Analysis of General Power Counting Rules in Effective Field Theory

TL;DR

This paper derives general power counting rules for effective field theories in arbitrary dimensions, clarifying how cross sections scale and connecting different counting schemes, with applications to various EFTs including chiral and Higgs EFT.

Contribution

It provides a unified set of counting rules valid for both strongly and weakly coupled EFTs across dimensions, clarifying the relation between different counting methods and their implications.

Findings

All kinetic terms are canonically normalized.

Cross section energy dependence is determined by $\\Lambda$ power counting.

Relation between $\\Lambda$ and $f$ is generalized to $d$ dimensions.

Abstract

We derive the general counting rules for a quantum effective field theory (EFT) in $\mathsf{d}$ dimensions. The rules are valid for strongly and weakly coupled theories, and predict that all kinetic energy terms are canonically normalized. They determine the energy dependence of scattering cross sections in the range of validity of the EFT expansion. We show that the size of cross sections is controlled by the $\Lambda$ power counting of EFT, not by chiral counting, even for chiral perturbation theory ($\chi$PT). The relation between $\Lambda$ and $f$ is generalized to $\mathsf{d}$ dimensions. We show that the naive dimensional analysis $4\pi$ counting is related to $\hbar$ counting. The EFT counting rules are applied to $\chi$PT, low-energy weak interactions, Standard Model EFT and the non-trivial case of Higgs EFT.