Naive dimensional analysis counting of gauge theory amplitudes and anomalous dimensions

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 Letter, 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 Letter 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. (C) 2013 Elsevier B.V. All rights reserved.