The Path to $0.01\%$ Theoretical Luminosity Precision for the FCC-ee

The current status of the theoretical precision for the Bhabha luminometry is critically reviewed and pathways are outlined to the requirement targeted by the FCC-ee precision studies. Various components of the pertinent error budget are discussed in detail -- starting from the context of the LEP experiments, through their current updates, up to prospects of their improvements for the sake of the FCC-ee. It is argued that with an appropriate upgrade of the Monte Carlo event generator BHLUMI and/or other similar MC programs calculating QED effects in the low angle Bhabha process, the total theoretical error of $0.01\%$ for the FCC-ee luminometry can be reached. A new study of the $Z$ and $s$-channel $\gamma$ exchanges within the angular range of the FCC-ee luminometer using the BHWIDE Monte Carlo was instrumental in obtaining the above result. Possible ways of BHLUMI upgrade are also discussed.
Comment: This work is based on the material presented at the FCC-ee-Z Theory Miniworkshop at CERN, January 2018", see Section C6 in arxiv:1809.018