Design of high brightness Plasma Wakefield Acceleration experiment at SPARC\_LAB test facility with particle-in-cell simulations.

The present numerical investigation of a Plasma Wakefield Acceleration scenario in the weakly non linear regime with external injection is motivated by the upcoming campaigns at the SPARC\_LAB test facility where the final goal is to demonstrate modest gradient acceleration ($\sim$1 GV/m) with no quality loss. The accelerated bunch can be envisioned to seed a free electron laser. The numerical study has been conducted with the particle-in-cell code ${\tt ALaDyn}$, an exhaustive description of the plasma-acceleration version is provided. The configuration consider a two bunches setup with parameters in the facility range, the bunches are generated and pre-accelerated up to 100 MeV by a high brightness photo-injector prior plasma injection. To verify the working point robustness we have considered case scenario where the driver bunch reaches the plasma or with a larger dimension or with large emittance. We also present an analytical approach based on the envelope equation that allows to reduce the matching condition in the presence of a ramp. Here, we limit our interest to a simplified theoretical case with a linear plasma ramp. As a final aspect we propose to combine classical integrated bunch diagnostics with the test by Shapiro-Wilk, a mathematical test to diagnose bunch deviation from a Gaussian distribution.