Perturbative formulation for nonlinear chromaticity of circular accelerators

The nonlinear chromaticity plays an important part in the dynamics of a particle at far off-momentum in a strong focusing circular accelerator. We derived the exact perturbative formula of the nonlinear dispersion function of a ring accelerator and gave explicit expressions of higher-order terms. Using the perturbative formula for the dispersion function, we derived the higher-order expressions for the nonlinear chromaticity up to the third order. We numerically estimated the nonlinear chromaticity of the SPring-8 storage ring, and it showed fairly good agreement with the measurement.