Bragg Likelihood Algorithm

The final algorithm which is investigated is the Bragg Likelihood algorithm, which has been developed as a part of this work.

Using the Landau and Gaussian widths outlined in Chapter 6.2, a probability map is con- structed for each particle species using the theoretical meand𝐸/d𝑥as a function of residual range

close to the Bragg peak (shown for muons, pions, protons, and kaons in figure 6.2), as well as for MIP-like particles which exit the detector and so do not have a Bragg peak. For a given residual range, the theoretical d𝐸/d𝑥 prediction is used to determine the mean of the Landau-Gaussian

distribution, and the width is taken from the results given in Tables 6.1 and 6.2. The resulting probability maps are shown in Figure 6.11.

These particle maps can be used to construct a likelihood under each particle hypothesis for each track, 𝐿𝑠 𝑇 𝑜𝑡 𝑎𝑙 = ∑𝑁ℎ𝑖𝑡 𝑠 𝑖=1 𝐿 𝑠 (d𝐸/d𝑥𝑖, 𝑅𝑖) 𝑁ℎ𝑖𝑡 𝑠 , (6.6)

where the sum is over each hit𝑖 associated to the track between 0 and 30 cm residual range.

The𝐿𝑠(d𝐸/d𝑥𝑖, 𝑅𝑖)corresponds to the evaluation of the likelihood map for the particle species,𝑠,

at the residual range (𝑅) andd𝐸/d𝑥 of the hit. As with previous algorithms, the first and last hits

of each track are neglected, as they are known to produce unreliable results.

This calculation is done for each plane separately, but results from multiple planes may be combined to produce a single result by taking the average of the likelihood across multiple planes. Currently, however, this is not performed, as there are known differences ind𝑄 /d𝑥 as a function

of track angle between data and simulation which are not currently understood. For this reason, only the collection plane is considered for calorimetry.

The algorithm is made to be more robust by the addition of two features:

• For each particle species, the likelihood is first calculated by assuming the track direction is correct, and then the direction is artificially reversed, and the likelihood is recalculated.

(a) Muon (b) MIP, no Bragg peak

(c) Proton (d) Pion

(e) Kaon

Figure 6.11: Probability maps for each particle species using the Landau and Gaussian widths for the simulated collection plane, as described in the text. Each x bin integrated over y is normalised to an integral of 1 in the range [0,100].

The result which maximises the likelihood is thought to be the correct direction, however both values are stored for future use.

• Additionally, the track end-point resolution is dynamically accounted for. This is done by allowing the hit residual range to float within±2 cm, (the estimated 1𝜎 envelope on the

end point resolution for tracks from the neutrino pass of the Pandora reconstruction suite, as shown in Figure 6.12), calculating the likelihood for many points with0.05cm spacing

within these bounds, and taking the maximised likelihood.

Figure 6.12: End point resolution for BNB-induced tracks reconstructed by neutrino pass of the Pandora reconstruction suite.

Both the data and the simulation suggest that as the the end of the track is approached, the Landau-Gaussian distributions become wider (see, for example, Figures 7.14 and 7.15), however it is believed that this is reconstruction-driven and not physics-driven. The explanation for this is that the end point resolution can have significant impact on the apparent width of the distribution and this is especially dominant at very low residual range, where thed𝐸/d𝑥 is rapidly changing.

This is shown in Figure 6.13. This effect could be captured by by modifying the width of the distribution as a function of residual range, however due to the dynamic floating of the end-point resolution, as outlined above, this is not necessary.

Figure 6.13: Demonstration of the impact of end point resolution on the effective width of the distribution. The black line here is the nominal muon theory as shown in Figure 6.2, the green line is what would be expected if the track was extended by 2 cm, and the red line is what might be expected if the track was truncated by 2 cm. It is clear here that reconstructing an incorrect end point for the track can significantly alter any PID variable which uses thed𝐸/d𝑥 and residual

range of each hit associated to a track unless this effect is specifically targeted.

Figure 6.14 shows a comparison of data and simulation for the bare likelihoods for muons, pions, protons, kaons, and MIPs. In general, the agreement between data and simulation here is unimpressive. The shape of the proton region is reasonably well approximated in the simulation, however there are significant shape differences in the muon region in each plot. Muons may stop in the detector and exhibit a Bragg peak, or may exit and look like a MIP. Part of the disagreement here may be due to different relative strengths of the different muon populations, although it seems unlikely that this captures the whole problem.

Taking the ratio of some of the variables produced by the BL algorithm can significantly im- prove the agreement between data and simulation, as shown in Figure 6.15. Here, each likelihood has been normalised by the sum of the likelihoods, therefore enforcing the demand that they

(a) Muon likelihood (b) Pion likelihood

(c) Proton likelihood (d) Kaon likelihood

(e) MIP likelihood

Figure 6.14: Template fit bare likelihoods for tracks under muon, pion, proton, kaon, and MIP assumptions.

range between 0 and 1. The improved agreement seems to indicate that the residual differences between data and simulation for the bare likelihoods in Figure 6.14 are caused by one or more effects which are not contained within the simulation, or where an improved implementation is necessary. Taking the ratio of the likelihood variables halps to mitigate the differences between data and simulation.

For the selection presented in Chapter 7, it is enough to identify tracks as MIP-like (either muon or pion), or proton-like. To maximise separation between these, the variable ln(𝐿_{𝑀 𝐼 𝑃}/𝐿_𝑝)

is constructed, as shown in Figure 6.16. In this variable we would expect protons to populate the region below 0, and MIP-like particles to populate the region above 0. Both the separation of muons and protons, and the agreement between the data and simulation in this variable are extremely good. The efficiency and purity plots presented in Figure 6.16 also indicate that this algorithm is reasonably insensitive to the detector effects which were investigated.

(a) Muon likelihood (b) Pion likelihood

(c) Proton likelihood (d) Kaon likelihood

(e) MIP likelihood

Figure 6.15: Template fit likelihoods for tracks under muon, pion, proton, kaon, and MIP assump- tions, normalised from 0 to 1.

(a) Without Calibration (b) Nominal simulation

(c) Dynamic induced charge (d) Birks model

(e) Efficiencies and purities