Naive dimensional analysis counting of gauge theory amplitudes and anomalous dimensions

Elizabeth E Jenkins,Aneesh V Manohar,Michael Trott

doi:10.1016/j.physletb.2013.09.020

Abstract

We show that naive dimensional analysis (NDA) is equivalent to the result that L-loop scattering amplitudes have perturbative order N=L+Δ, with a shift Δ 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.

Highlights

We show that naive dimensional analysis (NDA) is equivalent to the result that L-loop scattering amplitudes have perturbative order N = L+∆, with a shift ∆ that depends on the NDA-weight of operator insertions
The operators in the effective Lagrangian L(d) of dimension d can be normalized according to NDA and standard dimensional analysis, L(d) =
The formula has to be modified in corners of phase-space, such as near threshold, where one can have kinematic enhancements of 1/v, where the particle velocity v is a dimensionless number not controlled by NDA

Summary

The NDA Counting Formula

We derive the NDA-weight formula for a general EFT, i.e. a gauge theory with arbitrary higher dimension gauge invariant operators. The operators in the effective Lagrangian L(d) of dimension d can be normalized according to NDA and standard dimensional analysis, L(d) =. The standard dimensional analysis coefficients Cd,w are for the same operator Od,w normalized only using powers of Λ (i.e. with f → Λ), but still including a factor of g with each X. The NDA formula in Eq (2.13) can be converted to the usual perturbation expansion in terms of the coefficients Cd,w of standard dimensional analysis using the relation. We know that with the unrescaled theory, an L loop diagram is of order 1/(16π2)L Equating this power of 4π with the powers of 4π in Eq (2.15) gives our 4π-counting formula. The formula has to be modified in corners of phase-space, such as near threshold, where one can have kinematic enhancements of 1/v, where the particle velocity v is a dimensionless number not controlled by NDA

Application to the Standard Model

Conclusions