Spin response formalism in circular accelerators

Snowmass 2021 White Paper on Upgrading SuperKEKB with a Polarized Electron Beam: Discovery Potential and Proposed Implementation

“…Other parameters stands for: r -radius of curvature of the orbit at the point of emission of the photon, b is the unit vector indicating the direction of the magnetic field, n the unit vector along the equilibrium spin direction at a given azimuth, and d = γ(d n/dγ) the spin-orbit coupling vector, showing the direction of the jump of vector n when a photon is emitted, and the magnitude of this jump. Modulus distribution of vector d over the azimuth of the storage ring was calculated by the ASPIRRIN program, created in the 1990s [64,65]. Figure 22 shows the graph of | d(s)| calculated for the rotator optics option with the parameters in Table 12.…”

Polarized Electron and Hadron Beams at eRHIC

“…In addition, spin diffusion [5] leads to depolarization of both spin orientations. To maximize the time-averaged polarization, spin diffusion effects have to be minimized by proper spin matching [6], and entire bunches have to be replaced every few minutes. Assuming 85 percent polarization upon injection into the storage ring, and an equilibrium polarization of 30 percent as a result of Sokolov-Ternov self-polarization and depolarization due to spin diffusion, a time-averaged polarization of 80 percent is achieved by replacing every bunch with its polarization oriented parallel to the main dipole field after 1.2 minutes, and every bunch with its spins anti-parallel to the main dipole field every 3.2 minutes, as illustrated in Figure 3.…”

High Energy Polarized e+e− Beams

“…6 This vector value could be calculated either analytically or using the computer code ASPIRRIN. 7 It comes into formula for the spin relaxation time in the following way:…”