Naive Dimensional Analysis Counting of Gauge Theory Amplitudes and Anomalous Dimensions

TL;DR

The paper addresses how naive dimensional analysis (NDA) constrains the perturbative scaling of amplitudes and operator mixing in effective field theories. It introduces a general NDA-weight framework that yields the counting rule $N = L + w - \sum_k w_k$ (or $N = L + \Delta$) for $L$-loop amplitudes with operator insertions. Applied to the SM EFT, it explains why the one-loop anomalous-dimension matrix for dimension-six operators contains entries with $N$ from $0$ to $4$ and organizes these by operator-class weights. Beyond the SM, the result provides a general, model-agnostic constraint on coupling dependence in EFTs, with potential applicability to HQET and higher-loop operator mixing.

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.

Figures (1)

• Figure 1: Diagram contributing to the $\psi^2 X H- \psi^4$ anomalous dimension $\gamma_{68}$ given in Ref. Jenkins:2013zja. The solid square is a $\psi^4$ vertex from $\mathcal{L}^{(6)}$ and the dots are gauge and Yukawa vertices from $\mathcal{L}_{SM}$.