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Measuring $CP$ nature of $hτ{\barτ}$ coupling at $e^-p$ collider

The proposed future $e^- p$ collider provides sufficient energies to produce the Standard Model Higgs Boson ($h$) through $W^\pm$ and $Z$-Boson fusion in charged and neutral current modes, respectively and to measure its properties. We take this opportunity to investigate the prospect of measuring the $CP$ properties of $h$ through $h \to τ^+ τ^-$, where $τ^-\,\left(τ^+\right)$ decays to a charged pion $π^-\,\left(π^+\right)$ and a neutral pion $π^0$ in association with neutrino (anti-neutrino). An interesting $CP$ sensitive angular observable $α_{\rm CP}$ between the two $τ$-leptons decay plane in the $π^+π^-$ centre of mass frame is proposed and investigated in this work. For fixed electron energy of 150 GeV along with 7 (50) TeV of proton energy, the $CP$ phase can be measured approximately to $25^\circ$ ($14^\circ$) at integrated luminosity of 1~ab$^{-1}$ for $-$80\% polarised electron at 95\% confidence level.