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The first stage of the FCC (Future Circular Collider) is a high-luminosityelectron-positron collider (FCC-ee) with centre-of-mass energy ranging from 88to 365 GeV, to study with high precision the Z, W, Higgs and top particles,with samples of $5 \times 10^{12}$ Z bosons, $10^8$ W pairs, $10^6$ Higgsbosons and $10^6$ top quark pairs. A cornerstone of the physics program lays inthe precise (ppm) measurements of the W and Z masses and widths, as well asforward-backward asymmetries. To this effect the centre-of-mass energydistribution should be determined with the high precision. This documentdescribes the capacity offered by FCC-ee, starting with transverse polarizationof the beams around the Z pole and the W pair threshold. A running scheme basedon regular measurements of the beam energy by resonant depolarization of pilotbunches, during physics data taking, is proposed. The design for polarizationwigglers, polarimeter and depolarizer is outlined. The $e^\pm$ beam energieswill be monitored with a relative precision of $10^{-6}$. The centre-of-massenergy is derived subject to further corrections, related to the beamacceleration, synchrotron radiation and beamstrahlung; these effects areidentified and evaluated. Dimuon events $e^+e^- \to \mu^+ \mu^-$, recorded inthe detectors, provide with great precision the beam crossing angle, thecentre-of-mass energy spread, and the $e^+$ and $e^-$ energy difference.Monitoring methods to minimize absolute error and relative uncertainties arediscussed. The impact on the physics measurements is given. A programme offurther simulations, design, monitoring and R&D is outlined.

Polarization and Centre-of-mass Energy Calibration at FCC-ee