The main sources of systematic uncertainties have been stated in in the previous section. The size of the systematic uncertainties is determined by the accurate calculation of the following:
were thrown into and tracked throughout the
detector. By appropriately choosing the origin of the primary photons and
positrons, all the points (1)-(4) were addressed.
Figure: The
line-shapes of 
photon-induced showers in CsI as
calculated by GEANT, ITS and EGS4 (top panel). The calculations of the three
codes show excellent agreement, especially in the region above the threshold of
. The bottom panel shows the response of the plastic veto
counter to showers initiated by photons which originate in the target. Here
too, the results of the three different calculations agree remarkably well.
After a careful optimization of the tracking
medium parameters, a remarkable agreement was observed between the results of
the three simulation codes. The profiles of the shower spreadings as
calculated by GEANT, are shown in figure 
. The top panel of
figure shows the
line-shape in CsI of 
photon-induced showers. In the region
above the threshold of 
, the agreement between the
calculations of the three different codes is excellent. The integrated
detection efficiencies above 
agree to 
, which
is remarkable. Similar agreement is found for the energy leaked through the
back of the calorimeter.
The response of the plastic veto counter to showers initiated by photons
originating in the target is presented in the bottom panel of
figure for ITS, GEANT and EGS4. This response is a
combination of the processes (3) and (4) and affects the overall detection
efficiency for pion beta decays. The integrated yields above minimum ionizing
particle (MIP) threshold differ by 
of the total number of shower
histories. However, raising the threshold by 0.4 MeV improves the agreement
to about 
between ITS and GEANT results and 
between ITS
and EGS4 calculations.
Figure: The top panel is the total
shower backsplash as calculated by the three different codes. This is the total energy
flow back across the front face of the CsI detector. The response of the thin plastic
veto detector to this backsplash is displayed in bottom panel. A very small fraction of
the backsplash consists of charged particles.
To isolate the process (3), the monoenergetic 
photons were
thrown perpendicularly incident onto the CsI detector with their origin at the
front face of the CsI detector. A fraction of the showers proceed in the
direction opposite to that of the original photons and flows back across the
front face of the CsI detector. This shower backsplash is shown in top panel
of figure . ITS and Geant calculations of the energy flow
back across the front face of the CsI detector agree to better that 
in an integrated yield above 1 MeV. The EGS4 result is higher by almost one
percent. The plastic veto counter is the first detector encountered by the
shower backsplash; the response of this counter to the backsplash is shown
in the bottom panel of the same figure and is the more
relevant quantity from the standpoint of corrections to the detection
efficiency. A very small fraction of the backsplash consists of charged
particles, i.e. less than 
of the histories registered above the MIP
threshold. The integrated yields above the MIP threshold differ by less than
of the total number of histories for GEANT and EGS4 calculations.
However, the ITS result is slightly higher at
large deposited energies.
In conclusion, the processes which affect the detection efficiency of pion beta
decays occur at the level of few percent probability. A remarkable agreement
has been observed in the calculations of detection efficiencies using GEANT,
ITS and EGS4. Depending on the process considered, the three different
calculations agreed at the level of few per mill of better. Since these
calculations were carried out, GEANT has been the only code used for the
simulations of the various processes affecting the design of the apparatus.
This has been due to the fact that the GEANT simulation code includes the most
comprehensive geometry package and interfaces with CERN
library routines. These systematic uncertainties will have to be
evaluated experimentally during the set-up and calibration parts and monitored
throughout the data taking process.