Quantum field theory
Renormalization

Feynman rules can be used to directly evaluate tree-level diagrams. However, naïve computation of loop diagrams such as the one shown above will result in divergent momentum integrals, which seems to imply that almost all terms in the perturbative expansion are infinite. The renormalisation procedure is a systematic process for removing such infinities.

Parameters appearing in the Lagrangian, such as the mass m and the coupling constant λ, have no physical meaning — m, λ, and the field strength ϕ are not experimentally measurable quantities and are referred to here as the bare mass, bare coupling constant, and bare field, respectively. The physical mass and coupling constant are measured in some interaction process and are generally different from the bare quantities. While computing physical quantities from this interaction process, one may limit the domain of divergent momentum integrals to be below some momentum cut-off Λ, obtain expressions for the physical quantities, and then take the limit Λ → ∞. This is an example of regularization, a class of methods to treat divergences in QFT, with Λ being the regulator.

The approach illustrated above is called bare perturbation theory, as calculations involve only the bare quantities such as mass and coupling constant. A different approach, called renormalized perturbation theory, is to use physically meaningful quantities from the very beginning. In the case of ϕ theory, the field strength is first redefined:

where ϕ is the bare field, ϕ is the renormalized field, and Z is a constant to be determined. The Lagrangian density becomes:

where m and λ are the experimentally measurable, renormalized, mass and coupling constant, respectively, and

are constants to be determined. The first three terms are the ϕ Lagrangian density written in terms of the renormalized quantities, while the latter three terms are referred to as "counterterms". As the Lagrangian now contains more terms, so the Feynman diagrams should include additional elements, each with their own Feynman rules. The procedure is outlined as follows. First select a regularization scheme (such as the cut-off regularization introduced above or dimensional regularization); call the regulator Λ. Compute Feynman diagrams, in which divergent terms will depend on Λ. Then, define δ, δ, and δ such that Feynman diagrams for the counterterms will exactly cancel the divergent terms in the normal Feynman diagrams when the limit Λ → ∞ is taken. In this way, meaningful finite quantities are obtained.

It is only possible to eliminate all infinities to obtain a finite result in renormalizable theories, whereas in non-renormalizable theories infinities cannot be removed by the redefinition of a small number of parameters. The Standard Model of elementary particles is a renormalizable QFT, while quantum gravity is non-renormalizable.
detectable_format:number_highlighted_sections[3]