Renormalizability
Renormalizability is an important concept in quantum field theory, including gauge theories. In the context of gauge theories, such as quantum electrodynamics (QED) or the electroweak theory, renormalizability refers to the property of the theory to absorb or eliminate divergences that arise in perturbative calculations.
A gauge theory is a type of quantum field theory that incorporates local gauge symmetry. This symmetry is associated with the interaction between elementary particles, mediated by gauge bosons. Examples of gauge theories include QED, which describes the electromagnetic force, and the electroweak theory, which unifies the electromagnetic and weak forces.
In perturbative calculations of quantum field theories, one typically encounters integrals that diverge, leading to infinite results. These divergences arise due to the presence of ultraviolet (UV) divergences, which are associated with high-energy or short-distance physics. Renormalizability is the property of a theory to handle and remove these divergences in a systematic manner.
A renormalizable gauge theory possesses certain features:
1. Power counting: The theory must have a power counting property, which means that the divergent integrals in perturbation theory can be classified according to their degree of divergence. Renormalizable theories have a finite number of divergent diagrams at each order of perturbation theory.
2. Finite number of counterterms: The number of counterterms required to remove the divergences is finite. Counterterms are additional terms in the Lagrangian that absorb the divergences, effectively redefining the parameters of the theory. Renormalizable theories have only a finite number of such counterterms, which can be fixed by a finite number of experimental input parameters.
3. Consistency of loop diagrams: Renormalizable theories exhibit cancellations among loop diagrams, which ensure that the divergences are absorbed into the renormalized parameters in a consistent manner.
The renormalizability of a gauge theory is closely related to the behavior of the gauge bosons and fermions under renormalization. In renormalizable gauge theories, the gauge bosons and fermions acquire mass and wave function renormalization factors, which absorb the divergences. These factors can be computed systematically using renormalization group techniques.
It is worth noting that not all gauge theories are renormalizable. For example, non-Abelian gauge theories like quantum chromodynamics (QCD), which describes the strong nuclear force, are not renormalizable in the traditional sense. However, despite their non-renormalizability, such theories can still be used for practical calculations through methods like lattice gauge theory and effective field theories.
In summary, renormalizability is a crucial property of gauge theories that ensures the self-consistency and predictive power of the theory by handling and eliminating divergences.
