In the context of future electroweak precision measurements at the Future Circular Lepton Collider (FCC-ee), we consider recent proposals aimed at finding signatures of physics beyond the Standard Model. In particular, we focus on recent novel suggestions for very precise direct measurements of $\alpha_{\rm e m}(M_Z^2)$. It is shown that at a level of precision of order $10^{-5}$, the effects of a dark photon may be very significant.
Comment: 6 pages, 2 figures. Revised to match the published version