Superrenormalizability

Superrenormalizability is a concept in quantum field theory that describes theories where the coupling constants are dimensionless and the divergences encountered during renormalization are suppressed by increasing powers of the momentum, rather than increasing with the momentum. This is in contrast to asymptotically free theories where the coupling constants decrease with increasing momentum. In a superrenormalizable theory, the ultraviolet divergences are more severe than in a renormalizable theory but less severe than in a non-renormalizable theory.

The key characteristic of superrenormalizable theories is that their coupling constants have positive mass dimensions. This means that as energy scales increase, the effective coupling strength grows. This behavior is problematic because it leads to uncontrolled growth of loop corrections at high energies, often rendering the theory inconsistent.

Examples of superrenormalizable theories include the lambda phi to the fourth power theory in d=2 spatial dimensions and theories with higher-order derivative terms. While these theories are not typically considered fundamental descriptions of nature due to their high-energy behavior, they can sometimes arise as effective field theories valid within a certain energy range. Studying superrenormalizable theories can provide insights into the structure of quantum field theories and the behavior of divergences in different dimensions.