Theoretical Luminosity Precision for the FCC-ee: Overview of the Path to 0.01%

We present an overview of the pathways to the required theoretical precision forthe luminosity targeted by the FCC-ee precision studies. We put the discussionin context with a brief review of the situation at the time of LEP. We thenpresent the current status and an overview of routes to the desired 0.01%targeted by the FCC-ee (as well as by the ILC). We use the situation thatexisted at the end of LEP as our starting point. At the end of LEP, the errorbudget for the BHLUMI4.04 MC used by all LEP collaborations to simulate theluminosity process was calculated in Ref. [1]. For reference, we reproduce thisresult here in Table B.3. In this table, we show the published works upon whichthe various error estimates