Measuring $CP$ nature of $h\tau{\bar\tau}$ 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 $h \to \tau^+ \tau^-$, where $\tau^-\,\left(\tau^+\right)$ decays to a charged pion $\pi^-\,\left(\pi^+\right)$ and a neutral pion $\pi^0$ in association with neutrino (anti-neutrino). An interesting $CP$ sensitive angular observable $\alpha_{\rm CP}$ between the two $\tau$-leptons decay plane in the $\pi^+\pi^-$ 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.