The QED initial state corrections to the forward-backward asymmetry of e + e − → γ ⁎ / Z 0 ⁎ to higher orders

The QED initial state corrections are calculated to the forward–backward asymmetry for $e^{+}{e}^{-}\to {\gamma }^{⁎}/{Z^0}^{⁎}$ in the leading logarithmic approximation to $O\left({\alpha }^6{L}^6\right)$, with $L=\ln (s/m_e^2)$, extending the known corrections up to $O\left({\alpha }^2{L}^2\right)$ in analytic form. We use the method of massive on-shell operator matrix elements and present the radiators both in Mellin-N and momentum fraction z-space. Numerical results are presented for various energies around the Z-peak by also including energy cuts. These corrections are of relevance for the precision measurements at the FCC_ee.