On the Power Counting in Effective Field Theories

TL;DR

The paper develops a unified power-counting framework for effective field theories with strong dynamics, clarifying how loop order governs operator importance and how chiral dimensions encode this structure. It demonstrates that chiral dimensions reproduce a consistent, loop-based counting, while naive dimensional analysis requires generalization to remain valid in general EFTs. By applying the framework to chiral perturbation theory with dynamical photons and to the electroweak chiral Lagrangian with a light Higgs, the authors show how to correctly organize LO and NLO operators and counterterms and explain why certain operator classes cannot appear at leading order. The work thus provides a coherent foundation for constructing low-energy EFTs with strong sectors, resolves ambiguities in χDC and NDA, and offers practical guidance for systematic operator enumeration and power counting.

Abstract

We discuss the systematics of power counting in general effective field theories, focussing on those that are nonrenormalizable at leading order. As an illuminating example we consider chiral perturbation theory gauged under the electromagnetic $U(1)$ symmetry. This theory describes the low-energy interactions of the octet of pseudo-Goldstone bosons in QCD with photons and has been discussed extensively in the literature. Peculiarities of the standard approach are pointed out and it is shown how these are resolved within our scheme. The presentation follows closely our recent discussion of power counting for the electroweak chiral Lagrangian. The systematics of the latter is reviewed and shown to be consistent with the concept of chiral dimensions. The results imply that naive dimensional analysis (NDA) is incomplete in general effective field theories, while still reproducing the correct counting in special cases.