On the extraction of ${\alpha }_{\mathit{\text{em}}}(m_Z^2)$ at Tera-$Z$

The current projected sensitivity on the electromagnetic coupling ${\alpha }_{\mathit{\text{em}}}(m_Z^2)$ represents a bottleneck for the precision electroweak program at FCC-ee. We propose a novel methodology to extract this coupling directly from $Z$-pole data. By comparing the differential distribution of electrons, muons and positrons in the forward region, the approach achieves a projected statistical sensitivity below the ${10}^{-5}$ level, representing a significant improvement over other methods. We assess the impact of leading parametric uncertainties including that of the top quark mass.