renormalizable

Renormalizable is a term used to describe a quantum field theory whose ultraviolet divergences can be absorbed into a finite number of parameters of the theory, through the process of renormalization. In a renormalizable theory, the infinities that appear in perturbation theory can be canceled by counterterms that already exist in the original Lagrangian, so no new types of terms are required at higher orders.

Power-counting criteria in four spacetime dimensions provide a practical test. Fields have canonical dimensions (scalar fields 1, fermions 3/2, gauge fields 1). Interaction terms with total dimension four or less are renormalizable; those with dimension greater than four are non-renormalizable and typically generate an infinite series of new counterterms at higher orders.

Common renormalizable theories include quantum electrodynamics, non-Abelian gauge theories such as quantum chromodynamics, and the electroweak sector of the Standard Model. By contrast, the Fermi theory of weak interactions, which describes four-fermion interactions at low energy, is non-renormalizable in perturbation theory. Gravity, when treated perturbatively via the Einstein-Hilbert action, is also non-renormalizable in the traditional sense.

Modern perspective often describes theories as effective field theories (EFTs). In this view, non-renormalizable interactions are allowed but suppressed by powers of a high energy scale and organized in a controlled expansion. The theory remains predictive at energies below the cutoff, with a finite set of relevant and marginal operators determining low-energy physics.

Renormalizability is closely tied to the renormalization group flow: a finite set of couplings suffices to absorb divergences, yielding predictive relationships among observables. The Standard Model is renormalizable as a gauge theory, while gravity’s perturbative non-renormalizability motivates alternative approaches in quantum gravity.