  A critical element of the LHC physics program is the search for an additional
source of CP violation. This is largely unexplored in the context of non-linear
Higgs physics, which is naturally described in Higgs Effective Field Theory
(HEFT). Relevant new higher-dimensional operators modify the production rate
and branching ratios of the Higgs boson, de-correlating different Higgs
multiplicities. In this work, we consider single Higgs and Higgs pair
production via weak boson fusion from the perspective of gauge-Higgs CP
violation through the lens of Higgs non-linearity. This generalizes existing
rate-based searches and analyses by the ATLAS and CMS experiments. Particular
focus is given to the phenomenological differences in the expected BSM
sensitivity pattern when comparing HEFT constraints with Standard Model
Effective Field Theory (SMEFT) limits.
