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In a lepton storage ring of very high energy (e.g. in the $e^+e^-$ Higgsfactory) synchrotron radiation from quadrupoles constrains transverse dynamicaperture even in the absence of any magnetic nonlinearities. This was observedin tracking for LEP and the Future Circular $e^+e^-$ Collider (FCC-ee). Here wedescribe a new mechanism of instability created by modulation of the particleenergy at the double betatron frequency by synchrotron radiation in thequadrupoles. Energy modulation varies transverse focusing strength at the samefrequency and creates a parametric resonance of the betatron oscillations withunusual properties. It occurs at arbitrary betatron frequency (the resonantdetuning is always zero) and the magnitude of the parameter modulation of thebetatron oscillation (strength of the resonance driving term) depends on theoscillation amplitude. Equilibrium between the radiation damping and theresonant excitation gives the boundary of the stable motion. Starting from 6dequations of motion we derive and solve the relevant differential equationdescribing the resonance, and show good agreement between analytical resultsand numerical simulation.

Dynamic aperture limitation in $e^+e^-$ colliders due to synchrotron radiation in quadrupoles