Monte Carlo Simulations for Likelihood Analysis of the PEN experiment

The PEN collaboration performed a precision measurement of the $\pi^+\rightarrow e^+\nu_e(\gamma)$ branching ratio with the goal of obtaining a relative uncertainty of $5\times10^{-4}$ or better at the Paul Scherrer Institute. A precision measurement of the branching ratio $\Gamma(\pi\rightarrow e\bar{\nu}(\gamma))/\Gamma(\pi\rightarrow \mu \bar{\nu}(\gamma))$ can be used to give mass bounds on ``new'', or non V$-$A, particles and interactions. This ratio also proves to be one of the most sensitive tests for lepton universality. The PEN detector consists of beam counters, an active target, a mini-time projection chamber, multi-wire proportional chamber, a plastic scintillating hodoscope, and a CsI electromagnetic calorimeter. The Geant4 Monte Carlo simulation is used to construct ultra-realistic events by digitizing energies and times, creating synthetic target waveforms, and fully accounting for photo-electron statistics. We focus on the detailed detector response to specific decay and background processes in order to sharpen the discrimination between them in the data analysis.