The violation of charge conjugation (C) and parity (P) symmetries are a requirement for the observed dominance of matter over antimatter in the Universe. As an established effect of beyond the Standard Model physics, this could point towards additional CP violation in the Higgs-gauge sector. The phenomenological footprint of the associated anomalous couplings can be small, and designing measurement strategies with the highest sensitivity is therefore of the utmost importance in order to maximise the discovery potential of the Large Hadron Collider (LHC). There are, however, very few measurements of CP-sensitive observables in processes that probe the weak-boson self-interactions. In this article, we study the sensitivity to new sources of CP violation for a range of experimentally-accessible electroweak processes, including $Wγ$ production, $WW$ production via photon fusion, electroweak $Zjj$ production, electroweak $ZZjj$ production, and electroweak $W^\pm W^\pm jj$ production. We study simple angular observables as well CP-sensitive observables constructed using the outputs of machine-learning (ML) algorithms. We find that the ML-constructed CP-sensitive observables improve the sensitivity to CP-violating effects by up to a factor of five, depending on the process. We also find that inclusive $Wγ$ and electroweak $Zjj$ production have the potential to set the best possible constraints on certain CP-odd operators in the Higgs-gauge sector of dimension-six effective field theories.

11 pages, 7 figures, 4 tables, v2: journal version