# Introduction to Feynman Diagrams

# Feynman diagrams

A Feynman diagram is a picture that shows the interaction between particles. By studying Feynman diagrams, you can learn more about how particle physics really works. Physicists also use them to help them calculate the properties of different particle interactions.

A simple Feynman diagram looks like this:

[picture of e.g. electron-electron scattering Feynman diagram]

Feynman diagrams are made up of straight lines, that represent fermions, and wiggly or dashed lines that represent the different kinds of bosons. These lines represent the particles’ journey through space. The horizontal direction represents time, and the vertical direction represents position.

fermion: [straight line]

photon: [picture of photon propagator]

gluon: [picture of gluon propagator]

Three (or sometimes four) different lines can meet to form a “vertex”, which is nothing but an intersection point. When this happens, the particles can interact with each other. For example, they might cause each other to change direction, such as when two electrons repel each other due to their negative charge. This is shown in the diagram above. We can see that when the two electrons interact via their electric charges they exchange a photon. This is because the photon is the force carrying boson for the electromagnetic interaction.

Before we explore Feynman diagrams further, we need to take a short excursion to talk more about antimatter. We have seen that, for example, an anti-electron (usually called a positron) is exactly like an electron but with a positive charge rather than a negative charge. Now it turns out that a positron can also be thought of as an electron travelling backwards through time! This is a bit mind-bending, but it is absolutely true. Can you see how a positive charge travelling forwards through time is the same as a negative charge travelling backwards through time?

Now we are ready to return to Feynman diagrams. One cool thing about Feynman diagrams is that they can be rotated to form new diagrams for completely different processes. For example, we know that an antiparticle is just a particle travelling backwards through time. So when we rotate the Feynman diagram for electron-electron scattering by 90 degrees, we get the Feynman diagram for an electron and a positron (an anti-electron) annihilating to give a photon.

[electron positron pair annihilation diagram]

And if we rotate this diagram by 180 degrees, we find the diagram for a high energy photon turning into an electron and a positron:

[electron positron pair creation diagram]

When two quarks meet, they can also interact by exchanging a photon, as quarks carry electric charge. But quarks carry colour as well, so they can also interact with the strong interaction by exchanging a gluon:

[Feynman diagram of two quarks scatting via gluon exchange]

So far, we have seen Feynman diagrams for the electromagnetic and strong interactions. These interactions cannot cause one kind of fermion to change into a different kind of fermion. This is because the “allowed vertices” for the electromagnetic and strong interactions can only contain one kind of fermion. Of course, I could draw a vertex with, for example, an up quark, a down quark and a photon, but this would not represent a process that ever occurs in the real world. However, the weak interaction has many different allowed vertices in which one kind fermion turns into a different kind of fermion.

For example, in the radioactive decay of a nucleus (or more specifically beta decay), a neutron turns into a proton inside the nucleus. Remembering that the neutron is made of two down quarks and an up quark, and the proton is made of two up quarks and a down quark, we can see that a down quark must turn into an up quark. This cannot happen in the electromagnetic or the strong interactions, so beta decay must be to do with the weak interaction. The Feynman diagram for beta decay is:

[beta decay Feynman diagram]

We see that an up quark turns into a down quark, and emits an electron and an anti-electron neutrino. This process happens via the exchange of a W boson.

The mathematics of the standard model tells us that only certain vertices are allowed in Feynman diagrams. Any process that can be described by a Feynman diagram made only of these allowed vertices can occur in reality. If a particular process can only be described by a Feynman diagram with a not allowed vertex, it can never happen.

The Allowed Electromagnetic Vertices

There is only one type of allowed electromagnetic vertex - the meeting of two charged particles of the same type and a photon. For example:

[electron emits a photon Feynman diagram]

In this diagram, there are not actually two different electrons - this represents a single electron emitting or absorbing a photon, and changing direction as a result.

Remember that this vertex can be rotated to form a vertex with a charged particle, its antiparticle and a photon:

[pair annihilation Feynman diagram]

The Allowed Strong Vertices

The first allowed strong vertex is just like the electromagnetic vertex but with two important exceptions:

- It involves a gluon rather than a photon, as the gluon is the force carrier for the strong interaction

- It may involve just quarks rather than any charged particle. As leptons do not carry colour charge, they cannot participate in the strong interaction.

So the first allowed strong vertex is the meeting of two quarks of the same type and a gluon:

[Feynman diagram for up quark emitting a gluon]

As with the electromagnetic vertex, this can be rotated to form diagrams involving antiquarks;

[Feynman diagram for pair production of an up quark and an anti up quark from a gluon]

One important difference between the electromagnetic and strong interactions is that, while a photon does not carry electric charge, a gluon does carry colour. This means that gluons can interact with other gluons. Vertices where three or four gluons meet are allowed:

[Three and Four gluon self interaction vertices]

So what? Well, the gluon self-interaction is why the strong force is so strong. It is easy to separate an electron from a proton (just raise the temperature), but to split the proton we need the LHC! Even in the LHC, isolated quarks are never observed.

The Allowed Weak Vertices

In some senses, the weak interaction is much more complicated than the other two - there are several different allowed weak vertices. We will go through each of them in turn:

1) The meeting of a charged lepton (an electron, muon or tau lepton), a neutrino from the same generation as the charged lepton, and a W boson:

[Electron, electron neutrino, W vertex]

2) The meeting of any two quarks with different charges and a W boson:

[ u, d, W vertex]

3) The meeting of any two particles of the same type (except a photon or a gluon) and a Z boson:

[Feynman diagram for a neutrino emitting a Z boson]

4) A number of different vertices involving the W and Z bosons and the photon:

[Weak force self interaction vertices]

Can you see how the beta decay Feynman diagram above can be made from the allowed weak vertices?