The Bases of Effective Field Theories

TL;DR

This paper discusses how to select basis operators in effective field theories using the equivalence theorem, emphasizing the importance of potential-tree-generated operators for better modeling of beyond Standard Model physics.

Contribution

It introduces a classification scheme for basis operators in effective field theories based on their generation mechanism, applicable to the Standard Model and other models.

Findings

Basis selection can be guided by the equivalence theorem and operator classification.

The SM basis by Grzadkowski et al. satisfies the proposed selection criterion.

Dimension-six corrections to gauge couplings mainly arise from loop-generated operators.

Abstract

With reference to the equivalence theorem, we discuss the selection of basis operators for effective field theories in general. The equivalence relation can be used to partition operators into equivalence classes, from which inequivalent basis operators are selected. These classes can also be identified as containing Potential-Tree-Generated (PTG) operators, Loop-Generated (LG) operators, or both, independently of the specific dynamics of the underlying extended models, so long as it is perturbatively decoupling. For an equivalence class containing both, we argue that the basis operator should be chosen from among the PTG operators, because they may have the largest coefficients. We apply this classification scheme to dimension-six operators in an illustrative Yukawa model as well in the Standard Model (SM). We show that the basis chosen by Grzadkowski {\it et. al.} \cite{Grzadkowski:2010es} for the SM satisfies this criterion. In this light, we also revisit and verify our earlier result \cite{Arzt:1994gp} that the dimension-six corrections to the triple-gauge-boson couplings only arise from LG operators, so the magnitude of the coefficients should only be a few parts per thousand of the SM gauge coupling if BSM dynamics respects decoupling. The same is true of the quartic-gauge-boson couplings.