Systematics of the Extraction of the Elementary $\gamma $n$\to \pi ^{\mathrm{-}}$p Reaction Cross Sections beyond the Impulse Approximation for $\gamma $d$\to \pi^{\mathrm{-}}$pp

The radiative decay width of neutral baryons may be extracted from $\pi ^{\mathrm{-}}$ and $\pi^{\mathrm{0}}$ photo-production off the neutron, involving a bound neutron target, requiring the use of model-dependent nuclear final state interaction (FSI) corrections. The cross section for the processes $\gamma $n$\to \pi^{\mathrm{-}}$p will be extracted from recent CLAS (E $=$ 400 -- 2500 MeV) and MAX-lab (E $=$ 146 -- 166 MeV) measurements for $\gamma $d$\to \pi^{\mathrm{-}}$pp accounting for Fermi motion effects in the Impulse Approximation (IA) as well as nucleon-nucleon- and pion-nucleon-FSI effects beyond the IA. To test the GW-ITEP FSI code for $\gamma $n$\to \pi^{\mathrm{-}}$p in a reliable way to obtain information \quad on systematics of the extraction of the elementary $\gamma $n$\to \pi^{\mathrm{-}}$p reaction cross sections beyond the IA for $\gamma $d$\to \pi^{\mathrm{-}}$pp, three key factors were chosen and analyzed: (i) The sensitivity to the number of steps of integration for numerical calculations of the five-fold integrals in the determination of FSI amplitudes; (ii) The sensitivity to the alternative deuteron-wave functions. (iii) The sensitivity to the experimental kinematic cut-off of the detected protons (the experimental information is uncertain). Preliminary estimations show that the contribution of all three factors to the overall systematics is less than 4{\%}.