Systematic cross-checks

Results from the two analyses indicate good agreement for both HPDs with and without filters. A summary of systematic cross-checks and the methods used to ascribe systematic uncertainties follows :

• HPD geometry and alignment - By studying the dependence of the mean Cherenkov angle resolution in an event as a function of the number of photons, an estimate of the residual misalignment of the detector prototype is determined. This corresponds to between 0.1 and 0.3 mrad for HPDs (2,3,5,6) and (4,7) respectively.
• Radiator pressure variation - The resolutions given above are determined from a subset of the data within which the pressure variation of the mean Cherenkov angle is determined to be 0.02 mrad.
• Beam composition - The beam contamination cut on the upstream Cherenkov counter is varied widely by changing the cut applied on the output from the upstream Cherenkov counter. The maximum effect observed on the mean Cherenkov angle resolution is 0.1 mrad.
• Noise contamination from the hit selection - The effect of the default 4 sigma selection cut is studied by varying it from 3 to 5 sigma and determining the shift in the average Cherenkov resolution. The shift is asymmetric due to the increasing noise contamination when the cut is reduced to 3 sigma. As a conservative value, the systematic error is estimated to be 0.1 mrad.
• Use of the geometrical fitting procedure - To check for a systematic bias in the geometrical fitting method, a similar selection of HPDs (4,7) is applied to that for (2,3,5,6) in order to verify the stability of the measured resolutions. The procedure followed is to use hits from HPDs 3 & 6 only. This is the same but diametrically opposite configuration to HPDs 4 & 7. The hit multiplicities of (3,6) are then reduced to those of (4,7) by randomly deleting hits, to emulate the effects of the mylar coverings on HPDs (4,7). The results indicate that the effects of the elliptical fitting method induce a maximum change in the observed Cherenkov angle resolution of 0.08 mrad.

When combined with the total uncertainties, listed above, the systematic error on the measured Cherenkov resolution is expected to be of the order of between $\pm$0.2 and $\pm$0.3 mrad for HPDs (2,3,5,6) and (4,7) respectively.