  Using the production of a Higgs boson in association with a $W$ boson as a
test case, we assess the impact of dimension-8 operators within the context of
the Standard Model Effective Field Theory. Dimension-8--SM-interference and
dimension-6-squared terms appear at the same order in an expansion in
$1/\Lambda$, hence dimension-8 effects can be treated as a systematic
uncertainty on the new physics inferred from analyses using dimension-6
operators alone. To study the phenomenological consequences of dimension-8
operators, one must first determine the complete set of operators that can
contribute to a given process. We accomplish this through a combination of
Hilbert series methods, which yield the number of invariants and their field
content, and a step-by-step recipe to convert the Hilbert series output into a
phenomenologically useful format. The recipe we provide is general and applies
to any other process within the dimension $\le 8$ Standard Model Effective
Theory. We quantify the effects of dimension-8 by turning on one dimension-6
operator at a time and setting all dimension-8 operator coefficients to the
same magnitude. Under this procedure and given the current accuracy on
$\sigma(pp \to h\,W^+)$, we find the effect of dimension-8 operators on the
inferred new physics scale to be small, $\mathcal O(\text{few}\,\%)$, with some
variation depending on the relative signs of the dimension-8 coefficients and
on which dimension-6 operator is considered. The impact of the dimension-8
terms grows as $\sigma(pp \to h\,W^+)$ is measured more accurately or (more
significantly) in high-mass kinematic regions. We provide a FeynRules
implementation of our operator set to be used for further more detailed
analyses.
