Renormalization Group Treatment of Nonrenormalizable Interactions

TL;DR

This paper analyzes the ultraviolet divergences in higher-dimensional nonrenormalizable quantum field theories, showing that leading divergences are governed by one-loop diagrams and providing explicit formulas for their summation.

Contribution

It introduces a method to sum leading divergences in nonrenormalizable theories using one-loop diagrams and provides explicit expressions for counterterms in arbitrary dimensions.

Findings

Leading divergences are governed by one-loop diagrams.

Explicit formulas for counterterms are derived.

Results challenge the naive power-law running of couplings.

Abstract

The structure of the UV divergencies in higher dimensional nonrenormalizable theories is analysed. Based on renormalization operation and renormalization group theory it is shown that even in this case the leading divergencies (asymptotics) are governed by the one-loop diagrams the number of which, however, is infinite. Explicit expression for the one-loop counter term in an arbitrary D-dimensional quantum field theory without derivatives is suggested. This allows one to sum up the leading asymptotics which are independent of the arbitrariness in subtraction of higher order operators. Diagrammatic calculations in a number of scalar models in higher loops are performed to be in agreement with the above statements. These results do not support the idea of the na\"ive power-law running of couplings in nonrenormalizable theories and fail (with one exception) to reveal any simple closed formula for the leading terms.