  In a lepton storage ring of very high energy (e.g. in the $e^+e^-$ Higgs
factory) synchrotron radiation from quadrupoles constrains transverse dynamic
aperture even in the absence of any magnetic nonlinearities. This was observed
in tracking for LEP and the Future Circular $e^+e^-$ Collider (FCC-ee). Here we
describe a new mechanism of instability created by modulation of the particle
energy at the double betatron frequency by synchrotron radiation in the
quadrupoles. Energy modulation varies transverse focusing strength at the same
frequency and creates a parametric resonance of the betatron oscillations with
unusual properties. It occurs at arbitrary betatron frequency (the resonant
detuning is always zero) and the magnitude of the parameter modulation of the
betatron oscillation (strength of the resonance driving term) depends on the
oscillation amplitude. Equilibrium between the radiation damping and the
resonant excitation gives the boundary of the stable motion. Starting from 6d
equations of motion we derive and solve the relevant differential equation
describing the resonance, and show good agreement between analytical results
and numerical simulation.

Accelerator Physics