Feynman diagrams and perturbative amplitudes

Perturbation theory expands the transition amplitude in powers of the coupling: M = M₀ + M₁ + M₂ + …, each term two powers of e, one power of α, down on the one before. Writing those terms out algebraically is punishing, so Feynman's rules let you draw them instead. Each topologically distinct way of joining the given incoming and outgoing particles with vertices and internal lines is one term, and the rules translate the drawing back into an integral. What the drawing does not encode is where or when anything happened. The internal momenta are integrated over — every value, including ones no real particle could carry — so the position of a vertex on the page is a bookkeeping convenience. Deform the drawing however you like and it is the same term; change which external legs meet which vertex and it is a different term that must also be included. Every diagram at a given order contributes, and they are added, not chosen between.

Every diagram is built from three pieces. External lines are the real, on-shell particles you prepared and detected, each carrying a spinor or a polarisation vector. Vertices are the interaction points; each supplies a factor of the coupling — √(4πα) in QED — together with exact energy–momentum conservation at that point. Internal lines join two vertices and supply a propagator. The rules then say what is legal. QED has exactly one vertex, with two fermion legs and one photon leg, so charge conservation, lepton-number conservation and the continuity of fermion lines are automatic: an arrow runs unbroken along the fermion line, drawn against the reading direction for an antiparticle. There is no three-photon vertex, because the photon is uncharged. QCD differs precisely here — gluons carry colour, so three- and four-gluon vertices are in the rule book, and that single change is what eventually produces confinement.

Count the vertices. n vertices means n factors of e in the amplitude, so M ∝ α^(n/2), and probabilities go as |M|². Electron–positron annihilation to a muon pair has one tree diagram with two vertices, so M₀ ∝ α and σ ∝ α² ≈ 5.3×10⁻⁵. The next diagrams have four vertices — a photon exchanged between the outgoing muons, a vertex correction, a vacuum-polarisation loop — and each gives M₁ ∝ α². The trap is to square that and expect a change of order α² ≈ 5×10⁻⁵. Amplitudes add before they are squared: |M₀ + M₁|² = |M₀|² + 2Re(M₀*M₁) + |M₁|², and the interference term is only one power of α down on the tree. A four-vertex diagram therefore shifts the cross-section by of order α, and once the 1/π that a loop integral characteristically supplies is folded in, the working number for one more loop is α/π ≈ 0.2%. Treat that as a baseline, not a bound: logarithms of the collision energy can inflate a loop well beyond it, as the running of α below shows. Tree level before loops, and never square a correction on its own.

An internal line is not a particle: it is the factor 1/(q² − m²c²), with q fixed by conservation at the vertices it joins. Because q is whatever the external kinematics forces it to be, q² generally does not equal m²c² — the line is off the mass shell. The photon exchanged in electron scattering has q² < 0, a negative invariant mass squared, which no detectable photon can have. That denominator carries the steepest q-dependence of the answer. For a massless mediator it gives 1/q², and squaring it produces the 1/q⁴ that is Rutherford's sin⁻⁴(θ/2). For a heavy mediator at small transfer, q² is negligible beside M²c² and the propagator freezes to the constant −1/M²c²: exchange becomes a contact interaction, which is exactly Fermi's four-fermion theory of beta decay. That freezing, not a small coupling, is why the weak interaction is weak below the W mass — α_w ≈ 1/29.5 is 4.6 times α.

Tree diagrams are finite; many loops are not. A loop closes an internal momentum that the external kinematics does not fix, so the rules instruct you to integrate over it from zero to infinity, and that integral can diverge. Renormalisation is the observation that the divergences appear only in a fixed set of places — the electron's mass, its charge, the field normalisations — and those quantities were never predicted anyway; they are measured. Write the answer in terms of the measured charge rather than the bare one, and every remaining prediction comes out finite. The price is that the coupling now needs a scale attached to it. Vacuum-polarisation loops screen the bare charge, so α runs: α = 1/137.036 measured at large distance, but α(M_Z) ≈ 1/128. Quote a cross-section to a per cent and you must say which α you used, because at LEP energies the difference is about 7%.

Nothing guarantees that adding diagrams converges, and it does not. The number of diagrams at order n grows roughly like n!, and Dyson gave the reason convergence is impossible: a convergent series would converge in a disc around α = 0, and so would also describe α < 0 — a world in which like charges attract and the vacuum is unstable against making pairs. That physics is not analytic at zero, so the radius of convergence is zero. What you get instead is an asymptotic expansion: the terms shrink for a while, reach a minimum near n* ≈ 1/λ where λ is the cost of one order, then grow without bound, and the best accuracy available is the size of that smallest term, about e^(−1/λ). For QED with λ = α/π ≈ 2.3×10⁻³ the floor sits near 10⁻¹⁸⁷ and no experiment will ever meet it. For QCD, αₛ/π ≈ 0.038 at the Z still leaves many usable orders; by 1 GeV αₛ ≈ 0.5 and the turnaround arrives by the sixth term; near Λ_QCD ≈ 0.2 GeV there is no small number to expand in at all. That is why 99% of the proton's mass is computed on a lattice and not from diagrams.