Feynman Diagrams
Rosie Cavalier 2012
Edited by Mr Catchpole 2014
What is a Feynman diagram?
• A Feynman diagram is used by to illustrate and
calculate probabilities of reactions between
elementary particles.
• The diagram shows particle interactions and
the idea is to use interaction vertices* in
order to build up possible physical processes
Interaction and exchange particles
• To understand the Feynman diagram you must first look at
interactions and exchange particles
• A free electron can emit a (virtual) photon provided the
photon is very quickly absorbed
This diagram shows the exchange of
a virtual photon in the interaction
between electrons.
As the first electron emitted a
photon, it changes direction slightly
in order to conserve momentum.
The second photon also changed
direction, since it absorbed a
photon.
• The change in direction of the two electrons
can be interpreted as the result of a force or
interaction between the two electrons.
• The two electrons will exert repelling forces on
each other according to Coulomb’s law
• The particle physics view of the situation is
that Coulomb’s law is the exchange of a virtual
photon between the electrons.
• This electromagnetic interaction is the
exchange of a virtual photon between charged
particles. The exchanged photon is not
observable.
Basic Interaction Vertices
• There are four fundamental forces (or Interactions) in nature.
• The interactions and the particles involved, as well as their
relative strength are shown below.
Interaction
Interaction acts on
Exchange particle(s) Relative strength
Electromagnetic
Particles with
electric charge
Photon
1/137
Weak
Quarks and leptons
only
W and Z bosons
10-6
Strong (colour)
Quarks only
Gluons
1
Gravitational
Participles with
mass
Graviton
10-38
However, the electromagnetic and weak nuclear reactions have been shown to be
two faces of the same interactions, called the electroweak interaction.
Therefore there are in fact three fundamental interactions
Interaction
• The electroweak interaction
• The strong (colour interaction)
• The gravitational interaction
– Least relevant for particle physics as the masses of the
particles are so small.
Interaction vertices
• At a fundamental level, particle physics views
an interaction between two elementary
particles in terms of interaction vertices
The wavy line represents a photon
The arrow to the right represents an
electron
An arrow to the left would represent a
positron
-
This is an example of a Feynman diagram for electron-electron scattering.
It represents a mathematical expression called the amplitude of the process
The square of the amplitude gives the probability of the process actually taking place.
We assign a quantity to each vertex called the strength of the interaction
For the electromagnetic interaction , the basic vertex is assigned the value ∝ 𝐸𝑀 ,where
∝ 𝐸𝑀 ≈ 1/137 and is closely related to the charge of an electron.
- The amplitude of the diagram is the product of the ∝ 𝐸𝑀 for each vertex that appears.
In the electron-electron scattering process, there are two interaction vertices so the
amplitude of the diagram is proportional to:
∝ 𝐸𝑀 x ∝ 𝐸𝑀 = ∝ 𝐸𝑀
This is a similar diagram, built out of
the basic interaction vertex.
This is different, as it contains four
interaction vertices, therefore the
amplitude for this is proportional to
∝ 𝐸𝑀 x ∝ 𝐸𝑀 x ∝ 𝐸𝑀 x
∝ 𝐸𝑀 = ∝ 2𝐸𝑀
Calculating a Feynman diagram
• The introduction of Feynman diagrams has
made calculations of the probabilities much
simpler.
• ∝ 𝑬𝑴 ≈ 𝟏/𝟏𝟑𝟕
• Since 1/137 is a small number, the processes with
four interaction vertices are less likely to occur.
• To approximate you can examine the diagram
with two vertices only. If a better approximation
is required, more vertices must be used. The
larger number of vertices, the greater amount of
calculation required.
Building Feynman diagrams
You need:
• Basic interaction vertex
• Lines with arrows to represent electrons and
positrons (or “real” particles)
• Wavy lines to represent photons (or “virtual”
exchange particles)
Vertices examples
• Electromagnetic interaction is simple as it only
has one interaction vertex
• The weak and strong interaction
are more complex as they have
many vertices
Weak and Strong interactions
• Basic interaction vertices for the weak
interaction involve the W or Z boson along
with two fermions (quarks or leptons)
These vertices are given if
needed in the exam so
you don’t have to
remember them.
Weak Interactions
The d quark turns
into a u quark by
emitting a virtual
W- boson.
These are examples of interaction
vertices for weak interaction
The total charge is conserved as the charge going in is
-1/3e, the charge leaving the vertex is +2/3e for the u quark and
–e for the W- . Therefore the total charge is -1/3e.
Strong interaction
• One interaction vertex is similar to
electromagnetic vertex where electrons are
replaced by quarks and photons by gluons
• The flavour of the quark does not change so if
the incoming quark is a u quark then the
outgoing quark will also be u.
Calculating the range of interaction
The range of a particle is given
by:
R = h/4πmc
m is the rest mass of the virtual
particle which is why photons
have long range and W and Z
bosons have a very short range
Calculating the range of strong force
Since the individual gluons and
quarks are contained within the
proton or neutron, the masses
attributed to them cannot be used
in the range relationship to predict
the range of the force. When
something is viewed as emerging
from a proton or neutron, then it
must be at least a quark-antiquark
pair, so it is then plausible (Yukawa’a
prediction) that the pion as the
lightest meson should serve as a
predictor of the maximum range of
the strong force between nucleons.