A study of the charge conjugation and parity ($CP$) properties of the interaction between the Higgs boson and $\tau$-leptons is presented. The study is based on a measurement of $CP$-sensitive angular observables defined by the visible decay products of $\tau$-lepton decays, where at least one hadronic decay is required. The analysis uses 139 fb$^{-1}$ of proton$-$proton collision data recorded at a centre-of-mass energy of $\sqrt{s}= 13$ TeV with the ATLAS detector at the Large Hadron Collider. Contributions from $CP$-violating interactions between the Higgs boson and $\tau$-leptons are described by a single mixing angle parameter $\phi_{\tau}$ in the generalised Yukawa interaction. Without assuming the Standard Model hypothesis for the $H\rightarrow\tau\tau$ signal strength, the mixing angle $\phi_{\tau}$ is measured to be $9^{\circ} \pm 16^{\circ}$, with an expected value of $0^{\circ} \pm 28^{\circ}$ at the 68% confidence level. The pure $CP$-odd hypothesis is disfavoured at a level of 3.4 standard deviations. The results are compatible with the predictions for the Higgs boson in the Standard Model.
Observed 1-D likelihood scan of the $CP$-mixing angle $\phi_{\tau}$.
Expected 1-D likelihood scan of the $CP$-mixing angle $\phi_{\tau}$.
Observed 2-D likelihood scan of the signal strength $\mu_{\tau\tau}$ versus the $CP$-mixing angle $\phi_{\tau}$.
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