The vertex is where the bill comes due
In the previous guide you learned to read a [[feynman-diagram|Feynman diagram]] as an alphabet of three pieces: external legs (the real particles you start and end with), propagators (the internal lines, standing for virtual particles passing through), and vertices (the dots where lines meet). Now we turn that picture into a *size* — a number for how often the process actually happens. Here is the rule that makes everything else fall into place: the lines are free, and only the [[vertices-propagators-external-legs|vertices]] cost something. Each dot is a point where particles interact, and every interaction carries a price.
That price is the [[coupling-at-a-vertex|coupling at the vertex]] — a single number that says how strongly the particles meeting there actually talk to each other. In QED the vertex is always the same electron-photon dot, and its price is the QED coupling. The crucial point is that this is the *only* place strength enters the calculation. Two particles drifting along as lines, or a virtual particle ferrying force through a propagator, contribute their own factors, but they never decide how *likely* the interaction is. Likelihood lives entirely at the dots. So to gauge how big a diagram is, you go straight to its vertices and count them.
Counting powers of the coupling
Now the bookkeeping. Each vertex contributes one factor of the [[coupling-constant|coupling constant]] to the amplitude. In QED that per-vertex factor is the electron's charge e, and physicists prefer to package it as alpha, the fine-structure constant, where alpha is proportional to e-squared and works out to about 1/137. The arithmetic is simple: a diagram with two vertices carries e-squared in its amplitude, which is one power of alpha; a diagram with four vertices carries e-to-the-fourth, which is alpha-squared. Add two more vertices and you pay another factor of alpha. That is the entire scaling law.
QED vertex factor ~ e (and alpha ~ e^2 ~ 1/137 ~ 0.0073) 2 vertices : amplitude ~ e^2 probability ~ e^4 ~ alpha^2 4 vertices : amplitude ~ e^4 probability ~ e^8 ~ alpha^4 each EXTRA pair of vertices => one more factor of alpha ~ 1/137 so the 4-vertex correction is ~137x smaller than the 2-vertex one
Take a concrete case: two electrons scattering off each other. The simplest diagram has each electron sit at one vertex and toss a single virtual photon across to the other — two vertices, so the amplitude scales as alpha and the probability as alpha-squared. You can also draw a busier diagram where they swap *two* photons: now four vertices, and that contribution is smaller by a further factor of alpha, roughly a hundred-and-thirty-seven times feebler. The two-photon-exchange diagram is not wrong or forbidden; it is simply a small correction sitting on top of the dominant single-photon picture.
Tree level vs loops
Diagrams come in two great families, and the difference is purely topological — about whether the internal lines form a closed loop. A tree diagram has no closed loops: trace any internal line and it always leads somewhere, like branches of a tree that never circle back. The single-photon-exchange picture of two electrons is a tree. A [[tree-level-vs-loop-diagrams|loop diagram]], by contrast, contains at least one closed circuit of internal lines — a virtual particle (or a pair) that briefly comes into existence, goes around, and reconnects. The two-photon box you drew above is the simplest loop in electron scattering.
Why does the loop matter so much? Two reasons stack on top of each other. First, closing a loop requires extra vertices, so a loop diagram always carries more powers of the coupling than the tree it corrects — it is automatically suppressed by alpha, just as the vertex count predicts. Second, and more subtle: inside a loop the virtual particle's energy and momentum are not fixed by the external particles. Anything is allowed to flow around the loop, so the calculation must *sum over all possibilities* — an integral over every momentum the loop could carry. The tree just multiplies a few factors together; the loop makes you integrate.
Orders, and why a few diagrams suffice
Here is the strategy that ties it all together. To predict a process exactly you would have to add up *infinitely many* diagrams — tree, then one loop, then two loops, on and on forever. That sounds hopeless. But organize the diagrams by how many powers of the coupling they carry, and a miracle of accounting appears. The lowest power is the leading order (the tree); the next power up is the next-to-leading order (one loop); and so on. This ladder is the [[perturbation-theory-order|order of perturbation theory]], and each rung is smaller than the one below it by a factor of the coupling.
Because alpha is about 1/137, each successive order is roughly a hundred times smaller than the last. So the infinite sum behaves like 1 + 1/137 + 1/137-squared + ..., which converges fast: the leading term already gets you within about a percent, the first correction sharpens it to a fraction of a percent, and you stop when you reach the precision you need. *That* is why a handful of diagrams suffices. You are not ignoring the rest out of laziness — you have a quantitative guarantee that the diagrams you dropped are too small to matter at your target accuracy.
There is also a quieter rule that prunes the list before you even count orders: at a vertex, the conservation laws must hold. Charge, energy-momentum, and the other quantum numbers must balance at every dot, so most diagrams you might idly sketch are simply impossible and never get drawn. Between conservation forbidding many shapes and small couplings suppressing the survivors, the seemingly infinite calculation collapses to a short, manageable list — usually one or a few diagrams that carry essentially the whole answer.
When the trick breaks: the strong force
The whole scheme rests on one assumption: the coupling is small. Pull that out and it collapses. The strong force is exactly where it collapses. At everyday energies its coupling is not 1/137 but close to 1, so the series 1 + 1 + 1 + ... does not shrink — every order matters as much as the last, and adding more diagrams gets you nowhere. This is why you cannot draw a few Feynman diagrams to compute, say, how three quarks bind into a proton: there is no leading diagram, and the perturbation ladder has no bottom rung to stand on.
But here is the beautiful twist, one you have already met in disguise: the coupling is not a fixed number. It is a [[running-coupling|running coupling]] — its value depends on the energy of the interaction. The strong coupling is large at low energy (where quarks are locked inside hadrons) yet shrinks at very high energy, the property called asymptotic freedom. So even for the strong force, diagrams and perturbation theory work *beautifully* in violent, high-energy collisions, where the coupling has run small. The method has a domain of validity, set by whether the relevant coupling happens to be small at the energy you are probing.
One more honest wrinkle, and it is QED's own. Even the well-behaved electromagnetic series does not converge forever; pushed to enormously high order it eventually starts to grow again — it is what mathematicians call an asymptotic series. In practice this never bites, because the breakdown sets in far beyond the handful of orders anyone computes. The lesson to carry forward is not that perturbation theory is fragile, but that it is a *tool with a range*: superb where the coupling is small, silent where it is not, and always honest about which regime you are in.