The effects of O(α${}^2$) initial state QED corrections to e${}^{+}$e${}^{-}$ $\to$ γ${}^{\text{â}}\text{ˆ}\text{—}$/Z${}^{\text{â}}\text{ˆ}\text{—}$ at very high luminosity colliders

We present numerical results on the recently completed O(α${}^2$) initial state corrections to the process e${}^{+}$e${}^{-}$ $\to$ γ${}^{\text{â}}\text{ˆ}\text{—}$/Z${}^{\text{â}}\text{ˆ}\text{—}$, 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 [1] in the non-logarithmic contributions at O(α${}^2$). 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${}^{-}$ $\to$ γ${}^{\text{â}}\text{ˆ}\text{—}$/Z${}^{\text{â}}\text{ˆ}\text{—}$, e${}^{+}$e${}^{-}$ $\to$ ZH and e${}^{+}$e${}^{-}$ $\to$ t$\overline{t}$ 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.