The potential high orders of vertical electric field systematic effect due to hyperbolic/elliptical deformed electrode plates in the proton-EDM ring

To achieve high precision in a storage ring experiment, it is essential to eliminate field errors up to a certain order to ensure they do not contribute to systematic effect to the experiment. In this study, we modeled electrode plates of electrostatic deflector with hyperbolic/elliptical shape deformation schemes. We analyzed the resulting beam dynamics and spin effects caused by these higher-order electric field errors and explored the potential systematic effects introduced by such deformed electrostatic deflectors within the proton Electric Dipole Moment (pEDM) Symmetric-Hybrid ring design.