Naive Dimensional Analysis Counting of Gauge Theory Amplitudes and Anomalous Dimensions

Understand

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.