A diagram is a recipe, not a snapshot
By now in this rung you can read a Feynman diagram: you know the external legs are the real particles coming in and going out, the internal lines are virtual particles passing the force along, and each vertex is a meeting where particles couple. That is the picture. But a picture is not a number, and nature only ever hands experiments numbers — how often something happens. This guide is the bridge between the two: how a hand-drawn diagram becomes a probability you can actually go out and measure with a cross section or a decay rate.
The crucial reframing is this: a Feynman diagram is not a snapshot of what really happened inside the collision. It is a line in a recipe. Each piece of the drawing — every vertex, every internal line, every external leg — stands for a precise mathematical factor, and the rules that assign those factors are called the Feynman rules. Draw the diagram, read off the factors, multiply them together, and you have computed one contribution to a single complex number: the amplitude for the process.
From a diagram to the amplitude
The number a diagram computes is the matrix element, usually written with a script letter M. It is a probability amplitude in the same sense you met earlier in the ladder: a complex number whose size and phase encode 'how strongly' this process wants to happen. The single most important rule for reading it is that every vertex contributes a factor of the coupling constant — the number that sets how eagerly a force acts. For electromagnetism that strength is governed by the fine-structure constant, about 1/137, and each vertex pays one factor of its square root, which is why diagrams with fewer vertices dominate.
Internal lines contribute too. Each virtual particle carries a factor called the propagator — for a virtual photon it is the photon propagator — which is large when the virtual particle is 'almost real' and small when it is far off its natural energy-mass relation. This single factor is what makes the electric force fall off with distance and grow at short range, all hiding inside one term of the recipe. The external legs, the real incoming and outgoing particles, contribute their own simple factors that describe how they enter and leave.
M = (vertex) x (propagator) x (vertex) x (external legs)
e- ---<--- * * = vertex, factor ~ sqrt(alpha)
\ ~ = virtual photon, the propagator
~~~~~~ alpha ~ 1/137
/
e- ---<--- *Almost always, more than one diagram contributes to the same process. The simplest are the tree-level diagrams, with no closed loops; then come more elaborate diagrams with extra vertices and internal loops. Each extra pair of vertices costs another factor of the small coupling, so each is a smaller correction. This is perturbation theory: because the coupling is small, the simplest diagram captures most of the answer, and you add finer corrections only until you reach the precision you need. You sum the amplitudes of all relevant diagrams to get the total M.
Squaring it: amplitude becomes probability
An amplitude is not yet a probability — it is a complex number, and probabilities are real and positive. The bridge is the rule you first met in quantum mechanics: a probability is the amplitude's magnitude squared. So before you can compare with any experiment, you take your total M and form the quantity written |M|^2. This single step has a profound consequence: because you square the sum of diagrams, the diagrams can interfere. Two amplitudes that point in the same direction reinforce; two that point oppositely cancel.
Interference is not a footnote — it is where the deepest tests of the theory live. When two indistinguishable channels lead to the very same final particles, you must add their amplitudes before squaring, and the cross-terms reshape the result. The same idea explains a resonance: when the collision energy lands right on the natural energy of a short-lived particle, its propagator becomes large (peaks) and the rate spikes into a Breit-Wigner bump, whose height tells you the coupling and whose width tells you the decay width. The whole machinery of squaring a sum turns abstract amplitudes into shapes a detector can draw.
Fermi's golden rule: probability into a rate
Knowing |M|^2 tells you how strongly a process is favored, but not yet how often it happens per second. For that you need one more honest ingredient: there must actually be room for the outgoing particles to fly into. This is the idea behind Fermi's golden rule, a statement so general it long predates Feynman diagrams. In words, the rate of a process equals the strength of the coupling — the |M|^2 — multiplied by the amount of available final-state room.
That second factor is phase space: the count of all the ways the final particles can share out the available energy and momentum, while still respecting energy and momentum conservation. Phase space is pure geometry and bookkeeping — it knows nothing about forces — yet it shapes results powerfully. A decay that barely has enough energy to make its products has almost no phase space, and so proceeds slowly no matter how strong the coupling; open up more energy and the phase space grows, and the rate climbs with it. Many features of real data are phase space talking, not the interaction itself.
The two outputs experiments measure
Run the golden rule for two particles smashing together and the output is a cross section — the probability-dressed-as-an-area you met in the previous rung, now derived rather than just described. Run the very same machine for one unstable particle sitting at rest, and the output is a decay rate, which is exactly the inverse of the lifetime and the same thing as the decay width. One recipe, two faces: scattering when particles come together, decay when a particle comes apart.
Now the loop closes back onto the experiment. A cross section is a probability per unit beam exposure, so to turn it into a stream of events you multiply by the luminosity — how intensely the beams are aimed at each other. Count the events, divide back out, and you have measured the cross section. Compare it against the number the diagrams predicted, and you are doing the central act of the whole field: theory makes a number, the collider returns a number, and you check whether nature agrees.
Be honest about what this chain does and does not promise. The recipe is built on perturbation theory, so it works beautifully when the coupling is small — as in electromagnetism — and far less well for the strong force at low energies, where no single handful of diagrams suffices and physicists turn to other methods. And the agreement between predicted and measured numbers, stunning as it is, has so far confirmed the Standard Model at every turn: there is no firmly established discrepancy pointing beyond it yet. The triumph of this method is precisely how sharply it lets us say that — to test a theory, you must first be able to compute exactly what it predicts.