Energy Fit Window Study

Chapter IV: UCNA: Fierz Interference Analysis

4.4 Combining the Extraction Methods

4.4.1 Energy Fit Window Study

As a study prior to combining our Fierz interference results, we were interested in the systematics vs statistics error trade-off when different energy windows are chosen. Conceptually, the trade off proceeds as follows. We have poorer quality energy calibration error envelopes at lower energies8. So we reduce in systematic uncertainty if we increase the low-energy fit window cut off. However, in order to perform a spectral fit, the𝛽-decay spectrum must contain enough shape information and, in particular, the inflection point of a maximal Fierz spectrum, figure 3.4, which is around 250 𝑘 𝑒𝑉 must be retained for a high-quality fit. So from the spectrum perspective, we improve by raising the low-energy cut off region but not above the Fierz spectrum inflection point.

With the super-ratio asymmetry Fierz interference extraction method, we are limited by statistics. Due to the shape of the UCN𝛽-decay electron spectrum, we gain more statistics by including low-energy bins, up until we start running in to the trigger function at which point there are no more included events because the trigger function diminishes very quickly below 100−150𝑘 𝑒𝑉. Then from the asymmetry perspective, the optimal Fierz extraction comes from lowering the low-energy cut but not below 150 𝑘 𝑒𝑉.

Furthermore, this conceptual discussion does not consider the geometric, calibration, and statistics differences between the 2011-2012 and 2012-2013 datasets (recall that we are not considering the 2010 dataset for the time being since it was unblinded).

Of course, in principle, we can take different energy fit regions for the two different extraction methods. We note however that ultimately the asymmetry Fierz extraction would likely have to be fixed at the same asymmetry fix region that was used in the 𝐴0 extraction. This is because we took the asymmetry data from that analysis and the resulting calibration, analysis, and interpretations are performed for a certain energy fit region. So then the only interpretation from these energy fit region studies is whether or not there is another clearly optimal choice for asymmetry — if not, then we should default to that which was used in the asymmetry analysis.

The studies used both blinded super-sum spectrum data and blinded super-ratio

8We also have poorer energy calibration envelopes at high energies, but due to the decreased sensitivity to Fierz away from the𝛽-decay spectrum peak this is less of a concern.

asymmetry data. We held either the super-sum or super-ratio fit windows fixed and varied first the low-energy threshold of the other dataset, for both 2011-2012 and 2012-2013 datasets. Once we settled on an “optimal” low-energy threshold, we would hold that fixed and check the super-ratio or super-sum fits by varying the high-energy cutoff, for both year’s datasets. This iterative optimization order was chosen because the low-energy window had a more direct impact on the extracted Fierz interference results. Figures 4.16a and 4.16b show only a sample of one of these studies where we separately chose the low and high energy fit windows for the super-sum spectral extraction of the 2011-2012, 2012-2013 datasets. We confirmed that the resulting fitting uncertainties as a function of the low and high energy cutoffs is smooth and ultimately resulted in a fit window chosen from 195−645 𝑘 𝑒𝑉 for both years (no appreciable difference when examining the two datasets with different fit windows).

In addition, we also examine how the 𝜒2distribution varies with the energy region cuts. This becomes particularly significant in the context of statistical error driven (asymmetry) vs systematic error driven (spectrum). We wanted to see what cuts were needed on energy region to ensure consistency with a standard𝜒²distribution so that our fitted error could be approximated as statistical. Figure 4.17 gives a sample of one of the 𝜒² probability plots generated in this study. When we ultimately decided to only use the asymmetry data, we wanted to use this study as a guide to ensure consistency with a standard 𝜒²distribution so that our fitting error could be dominated by statistical fluctuations and not begin to fold in systematics considerations.

After the systematic studies on energy fit region were concluded, we decided to use energy fit regions of 195−645𝑘 𝑒𝑉 for the spectral fits and 190−740𝑘 𝑒𝑉 for the asymmetry fits, for both the 2011-2012 and 2012-2013 datasets. In addition, we examined the effects of choosing different energy fit regions. We fit the asymmetry data with𝐸_{𝑙 𝑜𝑤}−30𝑘 𝑒𝑉, 𝐸_{𝑙 𝑜𝑤}, and𝐸_{𝑙 𝑜𝑤}+30𝑘 𝑒𝑉, and with𝐸_{ℎ𝑖𝑔 ℎ}−60𝑘 𝑒𝑉, 𝐸_{ℎ𝑖𝑔 ℎ}, and 𝐸_{ℎ𝑖𝑔 ℎ} +60 𝑘 𝑒𝑉. We take the average of these fit values on the low and high end and compare them to the central chosen value for the energy cutoff. At low energy, the shift induced was Δ𝑏 ≈ 0.003. At high energy, the shift induced was Δ𝑏 ≈0.009.

Due to the statistics-driven uncertainty of the asymmetry Fierz extraction, these numbers only show the stability of the fit with respect to different energy region choices. In the end there was no optimized method to choose a particular fit

(a) Combined uncertainty on𝑏as a function of low energy fit window variation. The high energy cut off was originally fixed at≈735𝑘 𝑒𝑉.

(b) Combined uncertainty on𝑏 as a function of high energy fit window variation. The low energy cut off was originally fixed at≈195𝑘 𝑒𝑉.

Figure 4.16: A 2D histogram showing the resulting weighted error using a four-point weighted average, in the color bar, of𝑏 where the statistical error is the error used for the super-ratio extraction and the systematic error is the used for the super-sum extraction. The high energy cut off was originally fixed at≈ 740 𝑘 𝑒𝑉. In terms of minimizing the weighted average error on 𝑏, there is a large acceptable region for the low-energy cut off. The error is varied due to changing the low-energy cut offs for the 2011-2012 dataset against the 2012-2013 dataset.

Figure 4.17: Different low energy fit window’s 𝑝 values. A high energy cut off at 645𝑘 𝑒𝑉 is fixed. A dashed line is included at 10⁻²to represent the 1% probability that this fit was a statistical anomaly. We use 10⁻² as an approximate cut off before deciding a fit was dominated by uncertainties that were non-statistical (hence systematic). In general, we see the same behavior as discussed in section 4.2.2.4 with regards to the tin stitching: the 2012-2013 uncertainties are systematically shifted compared to the 2011-2012 uncertainties.

window over another once the decision to use only the asymmetry data in our final measurement was made. This was because trade-off between the systematic error on the spectral fit and the statistical error on the asymmetry fit is removed by omitting the spectral fit results.