The effects of $O({\alpha }^2)$ initial state QED corrections to $e^{+}{e}^{-}\to {\gamma }^{\ast }\mathrm{/}{Z}^{\ast }$ at very high luminosity colliders

We present numerical results on the recently completed $O({\alpha }^2)$ initial state corrections to the process $e^{+}{e}^{-}\to {\gamma }^{\ast }\mathrm{/}{Z}^{\ast }$, which is a central process at past and future high energy and high luminsoity colliders for precision measurements of the properties of the $Z$-boson, the Higgs boson, and the top quark. We observe differences to an earlier result \cite{Berends:1987ab} in the non-logarithmic contributions at $O({\alpha }^2)$. The new result leads to a 4 MeV shift in the $Z$ width considering the lower end $s_0=4m_{\tau }^2$ of the radiation region, which is larger than the present accuracy. We present predictions on the radiative corrections to the central processes $e^{+}{e}^{-}\to {\gamma }^{\ast }\mathrm{/}{Z}^{\ast }$, $e^{+}{e}^{-}\to ZH$ and $e^{+}{e}^{-}\to t\bar{t}$ planned at future colliders like the ILC, CLIC, Fcc_ee and CEPC to measure the mass and the width of the $Z$ boson, the Higgs boson and the top quark, for which the present corrections are significant.