High precision estimates of g − 2 and EDM in general spin precession in storage ring

We analyze the systematic uncertainties on the spin precession of particles in the storage ring, especially in the muon $g-2$/EDM project. Their particles have both anomalous magnetic moments ($g-2$) and electric dipole moments (EDM), and their positions and velocity directions have both $O\left({10}^{-3}\text{–}{10}^{-4}\right)\equiv O\left(ϵ\right)$ extensions. In order to measure these dipole moments up to 0.1 ppm or more we determine the precession frequency up to $O\left({ϵ}^2\right)$. Our analytical formulation includes the Farley's pitch correction in the special case and can be applied to more general experimental setups.