Precise treatment of pion decays in flight in the PEN experiment

The PEN experiment at Paul Scherrer Institute aims to measure the leptonic decay $\pi^+\to e^+\nu(\gamma)$ branching ratio with $5\cdot 10^{-4}$ relative uncertainty. The shower leakage for 70\,MeV monoenergetic positrons in the CsI electromagnetic calorimeter is of the order of 2\,\% for $E_{\rm CALO}<50\,$MeV and presents an important systematic uncertainty. We discuss the ways the low energy $e^+$ tail can be extracted from the measured data and distinguished from $\pi\to\mu\to e$ decays in flight (DIF) that results in a positron in the same $0-50\,$MeV range. We compare the measured $e^+$ energy tail spectra and DIF with realistic GEANT4 simulations done with the photonuclear processes included.