We will now generate a plot comparing the eccentricity of signal events from calibration data to background events. In addition another comparison will be made between photopeak photons and escape peak photons to show that events at different energies have different shapes, motivating the need for the X-ray tube data at different energies.

import ggplotnim, nimhdf5
import ingrid / [tos_helpers, ingrid_types]

proc read(f: string, run: int): DataFrame =
  withH5(f, "r"):
    result = h5f.readRunDsets(
      run = run,
      chipDsets = some((chip: 3, dsets: @["eccentricity", "centerX", "centerY", "energyFromCharge"]))
    )

proc main(calib, back: string) =

  # read data from each file, one fixed run with good statistics
  # cut on silver region
  let dfC = read(calib, 128)
  let dfB = read(back, 124)
  var df = bind_rows([("Calibration", dfC), ("Background", dfB)], "Type")
    .filter(f{float -> bool: inRegion(`centerX`, `centerY`, crSilver)},
            f{float: `eccentricity` < 10.0})
  ggplot(df, aes("eccentricity", fill = "Type")) +
    #geom_histogram(bins = 100,
    #               hdKind = hdOutline, 
    #               position = "identity",
    #               alpha = 0.5,
    #               density = true) +
    xlab("Eccentricity") + ylab("Density") +   
    geom_density(color = "black", size = 1.0, alpha = 0.7, normalize = true) + 
    ggtitle("Eccentricity of calibration and background data") +
    themeLatex(fWidth = 0.5, width = 600, baseTheme = sideBySide) +
    margin(left = 2.2, right = 5.2) +         
    ggsave("Figs/background/eccentricity_calibration_background.pdf", useTeX = true, standalone = true)

  proc splitPeaks(x: float): string =
    if x >= 2.75 and x <= 3.25:
      "Escapepeak"
    elif x >= 5.55 and x <= 6.25:
      "Photopeak"
    else:
      "Unclear"

  let dfP = dfC
      .mutate(f{float: "Peak" ~ splitPeaks(`energyFromCharge`)})
      .filter(f{string: `Peak` != "Unclear"},
              f{`eccentricity` <= 2.0})
  ggplot(dfP, aes("eccentricity", fill = "Peak")) +
    #geom_histogram(bins = 50,
    #               hdKind = hdOutline, 
    #               position = "identity",
    #               alpha = 0.5,
    #               density = true) +
    xlab("Eccentricity") + ylab("Density") + 
    geom_density(color = "black", size = 1.0, alpha = 0.7, normalize = true) +
    ggtitle(r"$^{55}\text{Fe}$ photopeak (5.9 keV) and escapepeak (3 keV)") +
    themeLatex(fWidth = 0.5, width = 600, baseTheme = sideBySide) +
    margin(left = 2.2, right = 5.2) +     
    ggsave("Figs/background/eccentricity_photo_escape_peak.pdf", useTeX = true, standalone = true)
    
when isMainModule:
  import cligen
  dispatch main

yielding and

The CAST detector lab provides a vacuum test stand, which contains an X-ray tube. A Micromegas-like detector can easily be mounted to the rear end of the test stand. An X-ray tube uses a filament to produce free electrons, which are then accelerated with a high voltage of the order of multiple \(\si{kV}\). The main part of the setup is a rotateable wheel inside the vacuum chamber, which contains a set of 18 different positions with 8 different target materials, as seen in tab. 17. The highly energetic electrons interacting with the target material generate a continuous Bremsstrahlung spectrum with characteristic lines depending on the target. A second rotateable wheel contains a set of 11 different filters, see tab. 18. These can be used to filter out undesired parts of the generated spectrum by choosing a filter that is opaque in the those energy ranges.

As mentioned previously in sec. 10.1, the detector was dismounted from CAST in Feb. 2019 and installed in the CAST detector lab on <2019-02-14 Thu>. The week from <2019-02-15 Fri> to <2019-02-21 Thu> X-ray tube data was taken in the CDL.

Fig. 2(a) shows the whole vacuum test stand, which contains the X-ray tube on the front left end and the Septemboard detector installed on the rear right, visible by the red HV cables and yellow HDMI cable. In fig. 2(b) we see the whole detector part from above, with the Septemboard detector installed to the vacuum test stand on the left side. The water cooling system is seen in the bottom right, with the power supply above that. The copper shielded box slightly right of the center is the Ortec pre-amplifier of the FADC, which is connected via a heavily shielded LEMO cable to the Septemboard detector. This cable was available in the CAST detector lab and proved invaluable for the measurements as it essentially removed any noise visible in the FADC signals (compared to the significant noise issues encountered at CAST, sec. 10.5.1). Given the activation threshold of the FADC with CAST amplifier settings (see sec. 11.4.2) at around \(\SI{2}{keV}\), the amplification needed to be adjusted in the CDL on a per-target basis. The shielded cable allowed the FADC to even act as a trigger for \(\SI{277}{eV}\) \(\ce{C}_{Kα}\) X-rays without any noise problems. ²

The measurements were performed with 8 different target and filter combinations, with a higher density towards lower energies due to the nature of the expected solar axion flux. For each target and filter at least two runs were measured, one with and one without using the FADC as a trigger. The latter was taken to collect more statistics in a shorter amount of time, as the FADC readout slows down the data taking speed due to increased dead time. Table 19 provides an overview of all data taking runs, the target, filter and HV setting, the main X-ray fluorescence line targeted and its energy. Finally, the mean position of the main fluorescence line in the charge spectrum and its width as determined by a fit is shown. As can be seen the position moves in some cases significantly between different runs, for example in case of the \(\ce{Cu}-\ce{Ni}\) measurements. Also within a single run some stronger variation can be seen, evident by the much larger line width for example in run \(\num{320}\) compared to \(\num{319}\). See appendix sec. 26.4 for all measured spectra (pixel and charge spectra), where each plot contains all runs for a target / filter combination. For example fig. #fig:appendix:cdl_charge_Cu-Ni-15kV_by_run shows runs 319, 320 and 345 of the \(\ce{Cu}-\ce{Ni}\) measurements at \(\SI{15}{kV}\) with the strong variation in charge position of the peak visible between the runs.

The variability both between runs for the same target and filter as well as within a run shows the detector was undergoing gas gain changes similar to the variations at CAST (sec. 11.2). With an understanding that the variation is correlated to the temperature this can be explained. The small laboratory underwent significant temperature changes due to the presence of 3 people, all running machines and freely opening and closing windows, in particular due to very warm temperatures relative to a typical February in Geneva. ³ Fig. 3 shows the calculated gas gains (based on \(\SI{90}{min}\) intervals) for each run colored by the target/filter combination. It undergoes a strong, almost exponential change over the course of the data taking campaign. Because of this variation between runs, each run is treated fully separately in contrast to (Christoph Krieger 2018), where all runs for one target / filter combination were combined.

Fortunately, this significant change of gas gain does not have an impact on the distributions of the cluster properties. See appendix 27.1 for comparisons of the cluster properties of the different runs (and thus different gas gains) for each target and filter combination.

As the main purpose is to use the CDL data to generate reference distributions for certain cluster properties, the relevant clusters that correspond to known X-ray energies must be extracted from the data. This is done in two different ways:

1. A set of fixed cuts (one set for each target/filter combination) is applied to each run, as presented in tab. 20. This is the same set as used in (Christoph Krieger 2018). Its main purpose is to remove events with multiple clusters and potential background contributions.
2. By cutting around the main fluorescence line in the charge spectrum in a \(3σ\) region, for which the spectrum needs to be fitted with the expected lines, see sec. 12.2.3. This is done on a run-by-run basis.

The remaining data after both sets of cuts can then be combined for each target/filter combination to make up the distributions for the cluster properties as needed for the likelihood method.

For a reference of the X-ray fluorescence lines (for more exact values and \(\alpha_1\), \(\alpha_2\) values etc.) see tab. 1. The used EPIC filter refers to a filter developed for the EPIC camera of the XMM-Newton telescope. It is a bilayer of \(\SI{1600}{\angstrom}\) polyimide and \(\SI{800}{\angstrom}\) aluminium. For more information about the EPIC filters see references (Strüder et al. 2001; Turner, M. J. L. et al. 2001; Barbera et al. 2003, 2016), in particular (Barbera et al. 2016) for an overview of the materials and production.

To the spectra of each run of the charge data a mixture of Gaussian functions is fitted. ⁴

Specifically the Gaussian expressed as

is used and will be referenced as \(G\) with possible arguments from here on. Note that while the physical X-ray transition lines are Lorentzian shaped (Weisskopf and Wigner 1997), the lines as detected by a gaseous detector are entirely dominated by detector resolution, resulting in Gaussian lines. For other types of detectors used in X-ray fluorescence (XRF) analysis, convolutions of Lorentzian and Gaussian distributions are used (Huang and Lim 1986; Heckel and Scholz 1987), called pseudo Voigt functions (Roberts, Riddle, and Squier 1975; Gunnink 1977).

The functions fitted to the different spectra then depend on which fluorescence lines are visible. The full list of all combinations is shown in tab. 22. Typically each line that is expected from the choice of target, filter and chosen voltage is fitted, if it can be visually identified in the data. ⁵ If no 'argument' is shown in the table to \(G\) it means each parameter (\(N, μ, σ\)) is fitted. Any specific argument given implies that parameter is fixed relative to another parameter. For example \(μ^{\ce{Ag}}_{Lα}·\left(\frac{3.151}{2.984}\right)\) fixes the \(Lβ\) line of silver to the fit parameter \(μ^{\ce{Ag}}_{Lα}\) of the \(Lα\) line with a multiplicative shift based on the relative eneries of \(Lα\) to \(Lβ\). In some cases the amplitude is fixed between different lines where relative amplitudes cannot be easily predicted or determined, e.g. in one of the \(\ce{Cu}-\text{EPIC}\) runs, the \(\ce{C}_{Kα}\) line is fixed to a tenth of the \(\ce{O}_{Kα}\) line. This is done to get a good fit based on trial and error.

Finally, in both \(\ce{Cu}-\text{EPIC}\) lines two 'unknown' Gaussians are added to cover the behavior of the data at higher charges. The physical origin of this additional contribution is not entirely clear. The used EPIC filter contains an aluminum coating (Barbera et al. 2016). As such it has the aluminum absorption edge at about \(\SI{1.5}{keV}\), possibly matching the additional contribution for the \(\SI{2}{kV}\) dataset. Whether it is from a continuous part of the spectrum or a form of aluminum fluorescence is not clear however. This explanation does not work in the \(\SI{0.9}{kV}\) case, which is why the line is deemed 'unknown'. It may also be a contribution of the specific polyimide used in the EPIC filter (Barbera et al. 2016). Another possibility is it is a case of multi-cluster events, which are too close to be split, but with properties close enough to a single X-ray as to not be removed by the cleaning cuts (which gets more likely the lower the energy is).

The full set of all fits (including the pixel spectra) is shown in appendix 26.4. Fig. 5 shows the charge spectrum of the \(\ce{Ti}\) target and \(\ce{Ti}\) filter at \(\SI{9}{kV}\) for one of the runs. These plots show the raw data in the green histogram and the data left after application of the cleaning cuts (tab. 20) in the purple histogram. The black line is the result of the fit as described in tab. 22 with the resulting parameters shown in the box (parameters that were fixed are not shown). The black straight lines with grey error bands represent the \(3σ\) region around the main fluorescence line, which is used to extract those clusters likely from the fluorescence line and therefore known energy.

\footnotesize

\normalsize

With the fits to the charge spectra performed on a run-by-run basis, they can be utilized to calibrate the energy of each cluster in the data. ⁶ This is done by using the linear relationship between charge and energy and therefore using the charge of the main fluorescence line as computed from the fit. Each run is therefore self-calibrated (in contrast to our normal energy calibration approach 11.1). Fig. 6 shows normalized histograms of all CDL data after applying basic cuts and performing said energy calibrations.

Having performed fits to all charge spectra of each run and the position of the main fluorescence line and its width determined, the reference distributions for the cluster properties entering the likelihood can be computed. By taking all clusters within the \(3σ\) bounds around the main fluorescence line of the data used for the fit, the dataset is selected. This guarantees to leave mainly X-rays of the targeted energy for each line in the dataset. As the fit is performed for each run separately, the \(3σ\) charge cut is performed run by run and then all data combined for each fluorescence line (target, filter & HV setting).

The desired reference distributions then are simply the normalized histograms of the clusters in each of the properties. With 8 targeted fluorescence lines and 3 properties this yields a total of 24 reference distributions. Each histogram is then interpreted as a probability density function (PDF) for clusters 'matching' (more on this in sec. 12.2.6) the energy of its fluorescence line. An overview of all the reference distributions is shown in fig. 7. We can see that all distributions tend to get wider towards lower energies (towards the top of the plot). This is expected and due to smaller clusters having less primary electrons and therefore statistical variations in geometric properties playing a more important role. ⁷ The binning shown in the figure is the exact binning used to define the PDF. In case of the fraction of pixels within a transverse RMS radius, bins with significantly higher counts are observed at low energies. This is not a binning artifact, but a result of the definition of the variable. The property computes the fraction of pixels that lie within a circle of the radius corresponding to the transverse RMS of the cluster around the cluster center (see fig. #fig:reco:property_explanations). At energies with few pixels in total, the integer nature of \(N\) or \(N+1\) primary electrons (active pixels) inside the radius becomes apparent.

The binning in the histograms is chosen by hand such that the binning is as fine as possible without leading to significant statistical fluctuations, as those would have a direct effect on the PDFs leading to unphysical effects on the probabilities. Ideally an approach of either an automatic bin selection algorithm or something like a kernel density estimation should be used. However, the latter is slightly problematic due to the integer effects in the low energies of the fraction in transverse RMS variable.

The summarized 'recipe' of the approach is therefore:

1. apply cleaning cuts according to tab. 20,
2. perform fits according to tab. 22,
3. cut to the \(3σ\) region around the main fluorescence line of the performed fit (i.e. the first term in tab. 22),
4. combine all remaining clusters for the same fluorescence line from each run,
5. compute a histogram for each desired cluster property and each fluorescence line,
6. normalize the histogram to define the reference distribution, \(\mathcal{P}_i\)

With our reference distributions defined it is time to look back at the equation for the definition of the likelihood, eq. \eqref{eq:background:likelihood_def}.

Note: To avoid numerical issues dealing with very small probabilities, the actual relation evaluated numerically is the negative log of the likelihood:

\[ -\ln \mathcal{L}(ε, f, l) = - \ln \mathcal{P}_ε(ε) - \ln \mathcal{P}_f(f) - \ln \mathcal{P}_l(l). \]

By considering a single fluorescence line, the three reference distributions \(\mathcal{P}_i(i)\) make up the likelihood \(\mathcal{L}(ε, l, f)\) function for that energy; a function of the variables \(ε, f, l\). In order to use the likelihood function as a classifier for X-ray like clusters we need a one dimensional expression. This is where the likelihood distribution \(\mathfrak{L}\) comes in. We reuse all the clusters used to define the reference distributions \(\mathcal{P}_{ε,f,l}\) and compute their likelihood values \(\{ \mathcal{L}_j \}\), where \(j\) is the index of the \(j\text{-th}\) cluster. By then computing the histogram of the set of all these likelihood values, we obtain the likelihood distribution

\[ \mathfrak{L} = \text{histogram}\left( \{ \mathcal{L}_j \} \right). \]

All these distributions are shown in fig. 8 as negative log likelihood distributions. ⁸ We see that the distributions change slightly in shape and move towards larger \(-\ln \mathcal{L}\) values towards the lower energy target/filter combinations. The shape change is most notably due to the significant integer nature of the 'fraction in transverse RMS' \(f\) variable, as seen in fig. 7. The shift to larger values expresses that the reference distributions \(\mathcal{P}_i\) become wider and thus each bin has lower probability.

To finally classify events as signal or background using the likelihood distribution, one sets a desired "software efficiency" \(ε_{\text{eff}}\), which is defined as:

The likelihood value \(\mathcal{L}'\) is the value corresponding to the \(ε_{\text{eff}}^{\text{th}}\) percentile of the likelihood distribution \(\mathfrak{L}\). In practical terms one computes the normalized cumulative sum of the log likelihood and searches for the point at which the desired \(ε_{\text{eff}}\) is reached. The typical software efficiency we aim for is \(\SI{80}{\percent}\). Classification as X-ray-like then is simply any cluster with a \(-\ln\mathcal{L}\) value smaller than the cut value \(-\ln\mathcal{L}'\). Note that this value \(\mathcal{L}'\) has to be determined for each likelihood distribution.

To summarize the derivation of the likelihood distribution \(\mathfrak{L}\) and its usage as a classifier as a 'recipe':

1. compute the reference distributions \(\mathcal{P}_i\) as described in sec. 12.2.5,
2. take the raw cluster data (unbinned data!) of those clusters that define the \(\mathcal{P}_i\) and feed each of these into eq. \eqref{eq:background:likelihood_def} for a single likelihood value \(\mathcal{L}_i\) each,
3. compute the histogram of the set of all these likelihood values \(\{ \mathcal{L}_i \}\) to define the likelihood distribution \(\mathfrak{L}\).
4. define a desired 'software efficiency' \(ε_{\text{eff}}\) and compute the corresponding likelihood value \(\mathcal{L}_c\) using eq. \eqref{eq:background:lnL:cut_condition},
5. any cluster with \(\mathcal{L}_i \leq \mathcal{L}_c\) is considered X-ray-like with efficiency \(ε_{\text{eff}}\).

Note that due to the usage of negative log likelihoods the raw data often contains infinities, which are just a side effect of picking up a zero probability from one of the reference distributions for a particular cluster. In reality the reference distributions should be a continuous distribution that is nowhere exactly zero. However, due to limited statistics there is a small range of non-zero probabilities (most bins are empty outside the main range). For all practical purposes this does not matter, but it does explain the rather hard cutoff from 'sensible' likelihood values to infinities in the raw data.

For the GridPix detector used in 2014/15, similar X-ray tube data was taken and each of the 8 X-ray tube energies were assigned an energy interval. The likelihood distribution to use for each cluster was chosen based on which energy interval the cluster's energy falls into. This leads to discontinuities of the properties at the interval boundaries due to the energy dependence of the cluster properties as seen in fig. 1(b). This change is of course continuous instead of discrete. It can then lead to jumps in the efficiency of the background suppression method and thus in the achieved background rate. It seems a safe assumption that the reference distributions undergo a continuous change for changing energies of the X-rays. Therefore, to avoid discontinuities we perform a linear interpolation for each cluster with energy \(E_β\) between the closest two neighboring X-ray tube energies \(E_α\) and \(E_γ\) in each probability density \(\mathcal{P}_i\) at the cluster's properties. With \(ΔE = E_α - E_γ\) the difference in energy between the closest two X-ray tube energies, each probability density is then interpolated to:

\[ \mathcal{P}_i(E_β, x_i) = \left(1 - \frac{|E_β - E_α|}{ΔE}\right) · \mathcal{P}_i(E_α, x_i) + \left( 1 - \frac{|E_γ - E_β|}{ΔE} \right) · \mathcal{P}_i(E_γ, x_i) . \]

Each probability density of the closest neighbors is evaluated at the cluster's property \(x_i\) and the linear interpolation weighted by the distance to each energy is computed.

The choice for a linear interpolation was made after different ideas were tried. Most importantly, a linear interpolation does not yield unphysical results (for example negative bin counts in interpolated data, which can happen in a spline interpolation) and yields very good results in the cases that can be tested, namely reconstructing a known likelihood distribution \(B\) by doing a linear interpolation between the two outer neighbors \(A\) and \(C\). 9(a) shows this idea using the fraction in transverse RMS variable. This variable is shown here as it has the strongest obvious shape differences going from line to line. The green histogram corresponds to interpolated bins based on the difference in energy between the line above and below the target (hence it is not done for the outer lines, as there is no 'partner' above / below). Despite the interpolation covering an energy range almost twice as large as used in practice, over- or underestimation of the interpolation (green) is minimal. To give an idea of what this results in for the interpolation, fig. 9(b) shows a heatmap of the same variable showing how the interpolation describes the energy and fraction in transverse RMS space continuously.

Given that such an interpolation works as well as it does on recovering a known line, implies that a linear interpolation on 'half' ⁹ the energy interval in practice should yield reasonably realistic distributions, certainly much better than allowing a discrete jump at specific boundaries.

See appendix 28 for a lot more information about the ideas considered and comparisons to the approach without interpolation. In particular fig. 140, which computes the \(\ln\mathcal{L}\) values for all of the (cleaned, cuts tab. 20 applied) CDL data comparing the case of no interpolation with interpolation showing a clear improvement in the smoothness of the point cloud.

The full study and notes about how we ended up at using a linear interpolation, can be found in appendix 28.

• [X] This is now in the appendix. Good enough? Link

Put here all our studies of how and why we ended up at linear interpolation!

The actual linear interpolation results (i.e. reproducing the "middle" one will be shown in the actual thesis).

On a slight tangent, with the main fluorescence lines fitted in all the charge spectra, the position and line width can be used to compute the energy resolution of the detector:

\[ ΔE = \frac{σ}{μ} \]

where \(σ\) is a measure of the line width and \(μ\) the position (in this case in total cluster charge). Note that in some cases the full width at half maximum (FWHM) is used and in others the standard deviation (for a normal distribution the FWHM is about \(2.35 σ\)). The energy resolutions obtained by our detector in the CDL dataset is seen in fig. 10. At the lowest energies below \(\SI{1}{keV}\) the resolution is about \(ΔE \approx \SI{30}{\%}\) and generally between \(\SIrange{10}{15}{\%}\) from \(\SI{2}{keV}\) on.

• [ ] NOTE: THE PLOT WE CURRENTLY GENERATE: DOES IT USE TRACKING INFO OR NOT?
• [ ] EXPLAIN AND SHOW HOW TO INSERT TRACKING INFO

./cast_log_reader tracking \
                  -p ../resources/LogFiles/tracking-logs \
                  --startTime 2018/05/01 \
                  --endTime 2018/12/31 \
                  --h5out ~/CastData/data/DataRuns2018_Reco.h5 \
                  --dryRun

With the dryRun option you are only presented with what would be written. Run without to actually add the data.

To generate the background rate plot we need:

1. the reconstructed data files for the background data at CAST
   • (optional for this part): the slow control and tracking logs of CAST
   • the tracking information added to the reconstructed background data files to compute the background rate only for tracking or only for signal candidate data, done by running the cast_log_reader with the reconstructed data files as input
2. the reconstructed CDL data and the calibration-cdl-2018.h5 file generated from it for the reference and likelihood distributions

To compute the background rate using these inputs we use the likelihood tool first in order to apply the likelihood cut method according to a desired software efficiency defined in the config.toml file. Afterwards we can use another tool to generate the plot of the background rate (which takes care of scaling the data to a rate etc.).

The relevant section of the config.toml file should look like this:

[Likelihood]
# the signal efficiency to be used for the logL cut (percentage of X-rays of the
# reference distributions that will be recovered with the corresponding cut value)
signalEfficiency = 0.8
# the CDL morphing technique to be used (see `MorphingKind` enum), none or linear
morphingKind = "Linear"
# clustering algorithm for septem veto
clusterAlgo = "dbscan" # choose from {"default", "dbscan"}
# the search radius for the cluster finding algorithm in pixel
searchRadius = 50 # for default clustering algorithm
epsilon = 65 # for DBSCAN algorithm

[CDL]
# whether to fit the CDL spectra by run or by target/filter combination.
# If `true` the resulting `calibration-cdl*.h5` file will contain sub groups
# for each run in each target/filter combination group!
fitByRun = true

(linear morphing, 80% efficiency, and CDL based on fits per run)

In total we'll want the following files:

• for years 2017 & 2018:
  • chip region crAll & crGold:
    • no vetoes
    • scinti veto
    • FADC veto
    • septem veto
    • line veto
• [X] USE createAllLikelihoodCombinations for it after short rewrite

In order to generate all likelihood output files (after having computed the likelihood datasets in the data files and added the tracking information!) we will use the createAllLikelihoodCombinations tool (rename please)

• [ ] INSERT TRACKING INFORMATION AND ADD THE --tracking FLAG TO LIKELIHOOD
• [ ] RERUN THE BELOW AND CHANGE PATHS TO DIRECTLY IN PHD DIRECTORY!

• UPDATE: The current files we will likely use are generated by:

All standard variants including the different FADC percentiles:

./createAllLikelihoodCombinations \
    --f2017 ~/CastData/data/DataRuns2017_Reco.h5 \
    --f2018 ~/CastData/data/DataRuns2018_Reco.h5 \
    --c2017 ~/CastData/data/CalibrationRuns2017_Reco.h5 \
    --c2018 ~/CastData/data/CalibrationRuns2018_Reco.h5 \
    --regions crGold --regions crAll \
    --vetoSets "{fkScinti, fkFadc, fkSeptem, fkLineVeto, fkExclusiveLineVeto}" \
    --fadcVetoPercentiles 0.9 --fadcVetoPercentiles 0.95 --fadcVetoPercentiles 0.99 \
    --out /t/lhood_outputs_adaptive_fadc \
    --cdlFile ~/CastData/data/CDL_2019/calibration-cdl-2018.h5 \
    --multiprocessing

The septem + line variants without the FADC:

./createAllLikelihoodCombinations \
    --f2017 ~/CastData/data/DataRuns2017_Reco.h5 \
    --f2018 ~/CastData/data/DataRuns2018_Reco.h5 \
    --c2017 ~/CastData/data/CalibrationRuns2017_Reco.h5 \
    --c2018 ~/CastData/data/CalibrationRuns2018_Reco.h5 \
    --regions crGold --regions crAll \
    --vetoSets "{+fkScinti, fkSeptem, fkLineVeto, fkExclusiveLineVeto}" \
    --out /t/lhood_outputs_adaptive_fadc \
    --cdlFile ~/CastData/data/CDL_2019/calibration-cdl-2018.h5 \
    --multiprocessing

and the lnL cut efficiency variants for 70 and 90% for the septem + line + FADC@90% variants:

./createAllLikelihoodCombinations \
    --f2017 ~/CastData/data/DataRuns2017_Reco.h5 \
    --f2018 ~/CastData/data/DataRuns2018_Reco.h5 \
    --c2017 ~/CastData/data/CalibrationRuns2017_Reco.h5 \
    --c2018 ~/CastData/data/CalibrationRuns2018_Reco.h5 \
    --regions crGold --regions crAll \
    --signalEfficiency 0.7 --signalEfficiency 0.9 \
    --vetoSets "{+fkScinti, +fkFadc, +fkSeptem, fkLineVeto}" \
    --fadcVetoPercentile 0.9 \
    --out /t/lhood_outputs_adaptive_fadc \
    --cdlFile ~/CastData/data/CDL_2019/calibration-cdl-2018.h5 \
    --multiprocessing \
    --dryRun

which for the time being are here: ./../org/resources/lhood_limits_automation_correct_duration/

cd $TPA/Analysis
./createAllLikelihoodCombinations --f2017 ~/CastData/data/DataRuns2017_Reco.h5 \
                                  --f2018 ~/CastData/data/DataRuns2018_Reco.h5 \
                                  --c2017 ~/CastData/data/CalibrationRuns2017_Reco.h5 \
                                  --c2018 ~/CastData/data/CalibrationRuns2018_Reco.h5 \
                                  --regions crGold \
                                  --regions crAll \
                                  --vetoes fkNoVeto \
                                  --vetoes fkScinti \
                                  --vetoes fkFadc \
                                  --vetoes fkSeptem \
                                  --vetoes fkLineVeto \
                                  --vetoes fkExclusiveLineVeto \
                                  --out ~/phd/resources/background/autoGen \
                                  --cdlFile ~/CastData/data/CDL_2019/calibration-cdl-2018.h5 \
                                  --multiprocessing \
                                  --dryRun 

using the --dryRun option first to see the generated commands that will be run.

We will run this over night <2023-02-13 Mon 00:37> now using the new default eccentricity cutoff for the line veto of 1.5 (motivated by the ratio of fraction passing real over fake!) as well as the lvRegularNoHLC line veto kind (although this should not matter, as we won't use the line veto without the septem veto!). The files are now in ./resources/background/autoGen/.

• [ ] WARNING: THE GENERATED FILES HERE STILL USE OUR HACKED IN CHANGED NUMBER OF BINS FOR THE REFERENCE DISTRIBUTIONS! -> TWICE AS MANY BINS!

• [ ] THESE ARE OUTDATED. WE USE OUR TOOL

First let's apply the likelihood tool to the 2017 data for the gold region:

likelihood -f ~/CastData/data/DataRuns2017_Reco.h5 --computeLogL \
           --region crGold \
           --cdlYear 2018 \
           --h5out /home/basti/phd/resources/background/lhood_2017_crGold_80eff_no_vetoes.h5 \
           --cdlFile ~/CastData/data/CDL_2019/calibration-cdl-2018.h5

and the same for the end of 2018 data:

likelihood -f ~/CastData/data/DataRuns2018_Reco.h5 --computeLogL \
           --region crGold \
           --cdlYear 2018 \
           --h5out /home/basti/phd/resources/background/lhood_2018_crGold_80eff_no_vetoes.h5 \
           --cdlFile ~/CastData/data/CDL_2019/calibration-cdl-2018.h5

and now the same for the whole chip:

likelihood -f ~/CastData/data/DataRuns2017_Reco.h5 --computeLogL \
           --region crAll \
           --cdlYear 2018 \
           --h5out /home/basti/phd/resources/background/lhood_2017_crAll_80eff_no_vetoes.h5 \
           --cdlFile ~/CastData/data/CDL_2019/calibration-cdl-2018.h5

and the same for the end of 2018 data:

likelihood -f ~/CastData/data/DataRuns2018_Reco.h5 --computeLogL \
           --region crAll \
           --cdlYear 2018 \
           --h5out /home/basti/phd/resources/background/lhood_2018_crAll_80eff_no_vetoes.h5 \
           --cdlFile ~/CastData/data/CDL_2019/calibration-cdl-2018.h5

With this we can first create the plot of the cluster centers from the all chip files using plotClusterCenters:

plotBackgroundClusters \
    ~/org/resources/lhood_lnL_17_11_23_septem_fixed/lhood_c18_R2_crAll_sEff_0.8_lnL.h5 \
    ~/org/resources/lhood_lnL_17_11_23_septem_fixed/lhood_c18_R3_crAll_sEff_0.8_lnL.h5 \
    --zMax 30 \
    --title "X-ray like clusters of CAST data" \
    --outpath ~/phd/Figs/backgroundClusters/ \
    --showGoldRegion \
    --backgroundSuppression \
    --energyMin 0.2 --energyMax 12.0 \
    --suffix "_lnL80_no_vetoes" \
    --filterNoisyPixels \
    --useTikZ

From the logL output files created by createAllLikelihoodCombinations, let's now create the 'classical' background rate in the gold region without any vetoes:

plotBackgroundRate \
    ~/org/resources/lhood_lnL_17_11_23_septem_fixed/lhood_c18_R2_crAll_sEff_0.8_lnL.h5 \
    ~/org/resources/lhood_lnL_17_11_23_septem_fixed/lhood_c18_R3_crAll_sEff_0.8_lnL.h5 \
    --combName 2017/18 \
    --combYear 2017 \
    --centerChip 3 \
    --region crGold \
    --title "Background rate from CAST data" \
    --showNumClusters \
    --showTotalTime \
    --topMargin 1.5 \
    --energyDset energyFromCharge \
    --outfile background_rate_crGold_no_vetoes.pdf \
    --outpath ~/phd/Figs/background/ \
    --useTeX \
    --quiet

which generates ./ Figs/background/background_rate_crGold_no_vetoes.pdf from the TikZ .tex file of the same name with the following output about the background rates.

• [ ] FINISH THIS

In tab. 14 we see that the total CAST data contains \(\num{984319} + \num{470188} = \num{1454507}\) events. Compared to our \(\num{94625}\) clusters left on the whole chip, this represents a:

let c17 = 984319'f64
let c18 = 470188'f64
let cCut = 94625'f64
echo (c17 + c18) / cCut

background suppression of about 15.3.

• [X] We could make a plot showing the background suppression? A tile map with text on taking X times Y squares and printing the suppression? Would be a nice way to give better understanding how the vetoes help to improve things over the chip.

Running plotBackgroundClusters now also produces a tile map of the background suppression over the total raw number of clusters on the chip! plots/background_suppression_tile_map.pdf which is in .

• [ ] WRITE ME -> Important because systematics.
• [ ] MAYBE INSTEAD OF EXTENDED, APPENDIX

• [ ] SHOW OUR VERIFICATION USING 55FE DATA -> Check and if not possibly do: Using fake energies (by leaving out pixel info) to generate arbitrary energies check the efficiency. I think this is what we did in that context. Right, that was it: ./../CastData/ExternCode/TimepixAnalysis/Tools/determineEffectiveEfficiency.nim -> This is "outdated" in the sense that we have good fake generation now. Probably good enough to just discuss efficiency that we use for systematic!
• [ ] CREATE PLOT OF THE LOGL CUT VALUES TO ACHIEVE THE USED EFFICIENCY! -> essentially take our morphed likelihood distributions and for each of those calculate the logL value for the cut value. Then create a plot of energy vs. cut value and show that. Should be a continuous function through the logL values that fixes the desired efficiency.
• [ ] THINK ABOUT OPTIMIZING EFFICIENCY -> We can think about writing a short script to optimize the efficiency
• [ ] AFAIR WE STILL COMPUTE LOGL CUT VALUE BASED ON BINNED HISTOGRAM. CHANGE THAT TO UNBINNED DATA

The simplest approach to using a neural network for data classification of CAST-like data, is an MLP that uses the pre-computed geometric properties for each cluster as an input.

The choice remains between a single or two output neurons. The more classical approach is treating signal (X-rays) and background events as two different classes that the classifier learns to predict, which we also use. By our convention the target outputs for signal and background events are

where the first entry of each \(\mathbf{\hat{y}}\) corresponds to output neuron 1 and the second to output neuron 2.

For the network to generalize to all real CAST data, the training dataset must be representative of the wide variety of the real data. For background-like clusters it can be sourced from the extensive non-tracking dataset recorded at CAST. For the signal-like X-ray data it is more problematic. The only source of X-rays with enough statistics are the \(\cefe\) calibration datasets from CAST, but these only describe X-rays of two energies. The CAST detector lab data from the X-ray tube are both limited in statistics as well as suffering from systematic differences to the CAST data due to different gas gains. ¹⁴

We will now describe how we generate X-ray clusters for MLP training by simulating them from a target energy, a specific transverse diffusion and a desired gas gain.

To generate simulated events we wish to use the least amount of inputs, which still yield events that represent the recorded data as well as possible. In particular, the systematic variations between different times and datasets should be reproducible based on these parameters. The idea is to generate events using the underlying gaseous detector physics (see sec. 6) and make as few heuristic modifications to better match the observed (imperfect) data as possible. This lead to an algorithm, which only uses three parameters: a target energy for the event (within the typical energy resolution). A gas gain to encode the gain variation seen. A gas diffusion coefficient to encode variations in the diffusion properties (also possibly a result of changing temperatures).

In contrast to approaches that simulate the interactions at the particle level, our simulation is based entirely on the emergent properties seen as a result of those. The basic Monte Carlo algorithm will now be described in a series of steps.

1. (optional) sample from different fluorescence lines for an element given their relative intensities to define a target energy \(E\).
2. sample from the exponential distribution of the absorption length (see sec. 6.1.1) for the used gas mixture and target photon energy to get the conversion point of the X-ray in the gas (Note: we only sample X-rays that convert; those that would traverse the whole chamber without conversion are ignored).
3. sample a target charge to achieve for the cluster, based on target energy by sampling from a normal distribution with a mean roughly matching detector energy resolution (target energy inverted to a charge).
4. sample center position of the cluster within a uniform radius of \(\SI{4.5}{mm}\) around the chip center.
5. begin the sampling of each electron in the cluster.
6. sample a charge (in number of electrons after amplification) for the electron based on a Pòlya distribution of input gas gain (and matching normalization & \(θ\) constant). Reject the electron if not crossing activation threshold (based on real data).
7. sample a radial distance from the cluster center based on gas diffusion constant \(D_T\) and diffusion after the remaining drift distance to the readout plane \(z_{\text{drift}}\) using eq. \eqref{eq:gas_physics:diffusion_after_drift}. Sample twice from the 1D version using a normal distribution \(\mathcal{N}(μ = 0, σ = D_T · \sqrt{z_{\text{drift}}})\) for each dimension. Combine to a radius from center, \(r = \sqrt{x² + y²}\).
8. sample a random angle, uniform in \((0, 2π]\).
9. convert radius and angle to an \((x, y)\) position of the electron, add to the cluster.
10. based on a linear approximation activate between 0 to 4 neighboring pixels with a slightly reduced gas gain. We never activate any neighbors at a charge of \(\SI{1000}{e⁻}\) and always activate at least one at \(\SI{10000}{e⁻}\). Number of activated pixels depends on a uniform random number being below one of 4 different thresholds: \[ N_{\text{neighbor}} = \text{rand}(0, 1) · N < \text{threshold} \] where the threshold is determined by the linear function described by the mentioned condition and \(\text{rand}(0, 1)\) is a uniform random number in \((0, 1)\).
11. continue sampling electrons until the total charge adds up to the target charge. We stop at the value closest to the target (before or after adding a final pixel that crosses the target charge) to avoid biasing us towards values always larger than the target.
12. The final cluster is reconstructed just like any other real cluster, using the same calibration functions as the real chip (depending on which dataset it should correspond to).

Note: Neighboring pixels are added to achieve matching eccentricity distributions between real data and simulated. Activating neighboring pixels increases the local pixel density randomly, which effectively increases the weight of some pixels, leading to a slight increase of eccentricity of a cluster. The approach of activating up to 4 neighbor pixels and giving them slightly lower charges stems from empirically matching the simulated data to real data. From a physical perspective the most likely cause of neighboring pixels is due to UV photons that are emitted in the amplification region and travel towards the grid and produce a new electron, which starts an avalanche from there. From that point of view neighboring pixels should see the full gas amplification and neighbors other than the direct { up, down, left, right } neighbors can be activated (depending on an exponential distribution due to possible absorption of the UV photons in the gas). See Markus Gruber's master thesis (Gruber 2018) for a related study on this for GridPix detectors.

Based on the above algorithm, fake events can be generated that either match the gas gain and gas diffusion of an existing data taking run (background or calibration) or any arbitrary combination of parameters. The former is important for verification of the validity of the generated events as well as to check the MLP cut efficiency (more on that in sec. 12.4.6). The latter is used to generate a wide variety of MLP training data. Event production is a very fast process ¹⁵, allowing to produce large amounts of statistics for MLP training in a reasonable time frame.

For other applications that require higher fidelity simulations, Degrad (Biagi 1995a), a sister program to Magboltz (Biagi 1995b), can be used, which simulates the particle interactions directly. This is utilized in a code by M. Gruber (Gruber 2023), which uses Degrad and Garfield++ (“Garfield++,” n.d.) to perform realistic event simulation for GridPix detectors (its data can also be analyzed by the software presented in sec. 9.1 and appendix 34).

To generate X-ray events with similar physical properties as those seen in a particular data taking run, we need the three inputs required: gas gain, gas diffusion and target energy. The determination of the gas gain is a well defined procedure, explained in sections 8.1.2 and 11.2.3. The target energy is a matter of the purpose the generated data is for. The gas diffusion however is more complicated and requires explanation.

In theory the gas diffusion is a fixed parameter for a specific gas mixture at fixed temperature, pressure and electromagnetic fields and can be computed using Monte Carlo tools as mentioned already in sec. 6.3.4. However, in practice MC tools suffer from significant uncertainty (in particular in the form of run-by-run RNG variation) especially at common numbers of MC samples. More importantly though, real detectors do not have perfectly stable and known temperatures, pressures and electromagnetic fields (for example the field inhomogeneity of the drift field in the 7-GridPix detector is not perfectly known), leading to important differences and variations from theoretical numbers.

Fortunately, similar to the gas gain, the effective numbers for the gas diffusion can be extracted from real data. The gas diffusion constant \(D_t(z)\) (introduced in sec. 6.3.4) describes the standard deviation of the transverse distance from the initial center after drifting a distance \(z\) along a homogeneous electric field. This parameter is one of our geometric properties, in the form of the transverse RMS '\rmst' (which is actually the transverse standard deviation of the electron positions ¹⁶) and thus it can be used to determine the gas diffusion coefficient. ¹⁷ As it is a statistical process and \(D_t\) is the standard deviation of the population a large ensemble of events is needed.

This is possible both for calibration data as well as background data. In both cases it is important to apply minor cuts to remove the majority of events in which \rmst contains contributions that are not due to gas diffusion (e.g. a double X-ray event which was not separated by the cluster finder). For calibration data the cuts should filter to single X-rays, whereas for background events clean muon tracks are the target.

In theory one could determine it by way of a fit to the \rmst distribution of all clusters in a data taking run as was done in (Krieger et al. 2018). This approach is problematic in practice due to the large variation in possible X-ray conversion points – and therefore drift distances – as a function of energy. Different energies lead to different convolutions of exponential distributions with the gas diffusion. This means a single fit does not describe the data well in general and the upper limit is not a good estimator for the real gas diffusion coefficient (because it is due to those X-rays converting directly behind the detector window by chance and/or undergoing statistically large amount of diffusion).

Instead of performing an analytical convolution of the exponential distribution and gas diffusion distributions to determine the correct fit, a Monte Carlo approach is used here as well. The idea here is to use a reasonable ¹⁸ gas diffusion constant to generate (simplified ¹⁹) fake events. Compute the \rmst distribution of this fake data and compare to the real \rmst distribution of the target run based on a test statistic. Next one computes the derivative of the test statistic as a form of loss function and adjusts the diffusion coefficient accordingly. Over a number of iterations the distribution of the simulated \rmst distribution will converge to the real \rmst distribution, if the choice of test statistic is suitable. In other words we use gradient descent to find that diffusion constant \(D_T\), which best reproduces the \rmst distribution of our target run.

One important point with regards to computing it for background data is the following: when generating fake events for the background dataset, the equivalent 'conversion point' for muons is a uniform distribution over all distances from the detector window to the readout. For X-rays it is given by the energy dependent attenuation length in the detector gas.

The test statistic chosen in practice for this purpose is the Cramér-von-Mises (CvM) criterion, defined by (Cramér 1928; von Mises 1936):

\[ ω² = ∫_{-∞}^∞ \left[ F_n(x) - F^*(x) \right]² \, \mathrm{d}F^*(x) \]

where \(F_n(x)\) is an empirical distribution function to test for and \(F^*(x)\) a cumulative distribution function to compare with. In case of the two-sample case it can be computed as follows (Anderson 1962):

\[ T = \frac{N M}{N + M} ω² = \frac{U}{N M (N + M)} - \frac{4 M N - 1}{6(M + N)} \]

with \(U\):

\[ U = N \sum_{i=1}^N \left( r_i - i \right)² + M \sum_{j = 1}^M \left( s_j - j \right)² \]

In contrast to – for example – the Kolmogorov-Smirnov test, it includes the entire (E)CDF into the statistic instead of just the largest deviation, which is a useful property to protect against outliers.

The iterative optimization process is therefore

\[ D_{T,i+1} = D_{T,i} - η \frac{∂ f(D_T)}{∂ D_T} \]

where \(f(D_T)\) is the Monte Carlo algorithm, which computes the test statistic for a given \(D_T\) and \(η\) is the step size along the gradient (a 'learning rate'). The derivative is computed using finite differences. ²⁰

Fig. 13 shows the values of the transverse gas diffusion constant \(D_T\) as they have been determined for all runs. The color scale is the value of the Cramér-von-Mises test criterion and indicates a measure of the uncertainty of the obtained parameter. We can see that for the CDL datasets (runs above 305) the uncertainty is larger. This is because reproducing the \rmst distribution for these datasets is more problematic than for the CAST \cefe datasets (due to more data impurity caused by X-ray backgrounds of energies other than the target fluorescence line). Generally we can see variation in the range from about \(\SIrange{620}{660}{μm.cm^{-1/2}}\) for the CAST data and larger variation for the CDL data. The theoretical value we expect is about \(\SI{670}{μm.cm^{-1/2}}\), implying the approach yields reasonable values.

Because this iterative calculation is not computationally negligible, the resulting \(D_T\) parameters for each run are cached in an HDF5 file. ²¹

Fig. 15 shows the comparison of all cluster properties of simulated and real data for one \cefe calibration run, number 241. The data is cut to the photopeak and normalized to fit into a single plot. Each ridge shows a kernel density estimation (KDE) of the data. Gas gain and gas diffusion are extracted from the data as explained in previous sections.

All distributions outside the number of hits agree extremely well. This is expected to not perfectly match. Only the total charge is used as an input to the MLP and it matches pretty well on its own. The reason is the neighboring activation logic in the MC algorithm, which is empirically modified such that it produces correct geometric behavior at the cost of slightly over- or underestimating the number of hits. The Pólya sampling and general neighboring logic is too simplistic to properly reproduce both aspects at the same time. Other runs show similar agreement. Plots for all runs like this can be found in the extended version of the thesis.

As a reference ²² an MLP was trained on a mix of simulated X-rays and a subset of the CAST background data. The implementation was done using Flambeau (SciNim contributors 2023), a wrapper for libtorch (Paszke et al. 2019) ²³, using the parameters and inputs as shown in tab. 24. The network layout consists of \(\num{14}\) input neurons, two hidden layers of only \(\num{30}\) neurons ²⁴ each and \(\num{2}\) neurons on the output layer. We use the Adam (Kingma and Ba 2014) optimizer and \(\tanh\) as an activation function on both hidden layers, with \(\text{sigmoid}\) used on the output layer. The mean squared error between output and target vectors is used as the loss function.

The \(\num{14}\) input neurons were fed with all geometric properties that do not scale directly with the energy (no number of active pixels or direct energy) or position on the chip (as the training data is skewed towards the center) with the exception of the total charge. For the background data we avoid 'contaminating' the training dataset with CAST clusters that will end up as part of our background rate for the limit calculation by only selecting clusters from chips other than the center chip. This way the MLP is trained entirely on data independent of the CAST data of interest. \(\num{250 000}\) synthetic X-rays and \(\num{288 000}\) background clusters are used for training and the same number of validation ²⁵.

The first \(\num{25000}\) epochs are trained on a synthetic X-ray dataset with a strong bias to low energy X-rays (clusters in the range from \(\SIrange{0}{3}{keV}\) with the frequency dropping linearly with increasing energy). This is because these are more difficult to separate from background. Afterwards we switch to a separate set of synthetic X-rays uniformly distributed in energies. Without this approach the network can drift towards a local minimum, in which low energy X-rays are classified as background, while still achieving good accuracy. At the cost of low energy X-rays, low energy background clusters are almost perfectly rejected. This is of course undesirable.

During the training process of \(\num{82 000}\) epochs, every \(\num{1000}\) epochs the accuracy and loss are evaluated based on the test sample and a model checkpoint is stored. The evolution of the loss function for this network is shown in fig. 16(a). We can see that performance for the test dataset is mostly stagnant from around epoch \(\num{10 000}\). The network develops minor overtraining, but this is not important, because no training data is ever used in practice.

Fig. 16(b) shows the output of neuron 0 (neuron 1 is a mirror image) for the test sample, with signal like data towards 1 and background like data towards 0. The separation is almost perfect, but a small amount of each type can be seen over the entire range. This is expected due to the presence of real X-rays in the background dataset, low energy events generally being similar and statistical outliers in the X-ray dataset.

Beyond the MLP prediction to indicate a measure of classification efficiency, we can look at the receiver operating characteristic (ROC) curves for different X-ray reference datasets of the CDL data. For this we read all real center GridPix background data and all CDL data. Both of these are split into the corresponding target/filter combinations based on the energy of each cluster.

We can then define the background rejection \(b_{\text{rej}}\) for different cut values \(c_i\) as

\[ b_{\text{rej}} = \frac{\text{card}(\{ b < c_i\})}{\text{card}(\{ b \})}, \]

where \(b\) are all the predictions of the MLP for background data and \(\text{card}\) is the cardinality of the set of numbers, i.e. the number of entries. \(i\) refers to the \(i^{\text{th}}\) cut value (as this is computed with discrete bins). In addition we define the signal efficiency by

\[ s_{\text{eff}} = \frac{\text{card}(\{ s \geq c_i\})}{\text{card}(\{ s \})}, \]

where similarly \(s\) are the MLP predictions for X-ray clusters.

If we plot the pairs of signal efficiency and background rejection values (one for each cut value), we produce a ROC curve. Done for each CDL target/filter combination and similarly for the likelihood cut by replacing the MLP prediction with the likelihood value of each cluster, we obtain fig. 17. The line style indicates the classifier method (\(\ln\mathcal{L}\) or MLP) and the color corresponds to the different targets and thus different fluorescence lines. The improvement in background rejection at a fixed signal efficiency exceeds \(\SI{10}{\percent}\) at multiple energies. The only region where the MLP is worse than the likelihood cut is for the Cu-EPIC \(\SI{0.9}{kV}\) target/filter combination at signal efficiencies above about \(\SI{92}{\%}\). Especially the shape of the ROC curve for the MLP at high signal efficiencies implies that it should produce comparable background rates to the likelihood cut at higher signal efficiencies.

Based on the output distributions of the MLP, fig. 16(b), it can be used to act as a cluster discriminator in the same way as for the likelihood cut method, following eq. \eqref{eq:background:lnL:cut_condition}. The likelihood distribution is replaced by the MLP prediction via either of the two output neurons. For example, as seen for one output neuron in fig. 16(b), the software efficiency would be determined based on the 'prediction' value along the x-axis such that the desired software efficiency is above the cut value. Any cluster below would be removed. In practice again the empirical distribution function of the signal-like output data is computed and the quantile of the target software efficiency \(ε_{\text{eff}}\) is determined. Similarly to the likelihood cut method this is done for \(\num{8}\) different energy ranges corresponding to the CDL fluorescence lines. However, no interpolation is performed because only the single output distribution is known. ²⁶

Due to the significant differences in gas gain between the CDL dataset and the CAST \cefe calibration data, we do not use a single cut value for each energy for all CAST data taking runs. Instead we use the X-ray cluster simulation machinery to provide run specific X-ray clusters from which to deduce cut values.

For each run we wish to apply the MLP to, we start by computing the mean gas gain based on all gas gain intervals. Then we determine the gas diffusion coefficient as explained in sec. 12.4.3. With two of the three required parameters for X-ray generation in hand, we then simulate X-rays following each X-ray fluorescence line measured in the CDL dataset, yielding 8 different simulated datasets for each run. Based on these we compute one cut value on the MLP output each. The so deduced cut value is applied as the MLP cut to that run in the valid energy range. The same approach is used for calibration runs as well as for background runs.

This means both the MLP training as well as determination of cut values is entirely based on synthetic data, only using aggregate parameters of the real data.

As the simulated X-ray clusters certainly differ from real X-rays, verification of the software efficiency using calibration data is required. For all \cefe calibration runs as well as all CDL data we produce cut values as explained in the previous section. Then we apply these to the real clusters of those runs and compute the effective efficiency as

\[ ε_{\text{effective}} = \frac{N_{\text{cut}}}{N_{\text{total}}} \]

where \(N_{\text{cut}}\) is the clusters remaining after applying the MLP cut. \(N_{\text{total}}\) is the total number of clusters after application of the standard cleaning cuts applied for the \cefe spectrum fit and the CDL fluorescence line fits.

The resulting efficiency is the effective efficiency the MLP produces. A close match with the target software efficiency implies the simulated X-ray data matches the real data well and the network learned to identify X-rays based on physical properties (and not due to overtraining). In the limit calculation later, the mean of all these effective efficiencies of the \cefe calibration runs is used in place of the target software efficiency as a realistic estimator for the signal efficiency. The standard deviation of all these effective efficiencies is one of the systematics used there.

Fig. 18 shows the effective efficiencies obtained for all \cefe calibration and CDL runs using the MLP introduced earlier. The marker symbol represents the energy of the target fluorescence line. Note that the \cefe calibration escape peak near \(\SI{3}{keV}\) is a separate data point "3.0" compared to the silver fluorescence line given as "2.98". That is because the two are fundamentally different things. The escape event is a \(\SI{5.9}{keV}\) photopeak X-ray entering the detector, which ends up 'losing' about \(\SI{3}{keV}\) due to excitation of an argon fluorescence Kα X-ray. This means the absorption length is that of a \(\SI{5.9}{keV}\) photon, whereas the silver fluorescence line corresponds to a real \(\SI{2.98}{keV}\) photon and thus corresponding absorption length. As such, the geometric properties on average are different.

The target efficiency in the plot was \(\SI{95}{\%}\) with an achieved mean effective efficiency close to the target. Values are slightly lower in Run-2 (runs < 200) and slightly above in Run-3. Variation is significantly larger in the CDL runs (runs above number 305), but often towards larger efficiencies. Only one CDL run is at significantly lower efficiency (\(\SI{83}{\%}\)), with a few other lower energy runs around \(\SI{91}{\%}\). The data quality in the CDL data is lower and it has larger variation of the common properties (gas gain, diffusion) as compared to the CAST dataset.

Applying the MLP cut as explained leads to background rates as presented in fig. 19, where the MLP is compared to the \(\ln\mathcal{L}\) cut (\(ε = \SI{80}{\%}\)) at a similar software efficiency \(ε = \SI{84.7}{\%}\) as well as at a significantly higher efficiency of \(ε = \SI{94.4}{\%}\). Note the efficiencies are effective efficiencies as explained in sec. 12.4.7.

The background suppression is significantly improved in particular at lower energies, implying other cluster properties provide better separation at those than the three inputs used for the likelihood cut. The mean background rates between \(\SIrange{0.2}{8}{keV}\) for each of these are ²⁷:

The most important aspect is that the MLP allows to achieve very comparable background rates at significantly higher efficiencies. Accepting lower signal efficiencies does not come with a noticeable improvement, implying that for the remaining clusters the distinction between background and X-ray properties is very small (and some remaining clusters surely are of X-ray origin).

As introduced in theory section 6.1.4 one problematic source of background is X-ray fluorescence created due to abundant muons interacting with material of – or close to – the detector. The resulting X-rays, if detected, are indistinguishable from those X-rays originating from an axion (or other particle of study). And given the required background levels, cosmic induced X-ray fluorescence plays a significant role in the datasets. Scintillators (ideally 4π around the whole detector setup) can be very helpful to reduce the influence of such background events by 'tagging' certain events. For a fully encapsulated detector, any (at least) muon induced X-ray would be preceded by a signal in one (or multiple) scintillators. As such, if the time \(t_s\) between the scintillator trigger and the time of activity recorded with the GridPix is small enough, the two are likely to be in real coincidence.

In the setup used at CAST with a \(\SI{42}{\cm}\) times \(\SI{82}{\cm}\) scintillator paddle (see sec. 7.7), about \(\sim\SI{30}{cm}\) above the detector center – and a \(\cos²(θ)\) distribution for muons – a significant fraction of muons should be tagged. Similarly, the second scintillator on the Septemboard detector, the small SiPM behind the readout area should trigger precisely in those cases where a muon traverses the detector from the front or back such that the muon track is orthogonal to the readout plane.

The term 'non trivial trigger' in the following indicates events in which a scintillator triggered and the number of clock cycles to the GridPix readout was larger than \(\num{0}\) (scintillator had a trigger in the first place) and smaller than \(\num{300}\). The latter cut is somewhat arbitrary, as the idea is two things: 1. no clock cycles of \(\num{4095}\) (indicates clock ran over) and 2. the physics involved leading to coincidences typically takes place on time scales shorter than \(\SI{100}{clock\;cycles} = \SI{2.5}{μs}\) (see sec. 7.7). Anything above should just be a random coincidence. \(\num{300}\) is therefore just chosen to have a buffer to where physical clock cycles start.

During the Run-3 data taking period a total of \(\num{69243}\) non trivial veto paddle triggers were recorded. ²⁸ The distribution of the clock cycles after which the GridPix readout happened is shown in fig. 20 on the right. The narrow peak at \(\SI{255}{clock\;cycles}\) seems to be some kind of artifact, potentially a firmware bug. The corresponding GridPix data was investigated and there is nothing unusual about it (neither cluster properties nor individual events). The source therefore remains unclear, but a physical origin is extremely unlikely as the peak is exactly one clock cycle wide (unrealistic for a physical process in this context) and coincidentally exactly \(\num{255}\) (0xFF in hexadecimal), hinting towards a counting issue in the firmware. The actual distribution looks as expected, being a more or less flat distribution corresponding to muons traversing at different distances to the readout. The real distribution does not start at exactly \(\SI{0}{clock\;cycles}\) due to inherent processing delays and even close tracks requiring some time to drift to the readout and activating the FADC. Further, geometric effects play a role. Very close to the grid, only perfectly parallel tracks can achieve low clock cycles, but the further away different angles contribute to the same times.

The SiPM recorded \(\num{4298}\) non trivial triggers in the same time, which are shown in fig. 20 on the left. ²⁹ Also this distribution looks more or less as expected, showing a peak towards low clock cycles (typical ionization and therefore similar times to accumulate enough charge to trigger) with a tail for less and less ionizing tracks. The same physical cutoff as in the veto paddle distribution is visible corresponding to the full \(\SI{3}{cm}\) of drift time.

Both of these distributions and the physically motivated cutoffs in clock cycles motivate a scintillator veto at clock cycles somewhere above \(\SIrange{60}{70}{clock\;cycles}\). To be on the conservative side and because random coincidences are very unlikely (on the time scales of several clock cycles; picking a value slightly larger implies a negligible dead time), a scintillator veto cut of \(\SI{150}{clock\;cycles}\) was chosen.

The resulting improvement of the background rate is shown in fig. 21, albeit only for the end of 2018 (Run-3) data as the scintillator trigger was not working correctly in Run-2 (as mentioned in sec. 10.5). The biggest improvements can be seen in the \(\SI{3}{keV}\) and \(\SI{8}{keV}\) peaks, both of which are likely X-ray fluorescence (\(\ce{Ar}_{Kα}\) \(\SI{3}{keV}\) & \(\ce{Cu}_{Kα}\) \(\SI{8}{keV}\)) and orthogonal muons (\(>\SI{8}{keV}\)). Arguably the improvements could be bigger, but the efficiency of the scintillator was not ideal, resulting in some likely muon induced X-rays to remain untagged. Lastly, the coverage of the scintillators is leaving large angular areas without a possibility to be tagged.

For a future detector an almost \(4π\) scintillator setup and correctly calibrated and tested scintillators are an extremely valuable upgrade.

As previously mentioned in 7.8 the FADC not only serves as a trigger for the readout and reference time for the scintillator triggers. Because of its high temporal resolution it can in principle act as a veto of its own by providing insight into the longitudinal cluster shape.

A cluster drifting towards the readout and finally through the grid induces a voltage measured by the FADC. As such the length of the FADC signal is a function of the time it takes the cluster to drift 'through' the grid. The kind of orthogonal muon events that should be triggered by the SiPM as explained in the previous section 12.5.1 for example should also be detectable by the FADC in the form of longer signal rise times than typical in an X-ray.

From gaseous detector physics theory we can estimate the typical sizes and therefore expected signal rise times for an X-ray if we know the gas of our detector. For the \(\SI{1050}{mbar}\), \(97.7/2.3\,\%\) \(\ce{Ar}/\ce{iC4H10}\) mixture used with a \(\SI{500}{V.cm⁻¹}\) drift field in the Septemboard detector at CAST, the relevant parameters are ³⁰

• drift velocity \(v = \SI{2.28}{cm.μs⁻¹}\)
• transverse diffusion \(σ_T = \SI{670}{μm.cm^{-1/2}}\)
• longitudinal diffusion \(σ_L = \SI{270}{μm.cm^{-1/2}}\)

As the detector has a height of \(\SI{3}{cm}\) we expect a typical X-ray interacting close to the cathode to form a cluster of \(\sqrt{\SI{3}{cm}}·\SI{670}{μm.cm^{-1/2}} \approx \SI{1160.5}{μm}\) transverse extent and \(\sqrt{\SI{3}{cm}}·\SI{270}{μm.cm^{-1/2}} \approx \SI{467.5}{μm}\) in longitudinal size, where this corresponds to a \(1 σ\) environment. To get a measure for the cluster size, a rough estimate for an upper limit is a \(3 σ\) distance away from the center in each direction. For the transverse size it leads to about \(\SI{7}{mm}\) and in longitudinal about \(\SI{2.8}{mm}\). From the CAST \cefe data we see a peak at around \(\SI{6}{mm}\) of transverse cluster size along the longer axis, which matches well with our expectation (see appendix 30.4 for the length data). From the drift velocity and the upper bound on the longitudinal cluster size we can therefore also compute an equivalent effective drift time seen by the FADC. This comes out to be

\[ t = \SI{2.8}{mm} / \SI{22.8}{mm.μs⁻¹} = \SI{0.123}{μs} \]

or about \(\SI{123}{ns}\), equivalent to \(\SI{123}{clock\;cycle}\) of the FADC clock. Alternatively, we can compute the most likely rise time based on the known peak of the cluster length, \(\SI{6}{mm}\) and the ratio of the transverse and longitudinal diffusion, \(σ_T / σ_L \approx 2.5\) to end up at a time of \(\frac{\SI{6}{mm}}{2.5} · \SI{22.8}{mm.μs⁻¹} = \SI{105}{ns}\).

Fig. 25 shows the measured rise time for the \cefe calibration data compared to the background data. The peak of the calibration rise time is at about \(\SI{55}{ns}\). While this is almost a factor of 2 smaller than the theoretical values mentioned before, this is expected as those first of all are an upper bound and secondly the "rise time" here is not the full time due to our definition starting from \(\SI{10}{\%}\) below the baseline and stopping \(\SI{2.5}{\%}\) before the peak, shortening our time (ref. sec. 9.5.3). At the same time we see that the background data has a much longer tail towards higher clock cycle counts, as one expects. This implies that we can perform a cut on the rise time and utilize it as an additional veto. The \cefe data allows us to define a cut based on a known and desired signal efficiency. This is done by identifying a desired percentile on both ends of the distribution of a cleaned X-ray dataset. It is important to note that the cut values need to be determined for each of the used FADC amplifier settings separately. This is done automatically based on the known runs corresponding to each setting.

For the signal decay time we do a simpler cut, with less of a physical motivation. As the decay time is highly dependent on the resistance and capacitance properties of the grid, high voltage and FADC amplifier settings, we simply introduce a conservative cut in such a range as to avoid cutting away any X-rays. Any background removed is just a bonus.

Finally, the FADC is only used as a veto if the individual spectrum is not considered noisy (see sec. 11.4.1). This is determined by 4 local dominant peaks based on a peak finding algorithm and in addition a general skewnees of the total signal larger than \(\num{-0.4}\). The skewness is an empirical cutoff as good FADC signals typically lie at larger negative skewness values (due to them having a signal towards negative values). See appendix 30.2 for a scatter plot of FADC rise times and skewness values. A very low percentage of false positives (good signals appearing 'noisy') is accepted for the certainty of not falsely rejecting noisy FADC events (as they would represent a random coincidence to the event on the center chip).

Applying the FADC veto with a target percentile of 1 from both the lower and upper end (\(1^{\text{st}}\) and \(99^{\text{th}}\) percentiles; thus a \(\SI{98}{\%}\) signal efficiency) in addition to the scintillator veto presented in the previous section, results in a background rate as seen in fig. 26. Different percentiles (and associated efficiencies) were be tested for their impact on the expected axion limit. Lower percentiles than 99 did not yield a significant improvement in the background suppression over the resulting signal efficiency loss. This is visible due to the sharp edge of the rise time for X-rays (green) in fig. 25 when comparing to its overlap with background (purple).

The achieved background rate of the methods leads to a background rate of \(\SI{9.0305(7277)e-06}{keV⁻¹.cm⁻².s⁻¹}\) in the energy range \(\SIrange{2}{8}{keV}\). And between \(\SIrange{4}{8}{keV}\) even \(\SI{4.5739(6343)e-06}{keV⁻¹.cm⁻².s⁻¹}\). The veto brings improvements across the whole energy range, in which the FADC triggers. Interestingly, in some cases even below that range. The events removed in the first two bins are events with a big spark on the upper right GridPix, which induced an X-ray like low energy cluster on the central chip and triggered the FADC. In the range at around \(\SI{8}{keV}\), likely candidates for removed events are cases of orthogonal muons that did not trigger the SiPM (but have a rise time incompatible with X-rays).

Generally, appendix 30.2 contains more figures of the FADC data. Rise times, fall times, how the different FADC settings affect these and an efficiency curve of a one-sided cut on the rise time for the signal efficiency and background suppression.

The final hardware feature that is used to improve the background rate is the outer ring of GridPixes. The size of large clusters is a significant fraction of a single GridPix. This means that depending on the cluster center position, parts of the cluster may very well be outside of the chip. While the most important area of the chip is the center area (due to the X-ray optics focusing the axion induced X-rays), misalignment and the extended nature of the 'axion image' mean that a significant portion of the chip should be as low in background as possible. Fig. 11(a) as we saw in sec. 12.3 shows a significant increase in cluster counts towards the edges and corners of the center GridPix. The GridPix ring can help us reduce this.

Normally the individual chips are treated separately in the analysis chain. The 'septem veto' is the name for an additional veto step, which can be optionally applied to the center chip. With it, each cluster that is considered signal-like based on the likelihood cut (or MLP), will be analyzed in a second step. The full event is reconstructed again from the beginning, but this time as an event covering all 7 chips. This allows the cluster finding algorithm to detect clusters beyond the center chip boundaries. After finding all clusters, the normal cluster reconstruction to compute all properties is done again. Finally, for each cluster in the event the likelihood method or MLP is applied. If now all clusters whose center is on the central chip are considered background-like, the event is vetoed, because the initial signal-like cluster turned out to be part of a larger cluster covering more than one chip.

There is a slight complication here. The layout of the septemboard includes spacing (see for example fig. 25 or 34 for the layout), which is required to mount the chips. This spacing, in addition to general lower efficiency towards the edges of a GridPix, means significant information is lost. When reconstructing the full 'septemboard events' this spacing would break the cluster finding algorithms, as the spacing might extend the distance over the cutoff criterion ³². For this reason the cluster finding algorithm is actually performed on a 'tight layout' where the spacing has been reduced to zero. For the calculation of the geometric properties however, the clusters are transformed into the real septemboard coordinates including spacing. ³³

In contrast to the regular single chip analysis performed before, which uses a simple, bespoke clustering algorithm (see sec. 9.4.1, 'Default'), the full 'septemboard reconstruction' uses the DBSCAN clustering algorithm. The equivalent search radius is extended a bit over the normal default (65 pixels compared to 50, unless changed in configuration). DBSCAN produces better results in terms of likely related clusters (over chip boundaries). ³⁴

An example that shows two clusters on the center chip, one of which was initially interpreted as a signal-like event before being vetoed by the 'septem veto', is shown in fig. 27. The colors indicate the clusters each pixel belongs to according to the cluster finding algorithm. As the chips are treated separately initially, there are two clusters found on the center chip. The green and purple cluster "portions" of the center chip. The cyan part passes the likelihood cut initially, which triggers the 'septem veto' (X-rays at such low energies are much less spherical on average; same diffusion, but less electrons). A full 7 GridPix event reconstruction shows the additional parts of the two clusters. The cyan cluster is finally rejected. It is a good example, as it shows a cluster that is still relatively close to the center, and yet still 'connects' to another chip.

The background rate with the septem veto is shown in fig. 28, where we see that most of the improvement is in the lower energy range \(< \SI{2}{keV}\). This is the most important region for the solar axion flux for the axion-electron coupling. Looking at the mean background rate in intervals of interest, between \(\SIrange{2}{8}{keV}\) it is \(\SI{8.0337(6864)e-06}{keV⁻¹.cm⁻².s⁻¹}\) and \(\SIrange{4}{8}{keV} = \SI{4.0462(5966)e-06}{keV⁻¹.cm⁻².s⁻¹}\). And even in the full range down to \(\SI{0}{keV}\) the mean is in the \(10⁻⁶\) range, \(\SI{9.5436(6479)e-06}{keV⁻¹.cm⁻².s⁻¹}\).

There is one more further optional veto, dubbed the 'line veto'. It checks whether there are clusters on the outer chips whose long axis "points at" clusters on the center chip (passing the likelihood or MLP cut). The idea being that there is a high chance that such clusters are correlated, especially because ionization is an inherently statistical process. It can be used in addition to the 'septem veto' or standalone. The approach uses the same general approach as the septem veto by reconstructing the full events as 'septemboard events' again. The difference to the septem veto is that it is not reliant on the cluster finding search radius.

The center cluster that initially passed the likelihood or MLP cut will be rejected if the long axis of the outer chip cluster cuts within \(3·\left(\frac{\text{RMS}_T + \text{RMS}_L}{2}\right)\). In other words within \(3 σ\) of the mean (along long and short axis) standard distribution of the pixels ³⁵.

If desired not every outer chip cluster is considered for the 'line veto'. An \(ε_{\text{cut}}\) parameter can be adjusted to only allow those clusters to contribute with an eccentricity larger than \(ε_{\text{cut}}\). The value of this cutoff impacts the efficiency, but also the expected random coincidence rate of the veto. More on that in sec. 12.5.5.

An example of an event being vetoed by the 'line veto' is shown in fig. 29. The black circle around the center chip cluster indicates the radius in which the orange line (extension of the long axis of the top, green cluster) needs to cut the center cluster.

The achieved background rate, see fig. 30, goes well into the \(10⁻⁶\) range with this veto over the whole energy range, as the influence is largest at low energies. The rate comes out to \(\SI{7.9164(5901)e-06}{keV⁻¹.cm⁻².s⁻¹}\) over the whole range up to \(\SI{8}{keV}\), to \(\SI{7.5645(6660)e-06}{keV⁻¹.cm⁻².s⁻¹}\) if starting from \(\SI{2}{keV}\) and \(\SI{3.7823(5768)e-06}{keV⁻¹.cm⁻².s⁻¹}\) if starting from \(\SI{4}{keV}\).

One potential issue with the septem and line veto is that the shutter times we ran with at CAST are very long (\(> \SI{2}{s}\)), but only the center chip is triggered by the FADC. This means that the outer chips can record cluster data not correlated to what the center chip sees. When applying one of these two vetoes the chance for random coincidence might be non negligible. This random coincidence rate needs to be treated as a dead time for the detector, reducing the effective solar tracking time available.

In order to estimate this we bootstrap fake events with guaranteed random coincidence. For each data taking run, we take events with clusters on the center chip, which pass the likelihood or MLP cuts and combine these with data from the outer chips from different events. The outer GridPix ring data is sampled from all events in that run, including empty events as otherwise we would bias the coincidence rate. The vetoes (either only septem veto, only line veto or both) are then applied to this bootstrapped dataset. The fraction of rejected events due to the veto is the random coincidence rate, because there is no physical correlation between center cluster and outer GridPix ring clusters.

By default \(\num{2000}\) such events are generated for each run. The random coincidence rate is the mean of all obtained values. Further, this can be used to study the effectiveness of the eccentricity cutoff \(ε_{\text{cut}}\) mentioned for the line veto. It is expected that a stricter eccentricity cut should yield less random coincidences. We can then compare the obtained random coincidence rates to the real fractions of events removed by the same veto setup. The ratio of real effectiveness to random coincidence rate is a measure of the potency of the veto setup.

Fig. 33 shows how the fraction of passing events changes for the line veto as a function of the eccentricity cluster cutoff. The cases of line veto only for fake and real data, as well as septem veto plus line veto for the same are shown. We can see that the fraction that passes the veto setups (y axis) drops the further we go towards a low eccentricity cut (x axis). For the real data (Real suffix in the legend) the drop is faster than for fake boostrapped data (Fake suffix in the legend), which means that we can use the lowest eccentricity cut possible (effectively disabling the cut at \(ε_\text{cut} = 1.0\)). In any case, the difference is very minor independent of the cutoff value. The septem veto without line veto is also shown in the plot (SeptemFake and SeptemReal) with only a single point at \(ε_{\text{cut}} = 1.0\) for completeness.

Table 25 shows the percentages of clusters left over after the line veto (at \(ε_\text{cut} = 1.0\)) and septem veto setups, both for real data (column 'Real') and bootstrapped fake data (column 'Fake'). For fake data this corresponds to the random coincidence rate of the veto setup (strictly speaking \(1 - x\) of course). For real data it is the real percentage of clusters left, meaning the vetoes achieve reduction to \(\frac{1}{10}\) of background, depending on setup. The random coincidence rates are between \(\SI{78.6}{\%}\) in case both outer GridPix vetoes are used, \(\SI{83.1}{\%}\) for the septem veto and \(\SI{85.4}{\%}\) for the line veto only. The line veto has lower random coincidence rate, but by itself is also less efficient.

This effect on the signal efficiency is one of the major reasons the choice of vetoes is not finalized outside the context of a limit calculation.

When dealing with the line veto without the septem veto there are multiple questions that come up of course.

First of all, what is the cluster that you're actually targeting with our 'line'? The original cluster (OC) that passed lnL or a hypothetical larger cluster that was found during the septem event reconstruction (HLC).

Assuming the former, the next question is whether we want to allow an HLC to veto our OC? In a naive implementation this is precisely what's happening, because in the regular use case of septem veto + line veto, the line veto would never have any effect anymore, as an HLC would almost certainly be vetoed by the septem veto! But without the septem veto, this decision is fully up to the line veto and the question becomes relevant. (we will implement a switch, maybe based on an environment variable or flag)

In the latter case the tricky part is mainly just identifying the correct cluster which to test for in order to find its center. However, this needs to be implemented to avoid the HLC in the above mentioned case. With it done, we then have 3 different ways to do the line veto:

1. 'regular' line veto. Every cluster checks the line to the center cluster. Without septem veto this includes HLC checking OC.
2. 'regular without HLC' line veto: Lines check the OC, but the HLC is explicitly not considered.
3. 'checking the HLC' line veto: In this case all clusters check the center of the HLC.

Thoughts on LvCheckHLC:

• The radii around the new HLC become so large that in practice this won't be a very good idea I think!
• The lineVetoRejected part of the title seems to be "true" in too many cases. What's going on here? See: for example "2882 and run 297" on page 31. Like huh? My first guess is that the distance calculation is off somehow? Similar page 33 and probably many more. Even worse is page 34: "event 30 and run 297"! -> Yeah, as it turns out the problem was just that our inRegionOfInterest check had become outdated due to our change of
• [ ] Select example events for each of the 'line veto kinds' to demonstrate their differences.

OC: Original Cluster (passing lnL cut on center chip) HCL: Hypothetical Large Cluster (new cluster that OC is part of after septemboard reco) Regular: is an example event in which we see the "regular" line veto without using the septem veto. Things to note:

• the black circle shows the 'radius' of the OC, not the HLC
• the OC is actually part of a HLC
• because of this and because the HLC is a nice track, the event is vetoed, not by the green track, but by the HLC itself!

This wouldn't be a problem if we also used the septem veto, as this event would already be removed due to the septem veto! (More plots: )

Regular no HLC: The reference cluster to check for is still the regular OC with the same radius. And again the OC is part of an HLC. However, in contrast to the 'regular' case, this event is not vetoed. The green and purple clusters simply don't point at the black circle and the HLC itself is not considered here. This defines the 'regular no HLC' veto. is just an example of an event that proves the method works & a nice example of a cluster barely hitting the radius of the OC. On the other hand though this is also a good example for why we should have an eccentricity cut on those clusters that we use to check for lines! The green cluster in this second event is not even remotely eccentric enough and indeed is actually part of the orange track! (More plots: )

Check HLC cluster: Is an example event where we can see how ridiculous the "check HLC" veto kind can become. There is a very large cluster that the OC is actually part of (in red). But because of that the radius is SO LARGE that it even encapsulates a whole other cluster (that technically should ideally be part of the 'lower' of the tracks!). For this reason I don't think this method is particularly useful. In other events of course it looks more reasonable, but still. There probably isn't a good way to make this work reliably. In any case though, for events that are significant in size, they would almost certainly never pass any lnL cuts anyhow. (More plots: )

The following is a broken event. THe purple cluster is not used for line veto. Why? /t/problem_{event12435run297.pdf}

• [X] Implement a cutoff for the eccentricity that a cluster must have in order to partake in the line veto. Currently this can only be set via an environment variable (ECC_LINE_VETO_CUT). A good value is around the 1.4 - 1.6 range I think (anything that rules out most X-ray like clusters!)

• [ ] TABLE OF FEATURE, EFFICIENCY, BACKGROUND RATE -> That table essentially motivates looking at different veto setups.
• [ ] Write a section about a) the motivation behind looking at different setups to begin with and b) show the kind of setups we will be looking at later in the limit part.

As a reference the random coincidence rates of:

• pure septem veto = 0.7841029411764704
• pure line veto = 0.8601764705882353
• septem + line veto = 0.732514705882353

Let's compute the total efficiencies of all the setups we look at.