Resolution-Optimizing Method

HVEV RUN 3 AT NEXUS

3.6 Relative channel weighting

3.6.1 Resolution-Optimizing Method

The first (flawed) method minimized the energy resolution in the 1𝑒⁻ℎ⁺quantized peak instead of directly removing partition dependence. The total signal (𝐴) and relative weighting factor (𝛼) were defined as in Eq. 3.1.

𝐴≡ 𝛼 𝐴_𝑖𝑛+𝐴_𝑜𝑢𝑡 (3.1)

where 𝐴_𝑖𝑛 and 𝐴_𝑜𝑢𝑡 are the inner and outer channel signals, respectively. For this method, we explicitly calculated the resolution on 𝐴 for a range of choices of 𝛼. Resolution was calculated by fitting a Gaussian to the 1 𝑒⁻ℎ⁺ peak in 𝐴. The resolution (with an approximate energy calibration) would then be defined by Eq.

3.2.

𝜎_eV ∼

𝜎𝐺 𝑎𝑢 𝑠 𝑠𝑖 𝑎𝑛 𝑓 𝑖𝑡

𝜇𝐺 𝑎𝑢 𝑠 𝑠𝑖 𝑎𝑛 𝑓 𝑖𝑡

100 eV (3.2)

This calculation was performed using OF0 science data from each day of data taking.

Laser data could not be used, since the large fraction of sidewall events altered the shape of the 1 𝑒⁻ℎ⁺ laser peaks and made them a poor estimator of detector resolution. Daily data was useful for checking that the resolution-optimizing𝛼did not vary significantly over time.

The calculation was performed before the analysis 𝜒²cut was finalized, so a basic energy-independent𝜒²cut was applied to the data instead. Another cut was applied to carefully select the 1 𝑒⁻ℎ⁺ peak events. The obvious choice would be to select these events using the quantized distribution of𝐴, but such a selection would rely on prior knowledge of𝛼. Instead, 1𝑒⁻ℎ⁺events were identified using a 2-dimensional cut dependent on both 𝐴_𝑖𝑛 and 𝐴_𝑜𝑢𝑡. The cut retained events within a 3𝜎 ellipse iteratively fit to data using principal-component analysis. The cut was 3𝜎 under the assumption that the underlying distribution was bivariate normal. In that case, a 3𝜎 cut would retain 98.9% of data from the underlying distribution. The 1𝑒⁻ℎ⁺ selection was performed separately for each dataset. An example of the iterative 1 𝑒⁻ℎ⁺ cut applied to NF-C data can be seen in Figure 3.20.

After applying the cuts,𝜎_𝑒𝑉 was calculated as a function of 𝛼for each dataset. An example result for NF-C can be seen in Figure 3.21. Each dataset’s result was then fit to an approximate model for the variance of the weighted and normalized sum of

Ain[μA]

A_out[μA]

Figure 3.20: The result of using primary component analysis to iteratively fit an ellipse to the 1-𝑒⁻ℎ⁺ peak in one NF-C dataset. The heat map displays the number of events in a given bin. Each blue ellipse is one iteration of the cut (starting from the largest ellipse). We see that the cut converges after about four iterations.

two random (and potentially dependent) variables. The model utilized is shown in Eq. 3.3.

𝜎²

eV =

100 eV 𝛼𝑚+1

2

𝛼²𝑎+𝑏+2𝛼𝑐

(3.3) The squared prefactor accounts for the normalization of the weighted sum to 100 eV. Specifically,𝑚accounts for the difference in average signal observed in the two channels. 𝑎, 𝑏, and 𝑐 are proportional to the inner-channel variance, the outer- channel variance, and the inner-outer-channel covariance, respectively. All four values were extracted via the fit and used to calculate 𝛼_{optimized𝜎} (the result of minimizing Eq. 3.3 with respect to𝛼).

𝛼_{optimized𝜎} = 𝑚 𝑏−𝑐 𝑎−𝑚 𝑐

(3.4) Figure 3.21 also displays the fit to this model (with residuals) with the resulting 𝛼_{optimized𝜎}in the legend.

NF-C

(Jan 11, 2021)

σeV[eV]

!

residuals [eV]

Figure 3.21: A typical result of applying the resolution-optimizing method to a dataset. (Top) The calculated resolution as a function of channel weighting (𝛼) is shown in orange (with grey uncertainties). The fit to the resolution as a function of𝛼 (using Eq. 3.3) is shown in black. The𝛼 that gives the minimum resolution (using Eq. 3.4) is marked with a vertical dotted line and labeled in the legend.

The𝛼_{optimized𝜎} for NF-C and NF-H were seen to be stable over the entirety of data taking (even going back to December 2020). For R1, 𝛼_{optimized𝜎} changed sharply on January 11th. This was likely caused by the change in noise observed on that day. Otherwise, the R1𝛼_{optimized𝜎}was stable but with relatively more variance than in the other two detectors. 𝛼_{optimized𝜎} for each detector and dataset can be seen in Figure 3.22.

Weighted means of𝛼_{optimized𝜎}were calculated for NF-C, NF-H, R1 (before January 11th), and R1 (after January 11th). These can also be seen in Figure 3.22. The NF-C and NF-H values were applied to calculate the total signal and associated RQs in the final analysis. The R1 𝛼_{optimized𝜎} was considered too uncertain and was not used.

Instead, an unweighted sum was used for R1. At the time this decision was made, it was already decided that the R1 detector would only be used as a veto detector (due to its inferior resolution and noise). Partition independence and energy calibration were not necessary in a veto detector.

!optimized σ

Figure 3.22: The result of calculating the resolution-minimizing channel weighting (𝛼_{optimized𝜎}) for each science dataset and each detector. Variance-weighted means were calculated for the NF-C and NF-H detectors (horizontal dashed lines). For the R1 detector, separate variance-weighted means were calculated for pre- and post-January 11th data (horizontal dashed and dotted lines). Variance weighting was used by mistake. Inverse variance weighting should have been used instead.

The NF-C and NF-H results were shown to be valid anyways (see Figure 3.24).

The misguided emphasis of this method on resolution optimization was caught during the early stages of the analysis’s inner-collaboration review. Even so, there was reason to believe that optimizing resolution may have also minimized the partition dependence. If the noise in both detector channels was subdominant to the variation in signal caused by varying energy partition, then partition variation would be the primary contributor to the signal variance of 1𝑒⁻ℎ⁺events. Figure 3.20 suggests this may be true for the NF-C detector. There are separate event clusters at either end of the distribution, each smaller than the total distribution. Presumably, the cluster size corresponds to the channel noise while the overall distribution (with a clear negative slope) is caused by the partition variation. In the case of partition- variation dominance, minimizing the 1 𝑒⁻ℎ⁺ resolution would also minimize the partition dependence.