTION is given by
with
The matrix element Mif is given by 2nd Order Perturbation Theory
where the sum is over all intermediate states.
( Factors of L3 cancel as before )

8. Consider 1 intermediate state described by
This describes a state with energy =
Rate of decay of Z:
Rate of formation of Z:

9. Hence,
pi is the C.O.M. momentum
natural units
BREIT-WIGNER
CROSS-SECTION

10. The factor g takes into account the SPIN
and is the ratio of the number of spin states for the resonant state to the total number of spin states for the a+b system.
Notes
Total cross-section
Replace Gf by G in the Breit-Wigner formula
Elastic cross-section so Gf = Gi
On peak of resonance (E=E0)
Thus,
From the measurement of stot and sel can infer g and hence spin of resonant state.

11. Example:
(GeV)
s (pb)
GeV s)

12. Section IV - Feynman Diagrams

13. Theoretical Framework
Macroscopic Microscopic
Slow Classical Mechanics Quantum Mechanics
Fast Special Relativity Quantum Field Theory
ELECTROMAGNETISM: QUANTUM ELECTRODYNAMICS (QED)
1948 Feynman, Schwinger, Tomonaga (1965 Nobel Prize)
ELECTROMAGNETISM: ELECTROWEAK UNIFICATION
+WEAK Glashow, Weinberg, Salam (1979 Nobel Prize)
STRONG: QUANTUM CHROMODYNAMICS (QCD)
1974 Politzer, Wilczek, Gross (2004 Nobel Prize)
CMK : 2008 Nobel prize

14. Feynman Diagrams
Results of calculations based on a single process in Time-Ordered Perturbation Theory 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
Time
Time +
Space =
FEYNMAN
DIAGRAM

15. Antimatter
The negative energy solution is equivalent to a negative energy particle state travelling backwards in time
 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 pair creation g e+ e+ g
< e- backwards in time =
> e+ forwards in time
but put e+ anyway !!
arrows show charge flow
All quantum numbers carried
into vertex by e+, same as if
viewed as outgoing e-.

16. 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.
ħ =c=1 natural units

17. 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
PHYSICAL (On-shell) mass
VIRTUAL (Off-shell) mass
PROPAGATOR

18. 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
Fermi’s Golden
Rule
Total rate

19. External Lines (visible particles)
Internal lines (propagators)
Spin ½
Particle
Incoming
Outgoing
Antiparticle
Spin 1
Spin ½
Particle
(antiparticle)
Spin 1
g, W and Z0 g
Each propagator gives
a factor of
Each vertex gives a factor of the coupling constant, g.

20. q q e e p p u u n d d p d u e Examples: ELECTROMAGNETIC STRONG
Electron-proton scattering
Quark-antiquark annihilation
WEAK u u n d d p d u e
Neutron decay

21. SECTION V - QED

22. 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
QED vertex always conserves PARITY

23. Pure QED Processes Compton Scattering (ge-ge-) e- g g e-
Bremsstrahlung (e-e-g)
Pair Production (g e+e-) e- g e- g e- e- g g e-
Nucleus
The processes e-e-g
and g e+e-cannot occur
for real e, g due to
energy, momentum
conservation. Ze
Nucleus

24. p0 u c J/Y e+e- 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 ~ s u p0 c
J/Y

25. Scattering in QED e e p p
Examples: Calculate the “spin-less” cross-sections for the two processes:
(1) (2)
Fermi’s Golden rule and Born Approximation
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. e e p p
Electron-proton scattering
Electron-positron annihilation

26. “Spin-less” e-p Scattering
q2 is the four-momentum transfer:
Neglecting electron mass: i.e and
Therefore for ELASTIC scattering p p e e
RUTHERFORD
SCATTERING

27. Discovery of Quarks Virtual g carries 4-momentum Large q 
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 E
q2 large
l<< size of proton
q2 > 1 (GeV)2
Elastic scattering from
quarks in proton

28. “Spin-less” e+e- Annihilation
Same formula, but different 4-momentum transfer:
Assuming we are in the centre-of-mass system
Integrating gives total cross-section:

29. This is not quite correct, because we have neglected spin
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)

30. 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. v v
   
O(1)
O(a)
O(a4)
12672 diagrams

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

32. +     +     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
+    

33. 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 e+e- pairs
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. a. -Q + - e+ e- g

34. a increases with increasing q2 (i.e. closer to the bare charge)
Measure a(q2) from etc
a increases with increasing q2
(i.e. closer to the bare charge)
At q2=0: a=1/137
At q2=(100 GeV)2: a=1/128
Can sum perturbation series and describe total interaction with an effective
three level coupling - infinite series of Feynman diagrams with fixed EM is equivalent to single diagram with effective running EM(q2)