Statistics and systematics of electron EDM searches with BaF

The NL-$e$EDM experiment searches for a non-zero electric dipole moment of the electron $d_e$ ($e$EDM) in the ground state of barium monofluoride (BaF). A beam of BaF from a supersonic expansion source is probed with the spin precession method presented in \cite{Boeschoten2024}. This method permits the extraction of an $e$EDM value as well as values for parameters causing a possible systematic bias leading to a false $e$EDM. The currently achievable sensitivity is limited by statistics collected in a period of 34 hours and yields an $d_e$ of $2(3) \times 10^{-25}$ $e\,$cm. Furthermore, from the same dataset sufficiently strong limits on parameters which can induce a false $e$EDM are extracted. These are mainly the electric field \textbf{E} and the intensity of the lasers fields in the fiducial volume of the experiment. We summarize the steps required to upgrade of the experiment to reach a competitive level on $d_e$, e.g. an intense laser-cooled beam from a cryogenic buffer gas source and the light collection efficiency of fluorescence.