Naive Dimensional Analysis Counting of Gauge Theory Amplitudes and Anomalous Dimensions

TL;DR

This paper establishes a link between naive dimensional analysis and the perturbative order of gauge theory amplitudes, providing a general formula that explains the structure of anomalous dimensions in effective field theories.

Contribution

It introduces the NDA weight of operators and derives a universal NDA formula for the perturbative order of scattering amplitudes and anomalous dimensions in effective field theories.

Findings

NDA weight of operators determines perturbative order shifts.

One-loop anomalous dimension matrix entries range from order 0 to 4.

Results apply broadly to arbitrary effective field theories.

Abstract

We show that naive dimensional analysis (NDA) is equivalent to the result that L-loop scattering amplitudes have perturbative order N=L+Delta, with a shift Delta that depends on the NDA-weight of operator insertions. The NDA weight of an operator is defined in this paper, and the general NDA formula for perturbative order N is derived. The formula is used to explain why the one-loop anomalous dimension matrix for dimension-six operators in the Standard Model effective field theory has entries with perturbative order ranging from 0 to 4. The results in this paper are valid for an arbitrary effective field theory, and they constrain the coupling constant dependence of anomalous dimensions and scattering amplitudes in a general effective field theory.