Renormalization of a class of non-renormalizable theories

TL;DR

This paper explores a simplified renormalization approach for certain non-renormalizable theories, revealing conditions for consistent renormalization with a manageable number of couplings, and demonstrating explicit solutions at low loop orders.

Contribution

It introduces a new expansion method for non-derivative functions in non-renormalizable theories, reducing the number of couplings needed for high-order predictions.

Findings

Theories can be renormalized with a finite or slowly growing set of couplings.

Explicit solutions are provided at one and two loops.

The approach applies to scalar and fermionic theories in various dimensions.

Abstract

Certain power-counting non-renormalizable theories, including the most general self-interacting scalar fields in four and three dimensions and fermions in two dimensions, have a simplified renormalization structure. For example, in four-dimensional scalar theories, 2n derivatives of the fields, n>1, do not appear before the nth loop. A new kind of expansion can be defined to treat functions of the fields (but not of their derivatives) non-perturbatively. I study the conditions under which these theories can be consistently renormalized with a reduced, eventually finite, set of independent couplings. I find that in common models the number of couplings sporadically grows together with the order of the expansion, but the growth is slow and a reasonably small number of couplings is sufficient to make predictions up to very high orders. Various examples are solved explicitly at one and two loops.