  We combine the NNLO QCD description of Higgs boson production in association
with an electroweak vector boson $V = W~{\rm or}~Z$ with a similarly-precise
description of Higgs boson decays into a pair of massive $b$ quarks and with
the anomalous couplings that modify interactions of the Higgs and electroweak
vector bosons. The resulting numerical code provides the most advanced
theoretical tool to investigate such anomalous couplings in the associated
Higgs boson production process. We study the impact of anomalous couplings on
fiducial cross sections and differential distributions and argue that, with
increased QCD precision, smaller anomalous couplings become accessible in
kinematic regions where the effects of higher-dimensional operators in the
Standard Model Effective Field Theory remain small and the EFT expansion is
under control.
