Determination of CP-violating $HZZ$ interaction with polarised beams at the ILC

We study possible CP-violation effects of the 125 GeV Higgs to $Z$ boson coupling at the 250 GeV ILC with transverse and longitudinal beam polarisation via the process $e^+ e^- \rightarrow HZ \rightarrow H \mu^-\mu^+$. We explore the azimuthal angular distribution of the muon pair from the $Z$ boson decay, and constructe CP-odd observables sensitive to CP-violation effects, where we derived this observable both by analytical calculations and by $\mathtt{Whizard}$ simulations. Particularly, we can construct two CP-odd observables with the help of transversely-polarised initial beams and improve the statistical significance of CP-violation effects by combining two measurements. We defined the asymmetries between the signal regions with different signs of the CP-odd observables, and determine the CP-violation effect by comparing with the SM 95% C.L. upper bound. In this paper, we setup a scenario which assumes that the total cross-section is always fixed while CP-violation is varying, and such a scenario helps us to determine the intrinsic CP-mixing angle limit around $|\xi_{CP}|\sim 0.03$ with (90%, 40%) polarised electron-positron beams and 5 ab$^{-1}$ integrated luminosity. In addition, we determine the CP-odd coupling limit $|\widetilde{c}_{HZZ}|\sim 0.01$ as well, where we suppose that the SM tree-level cross-section is fixed and the CP-violation is the varying additional contribution. Comparing with the analysis with unpolarised beams, the sensitivity to the CP-violation effect can be improved by transverse or longitudinal polarisation.