Scenario for Precision Beam Energy Calibration in FCC-ee

The resonance depolarization method was very successfully used in the experiments at LEP, where the mass of the Z-boson was determined with the relative uncertainty [1, 2]. In the future FCC-ee circular electron-positron collider the luminosity at Z-peak (beam energy 45.5 GeV) is expected be 4-5 orders of magnitude higher and one goal is to perform the same experiments as at LEP, but with much greater accuracy, approaching the level of [3]. Obviously this can be done only by measuring the spin precession frequency. But there are many problems which still need to be solved on the way towards a complete design. The first one: the self-polarization takes too long a time. The Sokolov-Ternov polarization time is about 250 hours at Z-peak. One approach is to install the special field-asymmetric polarizing wigglers to make the self-polarization time much shorter [4, 5] and to utilize only few percent of the polarization degree to measure the resonance spin precession frequency. But these very strong wigglers substantially increase the beam energy spread and, even including them, still a rather long time is required to get the beam polarized. Much more frequent energy measurements could be done if one can accelerate, in a booster ring, polarized bunches prepared beforehand in some low energy damping rings. These damping rings, one for positrons, one for electrons, shall operate at 1-2 GeV beam energy and use very strong bending field or polarizing wigglers. Straightforward estimations show that a polarization time of a few minutes could be achieved in such a ring. The second problem: by measuring the spin precession frequency one determines only the average beam energy around the ring circumference, while the experimenters are interested in determining the local energy at the collision point, which may significantly differ from the average. The deviation of the local beam energy at some azimuth from the average energy is described by the so-called saw-tooth graph. The latter represents the closed self-consistent energy distribution function along the ring, based on the theoretical or measured synchrotron radiation losses, the energy gains from RF cavities, and the energy loss caused by the longitudinal impedance. While the SR losses can be calculated rather well taking into account the real bending field azimuth dependence, the other sources of the energy uncertainty are much less predictable and need to be experimentally confirmed. We will discuss these problems in the next chapters.