Focusing of a Long Relativistic Proton Bunch in Underdense Plasma

We discuss the focusing effect experienced by a long relativistic proton bunch, when propagating in underdense plasma. We show with 2D quasi-static simulations that the response of the plasma to the presence of the bunch provides a focusing force for the protons. We discuss the impact of the finite transverse size of the plasma on the dynamics of the process and we introduce the measurements performed at the AWAKE experiment at CERN.