Cherenkov angle resolutions for C₄F₁₀ and silica aerogel

The small number of Cherenkov photons from the aerogel, and the absence of the silicon telescope in Configuration 1, prevents use of the improved resolution fitting methods detailed in Section 7.2. In the absence of any beam trajectory information, beyond that implied by the scintillator acceptance, this method is applied directly to data, using the nominal (0, 0, - 1) beam direction. It serves as a cross-check to allow comparisons with more precise determinations explained later. These ``raw'' $\theta_{c}^{}$ distributions are shown in Figure 12 for the case of Configuration 2 with 1 m of C₄F₁₀.

Figure: Raw Cherenkov angle distributions for hits detected in each of the outer HPD detectors. The x axes units are in milliradians.

The pixellated nature of the HPD is apparent, as it induces a multi-peak structure of the $\theta_{c}^{}$ distribution. In addition, there is a shift in the mean $\theta_{c}^{}$ , and a clear reduction in the raw resolutions for orientations of HPDs 2 and 5 where the Cherenkov ring lies parallel to a single row of pixels.

Raw Cherenkov resolutions are determined from data using HPDs (2,3,5,6), without filters, in Figure 12, and which average 1.76 mrad. The RMS of the observed $\theta_{\rm c}^{}$ distribution is used as an estimate of the width, as pixellisation effects makes the distributions difficult to interpret using Gaussian fits. Similarly, the raw $\theta_{c}^{}$ distributions are determined for runs taken with aerogel in Configuration 1. The measured RMS values of the raw Cherenkov resolutions average 4.78 mrad for HPDs (2,3,5,6). No aerogel data was collected using Configuration 2.

Simulation is used to determine resolution contributions due to chromatic aberration, discrete pixel sizes and the emission point uncertainty. The input beam trajectory uncertainty for this configuration is that implied by the trigger scintillator acceptance. The expected contributions are summarised in Table 6 and, when added in quadrature, total 1.72 mrad for C₄F₁₀ and 3.20 mrad for silica aerogel.

Table 6: Summary of expected resolution contributions for the analysis of the Cherenkov angles using the nominal beam direction for C₄F₁₀ without aerogel, with a mylar window, and and silica aerogel in Configurations 2 and 1 respectively.

	$\sigma$($\theta_{\rm c}^{}$) (mrad)
Contribution Source	C4F10	Aerogel
Pressure Variation	0.70	0.70
Emission Point Error	0.58	0.66
Chromatic Aberration	1.03	1.06
Finite Pixel Size	0.56	2.73
Beam Trajectory Error	0.87	0.87
Total in Quadrature	1.72	3.20

The aerogel Cherenkov angle resolution is dominated by the finite pixel size, because of the shorter focal length mirror used in Configuration 1. The agreement between observed and expected raw resolutions is reasonable considering systematic uncertainties due to estimating the widths of the discretised distributions.

In order to improve the Cherenkov angle resolution, it is necessary to reduce the dominant contributions listed in Table 6. Effects of radiator pressure variation are reduced by deactivating the recirculation system, and measuring the mean Cherenkov angles in blocks of events recorded sequentially in time. Extrapolating the observed time variation of the reconstructed Cherenkov angles allows remaining uncertainties to be extracted from data. The chromatic aberration contribution is reduced for HPDs (4,7) by using mylar filters to prevent photons with wavelengths below $\sim$ 350 nm from entering the HPD photo-cathode. The beam trajectory is estimated more accurately using two methods:

• Silicon telescope  data, when available, allows precise constraints on the input particle trajectory. A single specific pixel, on each of the three silicon planes, is demanded as being hit for the beam vector to be used⁵.
• Geometrical reconstruction uses an elliptical fit to events with $\geq$ 4 hit pixels from each of HPDs (2,3,5,6) and (4,7). The centre of each ellipse is used to estimate the beam trajectory for the Cherenkov angle reconstruction on that HPD. A vector, connecting the ellipse centre to the mirror centre, is reflected in the mirror to give an estimate of the incoming beam direction. Systematic checks are necessary to ensure the fit procedure does not bias the results. Resolution contribution from uncertainties in the beam trajectory can be determined from the observed error distributions from the fit. They correspond to values of 0.28 mrad for HPDs (2,3,5,6) and 0.58 mrad for HPDs(4,7).

The results from data and simulated expectations are summarised separately for the two types of improved beam direction estimates.