Proton beam defocusing in AWAKE: comparison of simulations and measurements

In 2017, AWAKE demonstrated the seeded self-modulation (SSM) of a 400 GeV proton beam from the Super Proton Synchrotron (SPS) at CERN. The angular distribution of the protons deflected due to SSM is a quantitative measure of the process, which agrees with simulations by the two-dimensional (axisymmetric) particle-in-cell code LCODE. Agreement is achieved for beam populations between $10^{11}$ and $3 \times 10^{11}$ particles, various plasma density gradients ($-20 \div 20\%$) and two plasma densities ($2\times 10^{14} \text{cm}^{-3}$ and $7 \times 10^{14} \text{cm}^{-3}$). The agreement is reached only in the case of a wide enough simulation box (at least five plasma wavelengths).