Berry phases in an electric-dipole-moment experiment in an all-electric storage ring

Systematic effects caused by the Berry (geometric) phases in an electric-dipole-moment experiment in an all-electric storage ring are considered. We analyze the experimental setup when the spin is frozen and local longitudinal and vertical electric fields alternate. Due to the Berry phases, the spin rotates about the radial axis. The corresponding systematic error is rather important while it can be canceled with clockwise and counterclockwise beams. The Berry phases also lead to the spin rotation about the radial axis. This effect can be canceled with clockwise and counterclockwise beams as well. The sign of the azimuthal component of the angular velocity of the spin precession depends on the starting point where the spin orientation is perfect. The radial component of this quantity keeps its value and sign for each starting point. When the longitudinal and vertical electric fields are joined in the same sections without any alternation, the systematic error due to the geometric phases does not appear. However, another systematic effect of the spin rotation about the azimuthal axis takes place and has opposite signs for clockwise and counterclockwise beams.