TeV-scale Higgs-portal dark matter produces Higgs mass corrections larger than the measured Higgs mass, which excludes most such WIMPs except near half the Higgs mass.
This means that the tree-level value mh is changed appropriately with changing DM mass
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Dark Matter Escaping Direct Detection Runs into Higgs Mass Hierarchy Problem
TeV-scale Higgs-portal dark matter produces Higgs mass corrections larger than the measured Higgs mass, which excludes most such WIMPs except near half the Higgs mass.