renormalizability

Renormalizability is a property of quantum field theories that describes how their ultraviolet (high-energy) divergences can be controlled. In perturbation theory, many quantities are computed as series in coupling constants and often diverge as the regulator is removed. A renormalizable theory is one in which these divergences can be absorbed into a finite number of redefinitions of the parameters and fields that already appear in the original Lagrangian, leaving a finite set of measurable predictions that do not require an infinite number of new parameters at each order.

Technically, this concept is connected to the operator content of the theory. In four spacetime dimensions, interactions built from operators of dimension four or less are typically considered renormalizable, while operators of higher dimension generate an infinite tower of counterterms and are nonrenormalizable in the traditional sense. In practice, nonrenormalizable interactions can still be meaningful as effective field theories valid up to a finite cutoff, with their effects suppressed by powers of energy over the cutoff.

Renormalization group ideas describe how couplings run with energy, encoded in beta functions. The existence of a finite set of renormalizable couplings means a theory has a finite number of essential parameters and therefore predictive power. Classic examples include quantum electrodynamics, which is renormalizable, and Fermi theory of weak interactions, which is nonrenormalizable and effectively replaced by the renormalizable electroweak theory. Gravity, treated as a quantum field theory, is nonrenormalizable in perturbation theory but can be used as an effective field theory at low energies.