Anomalous couplings in associated $VH$ production with Higgs decay to massive $b$ quarks at NNLO in QCD

We combine the NNLO QCD description of Higgs boson production in association with an electroweak vector boson V=W 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 effective field theory expansion is under control.