Higher-Order Spin Resonances in RHIC and the EIC

Investigations of proton spin structure have shown that its spin is not nearly accounted for entirely by just 3 valence quarks, and that virtual quark-antiquark pairs and gluons must carry much of its spin. This lead to the "Proton Spin Crisis" in the late ‘80s and remains an important topic in nuclear structure. The Electron Ion Collider being built at Brookhaven National Laboratory plans to collide polarized electron and proton beams in order to investigate this spin crisis, amongst other mysteries, as spin-polarized beams are the only tool available to investigate the spin structure of nucleons. However, beam polarization is notoriously difficult to maintain. This is due to depolarizing spin-orbit resonances that naturally emerge in the context of magnetic focusing systems in particle accelerators. In this presentation, I will discuss magnetic devices known as Siberian snakes, designed to circumvent the depolarizing nature of traversing spin-orbit resonances, and how to understand the subsequent higher-order resonances that become apparent after installing Siberian snakes. I introduce a critical mathematical object known as the "Invariant Spin Field." This is used to understand spin dynamics in accelerators and is a powerful tool for investigating spin behavior in the vicinity of resonances. In particular, I show how to use the ISF for calculating linear and higher-order resonance strengths, as well as calculating equilibrium beam polarization.