Probing [Formula: see text] -violating Higgs and gauge-boson couplings in the Standard Model effective field theory

We study the phenomenological consequences of several CP-violating structures that could arise in the Standard Model effective field theory framework. Focusing on operators involving electroweak gauge and/or Higgs bosons, we derive constraints originating from Run I LHC data. We then study the capabilities of the present and future LHC runs at higher energies to further probe associated CP-violating phenomena and we demonstrate how differential information can play a key role. We consider both traditional four-lepton probes of CP-violation in the Higgs sector and novel new physics handles based on varied angular and non-angular observables.