One possible evidence for CP-violation beyond the Standard Model would be a discovery of non-vanishing Electric Dipole Moments (EDM) of elementary particles. To search for the EDM of charged particles one can store them in a circular accelerator and observe the EDM effect on the beam polarization. The necessary condition for a build-up of the observable EDM-effect is a coherent spin motion. Possible sources of spin decoherence include spin chromaticity, orbit lengthening and spin resonances. In this regard of special interest are novel features of the so-called “frozen spin” storage rings with electrostatic and hybrid E+B bending. The first step to increase the Spin Coherence Time (SCT) is to turn on a radiofrequency cavity. The next step is to manipulate equilibrium energy levels associated with betatron orbit lengthening and nonlinear momentum compaction factor, as suggested by a solution of nonlinear equations of longitudinal motion. We demonstrated that the effective equilibrium energy is a scalar characteristic of the spin motion of a beam with a distribution in a 6D phase space. It has to be the same for all particles in the beam to achieve a high SCT. Spin resonances act as another source of spin-decoherence. Their impact needs to be taken care of especially for the proton beam in the entire energy range of the machine.

Phase trajectory in longitudinal plane for initial coordinates x = 0, y = 0 (a) and x = 3 mm, y = 0 (b); initial dp/p = 1.2 · $10^{−4}$

Melnikov A.A., Senichev Yu.V., Aksentyev A.E., Kolokolchikov S.D.
JETP Letters 118, issue 10 (2023)