Perturbative quantum field theory

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

Feynman rules translate each diagram into momentum-space factors for propagators, vertices, external states, momentum conservation and loop integration.

A Feynman parameter combines propagator denominators; for example,After a momentum shift, this often converts a loop integral into a rotationally symmetric one.

A tree-level diagram has no loops and gives the leading classical contribution allowed by the interaction vertices.

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.

A connected Feynman diagram has a path between every pair of its vertices. Normalized correlation functions discard disconnected vacuum bubbles, while full correlators can still factor into disconnected components carrying external insertions.

A one-particle-irreducible Feynman diagram is a connected Feynman diagram that remains connected after any one internal line is cut. The vertices of the quantum effective action generate precisely these diagrams.

A Schwinger-Dyson equation follows from invariance of a path integral under a change of integration variable and relates correlation functions through the field equations and contact terms.

The quantum effective action is the Legendre transform of the generating functional of connected correlation functions. Its functional derivatives are the one-particle-irreducible vertex functions.

Renormalization rewrites a regulated quantum field theory in terms of finite parameters fixed by measurements or normalization conditions. Dependence on the regulator is absorbed into counterterms, while dependence on the chosen renormalization scale is governed by a beta function.

A counterterm is a local term added to a regulated Lagrangian density to cancel ultraviolet divergences and impose chosen renormalization conditions. Field-strength, mass and coupling counterterms respectively adjust propagator normalization, pole position and interaction strength.

A renormalization condition defines a renormalized field or parameter by prescribing a correlation function at a chosen kinematic point. Changing that point changes the renormalized parameters while leaving physical predictions invariant.

The renormalization scale is the auxiliary momentum or energy scale at which renormalization conditions define the parameters of a quantum field theory.

A running coupling is a renormalized coupling regarded as a function of the renormalization scale. Its scale derivative is its beta function.

The beta function of a coupling $g$ isIt describes how the running coupling changes when the renormalization scale changes.

Regularization modifies divergent loop integrals by introducing an auxiliary parameter. The regulator is removed after its dependence has been absorbed into counterterms.

Cutoff regularization restricts loop momenta to $|\ell |<\Lambda$. The ultraviolet cutoff $\Lambda$ makes individual integrals finite while displaying power and logarithmic ultraviolet divergences explicitly.

Dimensional regularization analytically continues loop integrals from an integer spacetime dimension to $d=d_0-ϵ$. Ultraviolet logarithms then appear as poles in $1/ϵ$.

The minimal subtraction scheme chooses counterterms that remove only the poles in the dimensional regulator $ϵ$, without additional finite terms.

Quantum electrodynamics couples a Dirac field to an Abelian gauge field through $-e\bar{\psi }{\gamma }^{\mu }\psi A_{\mu }$.

Photon vacuum polarization is the one-particle-irreducible photon two-point function generated by charged-particle loops.

Scalar quantum electrodynamics minimally couples a complex scalar field to the electromagnetic gauge field through $D_{\mu }={\partial }_{\mu }+ie{A}_{\mu }$.

The scalar-QED term $e^2{A}_{\mu }{A}^{\mu }{\varphi }^{\ast }\varphi$ produces a local two-photon two-scalar vertex, conventionally called the seagull vertex.

For an on-shell amplitude $\mathcal{M}={ϵ}_{\mu }(k){\mathcal{M}}^{\mu }$ with an external photon, gauge invariance implies $k_{\mu }{\mathcal{M}}^{\mu }=0$.

Power counting assigns mass dimensions to fields and couplings. An interaction is relevant, marginal or irrelevant when its coupling has positive, zero or negative mass dimension, respectively.

The superficial degree of divergence $D$ is the ultraviolet power predicted by rescaling all independent loop momenta together. For a scalar graph with $L$ loops and $I$ propagators behaving as $p^{-2}$ in $d$ dimensions, $D=dL-2I$ before cancellations and subdivergences are considered.

A marginal coupling is dimensionless under classical power counting.

A relevant coupling has positive mass dimension and grows under scaling toward long distances.

An irrelevant coupling has negative mass dimension and is suppressed toward long distances.