Feynman Diagrams

Rosie Cavalier 2012Edited 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,where and is

closely related to the charge of an electron. - The amplitude of the diagram is the product of the 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:

x =

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 x x x =

Calculating a Feynman diagram

• The introduction of Feynman diagrams has made calculations of the probabilities much simpler. • • 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 interactionare more complex as they havemany 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

These are examples of interaction vertices for weak interaction

The d quark turns into a u quark by emitting a virtual W- boson.

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/4πmcm 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 forceSince 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.