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A search for CP violation in the decay kinematics and vector-boson productionof the Higgs boson is performed in the $H \to ZZ^{*} \to 4\ell$ ($\ell =e,\mu$) decay channel. The results are based on proton-proton collision dataproduced at the LHC at a centre-of-mass energy of 13 TeV and recorded by theATLAS detector from 2015 to 2018, corresponding to an integrated luminosity of139 $\mbox{fb$^{-1}$}$. Matrix element-based optimal observables are used toconstrain CP-odd couplings beyond the Standard Model in the framework ofStandard Model effective field theory expressed in the Warsaw and Higgs bases.Differential fiducial cross-section measurements of the optimal observables arealso performed, and a new fiducial cross-section measurement forvector-boson-fusion production is provided. All measurements are in agreementwith the Standard Model prediction of a CP-even Higgs boson.

Test of CP-invariance of the Higgs boson in vector-boson fusion production and its decay into four leptons