Feynman diagrams

A Feynman diagram is a simple picture of a particle interaction. Straight lines show the particles going in and coming out, a wavy line shows the exchange particle that carries the force, and the points where lines meet are called vertices. At A-level you draw them for electron repulsion (a virtual photon) and for the weak interactions: beta minus decay, beta plus decay, electron capture and electron proton collisions (a W boson).

What is a Feynman diagram?

Forces between particles are explained by exchange particles. Two electrons repel because they swap a virtual photon; a neutron turns into a proton by emitting a W boson. A Feynman diagram shows that exchange as a sketch, named after Richard Feynman, who shared the 1965 Nobel Prize in Physics for his work on quantum electrodynamics, the theory these diagrams were invented for.

The diagram is not a map of where the particles go. It only records which particles go in, which come out, and what is exchanged between them. Every Feynman diagram is built from the same parts:

Feynman diagram rules

1. Time runs up the page. Incoming particles start at the bottom and outgoing particles leave at the top. Some books draw time from left to right instead; either is fine if you keep to one.
2. Particles are straight lines with arrows pointing the way they travel. Label every line with its symbol.
3. The exchange particle is a wavy line between two vertices, labelled γ, W⁺, W⁻ or Z⁰. It never starts or ends in mid air.
4. Two lines meet the exchange particle at each vertex: one going in, one coming out.
5. Charge is conserved at every vertex, not just overall. This is how you decide whether the W boson is W⁺ or W⁻.
6. Baryon number and lepton number are conserved overall. Hadrons such as protons and neutrons stay on one side of the diagram; leptons such as electrons and neutrinos stay on the other.

Exchange particles for each force

Each fundamental interaction has its own exchange particle. Learn which one goes with which force before you draw anything.

AQA A-level Physics names the virtual photon for the electromagnetic force, W⁺ and W⁻ for the weak interaction, and the pion as the exchange particle of the strong nuclear force between nucleons. It limits the weak interactions you need to beta minus decay, beta plus decay, electron capture and electron proton collisions. Match each force to its carrier in this question from the Particles course.

Electromagnetic repulsion: the virtual photon

Two electrons come in at the bottom. At the left vertex one electron emits a virtual photon; at the right vertex the other absorbs it. Both electrons carry on, now moving apart. The photon has no charge, so charge balances at each vertex: an electron goes in and an electron comes out. The same diagram works for any two charged particles, such as an electron and a proton attracting.

Feynman diagram for beta minus decay

In beta minus decay a neutron turns into a proton and emits an electron and an electron antineutrino:

Draw the neutron coming in on the left and leaving as a proton. At that vertex the charge goes from 0 to +1, so the W boson must carry away −1: it is a W⁻. The W⁻ then decays at a second vertex into an electron (−1) and an electron antineutrino (0), which leave at the top. At quark level, a down quark changes to an up quark: udd becomes uud.

Feynman diagram for beta plus decay

In beta plus decay a proton inside a nucleus turns into a neutron and emits a positron and an electron neutrino:

The proton (+1) leaves as a neutron (0), so the boson takes away +1: a W⁺. It decays into a positron (+1) and an electron neutrino (0). At quark level an up quark changes to a down quark. A free proton can't do this, because a neutron has more mass than a proton; it only happens in nuclei with too many protons.

Feynman diagram for electron capture

In electron capture a proton in the nucleus captures one of the atom's inner electrons:

This time two particles go in, so draw a proton on the left becoming a neutron, and an electron on the right becoming an electron neutrino. The proton (+1) becomes a neutron (0) and emits a W⁺; the electron (−1) absorbs it and becomes a neutrino (0). Charge balances at both vertices.

You can also draw it the other way round: the electron emits a W⁻ and turns into a neutrino, and the proton absorbs the W⁻ and becomes a neutron. Both versions are correct, as long as the charge balances at each vertex.

Electron proton collision Feynman diagram

An electron fired at a proton at high speed can produce the same result as electron capture: a neutron and an electron neutrino. The equation is identical; the difference is that the electron comes from outside the atom, from an accelerator, rather than from an inner shell. It is usually drawn with the electron emitting a W⁻, which the proton absorbs.

How to draw a Feynman diagram, step by step

1. Write the equation and check charge, baryon number and lepton number balance.
2. Put the hadron (proton or neutron) on the left: in at the bottom, out at the top.
3. Put the leptons on the right. If one lepton goes in, it comes out as a different lepton; if none go in, two leave from the second vertex.
4. Join the two vertices with a wavy line.
5. Find the charge of the boson from one vertex: charge in = charge out + charge of the boson. Label it γ, W⁺ or W⁻.
6. Check the other vertex balances too, and label every line.

Worked examples

Example 1: which boson?

A neutron absorbs an electron neutrino and becomes a proton, giving out an electron: νₑ + n → p + e⁻. Which exchange particle is involved?

• It turns a neutron into a proton, so it is a weak interaction and the boson is a W.
• At the neutron vertex the charge goes from 0 to +1, so the neutron must absorb a W⁺ (0 + 1 = +1).
• At the lepton vertex the neutrino (0) emits that W⁺ and becomes an electron: 0 = −1 + 1.

Charge balances at both vertices, so the answer is a W⁺ passed from the neutrino to the neutron. Drawing a W⁻ going the other way, from the neutron to the neutrino, is also correct.

Example 2: detecting the neutrino

In 1956 Clyde Cowan and Frederick Reines first detected neutrinos, using antineutrinos from a nuclear reactor hitting protons in a tank of water. Reines shared the 1995 Nobel Prize for it. The reaction was:

Check it: charge 0 + 1 = 0 + 1; baryon number 0 + 1 = 1 + 0; lepton number −1 + 0 = 0 + (−1). To draw it, the proton (+1) becomes a neutron (0) and emits a W⁺, which the antineutrino (0) absorbs to become a positron (+1).

Example 3: spot the mistake

A student draws beta minus decay with a photon as the exchange particle. What is wrong?

A photon carries the electromagnetic force, which cannot change a neutron into a proton. Changing the type of a quark (down to up) needs the weak interaction, so the exchange particle must be a W boson, and charge at the neutron vertex makes it a W⁻.

Check your understanding.

Common mistakes

• Balancing charge only overall. Charge must balance at each vertex, which is what fixes W⁺ or W⁻.
• Using a photon in a weak interaction. Anything that changes a proton into a neutron, or the reverse, needs a W boson.
• Mixing up neutrinos and antineutrinos. Beta minus gives an electron antineutrino; beta plus and electron capture give an electron neutrino. Check lepton number.
• Leaving lines unlabelled, or leaving the exchange particle unlabelled.
• Crossing hadrons and leptons. A proton line never turns into an electron line. Keep hadrons on one side and leptons on the other.
• Thinking the lines are paths. The angles mean nothing; only which lines meet at which vertex matters.

Beta decay equations and electron capture are covered in more depth in alpha, beta and gamma decay. For a course that builds up from quarks to the Standard Model, see the Particles course.

Practise with a short timed paper

Particles and their interactions are in the first year of A-level Physics. The daily A-level Physics: Year 1 paper is a short timed paper of 8 questions, with a new one every day.

Read next

Physics · Atoms and particlesAlpha, beta and gamma decay: equations and propertiesAlpha, beta and gamma radiation compared, with decay equations for alpha, beta minus, beta plus and electron capture, Feynman diagrams and worked examples.