Global analysis

Chapter III: Extraction of the neutron lifetime

3.9 Global analysis

Section 3.8.1 demonstrated that excluding fixed lengths of time from the holding length induced a bias in the extracted lifetimes. In order to quantify the effect of just the variation of the mean arrival times on the extracted lifetime, each run’s mean arrival time was decomposed into two parts:

¯

𝑡_𝑖 =h¯𝑡_𝑖i + (𝑡¯_𝑖− h¯𝑡_𝑖i).

The h¯𝑡_𝑖is were calculated separately for 2017 and 2018 because of the significant change in mean arrival times between the two years. Two lifetimes were extracted:

the first included h¯𝑡_𝑖i in each holding length, and the second included ¯𝑡_𝑖 in each holding length. The second extracted lifetime was 0.04± 0.01 s greater than the first extracted lifetime, which roughly agreed with the results of the Monte Carlo simulation developed in this section.

3.9 Global analysis

The process from Section 3.7.9 that found the optimal parameters of the normaliza- tion model required a value for the lifetime as an input. Extracting a global lifetime required the normalization estimates as inputs. The MLE of 𝜏 and the optimal parameters of the normalization model were simultaneously solved for. Golden- section search [52] was used to search for the MLE of 𝜏 between 800 and 1000 s. Golden-section search is a convex optimization algorithm that can optimize a convex function to arbitrary precision without calculating any derivatives. In this analysis, the MLE of𝜏was found to within a precision of 10⁻⁴s. For a fixed value of 𝜏,the MLE of the normalization parameters were found using the method described in Section 3.7.9. For each trial value of 𝜏,the log-likelihood from Equation 3.14 was summed over only the long and extra-long runs to reduce correlation between 𝜏 and the normalization parameters. After the MLE of 𝜏 and the normalization parameters were found andL_minwas calculated, the uncertainty of𝜏was estimated using the following method:

1. 𝜏was displaced above the MLE of𝜏;

2. Using the displaced value of𝜏,the normalization parameters were optimized using the method described in Section 3.7.9;

3. L⁰was calculated by summing over only the long and extra-long runs, using the displaced value of𝜏and the normalization parameters found in Step 2;

4. The displacement of 𝜏 was adjusted and Steps 2 and 3 were repeated until L⁰=L_min+ ¹

2;

5. Steps 1 through 4 were repeated with an initial displacement of𝜏 below the MLE of𝜏 .

3.9.3 Statistical bias

Many simplifying assumptions were made in Section 3.7 while constructing Equa- tion 3.14. Some of these assumptions could have introduced a bias into the extraction of a global lifetime. A Monte Carlo data set was produced using the first five steps of the Monte Carlo simulation detailed in Section 3.7.9. The inputs to the simulation (the estimate of the number of UCN loaded into the trap and associated uncertainty, the holding length of each run, the estimate of the efficiency of the primary detector and associated uncertainty, the estimate of the software-dead-time and UCN-event- tail corrections, and the estimate of the background CCs) were chosen to match the

values from the analysis of the real data. However, this process did not simulate values for the number of counts recorded in the normalization monitors during the filling process. The simulated data were combined with the real values for the hold- ing lengths and the counts in the normalization monitors to produce a combined data set with characteristics that were similar to the real data set.

The global analysis from Section 3.9.2 was used to extract a lifetime from the combined data set. This procedure was repeated 1000 times to measure any statistical bias to high precision. Each extracted lifetime had a statistical uncertainty that was comparable to the statistical uncertainty from the analysis of the real data. The mean of the lifetimes extracted from the 1000 iterations of the Monte Carlo simulation differed from the true Monte Carlo lifetime by 7±10 ms (p-value = 0.48), so no evidence of a bias was found.

3.9.4 Results and conclusion

The global extracted lifetime was𝜏_𝑔 =877.62±0.27 s. When a spectral correction was not used in the normalization estimate (see Equation 3.6), the global lifetime was 𝜏⁰

𝑔 =877.63±0.29 s. Figure 3.29 shows the results of a global analysis performed on 3853 runs.

These lifetimes were extracted using the method developed in Section 3.9.2, and Section 3.9.3 used Monte Carlo simulations to confirm that this method did not induce a bias in the extracted lifetimes. The Cramér-Rao bound confirmed that using the method of maximum likelihood estimation was the most efficient way to use the available data to extract the lifetime and normalization estimates [53]. The analysis developed in this section was close to, but was not exactly, a full maximum likelihood estimation over the entire data set. All runs contained some information about both the normalization and the lifetime, but the short runs were used to only estimate the normalization and the long runs were used to only estimate the lifetime.

Section 3.10.7 will demonstrate that the information about the lifetime contained in the short runs was insignificant to this analysis.

Figure 3.29: The results of a global lifetime fit of all runs. Top: average yields for each holding length, as well as residuals for each average yield. Bottom: differences between the measured and expected number of CCs recorded during the unloaded process of each run, divided by the uncertainty of that difference. In the top plot, all average yields are represented with central values and statistical uncertainty bars, but the bars are too small to see in all but the 5000 s runs. The bottom two plots display the same data, but the right plot uses a logarithmic vertical scale to better demonstrate the behavior of the tails of the distribution. The orange curve is a Gaussian distribution with a mean of 0 and a standard deviation of 1. Both plots in the bottom section share the same value of 𝜒²/𝜈 =3657.2/3549 (p-value=0.10).

3.10 Paired analysis