On the consistent use of Constructed Observables

TL;DR

This work exposes subtle inconsistencies that arise when constraining the SM EFT with constructed observables that assume SM-like deviations. It shows that equations of motion map between operator bases, so naive constraints can create functionally redundant parameter directions unless the defining conditions are applied in a basis-independent, EoM-consistent way. Through analysis of the S parameter, off-shell triple gauge couplings, LEP data, and h → V F spectra, the authors demonstrate how strong LEP bounds can sever connections between certain observables, and how finite renormalization can obscure direct relationships unless properly accounted for. The paper advocates using at least two operator bases and careful EoM mappings to obtain consistent, basis-independent SMEFT constraints as LHC analyses move toward more complex final states.

Abstract

We define "constructed observables" as relating experimental measurements to terms in a Lagrangian while simultaneously making assumptions about possible deviations from the Standard Model (SM), in other Lagrangian terms. Ensuring that the SM effective field theory (EFT) is constrained correctly when using constructed observables requires that their defining conditions are imposed on the EFT in a manner that is consistent with the equations of motion. Failing to do so can result in a "functionally redundant" operator basis and the wrong expectation as to how experimental quantities are related in the EFT. We illustrate the issues involved considering the $\rm S$ parameter and the off shell triple gauge coupling (TGC) verticies. We show that the relationships between $h \rightarrow V \bar{f} \, f$ decay and the off shell TGC verticies are subject to these subtleties, and how the connections between these observables vanish in the limit of strong bounds due to LEP. The challenge of using constructed observables to consistently constrain the Standard Model EFT is only expected to grow with future LHC data, as more complex processes are studied.

Figures (1)

• Figure 1: An example of a functional redundancy. An operator basis can be chosen that maps parameters characterizing differences in the coupling of the $W$ and $Z$ to leptons (compared to the SM) into another sector of the field theory, where these parameters contribute to an anomalous TGC vertex. (Parameters can be mapped from the dot in the diagrams above to the box with the EoM.) Subsequently, using a TGC vertex bound, naively constrains these parameters in the SMEFT. Some of the parameters apparently constrained in this manner are functionally redundant, as in the middle two diagrams the production and decay of the $W,Z$ is simultaneously assumed to be SM like. (When experimental bounds are constructed on the parameters in the box, the dot is assumed to be SM like.) This procedure is inconsistent and does not constrain a flat direction due to LEP $Z$ pole data that can modify $h \rightarrow V \mathcal{F}$ decay, when $V = W$. Unphysical field redefinitions, or an operator basis choice, do not make this procedure consistent.