At a glance
A Feynman diagram is a picture of one term in the quantum amplitude for a process. This sandbox checks every vertex you build and, for e⁻e⁺ → ℓ⁻ℓ⁺, turns the diagram into a cross section σ.
What you'll learn
- What the lines, the vertices and the time direction of a diagram stand for.
- Which vertices are allowed, and which boson each process needs.
- How two diagrams for the same process combine.
- How σ for e⁻e⁺ → μ⁻μ⁺ depends on √s, and why.
Time runs left to right. Two particles come in, two go out, and a boson line joins the vertices.
What's on screen
- Legs and slots: two incoming particles on the left, two outgoing on the right, coloured by species (e blue, μ amber, τ purple).
- Vertices V₁, V₂ and the wavy line: the exchanged boson, labelled with its q². A dashed "now" line sweeps across.
- Status chip (top right): ✓ when every vertex is allowed, ✗ with the failed rule, amber when √s is too low.
- Result panel and σ(√s) plot: σ, s·σ, and σ against √s on log axes.
See itLoad the default build, e⁻e⁺ → μ⁻μ⁺. Watch for: dots running in along the legs, the dashed line sweeping across, and a green ✓ in the status chip.
The controls
Press ⌖ on a row to see where that control sits in the sim.
| Control | What it changes | Range | |
|---|---|---|---|
| Puzzles | Pick a target process and build its lowest-order diagram yourself. Picking one clears the legs. A ✓ marks a solved puzzle. | 5 processes | |
| Topology | Annihilation (s): the incoming pair meets at V₁. Exchange (t): the two particles pass a boson between them. | s / t opens on s | |
| Exchanged boson | The boson on the wavy line. | γ, Z⁰, W±, g opens on γ | |
| Particles | Drag a chip onto a leg, or click a chip and then a leg. Click a filled leg to clear it. | 16 particles | |
| Auto-fill / Clear legs | Auto-fill loads the default build, e⁻e⁺ → μ⁻μ⁺ through a γ. Clear legs empties all four slots. | two buttons | |
| √s (Collider) | The collision energy in the centre-of-mass frame. The slider is logarithmic and snaps to 1, 3 and 10 GeV. Moving it pauses the run. | 0.5–30 GeV opens at 3.00 | |
| Show σ(√s) plot | Opens the log–log plot of σ against √s for the built final state, with a marker at your √s. | on / off opens off | |
| Result | The process, σ for this diagram, and s·σ. It appears once all four legs are filled. In the Guided Inquiry, σ and s·σ appear after you commit your σ prediction. | panel | |
| ▶ Play / ⏸ Pause | Runs the dashed "now" line along the diagram, or freezes it. | top bar | |
| Speed | How fast the dashed line sweeps. It does not change any physics. | 0.25×–4× opens at 1× | |
| ↻ Reset | Restores the default build and √s = 3.00 GeV. | top bar |
Fixed: lowest-order QED, α = 1/137.036 (no running), and massless beams.
A diagram is a recipe for an amplitude
Each piece of the diagram stands for one factor in the amplitude 𝓜 of the process, and the factors multiply:
The amplitude of e⁻e⁺ → μ⁻μ⁺ through a γ*.
- Legs are the particles that go in and come out. Each gives one spinor (u, v̄ and so on).
- Vertices V₁ and V₂ each give one coupling factor. For a photon it is −ieγμ.
- The wavy line is the boson's propagator, −i gμν/q², where q is the momentum it carries.
There are two topologies. In annihilation the incoming pair meets at V₁ and the boson carries everything to V₂. In exchange each incoming particle continues as an outgoing one, and the boson passes momentum between them.
Annihilation (top) and exchange (bottom), with the sim's leg colours.
See itBuild annihilation, e⁻e⁺ → μ⁻μ⁺. Watch for: the incoming pair meeting at V₁, and the wavy line carrying everything across to V₂.
See itBuild exchange, νμe⁻ → μ⁻νe. Watch for: both fermion lines running straight through, joined by a vertical wavy line.
Each vertex follows a coupling rule
Before it writes ✓, the sandbox checks both vertices against these rules:
The rules the sandbox applies at a vertex.
- A γ, Z⁰ or g line does not change species. An e⁻ cannot become a μ⁻ at a photon vertex.
- Only the W changes species, and only between doublet partners. It carries the charge away: νμ → μ⁻ emits a W⁺.
- The W emitted at one vertex must be the one absorbed at the other. The sandbox flags a mismatch with ✗.
Left: allowed. Middle: a photon cannot turn e⁻ into μ⁻. Right: a W links doublet partners.
Real W vertices also mix quark generations. The sandbox leaves this out.
See itBuild e⁻e⁺ → μ⁻μ⁺ by exchange with a photon. Watch for: a red ✗ at V₁ and a status chip that names the rule it broke.
See itBuild νμe⁺ → μ⁻ν̄e through a W. Watch for: a red ✗ saying the emitted W does not match the one absorbed.
Diagrams are terms, not events
The wavy line is an internal line, and its q² is fixed by the legs. A real photon has q² = 0. The internal line never does.
q² of the internal line in each topology.
Nothing observable travels between the vertices. The line is a factor in 𝓜, not a particle on a path.
When the same legs fit two diagrams, as in e⁻e⁺ → e⁻e⁺, nature adds the amplitudes first and squares the sum:
The cross term belongs to neither diagram alone.
Treating the diagrams as two separate events and adding probabilities would leave out the interference. In the sandbox, σ for e⁻e⁺ → e⁻e⁺ is the s-channel piece only, and the plot header says so.
See itBuild e⁻μ⁻ → e⁻μ⁻ by photon exchange. Watch for: the label on the wavy line reading q² < 0.
See itBuild e⁻e⁺ → e⁻e⁺ by annihilation, then by exchange. Watch for: the status chip staying green ✓ when the topology changes.
The cross section falls as 1/s
For e⁻e⁺ → μ⁻μ⁺ through one γ*, the squared amplitude averaged over spins is e⁴(1 + cos²θ), with no s in it. The 1/s comes from the flux and phase-space factor:
In the massless limit. θ is the angle between the μ⁻ and the e⁻ beam.
- The propagator's 1/s² is cancelled by the factors of s in the spinor currents. It is not the source of the fall.
- So σ drops by 4 each time √s doubles. In the massless limit s·σ = 86.85 nb·GeV², a constant.
- The plot labels the massless values 86.85, 9.65 and 0.87 nb at √s = 1, 3 and 10 GeV. The μ mass changes the 1 GeV value by under 0.1%.
See itSweep √s from 1 to 10 GeV with the σ plot open. Watch for: σ falling by a factor of 100 while s·σ barely moves.
Mass sets a threshold
A final pair of mass m adds a factor to σ. Here β is the speed of each produced particle in the centre-of-mass frame, in units of c:
The cross section the sandbox reports, in nanobarns.
- Below √s = 2m, no real pair can be made. σ = 0 and the sandbox calls the channel closed. The diagram itself is still legal.
- Above it, σ climbs steeply, then joins the 1/s line as β → 1. At 10 GeV, τ⁻τ⁺ is within 1% of it.
- In the sim's range (0.5–30 GeV) only τ⁻τ⁺ has a threshold, at 2mτ = 3.554 GeV.
√s
out
These sliders move the sim too.
σ(√s) from the sim's own formula. τ⁻τ⁺ (violet) joins the μ⁻μ⁺ line above its threshold.
See itBuild e⁻e⁺ → τ⁻τ⁺ at 3 GeV, then raise √s to 6 GeV. Watch for: σ reading 0 below 3.554 GeV, then rising from zero.
Where it matters
- The yardstick at e⁺e⁻ colliders. σ = 4πα²/3s is the reference cross section for pair production. Measured rates for e⁺e⁻ → hadrons are quoted as a ratio R to it.
- Luminosity. Bhabha scattering, e⁻e⁺ → e⁻e⁺, is common and precisely calculable, so colliders use it to measure how many collisions they delivered.
- Inverse muon decay. νμe⁻ → μ⁻νe, the W-exchange diagram in the sim, is observed in neutrino beams. It needs a neutrino energy above about 10.9 GeV to make the μ⁻ from an electron at rest.
Go deeper: what the sandbox leaves out
- R. At lowest order, R = 3 Σ q² over the quarks that can be made, which is 10/3 for u, d, s and c. Measured values lie a little higher because of gluon radiation.
- The Z⁰. The σ formula has only a photon. The Z⁰ matters near 91 GeV, far above the sim's 30 GeV maximum.
- α runs. The sim fixes α = 1/137.036. Near the Z⁰ mass it is about 1/129.
Summary
In one line. A diagram is a recipe for one term of an amplitude, the vertex rules decide which diagrams exist, and for e⁻e⁺ → ℓ⁻ℓ⁺ the result is σ = 4πα²/3s times a threshold factor.
Key points
- Legs, vertices and the wavy line each stand for a factor in 𝓜.
- γ, Z⁰ and g keep the species. Only the W changes it, between doublet partners.
- When two diagrams fit the same legs, add the amplitudes before squaring.
- σ ∝ 1/s comes from flux and phase space, and a pair of mass m needs √s > 2m.
Common mix-ups
- "The wavy line is a photon travelling between the particles." It is off shell and exists only inside the amplitude.
- "Two diagrams for one process are two separate events." They are two terms of one amplitude, and they interfere.
- "σ falls because the photon propagator weakens." Its 1/s² is cancelled. The 1/s comes from flux and phase space.
Things to try on your own
- Solve all five puzzles, finding both diagrams for e⁻e⁺ → e⁻e⁺.
- Build e⁻e⁺ → τ⁻τ⁺ and step √s across 3.554 GeV.
- With the plot open, multiply σ by s at 1, 3 and 10 GeV and compare the three products.