1. STAR SVT Self Alignment V. Perevoztchikov Brookhaven National Laboratory,USA

2. STAR Victor Perevoztchikov, BNL SVT Review What is the PROBLEM? Right now the SVT alignment is based on TPC tracks. But TPC has less precision than the SVT. Now it is solved by big statistics of TPC data. Then our accuracy depends on systematic errors of TPC, which can not be improved by statistics. Hence, alignment based on TPC could be considered only as preliminary. What is the best possible strategy?  Direct measurement alignment;  Alignment based on big TPC statistics;  Self alignment, based on good tracks, selected by TPC+SVT. Only SVT hits,selected by these tracks plus condition that tracks are primaries, and hence have one common point, are used for alignment. Measurements from TPC are ignored now;  Again TPC+SVT track are used for alignment of SVT global position. To avoid time dependency of TPC space charge, we need to select all runs, where we believe, SVT in the same position. Then influence of TPC systematic errors to some extent is diminished;  It is still not very good, but we do not have any additional information to improve the situation.

3. STAR Victor Perevoztchikov, BNL SVT Review Simple example vertex

4. STAR Victor Perevoztchikov, BNL SVT Review Drift velocity. SVT alignment consists of SVT geometry and SVT drift velocities. It is practically impossible to align them separately. How to account drift velocity?  Self alignment based on drift velocities from the previous TPC+SVT alignment.  Improve velocities using: l Known size of hybrid; l Distribution of hits on hybrid must be flat;  Self alignment again;  Iterations; Non flat noise makes the above procedure as a non trivial task.

5. STAR Victor Perevoztchikov, BNL SVT Review Self Alignment assumptions. Basic assumptions:  Selection of minimal geometric entities. Right now we selected the ladders as a minimal geometrical part of SVT. When they will be successfully aligned, then wafers or hybrids will be selected;  Ideal local frame. This is rather arbitrary frame, but must be close to real one. The most calculations are made in this frame. This frame is not changing during alignment. As ideal local frame we selected an Sti local frame. In this frame all ladders are placed exactly along Z axis and rotated around Z by fixed angles ;  The real ladders have 3 rotations and 3 space shifts wrt to ideal ladders; These 6 parameters for each ladder we must to find. The ideal local frame is selected in such a way, that these 6 parameters are small enough, to use linear approximation;  Projected hits: Real hits transformed into ideal local frame are not sitting on measurement plane, but not too far from it. We move hit point along the track direction up to crossing of ideal measurement plane. This new point is a “projected hit”.

6. STAR Victor Perevoztchikov, BNL SVT Review Self Alignment assumptions (continue)  6 parameters transformation is applied not to ideal local frame, but to projected hits. As a result, these hits are not anymore on the ideal measurement plane. Then we repeat projection of them along the track direction to the ideal measurement plane. So, transformation of projected hits consists of two parts: l Geometrical transformation, which is the same for all events; l Plus projection along track to ideal measurement plane, which is different for each event.  As a result, we got 6 parameters geometrical transformation for each ladder, transform them into standard SVT transformation format. Which will be used in geometry reconstruction.

7. STAR Victor Perevoztchikov, BNL SVT Review A simplified fitting algorithm There are two approaches for fitting. A simplified one and the complex one. The simplified algorithm does not need complex calculation of derivatives. A simplified algorithm:  Reading Sti reconstruction result;  Selects tracks with 3 or more Svt hits plus some additional cuts;  If event has less than 5 such tracks, discard it and read the next event;  Keep only SVT (or SSD) hits;  Transform hits into ideal local frame;  Apply the 6 parameters geometrical transformation from the previous iteration, for these hits;  Transform updated hits back into global system;  Fit these hits by simple helix and create SelfTracks;

8. STAR Victor Perevoztchikov, BNL SVT Review A simplified fitting algorithm (continue)  Evaluate primary vertex as a common point of all tracks;  Refit SelfTracks with condition to go exactly through the vertex;  Project hits into projected hits;  Calculate residuals, as a difference between projected hits and track crossing points in ideal measurement plane.  Update specific matrix by residuals and read the next event.  Events are finished. Use the matrix, filled during the reading, to calculate all 6 parameters geometrical transformation, which minimizes the sum of squares of all residuals.  Go to next iteration;

9. STAR Victor Perevoztchikov, BNL SVT Review Why simplified algorithm must work?  For each event we choose the vertex point, for which sum of square of residuals for this event is minimal;  At the end of iteration we apply transformation which decreases the total sum of residuals  During the next iteration, updating the vertex, residuals for each event could be only improved.  As a result, each iteration could only improve total residuals or leave it as it is.

10. STAR Victor Perevoztchikov, BNL SVT Review Complex algorithm This algorithm is very similar to the previous one. The difference is:  Simple algorithm is trying to minimize residuals without accounting that after moving the hits, track will move as well.  Complex algorithm minimizes the sum of chi-squares of tracks, taking into account track changing. Also it accounts condition that all tracks have the common point. To do this, you must evaluate derivatives of track parameters by hit. This algorithm must need less iterations, but result should be the same.

11. STAR Victor Perevoztchikov, BNL SVT Review Current status Work is not finished yet, but close to the end. Right now both algorithms are working well, when the solution is far from the minimum. Near the minimum some instability occurs and no convergence happened. This behavior is not yet understood. But we believe, it will be fixed soon.