Anomalous Diffusion Near Resonances

Synchro-betatron resonances can lead to emittance growth and the loss of luminosity. We consider the detailed dynamics of a bunch near such a low order resonance driven by crossing angles at the collision points. We characterize the nature of diffusion and find that it is anomalous and sub-diffusive. This affects both the shape of the beam distribution and the time scales for growth. Predictions of a simplified anomalous diffusion model are compared with direct simulations. Transport of particles near resonances is still not a well understood phenomenon. Often, without justification, phase space motion is assumed to be a normal diffusion process although at least one case of anomalous diffusion in beam dynamics has been reported [1]. Here we will focus on the motion near synchro-betatron resonances which can be excited by several means, including beams crossing at an angle at the collision points as in the LHC. We will consider low order resonances which couple the horizontal and longitudinal planes, both for simplicity and to observe large effects over short time scales. While the tunes we consider are not practical for a collider, nonetheless the transport mechanisms we uncover are also likely to operate at higher order resonances