  The first part of the physics programme of the integrated FCC (Future
Circular Colliders) proposal includes measurements of Standard Model processes
in $e^+e^-$ collisions (FCC-ee) with an unprecedented precision. In particular,
the potential precision of the Z lineshape determination calls for a very
precise measurement of the absolute luminosity, at the level of 1E-4, and the
precision on the relative luminosity between energy scan points around the Z
pole should be an order of magnitude better. The luminosity is principally
determined from the rate of low-angle Bhabha interactions, $e^+e^- \to e^+e^-$,
where the final state electrons and positrons are detected in dedicated
calorimeters covering small angles from the outgoing beam directions.
Electromagnetic effects caused by the very large charge density of the beam
bunches affect the effective acceptance of these luminometers in a nontrivial
way. If not corrected for, these effects would lead, at the Z pole, to a
systematic bias of the measured luminosity that is more than one order of
magnitude larger than the desired precision. In this note, these effects are
studied in detail, and methods to measure and correct for them are proposed.
