# On solving the Thomas Bargman-Michel-Telegdi equation using the Bogoliubov Krylov method of averages and the calculation of the Berry phases

## TL;DR

This paper applies the Bogoliubov-Krylov-Mitropolski method to solve the Thomas Bargman-Michel-Telegdi equation and compute Berry phases, aiding precise spin evolution simulations for charged particle EDM measurements.

## Contribution

It introduces a novel application of the Bogoliubov-Krylov-Mitropolski method to the T-BMT equation and Berry phase calculations in proton EDM storage rings.

## Key findings

- Validated the numerical method against analytical estimates.
- Demonstrated convergence of simulations with lattice imperfections.
- Enhanced understanding of systematic errors in EDM experiments.

## Abstract

Several proposals aimed at measuring the Electric Dipole Moment (EDM) for charged particles require very precise simulations and understanding of the systematic errors that can contribute to a spin buildup mimicking the EDM signal to be detected. In what follows, one used the Bogoliubov-Krylov-Mitropolski method of averages to solve the T-BMT equation and calculate the Berry phases arising for a proton EDM storage ring. The formalism employed proved to be particularly useful to determine the evolution of the spin at the observation point, i.e. at the location of the polarimeter. Several selected cases of lattice imperfections were simulated and benchmarked with the analytical estimates. This allowed the proof of the convergence of the numerical simulations and helped gain better understanding of the systematic errors.