Superrenormalizable

Superrenormalizable is a term used in quantum field theory to describe a theory where the coupling constants, which dictate the strength of interactions, have negative mass dimensions. This property has significant implications for how the theory behaves at different energy scales. In a superrenormalizable theory, divergences that arise in calculations, typically when considering interactions at very high energies or very short distances, can be absorbed into the redefinition of a finite number of parameters. This process is known as renormalization.

Unlike asymptotically free theories, where the coupling constant decreases at high energies, or trivial theories, where the coupling effectively vanishes, superrenormalizable theories exhibit a behavior where the coupling constant tends to increase at high energies. However, this increase is controlled and does not lead to an uncontrollable breakdown of the theory at arbitrarily high energies, provided the theory is considered within a certain energy range.

A key characteristic of superrenormalizable theories is that the number of divergent diagrams that require renormalization is finite and generally small. This makes them conceptually simpler to handle than non-renormalizable theories, where an infinite number of counterterms would be needed to absorb all divergences. Examples of theories that exhibit superrenormalizable behavior include certain topological quantum field theories and some toy models used to illustrate renormalization group flow. The presence of negative mass dimensions for coupling constants is the defining feature that classifies a theory as superrenormalizable.