Feynman Diagram Studio Mathematical meaning
Quantum field theory describes particles as excitations of fields. For weak couplings, a transition amplitude can be expanded in powers of the interaction strength. Each diagram records one term in that expansion, including its external states, interaction vertices, and internal propagators.
In a momentum-space diagram, page position organizes the graph and its algebra. It generally carries no measured distance or duration. Authors may adopt a left-to-right time convention to make scattering processes easier to read.
An internal four-momentum can be off shell, so it need not satisfy the free-particle relation p² = m². External states correspond to the particles prepared or measured in the process.
Lines, vertices, and momentum
External lines
Incoming and outgoing lines label the prepared initial state and measured final state. Their momenta are on shell, and their spin or polarization states enter the amplitude.
Internal lines
An internal line connects two vertices and contributes a propagator. In momentum space, a scalar propagator has the schematic form
Vertices
A vertex comes from an interaction term in the Lagrangian. In quantum electrodynamics, the electron-photon interaction contributes a factor proportional to the electric charge e and a Dirac matrix γμ. The allowed combinations of fields determine which diagrams exist.
Momentum arrows
A momentum arrow assigns an orientation to a four-momentum variable such as p, q, or k. The chosen orientation fixes the sign convention used in vertex conservation equations. For a vertex with all momenta defined as incoming, conservation is written
The momentum arrow and the fermion-flow arrow serve different purposes. Fermion-flow arrows track the orientation of a fermion line; momentum arrows record a momentum convention and can be added to any propagator style.
Quarks and leptons; arrow direction carries fermion flow.
Electromagnetic gauge boson.
Strong-force gauge boson.
Spin-0 field, often the Higgs or a model-dependent scalar.
Auxiliary field used in gauge-theory calculations.
From a diagram to a probability
Feynman rules translate each external line, internal line, and vertex into an algebraic factor. You multiply those factors, conserve four-momentum at every vertex, integrate over undetermined internal momenta, and include the sign and symmetry factor.
Several diagrams can lead to the same initial and final states. Add their complex amplitudes before taking the magnitude squared:
Phase-space factors and flux convert |ℳ|² into a decay rate or scattering cross section. The calculation therefore requires the amplitude and the relevant kinematic factors.
Orders, loops, and corrections
Each interaction vertex brings a power of a coupling. In QED, the tree diagram for e⁻e⁺ → μ⁻μ⁺ has two electromagnetic vertices, so its amplitude starts at order e². Squaring the amplitude produces a leading cross section of order e⁴, often written in terms of α = e²/(4π).
No closed momentum loop
The leading contribution commonly gives the first useful prediction. “Tree” describes graph topology.
Undetermined internal momentum
Each independent loop introduces an integral. Loop diagrams provide quantum or radiative corrections and may require regularization and renormalization.
Coupling size, kinematics, symmetries, and cancellations determine the numerical importance of a higher-order correction.
Conservation laws at vertices
Every vertex conserves four-momentum. The interaction also enforces the charges and quantum numbers respected by the theory. Check the following conditions while constructing a diagram:
- Electric charge: the algebraic sum entering a vertex equals the sum leaving it.
- Color: QCD vertices connect color flow according to the gauge-group rules; a gluon carries color and anticolor.
- Fermion flow: arrows form continuous lines through allowed vertices. Antifermion arrows point opposite the common left-to-right particle-flow reading.
- Angular momentum: spin and orbital-angular-momentum constraints follow from the amplitude. The drawn angle between two lines has no direct physical value.
Specific interactions may conserve, approximate, or violate additional quantum numbers. Each drawn vertex must correspond to an interaction term in the Lagrangian.
Standard Model context
The Standard Model combines quantum electrodynamics, the weak interaction, and quantum chromodynamics. Matter fields include quarks and leptons. Gauge bosons mediate the interactions: photons for electromagnetism, gluons for the strong interaction, and W and Z bosons for the weak interaction. The Higgs field supplies a scalar particle and participates in mass generation.
The editor’s five line styles follow common visual conventions. The particle label and the interaction Lagrangian supply the physical meaning. A wavy line can represent γ, Z, or W; a scalar line can represent h or another scalar field. Ghost lines represent auxiliary fields used in gauge-fixed perturbative calculations and appear on internal lines.
CERN’s Standard Model overview (opens in a new tab) introduces the particles and three included forces. The Particle Data Group reviews (opens in a new tab) provide technical reference material.