In order to quantify the systematic uncertainty of the precise half-life measurement of 215Po, we take into account both the instrumental and analytic uncertainties, as mentioned before.

We performed dedicated measurements to estimate the instrumental uncertainty.

Regarding the uncertainty from the analysis, which follows a blind analysis procedure. Firstly, we use one day of data, making up 5% of the total data, to estimate the event selections, the resulting signal and background events. Then, we produce the toy data from the signal and background counts with respect to 95% of the data. This step is to estimate the fit bias and optimize the binning and the fitting range. Lastly, we evaluate the uncertainty with the 95% physics data.

The structure is divided into two parts. The first part is to cross-check with Silvia's analysis; Secondly, I summarize the individual uncertainties contributed by several factors.

Due to the limited energy resolution of the scintillation detector, the alpha energies cannot be well separated. Optimizing the events selection is necessary to maximize the signal strength while keeping a low background. This section is to evaluate the optimal energy intervals. For this purpose, a significance formula is used, which should be more robust than the signal-to-background ratio (s/√b)) in the case of prominent signal events. The formula is shown below,

where z is the significance, s represents the signal counts, and b is the background events in the ROI.

1) Events selection

In the energy spectrum, the first α-decay (Q = 6946 keV) candidates do not show a single Gaussian peak due to the accompanying γ emissions after the preceding decay of 219Rn. Therefore, the energy interval is adjusted with 500 integrals on the left and right sides. The second α-decay (Q = 7526 keV) events are fitted with a Gaussian function, and the selection is tuned between (mean - i*sigma) and (mean + i*sigma), as shown in the figure below on the left. The right plot shows the resulting selection.

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The optimal energy interval is E₁∈[56500, 64000], E₂∈[64100, 75800], which is consistent with Silvia's selection (57000, 64300, 76000), so are the resulting signal and background counts.

2) Toy data

The toy data are produced using the numbers (s, b, and T_1/2 ) provided by Silvia.

In this simulation, a maximum time difference of 0.05 s is used, which is around 28 times the expected half-life of 1.781 ms. Considering the negative region, the total Δt range is [-0.05s, 0.05]. The positive side covers more than 99.99% of the 215Po decay. In each bin, the signal and background follow Poisson statistics.

The parameters to be studied are:

• bin width
• fit range Δt
• fitting method

1) bin width

The optimal bin width shall be compromised between the number of bins for the chi2 test and the Poisson uncertainty per bin. In this case, the bin width is varied from 5e-6 s to 1.5e-4 s in different steps. Each bin has 1e4 toys.

The result is shown below. With the increased bin width, the T_1/2 deviates from the true value of 1.781 ms, and the error bar, indicating the standard deviation of T_1/2, is getting larger.

The optimal value shall be the one with minimal deviation on the extracted T_1/2 and reasonable statistics per bin, which is 0.02 ms. This value is different from Silvia's 0.05 ms.

2) fit range Δt

According to the previous step, the bin width is fixed at 0.02 ms. The fit range also affects the chi2 test. Meanwhile, tuning Δt should not significantly change the statistics (signals). Therefore, the Δt range is adjusted from 0.01 s to 0.05 s, and 1e4 toys are produced for each Δt.

Unlike the bin width, adjusting Δt does not change the standard deviation dramatically, as shown below. A finer-tuning (right plot) gives the same result.

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As previously, the optimal Δt is the value with the minimal standard deviation, 0.03 s, which is also different from Silvia's 0.1 s.

3) fit method

The binning and the range are fixed accordingly. This step checks whether the fitting method leads to any bias. The previous steps have used the “RLE” option in root to fit the spectrum, which uses the log likelihood method and estimates errors better with the Minos technique within a specified range. This would be the default fitting method.

Alternatively, one can use Bayesian analysis that predicts which model and parameter are favored by the data. A software named Bat is used. The fit is performed with 1e4 toys, and the result is shown below. No significant bias is observed.

With the optimized parameters, 95% of the data are fitted using root. The first bins are ignored due to the dead time.

The residue plot shows an excess of events on the negative side. This is already explained in Silvia's note. The reason is that the two α candidates are classified into the wrong energy region, meaning the first candidates E₁ are accidentally seen as the second candidates E₂, and E₂ as E₁.

One can avoid this by a stricter selection, increasing the energy difference of the two candidates. The excess is disappeared, as shown below, but at the price of losing signal strength.

Alternatively, only the positive side is considered, while the negative region is used for background estimation. In this case, as pointed out by Silvia, the fit is slightly better than that without using the negative side to estimate the background.

Another way to fit the data is to reject events within [-0.01 s, 0 s]. As seen below, the obtained result is,

T_1/2 = (1.77809 ± 9.12e-4)ms.

Update: Add an exponential function at the negative side

To describe the excess events at the negative side, we could add an exponential function. So, the fitting function is,

A*exp(x*log(2)/T_1/2) + B; x < negative_first_bin

C*exp(-x*log(2)/T_1/2) + B; x > first_bin

The obtained half-life is (1.77807 ± 9.07808e-04)ms,

which is the default half-life. The following sections are to estimate each systematic contribution with respect to this value.

What if the half-life value is different from the one used at the positive side? The following graph shows the fit with the half-life at the negative side set as a free parameter.

The half-life is (1.77808 ± 9.07486e-04)ms, which is very similar to the above mentioned value. Therefore, the half-life parameter does not affect the fit. The following analytic uncertainties have been updated with this new fit.

The factors we consider are:

• time resolution
• dead time
• trigger delay
• time stability

In the previous pulser test section, we already measured the time resolution, which is 107 ns, contributing to 0.006% of uncertainty.

Regarding the dead time, events within 500 ns are affected. For this reason, we skipped events in the first bins during the fit, as mentioned previously. To estimate the uncertainty, the simulated toy data are fitted by including and excluding the first bins. The results with the standard deviations are,

T_1/2 = (1.781 ± 6.14e-5)ms, with first bins;

T_1/2 = (1.781 ± 6.17e-5)ms, without first bins;

which gives the same half-life. One can say that the dead time does not contribute to the systematic uncertainty.

The trigger delay, caused by baseline fluctuations, can affect the accuracy of the timestamp. Basically, the time difference between two events is calculated by

(t₁ + t_delay) - (t₂ + t_delay).

The trigger delay distribution of α events is shown below. The standard deviation of the trigger delay is 4.73e-9 s, meaning the systematic contribution is .

As for the time stability, 95% of the data are equally partitioned into nine subsets according to the time periods. For each period, the obtained T_1/2 is 1.77753 ms, 1.77594 ms, 1.77727 ms, 1.78302 ms, 1.77643 ms,1.77787 ms, 1.77975 ms, 1.77951 ms, 1.77533 ms. The standard deviation (0.002236) results in 0.129% of uncertainty.

Update: Investigating the possible gain shift that may cause this large uncertainty

Firstly, the non-calibrated energy spectra of the nine subsets are shown below. All the spectra are overlapped, and no large shift is seen. The time difference distribution is also shown, together with the respective signal and background counts.

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The relative maximum difference of signal events is 0.5%, resulted from (max-minus)/minus*100% between subset 4 and the subset 7. The relative difference of background counts is around 1.2%. These are probably negligible given the large statistics.

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The following plots show the fit with the subset of data.

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The parameters to be investigated are:

• event selections
• background
• bin width
• fit range Δt
• fitting method

Firstly, we consider the uncertainty from event selections. The same selection rule used for the 5% of data is applied to 95% of physics data. The resulting energy interval is shown below, considering selections with significance larger than 3590, one can get an uncertainty of 0.006%.

The background fluctuation contributed by random coincidences can also lead to uncertainties in the half-life. The flat background is adjusted accordingly to see the changes in the half-life. Firstly, it is set as a free parameter, and the resulting half-life is (1.77806±9.10e-4) ms. Alternatively, one can fix the background extracted from the region of [-0.05 s, -0.04 s], resulting in the half-life of (1.77818±8.80e-4) ms as shown below, which has a larger difference of 0.00009 ms with respect to the default half-life. To be conservative, the fixed background is used to estimate the uncertainty, contributing to 0.005%.

As mentioned earlier, the bin width affects the chi2 test. In this regard, the chi2 is taken into account when the binning is varied. With a chi2 variance of 0.1, the maximum T_1/2 is (1.77866±9.10e-4) ms, leading to an uncertainty of 0.03% to be conservative.

Like the bin width, the fit range also affects the chi2 test. The maximum T_1/2 is (1.77822±9.10e-4) ms, leading to an uncertainty of 0.007%.

The following factor is the fit method. Besides the root fit, one can use the Bayesian analysis to estimate the half-life by maximizing the likelihood. The posterior distribution and the corresponding quantile are shown below.

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Summarizing all the factors, the table shows the total systematic uncertainty.

So, the final measured half-life of 215Po is T_1/2 = (1.77809 ± 9.12e-4 (stat.) ± 0.002 (syst.))ms.

The last step is to cross-check data with other selections.

a) 'Golden dataset'

A stricter selection is applied to keep as small number of background as possible. For example, the energy intervals for the two candidates are [58000, 62000] and [66000, 72000]. The obtained half-life is

T_1/2 = (1.7796 ± 1.24e-3)ms.

b) Another selection

This selection uses the bin width of 0.05 ms, and fit range of [-0.1 s, 0.1 s], as used by Silvia, and the obtained half-life is

T_1/2 = (1.77856 ± 8.83e-4)ms.

The results from the two selections agree well with the default value within the uncertainty range.