Results with DUT

In order for GBL to be used with a DUT, it needs an improved performance compared to previous track fit implementations. There are several technical reasons this is the case, which are difficult to assess in track fit performance, mainly on the implementation details. The GBL library can be easily included into the project and is in the imple-mentation sense a very lightweight approach, compared to the entire impleimple-mentation of a Deterministic Annealing Filter (DAF). The direct interface between GBL and

Mille-pedeII is also desired as it again keeps the implementation simple and clean. There are less errors to be made as a lot of functionality is implemented by GBL and does not have to be rewritten in EUTelescope. In the light of maintainability, this is a big advantage as it keeps implementations separated and easy to understand.

In order to justify the implementation of GBL, there need to be advantages over the existing track fits aside from maintainability. The GBL implementation is more powerful in terms of the material description. This means, that both scattering in air, but also due to other material in the beam are correctly described. Furthermore, the GBL implementation allows to set the measurement uncertainty on the measurement for each hit independently depending on cluster size in both directions in contrast to the DAF fitter where an average resolution is used.

In order to compare the two track fits, data of a run taken at the DESY test beam facility at 4 GeV are reconstructed and compared. The set-up was a standard ATLAS pixel measurement with the Mimosa26 telescope and two LHC-type devices, in particular two FE-I4 modules, between the upstream and downstream arm of the telescope.

Shown in Figure 8.8 are the obtained residuals of the track fits on the second telescope module, i.e. plane 1. A fit of a Gaussian is used to quantify the width of the residuals.

The assumption is, that the GBL track fit will provide a better residual due to the more precise description of the scattering system and as the results will shown, this indeed is the case.

The Kálmán filter in the DAF implementation uses state predictions in both directions, combining them via a smoother. As a consequence, the predicted state on detector is not biased by the measurement on this detector. Therefore, the DAF predictions are compared to the unbiased residuals of the GBL track fit.

In detail, the GBL fit yields an unbiased residual width of (7.7±0.1) µm whereas the DAF implementation yields a biased residual width of (8.9±0.1) µm as shown in Figure 8.8. The other direction yields consistent results.

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(a) Residual distribution on the second tele-scope module with the GBL track fit.

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(b) The residual distribution on the same de-tector module with the DAF track fit.

Figure 8.8.: Comparison of the residuals of the two track fits.

In order to compare the tracking resolution at the location of the DUT, the residual

distributions at one of the DUTs is investigated. The long pixel edge of the FE-I4 measuring 250 µm is used to fit a box function smeared with a Gaussian. The results are shown in Figure 8.9, where again the GBL and DAF algorithm are compared. The rather simple attaching of the DUT to the triplet pair, as discussed in the previous Section, does not reject all background. Still, the derived σ determining the rising and falling behaviour of the box was fitted to (7.7±0.4) µm for the GBL track fit and (11.5±0.3) µm for the DAF track fit.

(a) DUT residual distribution in x-direction with the GBL track fit.

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(b) The same distribution but with the DAF fitter used.

Figure 8.9.: The same comparison as in Figure 8.8 but here for the long pixel direction of a FE-I4 module.

Another important result of the GBL track fit is the obtained χ²-distribution. To show results of a DUT included in the track fit, the χ²-distribution obtained during alignment when the DUTs are included in the track fit, is shown. A cut on the degrees of freedom is imposed so that onlyχ²-values for ten degrees of freedom are shown. This reflects the case where there is one measurement from the DUTs contributing to the track fit during alignment. The results are shown in Figure 8.10.

Indicated by the red fit is again aχ²-distribution with ten degrees of freedom, i.e. the theoretical expectation. There is an even more pronounced tail towards highχ²-values.

As discussed, this is due to more extreme scattering events which are present as non-Gaussian tails in the angular scattering distribution. However, in addition, the FE-I4 is a hybrid pixel sensor. Tracks propagating through the interconnecting bumps of at least one of the DUTs in the beam will experience significantly more material than described by the scattering algorithm.

As a comparison, the run was also reconstructed with the DAF fitter and entries with ten degrees of freedom selected. Results are shown in Figure 8.11, where a large deviation between the expected tail in theχ² distribution and the obtained results from data are visible.

It must be noted, that the DAF fitter is typically not used to incorporate DUT mea-surements and also lacks the functionality to input precise measurement uncertainty, hence the deviations in the high-χ² can also be attributed to an incorrect DUT

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Figure 8.10.: The measured (black) and expected (red) χ²-distribution for events with ten degrees of freedom for the GBL track fit.

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Figure 8.11.: The measured (black) and expected (red) χ²-distribution for events with ten degrees of freedom as obtained by the DAF track fit.

scription. Summarising, the GBL track-fit given in Figure 8.10, yields a reasonable χ²-distribution with expected deviations due to non-Gaussian scattering, whereas, im-plementation details limit the accuracy of the track fit when using the DAF fitter, as seen in Figure 8.11.

It is important to keep in mind that the obtained estimates for the tracking resolution, i.e. theσfrom the box-fit, andχ²-distributions, are for this sample and set-up. Different locations of the sensors, different tunings and other beam energies will impact the actual performance. It is, therefore, necessary that these parameters are validated at each test beam campaign.

In order to validate the tracking algorithms even more precisely, a Monte-Carlo frame-work is used. For the purpose of further studies, Allpix²has been modified to incorporate needed functionality, as described in the next Section.