A generalization of the Froissart-Stora formula to piecewise-linear spin-orbit resonance crossings

TL;DR

This paper generalizes the Froissart-Stora formula to account for changing slopes in higher-order spin-orbit resonance crossings, with applications demonstrated through simulations and connections to the Landau-Zener formula.

Contribution

It introduces a new version of the Froissart-Stora formula applicable when the slope of the spin tune changes at resonance crossing, extending its use to more complex scenarios.

Findings

The generalized formula accurately predicts depolarization in simulations.

The work shows mathematical equivalence between Froissart-Stora and Landau-Zener formulas.

Application to RHIC demonstrates practical relevance.

Abstract

Spin-polarized beams are important for some nuclear and high-energy physics experiments, such as those planned for the future Electron-Ion Collider (EIC). However, maintaining polarization during the acceleration of a charged-particle beam is difficult because the periodic nature of circular accelerators leads to spin-orbit resonances where the spin-precession frequency is a sum of integer multiples of the orbital frequencies. Usually, the dominant depolarization mechanisms are first-order spin-orbit resonances and the depolarization associated with crossing such a resonance can be computed using the Froissart-Stora formula. However, accelerating polarized hadron beams to high energy requires special magnet structures called Siberian snakes. When these are implemented to maintain a spin-precession frequency of one-half the revolution frequency, there will be no first-order spin-orbit resonance crossings. The dominant depolarization mechanisms are then higher-order spin-orbit resonances. The Froissart-Stora formula can be applied to higher-order resonances when the slope of the amplitude-dependent spin tune is constant. However, the slope of the amplitude-dependent spin tune often changes at the moment of resonance crossing. This work introduces a generalization of the Froissart-Stora formula which is applicable when the slope changes in this manner. The applicability of this formula is demonstrated through tracking simulations of a higher-order resonance crossing in both a toy model and the Relativistic Heavy Ion Collider (RHIC). It is additionally shown that the Froissart-Stora formula is mathematically equivalent to the Landau-Zener formula for the diabatic transition probability in two-level systems with a linearly increasing energy gap and constant coupling. This work therefore also extends the Landau-Zener formula to the case of changing slope.