Impact on EWPOs of the \(\alpha_{\text{QED}}(m_Z^2)\) uncertainty; possible mitigation

We consider the uncertainties on the prediction of most important Electroweak quantities at the FCC-ee upon the Electromagnetic coupling constant \(\alpha_{\text{QED}}(m_Z^2)\). 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-$\mathrm \gamma$ 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 \(\alpha_{\text{QED}}(m_Z^2)\) as alternative input for the prediction of the other observables. It remains essential to improve the uncertainty on \(\alpha_{\text{QED}}(m_Z^2)\), since the  relationship between the measured effective weak mixing angle and \(\alpha_{\text{QED}}(m_Z^2)\)  concentrates the sensitivity to the 'S' parameter of electroweak corrections.