The point-to-point uncertainty on the centre-of-mass energy and the Z width at FCC-ee

The Tera-Z run at the FCC-ee, with a total of about 6e12 Z bosons produced, offers the potential for electroweak measurements at an unprecedented level of accuracy. That is the case in particular for the Z width, which would be extracted from the Z lineshape scan with a statistical uncertainty of 4 keV. One of the leading sources of systematic uncertainty for  this measurement is expected to be the relative, point-to-point uncertainty of the measurement of the centre-of-mass energy (sqrts), namely the uncertainty of the difference between the energies at which the machine will run for the lineshape scan. For the corresponding systematic uncertainty on the Z width to be commensurate with the statistical uncertainty, this difference will have to be known to much better than what will be derived from the absolute measurements of sqrts resulting from the resonant depolarisation method. This note explores the potential for controlling this systematic uncertainty in-situ using dimuon events. An excellent track momentum resolution is a key requirement for the proposed method.