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1. VELO Residuals
2. Problem & Strategy
3. Internal Alignment
4. Box Alignment
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Application to the VELO

2. VELO algorithm (Part III)

D. Global Alignment (STEP 2)

D1. Introduction

At the end of Step 1, the VELO halves are internally aligned. Boxes offsets and tilts are still to be determined. Classic tracks become useless to perform this task, as we now need parameters involving both parts of the VELO.

Having said that, two possibilities are available: The whole method and results presented briefly in this part are extensively described in LHCb-note-2007-067.

D2. Using the primary vertex: definitions

The idea is to use again Millepede, so a generalized least square minimization, we will just fit primary vertices instead of track parameters.

Xi = slx.Zi + Xo + Globals

Yi = sly.Zi + Yo + Globals

During Step 1, we use the coordinates (Xi,Yi,Zi) in order to fit global and local parameters. In Step 2, we use the tracks parameters as the coordinates, in order to determine the primary vertex corrdinates (Vx,Vy,Vz):

Vx = slxj.Vz + Xoj + Globals

Vy = slyj.Vz + Yoj + Globals

The track parameters are corrected from the individual misalignments di', and the remaining global parameters are just the one we are looking for (ie the global offsets and tilts).

The simplicity of this method is that the global derivatives have nearly the same form for the primary vertex than for the track hits. One of the (substantial) difference is between Vz and Zi. The last one is precisely defined, but it's clearly not the case of the vertex Z position. The other difference is that primary vertex position explicitely depends on tracks slopes. But track slopes are biased by box tilts. So it means that we will have to apply the alignment two times: first in order to get the boxes tilts, and so correct the track slopes. And then, appply the alignment again in order to get the offsets.

D3. Using the overlap tracks: definitions

Once again we will try to use Millepede here, but it's even more simpler than with primary vertices. Idea is to fit overlaping tracks as we are fitting the classic ones. The only difference is that those tracks are corrected from internal misalignments, and the only misalignment are the one between the two boxes. So at the end of the day, we measure the offsets and tilts of one box w.r.t.the other one, i.e. only 6 constants.

The only difficulty here is to get the overlaping tracks, and that's a tough point... Due to VELO geometry, classic (R,Z) linear overlap tracks are rare and difficult to detect via the pattern recognition. However, using only high momentum tracks, and relying on the fact that we have a large amount of them, we could expect to use them into our code.

A more simple solution however, would be to use overlapping halo tracks. They should be cleaner and easier to detect. This is the path we have followed for our MC studies.

D4. Step 2 results with minimum bias and particle gun events

Results presented here have been obtained via the same code and datasets than the ones used for internal alignment studies. To only difference is the adjunction of 20000 ParticleGun events to each datasets: charged pions (1.0<P<30GeV/c) shooted over the whole VELO sensitive area.

These events were of course generated and digitized using the same geometries than the min.bias ones. STEP1 is performed, then track parameters are corrected from misalignments, and STEP2, for both method, is performed.

D4.1 Primary vertices only

The primary vertex selection prior to the Millepede fit is largely based on LHCb PrimVtxFinder.cpp method. We first look for vertex seeds on the Z axis, then a vertex candidate is fitted, outlying tracks are rejected if necessary, and if the candidate satisfies the selection criteria, it is feed into Millepede. This is a constrained fit, as we expect all the primary vertices to come from the same (X,Y) position. However, it should be noticed that this (X,Y) position hasn't to be known.

Then, once all the candidate are selected, we perform the global fit retrieve the 8 global parameters we are sensitive to: X/Y tilts and X/Y offsets of left and right boxes. Preliminary results of this process are shown on Fig.1 for both boxes offsets and tilts.

1. Box alignment with primary vertices and vertex transverse position constrained. This give the position of both boxes w.r.t. the beam. Top plots show the results obtained for boxes offsets and tilts. Bottom plots show the corresponding resolutions.

A red dotted line on each of the top plots shows that the results doesn't lie on the diagonal as expected. A slight bias appears, particularly for large misalignments. This bias comes from the fact that for large misalignments the linear relation we are using for Millepede becomes less accurate. The best resolutions obtained if one uses only the PV information are 36 micrometers for x and y translations and 190 microradians for x and y tilts.

D4.2 Overlapping tracks

Using the overlapping tracks, one didn't get the boxes position w.r.t. the beam, but the position of one box w.r.t. the other. Indeed left box is fixed and we move only the right one. The results obtained via this process are shown on Fig.2, once again for both tilts and offsets.

2. Box alignment with overlap tracks. This give the position of one box w.r.t. the other. Top plots show the results obtained for boxes relative offsets and tilts. Bottom plots show the corresponding resolutions.

As expected, results with this constraining technique are much more accurate the PV method. Resolutions obtained are 18 micrometers for x and y translations and 61 microradians for x and y tilts.

D4.3 Improvment of the PV method

In order to suppress the bias seen on Fig.1, the idea is to run in parallel PV and overlap techniques, and applying some very simple constraint equations linking the two method. Results obtained in this case are shown on Fig.3. In particular, the resolution plots show that using the constraints not only improves the resolution of the absolute box positions w.r.t. the beam (O(10%) improvment), but also provides a good correction for all the parameters (chi-square of the distributions is much better), thus removing the outliers.

3. Box alignment with primary vertices and vertex transverse position constrained and using overlap tracks information. This give the position of both boxes w.r.t. the beam. Top plots show the results obtained for boxes offsets and tilts. Bottom plots show the corresponding resolutions.

The resolutions obtained, i.e. the ones using the combined method are 33 micrometers for x and y translations and 173 microradians for x and y tilts. Relative box position resolutions stays unchanged, as they are obtained with the overlaps which is a much stronger constraint and thus don't take any advantage of the PV addition. It should be precised that all these results have been obtained with low statistics, and a clear improvment could be expected with more events. For example, a suitable aim might be to select ~1000 overlap tracks in LHCb and combine with ~5000 primary vertex events.