ratio of the detector-wide measured and predicted content ofEreco bin j[85]:
χ2 =
Here, the rescaling of the prediction to the measurement for each energy bin j, reduces the dependence on the reactor spectrum and rate of the model. In addition, possible relative energy response variations between detector cells are directly corrected for.
6.4 SoLid oscillation fit
For the first oscillation analysis result, the SoLid collaboration will use a rel-ative fit similar to the principle of the PROSPECT- and STEREO-like fits. The difference between these two is mostly determined by the treatment of the uncertainties, where the PROSPECT-like fit uses more a priori information on the uncertainties, summarised in covariance matrices, and the STEREO-fit introduces free nuisance parameters with pull terms.
As the studies on the systematic uncertainties of the current SoLid dataset are still ongoing at the time of writing (see chapter 7), it is left undecided how these uncertainties will be propagated in the final fit. In this thesis, we choose to present the example contours based on the PROSPECT-like fit, with covari-ance matrices representing the main uncertainties.
6.4.1 Fake dataset
A first contour based on real SoLid data, albeit for only a small data sample, will be presented in chapter 8. To illustrate the concepts presented in the following sections of this chapter, however, we will use afakedataset that has roughly the same size as the total SoLid Phase I dataset.
The predicted IBD events are generated with the neutrino generator, pre-sented in chapter 3, and the migration matrix, prepre-sented in chapter 4:
• The simulated IBD interactions are generated requiring a normalisation corresponding to 350 days of reactor-ON data with a thermal power
146 CHAPTER 6. OSCILLATION ANALYSIS of 55 MW. This roughly corresponds to the total amount of reactor-ON data gathered by the Phase I detector (see table 5.1), considering a fraction of the days will probably be removed from the analysis due to instabilities. The reactor spectrum is simulated according to the Huber-Mueller prediction, without sterile oscillation, and the antineu-trino events are generated based on the average reactor settings and not per individual reactor cycle. The latter choice is motivated by the fact that the shapes of the Etrue- and Ltrue-distributions are practically the same for each reactor cycle, as illustrated in figure 6.3. The num-ber of events generated in the simulation is increased with a factor 100 with respect to the expected signal normalisation, and then scaled back by dividing the resulting event distributions with 100. This is done to reduce statistical fluctuations in the predicted distributions.
• The reconstructed event distributions are found by applying a simpli-fied detector model to the predicted true distribution, that shifts and smears the energy corresponding with the loss of both annihilation gammas and an energy resolution at 1 MeV of 18%, respectively. In addition, the number of events was reduced according to an IBD se-lection efficiency of 14%. This simple model was used instead of the migration matrix presented in chapter 4, because it significantly speeds up the fit procedures described below.
The fake dataset is then generated by adding fluctuations to the predicted event distributions:
• The fake signal distribution is generated by randomising the (Lreco,Ereco )-distribution bin counts, in agreement with the uncertainty matrix of the prediction. In practice, this construction of the fake experiment happens as follows:
1. A lower triangular matrixLis derived from the Cholesky-decom-position1of the full covariance matrix for the prediction.
2. Random numbers are drawn from a Gaussian distribution with mean equal to 0 and a standard deviation of 1, to construct a random number vector of the same length asL.
3. The vector of random values is multiplied with Land the result is added to the original prediction distribution.
1Cholesky-decomposition is the operation in which a Hermitian, positive-definite matrix Mis decomposed into the product of a lower triangular matrix and its conjugate transpose;
M = L(L)T. This makes its use in numerical operations, such as MC simulations, more efficient. [138]
6.4. SOLID OSCILLATION FIT 147
5 5.5 6 6.5 7 7.5 8 8.5 9 9.5 10
true (m) L 0
0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045
A.U. Average
Individual cycles
0 1 2 3 4 5 6 7 8 9 10
(MeV) Etrue
0 0.02 0.04 0.06 0.08 0.1 A.U. 0.12
Average Individual cycles
Figure 6.3: The normalised Ltrue (top) and Etrue (bottom) distributions for simula-tions of 8 different reactor cycles, shown in blue. The average shape is shown in black.
148 CHAPTER 6. OSCILLATION ANALYSIS
• The number of background events is determined according to the sig-nal normalisation, scaled with the sigsig-nal-to-background ratio of 1:5 that was derived with the uBDT selection. The energy distribution of the background events is simulated based on the detected reactor-OFF en-ergy spectrum and the events are assumed to be uniformly distributed over the detector modules after quality cuts.
A plot of the resulting fake data spectrum in the first detector module is shown in figure 6.4. The ratio of this fake data spectrum to the no-oscillation prediction is also shown for each detector module in figure 6.5. This fig-ure also shows this ratio for an oscillated prediction using the RAA best fit oscillation parameters: sin2(2θ14) =0.17,∆m241 =2.3 eV2.
SoLid module 1 - 6 cycles Predicted spectrum
Figure 6.4: Top: Reconstructed energy spectrum of the example dataset in detector module 1 (black points) and the predicted reconstructed energy spectrum in the case of no sterile oscillations (blue). Bottom: Ratio of the fake data spectrum to the predicted no-oscillation spectrum.
We note that only the statistical uncertainties are used for the generation of the toy experiments and for the example plots generated with the fake dataset. This will be the case for all figures shown below. The determination
6.4. SOLID OSCILLATION FIT 149
Figure 6.5:Ratio of the reconstructed energy distributions for the fake dataset in each of the 5 detector modules to the prediction containing no sterile oscillation (black points). The ratio for an oscillation prediction according to the reactor antineutrino anomaly best fit oscillation parameters is also shown (pink).
150 CHAPTER 6. OSCILLATION ANALYSIS of these statistical uncertainties and of additional systematic uncertainties, as well as their expected effects, will be discussed in chapter 7.
Furthermore, we note that the data binning for the fit is chosen in 10 bins of 0.5 MeV in Ereco and per detector module of 10 planes in Lreco. Given the 1/L2 reduction of IBD signal events with baseline, the event statistics will decrease towards the higher position bins. By segmenting the detector in length such that each position bin has roughly similar IBD signal statistics, one could provide a better statistical coverage and improve the overall os-cillation sensitivity. However, this would also require an evaluation of the relative performance and uncertainties per detector plane, rather than per detector module, and a variable Lreco-binwidth has therefore not been imple-mented yet.
Performing a fit of equation 6.17 using this fake dataset, results in a χ2 -value of 43.8. We note that for our relative fit, the number of degrees of freedom is reduced from the 50 (Lreco,Ereco)-bins to 40, because of the use of the ratiosDj/Pj for each of the 10Ereco-bins. We thus find a reducedχ2-value of 43.8/40 = 1.095, which implies a good agreement of the null hypothesis with the (fake) data. This result is as expected, indeed, since the fake dataset was generated based on a no-oscillation prediction. However, it gives us confidence in the implementation of the fit. In addition, one can calculate the χ2-result for a large number of toys, each generated in the same way as the presented fake dataset, and look at the resulting distribution. The χ2 -distribution made from 10 000 toys is shown in figure 6.6 and agrees well with theχ2 p.d.f. of 40 degrees of freedom.