Measurements of $H \to \tau\tau$ properties at FCC-ee in the ZH channel at 240 GeV Updated

We consider the uncertainties on the prediction of most important Electroweak quantities at the FCC-ee upon the Electromagnetic coupling constant $\alpha$QEDmz. A direct and precise measurement of \(\alpha_{\text{QED}}(m_Z^2)\) is possible at FCC-ee using the variation of the  forward-backward muon pair asymmetry across the Z resonance peak,  which is sensitive to the Z-γ interference; However the achievable precision leads to uncertainties that exceed the target experimental precision for the prediction of the most important EW observables, e.g. the effective weak mixing angle, the W mass, the Z width and the Z leptonic patial width. It is shown however that this part of the prediction uncertainties could be reduced to a more acceptable level by substituting the measured effective weak mixing angle to  $\alpha$QEDmz as alternative input for the prediction of the other observables. It remains essential to improve the uncertainty on   $\alpha$QEDmz, since the  relationship between the measured effective weak mixing angle and   $\alpha$QEDmz  concentrates the sensitivity to the 'S' parameter of electroweak corrections.