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The antiparticle of a fermion. In a Dirac field, its external wavefunctions are the negative-frequency $v$ spinors, while its physical energy is positive. Fermion-number arrows on Feynman diagrams run opposite an antifermion's physical momentum.

A one-loop Feynman diagram with four vertices joined into a box. Two W bosons and two internal quarks give a short-distance contribution to neutral D-meson mixing.

A perturbative correlator factors into an exponential of connected Vacuum Feynman diagrams times a sum of Feynman diagrams attached to external insertions. Dividing its path integral by the vacuum generating functional at zero source cancels that exponential, including all disconnected products of vacuum components. The corresponding operator formalism uses the same denominator in the vacuum expectation of the Dyson series. This leaves externally disconnected correlators intact; the connected generating functional removes their products.

The group-theoretic tensor multiplying the kinematic part of a Feynman diagram involving color charge. It is obtained by contracting internal indices among gauge generators, invariant tensors and structure constants of a Lie algebra. For instance, one gluon exchange on a single quark line contributes ${\sum }_a{T}^a{T}^a=C_{F}1$; averaging or summing the squared tensor over external colour labels gives the corresponding color average in a squared scattering amplitude.

The achievement depended on complementary ways to organize the same physics. In his Nobel lecture, delivered in 1966 for the 1965 prize, Sin-Itiro Tomonaga described covariant contact transformations and replacing the unobservable bare mass and electric charge by experimental values. He also credited Julian Seymour Schwinger's magnetic-moment work and Richard Feynman's methods for extending the approximation.^[ref]

Richard Feynman's 1965 lecture traced his spacetime and path integral viewpoint and the Feynman diagrams that made perturbation theory easier to organize. He emphasized that the path integral viewpoint helped him find expressions without being essential to every calculation. These accounts support shared credit for workable renormalized QED, not attribution of the whole theory to a single person.^[ref]

A Feynman diagram with more than one connected component. Products of component contributions exponentiate; taking the logarithm of a Gaussian perturbation expansion removes disconnected products and keeps connected Feynman diagrams.

At tree level, the Feynman diagram has an electron-positron current and a muon-antimuon current joined by one virtual photon. In the massless limit, the spin average and final-state spin sum give $\overline{|\mathcal{M}{|}^2}=e^4(1+{\cos }^2\theta )$, and the integrated relativistic scattering cross-section is $4\pi {\alpha }^2/(3s)$.

The arrow on a charged-fermion line in a Feynman diagram tracks fermion-number flow and fixes the order of the gamma matrices between external Dirac spinors. For an external particle it agrees with the particle direction; for an external antiparticle it is opposite to the antiparticle direction. It therefore need not point along the assigned four-momentum of every external leg.

A Feynman diagram records a term in the perturbative expansion: edges represent propagators, vertices represent interactions, and loops represent unconstrained momentum integrals.

A Feynman integral is the momentum-space integral associated with a Feynman diagram: each independent loop momentum is integrated with measure $d^{d}k/(2\pi {)}^d$, and propagators, vertex numerators and momentum conservation determine the integrand. Its definition also requires a pole prescription and, when divergent, a regularization in quantum field theory. Feynman parameters combine propagator denominators and permit shifts of loop momenta. A massless two-propagator example is $\int d^{d}k/[(2\pi {)}^d{k}^2(p-k{)}^2]$, with its causal prescriptions understood. Subtracting local counterterms turns a regulated diagram into a finite contribution to a renormalized amplitude.

Feynman rules translate a Feynman diagram into a contribution to a scattering amplitude or correlation function in perturbation theory. They specify factors for external states, interaction vertices, internal propagators, momentum-conservation constraints and loop integrals, with statistics-dependent signs and symmetry factors. For quantum electrodynamics, a photon-fermion vertex contains the coupling $e$, while a photon propagator supplies an inverse squared four-momentum. The complete amplitude also contains currents, so propagator powers alone do not determine the energy dependence of a scattering cross-section.

A ghost loop is a closed cycle of ghost propagators in a Feynman diagram. Its Grassmann statistics contribute an additional minus sign.

An interaction vertex joins the field insertions from one interaction term in a Feynman diagram. Its number of incident lines is the number of fields in that term, and its momentum-space Feynman rule includes the interaction coefficient and momentum conservation.

A loop momentum is an independent momentum integrated over in a Feynman diagram. Momentum conservation leaves one independent loop momentum for each cycle counted by the loop order.

The loop order of a connected Feynman diagram is the number $L=I-V+1$ of independent momentum cycles, where $I$ is the number of internal lines and $V$ the number of vertices. Each loop introduces an unconstrained momentum integral.

An integral over momentum space. For example, state counting in $d$ spatial dimensions uses a measure proportional to $d^{d}p/(2\pi \hslash {)}^d$, whereas a relativistic Feynman integral commonly uses $d^{d}p/(2\pi {)}^d$ in natural units. The convention and any regulator must be specified. Independent loop momenta supply separate integration variables in a loop Feynman diagram; ultraviolet divergences and infrared divergences can require different regularizations.

A weak interaction decay of a baryon into final hadrons without final leptons. A weak charged current changes quark flavour, while spectator quarks and the produced quarks undergo hadronization. A quark-flow Feynman diagram specifies a possible weak topology; it does not calculate the full hadronic decay amplitude.

For a connected real-scalar Feynman diagram, degree counting and $L=I-V+1$ give $V_3+2V_4=E+2L-2$. A one-loop one-particle-irreducible Feynman diagram is a cycle: every vertex has two internal half-edges. For three labelled external legs, the possibilities are one cubic triangle with symmetry factor one and three mixed cubic-quartic bubbles with symmetry factor two. For four labelled external legs, there are three quartic bubbles with symmetry factor two, six mixed triangles with symmetry factor one, and three cubic boxes with symmetry factor one. Cyclic ordering modulo reversal explains the three boxes.

A Feynman diagram with one independent closed cycle and hence one independent loop momentum. An $n$-vertex polygon of Dirac propagators in four dimensions has superficial degree of divergence $4-n$ before symmetry cancellations. For a fermion loop, the closed fermion loop sign is separate from any symmetry factor or pairwise charge conjugation cancellation.

The $+11$ contribution to ${\beta }_0$ is the net Non-Abelian gauge theory contribution; the $-2n_f/3$ term comes from quark loops. Thus quarks screen color charge, opposing asymptotic freedom. It is the net gluon contribution that produces antiscreening: the inferred QCD coupling decreases toward shorter distances. Unlike the electrically neutral photon, a gluon carries color charge and has self-interactions. Describing this as vacuum polarization must not imply that all vacuum contributions have the screening sign of quantum electrodynamics.^[ref]

Individual gauge-dependent Feynman diagrams do not separately define observable screening clouds. A proof of the coefficient sums the gauge fixing, Faddeev-Popov ghost and matter contributions and extracts the charge renormalization; the final one-loop order coefficient is independent of the gauge choice. The beta function is negative near zero for integer $0\le n_f\le 16$. This is a statement about weak coupling, not a determination of the low-energy particle spectrum.^[ref]