Quantum Excitation due to Classical Beamstrahlung in Circular Colliders

In the collisions of proposed future circular colliders, like FCC-ee and CEPC, the beamstrahlung regime is classical, i.e. with an "Upsilon parameter" much smaller than 1. In the classical regime, for a constant electromagnetic field a simple relation exists between the average photon energy u and the average squared photon energy u², which is the same as for standard synchrotron radiation in storage rings. This relation breaks down, however, if the electromagnetic field is not constant in time and position, as is the case for a beam-beam collision. We derive an analytical expression for u²/u², considering the case of Gaussian-bunch collisions with crossing angle (and possibly crab waist). We compare our result with the photon energies obtained in beam-beam simulation for FCC-ee at beam energies of 45.6 GeV and 175 GeV, using the two independent codes BBWS and Guineapig. Finally, we re-optimize the FCC-ee parameters of a possible mono-chromatization scheme for direct Higgs production at 125 GeV, derived previously, by applying the refined expression for the rms photon energy.