Lévy–Student processes for a stochastic model of beam halos

Abstract

We describe the transverse beam distribution in particle accelerators within the controlled, stochastic dynamical scheme of the stochastic mechanics which produces time reversal invariant diffusion processes. In this paper we analyze the consequences of introducing the generalized Student laws, namely non-Gaussian, Lévy infinitely divisible (but not stable) distributions. We will analyze this idea from two different standpoints: (a) first by supposing that the stationary distribution of our (Wiener powered) stochastic model is a Student distribution; (b) by supposing that our model is based on a (non-Gaussian) Lévy process whose increments are Student distributed. In the case (a) the longer tails of the power decay of the Student laws, and in the case (b) the discontinuities of the Lévy–Student process can well account for the rare escape of particles from the beam core, and hence for the formation of a halo in intense beams.