Quantum Field Theory

Feynman Diagrams

Each diagram = one term in a perturbative expansion of the scattering amplitude

Feynman diagrams (Richard Feynman, 1948) are graphical representations of terms in the perturbative expansion of scattering amplitudes in QED, QCD, and QFT generally. Each diagram has: external legs (incoming/outgoing real particles), internal lines (virtual particles, propagators), and vertices (interaction points labeled by coupling constants). The Feynman rules translate diagrams into mathematical expressions: a Dirac fermion line gives 1/(p̸−m); a photon propagator −i g_μν/q²; a QED vertex −ieγ^μ. Each loop contributes a factor of the coupling — α ≈ 1/137 in QED, allowing perturbation. Lamb shift and anomalous magnetic moment of the electron computed to 12+ digits via thousands of diagrams (Kinoshita 2018: 891 5-loop diagrams). Earned Feynman, Schwinger, and Tomonaga the 1965 Nobel Prize. Foundation of LHC predictions.

  • AuthorFeynman 1948
  • ComponentsLegs, lines, vertices
  • QED couplingα ≈ 1/137
  • Anomalous magnetic moment12 digits
  • 5-loop QED891 diagrams
  • Nobel 1965Feynman / Schwinger / Tomonaga

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Why Feynman diagrams matter

  • LHC predictions. Matrix-element generators (MadGraph, Sherpa) automatically sum thousands of diagrams to predict proton-proton cross sections to 1 to 10% precision. Higgs discovery and top measurements rely on this pipeline.
  • QED precision tests. The electron's anomalous magnetic moment matches theory to 12 digits — the most precise theory-experiment agreement in physics. Tests for new physics live in residual disagreements at the 13th digit.
  • Theoretical particle physics. Every paper proposing a beyond-Standard-Model theory writes new vertices and computes diagrams to predict signatures at colliders or low-energy experiments.
  • Renormalization. The systematic redefinition of mass and charge order-by-order in perturbation theory is articulated diagrammatically. Wilson's modern view interprets the running coupling as a flow in the space of theories.
  • Pedagogy. Diagrams turn abstract field-theory calculations into picture-driven bookkeeping. Graduate quantum field theory is essentially the art of drawing, evaluating, and summing diagrams.
  • Beyond particle physics. Diagrammatic methods power condensed matter (Green's functions, many-body theory), nuclear physics (chiral effective theory), and even lattice QCD simulations.
  • Loop expansion. Tree-level diagrams give classical predictions; loops encode quantum corrections. The hierarchy translates directly to powers of ℏ.

Anatomy of a diagram

A QED scattering process — say electron-electron scattering (Moller scattering) — has external lines: two incoming electrons, two outgoing electrons. The simplest (tree-level) diagram joins these via one internal photon line, with two vertices. Reading the diagram top-to-bottom or side-to-side, vertices represent points where particles meet and exchange a virtual photon. Each vertex contributes the QED coupling −ieγ^μ. The internal photon contributes its propagator. Multiplying these gives one term in the scattering amplitude.

Higher-order diagrams add loops: the photon temporarily fluctuates into an electron-positron pair (vacuum polarization), or each external line gets self-energy corrections. Every loop closes back on itself, contributing an integral over an unconstrained four-momentum. These integrals can diverge; regularization and renormalization tame them.

The Feynman rules in QED

  • Fermion propagator. i(p̸ + m) / (p² − m² + iε). Represents a virtual electron or positron.
  • Photon propagator. −i g_μν / (q² + iε) in Feynman gauge. Represents a virtual photon.
  • Vertex factor. −ieγ^μ. The basic QED interaction: an electron line emits or absorbs a photon.
  • External lines. Spinors u(p), v(p) for incoming/outgoing fermions; polarization vectors ε^μ(k) for photons.
  • Loop integration. Integrate over each undetermined internal momentum: ∫d&sup4;k/(2π)&sup4;.
  • Symmetry factors and signs. Identical particles, fermion loops (factor of −1), and combinatoric multiplicities.

QED precision: the electron g-2

Schwinger's 1948 one-loop calculation gave a_e = α/(2π) ≈ 0.00116. Successive multiloop corrections by Kinoshita and collaborators evaluated all diagrams up through five loops. The five-loop computation alone (completed 2018) involved 891 distinct diagrams; many demanded numerical integration on supercomputers. The theoretical prediction agrees with measurement to twelve significant figures — an extraordinary validation of QED and a stress-test for the calculation machinery.

Common misconceptions

  • "Particles really travel paths." Lines in a diagram are perturbative bookkeeping, not literal trajectories. Different diagrams contribute different amplitudes that must be summed before squaring.
  • "Only QED." Feynman-diagram methods extend to all renormalizable QFTs: QCD, electroweak, scalar field theories, and most BSM models.
  • "Loops trivial." Loop integrals diverge and require regularization plus renormalization. Multiloop calculations require sophisticated mathematical techniques: integration by parts, master integrals, sector decomposition.
  • "Virtual particles are real." Virtual particles are internal lines; they are off-shell mathematical objects, not directly observable. Vacuum fluctuations are sometimes spoken of in this language but are not truly localized particles.
  • "Sum is finite by inspection." Some QFTs have asymptotic perturbation series that ultimately diverge. QED is believed to be asymptotic; the series is meaningful at small orders but not literally convergent.
  • "Diagrams are unique." The set of diagrams depends on the gauge fixing, the choice of effective field theory, and chosen power counting.

Frequently asked questions

What does a Feynman diagram represent mathematically?

Each Feynman diagram represents one term in a perturbative expansion of a scattering amplitude. The Feynman rules assign a mathematical expression to each piece of the diagram: a Dirac fermion propagator gives 1/(p-slash minus m), a photon propagator gives minus i times g_mu_nu divided by q-squared, and a QED vertex gives minus i times e times gamma-mu. Multiplying these factors and integrating over loop momenta yields a contribution to the amplitude. Squaring the total amplitude gives the cross section or decay rate measured in experiments.

What are virtual particles in Feynman diagrams?

Virtual particles are internal lines in a diagram. They do not satisfy the on-shell condition E-squared equals p-squared c-squared plus m-squared c-fourth — their four-momentum is off-shell because they are intermediate states integrated over in the calculation. Virtual particles are bookkeeping devices for terms in perturbation theory, not literal particles. They cannot be directly detected. Their effects appear in measured quantities like the Lamb shift, the running of coupling constants, and the anomalous magnetic moment.

Why does perturbation theory work for QED but not QCD at low energy?

Perturbative expansions converge when the coupling is small. In QED the fine-structure constant alpha is approximately 1/137, so each additional vertex pair suppresses a diagram by about 1%. In QCD the strong coupling alpha-s runs with energy: at high energies it is small (asymptotic freedom) and perturbation works, but at low energies (typical hadron scales) alpha-s approaches one and perturbation fails. Low-energy QCD requires nonperturbative methods like lattice QCD.

What is the anomalous magnetic moment calculation?

The electron magnetic moment ratio g is two at tree level. Quantum corrections shift it: the famous a_e = (g minus 2)/2 has been computed via thousands of Feynman diagrams. Schwinger's 1948 one-loop result gave alpha/(2 pi). Kinoshita and collaborators completed the five-loop QED calculation in 2018, evaluating 891 distinct diagrams. The theoretical prediction agrees with experiment to about twelve significant digits, making it one of the most precise theory-experiment comparisons in all of physics.

How are loop integrals regularized?

Loop integrals over virtual momenta typically diverge at large momentum (ultraviolet) or small momentum (infrared). Regularization makes the divergent integrals temporarily finite via a cutoff: dimensional regularization works in 4 minus epsilon dimensions; Pauli-Villars adds heavy fictitious particles; lattice discretizes spacetime. Renormalization then absorbs the divergences into redefinitions of physical parameters (mass, charge). Renormalizable theories like QED and QCD make finite predictions for observables despite divergent intermediate steps.

What is the LHC connection to Feynman diagrams?

LHC predictions rely on matrix element generators (MadGraph, Sherpa, OpenLoops) that automatically generate, evaluate, and sum thousands of Feynman diagrams per process at tree level, one-loop, and beyond. Combined with parton distribution functions and parton showers, these diagrams predict cross sections for proton-proton collisions to 1 to 10% precision. Higgs discovery, top quark measurements, and beyond-Standard-Model search limits all hinge on Feynman-diagram-based predictions agreeing with measured event rates.