the order of perturbation theory

Suppose you want to add up a long list of numbers where each one is a hundred times smaller than the last: 1, then 0.01, then 0.0001, and so on. You do not need the whole infinite list to get an excellent answer; the first few terms already nail it, because the rest are negligible. This is the spirit of perturbation theory: when the coupling is small, the full answer is a sum of contributions that get rapidly smaller, and you keep only the first few. The order is simply how many terms deep you go.

In Feynman-diagram language, the order is tied to how many vertices (and hence how many powers of the coupling) a diagram carries. The simplest diagrams, with the fewest vertices, are called leading order — the dominant contribution. Add the next batch of slightly more complex diagrams and you have next-to-leading order, a refinement; then next-to-next-to-leading order, and so on. Each step adds smaller corrections and more diagrams, sharpening the prediction. Going to higher order is how physicists squeeze a calculation to match ever more precise measurements.

Perturbation theory is the workhorse of particle physics, but it has an honest limit: it only works when the coupling is small enough that later terms really are tiny. For electromagnetism and the weak force this works beautifully. For the strong force at low energies the coupling is large, the series does not settle down, and physicists must turn to other methods such as lattice calculations. So the order of perturbation theory is both a measure of how hard you have worked and a reminder that the method itself only applies where the dial is turned low.