tree-level vs loop diagrams
Picture a family tree: it branches outward and never loops back on itself. Some Feynman diagrams are like that — lines split and join but never form a closed circuit. These are called tree-level diagrams, and they give the simplest, leading estimate of how likely a process is. Other diagrams contain a closed loop, where an internal particle line curls around and comes back, like a road that briefly circles before continuing. These are loop diagrams, and they are the quantum corrections that fine-tune the answer.
The difference is not cosmetic. A tree diagram has no closed loops, so its value is found by straightforward multiplication of the pieces. A loop diagram includes a virtual particle (or several) that pops into existence, goes around, and vanishes; because the loop's energy and momentum are not fixed by the incoming particles, you must add up over all possible values the loop could carry — a sum that becomes an integral. Each additional loop also adds vertices, so loop diagrams carry extra powers of the coupling and are usually smaller corrections on top of the tree result. One loop is a small correction; two loops, smaller still.
Loop diagrams are where quantum field theory gets both its precision and its headaches. They produce the tiny, measurable shifts — like the electron's anomalous magnetic moment, predicted and measured to more than ten decimal places — that are among the greatest triumphs in science. But the loop integrals often come out infinite at first, and taming those infinities requires the careful machinery of renormalization. Honestly, computing many loops is extraordinarily hard, which is why higher-order predictions are a major research effort in themselves.
At tree level, an electron's magnetism is predicted by a simple value. Add one loop — where the electron briefly emits and reabsorbs a virtual photon — and the prediction shifts by about one part in a thousand. That tiny loop correction, computed and measured to extraordinary precision, is one of the strongest confirmations of quantum electrodynamics.
One loop, one part in a thousand — and a triumph of theory.
Loop integrals often come out infinite before renormalization removes the infinities in a well-defined way; this is a feature of the calculation, not a sign that loops are unphysical.