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Loop diagrams arise at higher orders in perturbation theory and involve integrals over undetermined internal momenta. These integrals often diverge: ultraviolet (UV) divergences come from high momenta (short distances) and infrared (IR) divergences from low momenta (long distances). Understanding and controlling these divergences is essential for extracting finite, physical predictions.
Tree-level Feynman diagrams give the leading-order predictions of quantum field theory. The next level of precision requires loop diagrams, in which one or more internal propagators form closed loops. Each loop introduces an integral over an undetermined four-momentum, and these integrals frequently diverge. Understanding the nature and origin of these divergences is the gateway to renormalization.
Ultraviolet (UV) divergences arise from the high-momentum (short-distance) behavior of loop integrals. In four dimensions, the integration measure d4k grows as k3 dk, while propagators fall off as powers of 1/k. If the measure grows faster than the propagators fall, the integral diverges. Simple power counting determines the degree of divergence: for a diagram with L loops, I internal lines, and V vertices, the superficial degree of divergence is D = 4L - 2I_B - I_F (where I_B and I_F count boson and fermion propagators). If D >= 0, the diagram diverges (quadratically for D = 2, logarithmically for D = 0). Only a finite number of diagram types are divergent in renormalizable theories — this is what makes renormalization possible.
Infrared (IR) divergences arise from the low-momentum (long-distance) behavior and are associated with massless particles. In QED, virtual photons with very low momentum give logarithmic divergences in loop integrals. These are not handled by renormalization but instead cancel when you ask the right physical question. No detector can distinguish between an electron and an electron accompanied by a very soft photon, so the physically measurable quantity includes both virtual and real soft photon contributions. The Bloch-Nordsieck theorem guarantees that IR divergences from virtual loops cancel against those from real soft photon emission in any inclusive cross section.
The three UV-divergent diagrams in QED are the electron self-energy (fermion loop correction to the electron propagator), the vacuum polarization (fermion loop correction to the photon propagator), and the vertex correction (photon-electron vertex with a loop). These three diagrams generate all the divergences of QED at every order in perturbation theory. The self-energy renormalizes the electron mass and wave function, the vacuum polarization renormalizes the electric charge, and the vertex correction renormalizes the coupling. All other diagrams either are finite or contain these three as subdiagrams. This finiteness of the set of divergent structures is the hallmark of a renormalizable theory.