1. STAR Kalman Track Fit V. Perevoztchikov Brookhaven National Laboratory,USA

2. STAR Victor Perevoztchikov, BNL StarSoft local Track seeds Track seeds are the short initial tracks. They are made from a few hits (<=8). Searching of appropriate hits started in the region far from the collision, where track density is low. Used hits are marked to avoid reusing. The algorithm, used in Sti, is traditional. Nothing special.

3. STAR Victor Perevoztchikov, BNL StarSoft local What is Kalman The simple example. We have 4 measurements of the same parameter. To estimate the value we search the minimum of: The solution is trivial, this is a “global fit”:

4. STAR Victor Perevoztchikov, BNL StarSoft local Kalman continue It was a global solution. The Kalman approach is:

5. STAR Victor Perevoztchikov, BNL StarSoft local Kalman continue Kalman fit features:  The result of Kalman fit is exactly the same as for a global one. No mystics;  In global fit, size of matrix to invert, depends on number of measurements. In Kalman fit not. As a result, machine accuracy for Kalman fit are more critical;  In Kalman fit an estimation is updated with each new measurement. This allows to make a decision: is this new measurement belong to our object, or not? So we can use not only Kalman fit but Kalman filter. Not possible in global;  By the points above, Kalman fit and filter are so popular.

6. STAR Victor Perevoztchikov, BNL StarSoft local Smoother In real life Kalman of course works for much more complicated case. During track fit, information is added hit by hit. So the end point is reconstructed on the base of all hits in track. After that it is possible to go in the backward direction. Then the first point also accumulates information from all hits. But all intermediate points have “intermediate” accuracy. This could be solved by smoother. Smoother is a part of Kalman algorithm, which provides equal accuracy of all points. Smoother is implemented in Sti. From the first glance it is enough. But it is not. The problem is, that Kalman is the algorithm of finding exact minimum of quadratic functional. For such functional, after fit in one direction and smoother in backward direction, we have exact solution. But our functional is not quadratic at all. That means we must use “quadratisation” of our functional, or,which is the same, linearisation of according equations. If we use such approximation, evident, we must use iterations.

7. STAR Victor Perevoztchikov, BNL StarSoft local Refit For the “quadratic” functional, refit is senseless. Each iteration gives the same result. Our track is approximated by helices near the hits, with some boundary conditions between neighboring helices. It is typical non linear case. A linearisation is based on derivatives. If these derivatives taken in a proper points, then Kalman will give the exact result. But Kalman is started with the infinite errors, hence the first point is the first hit. Derivatives in this point are wrong. Hence second point also is wrong, and so on. To fix the problem, the evident choice is iterations, refit. Evident, when parameters of track do not mach different from the previous iteration, refit stopped

8. STAR Victor Perevoztchikov, BNL StarSoft local Bad seed and global helix approximation

9. STAR Victor Perevoztchikov, BNL StarSoft local Bad track and global helix approximation

10. STAR Victor Perevoztchikov, BNL StarSoft local Global Helix Kalman follows it’s nose. It is good and bad in the same time. As you see on the previous slides, some times it is very bad. To improve it, Simple Global Helix was implemented.  It does not account hit errors for seeds  It does not account energy loss  Simplified fit: by X,Y and Z,L separately So it allows to make “global vision” of the track an addition to Kalman’s “local vision”. It is used just after seed finder, and just before refit.

11. STAR Victor Perevoztchikov, BNL StarSoft local Fit to Primary vertex Mathematically the primary vertex is a common point of all primary tracks. More precise, of all models of primary track. The primary vertex is calculated from the tracks using this constrain. The classical way to do it, is fitting all the tracks with common point constrain. Then all the primary tracks have exactly the same common point and have improved errors. But in practice:  To obtain primary we selected the best tracks. We must anyway to assign this point to other non selected but probably primary tracks;  We fit the primary vertex as a point closest to all tracks, without touching the track parameters;  So we got a primary vertex and them try to assign this point to all the tracks; But how to do it mathematically correct?

12. STAR Victor Perevoztchikov, BNL StarSoft local Fit to Primary vertex(2) How to do it mathematically correct? We can:  Consider the vertex as an additional hit with it’s errors. But then all the tracks will have the different coordinates of primary vertex. It contradicts to our main assumption;  Consider the vertex as an additional hit with zero errors. Then the point is the same for all tracks. But the vertex errors are not accounted and track errors are underestimated.  The combination of two above. Fit track with constrain to go through the primary vertex, which is equivalent to zero errors, and then, propagate vertex errors to track. The last approach was implemented. It works well, but… Always “but”.

13. STAR Victor Perevoztchikov, BNL StarSoft local Fit to Primary vertex problems How it is implemented:  Check chi-square of track to vertex.  If it is too big, it is not a primary. Due to non linearity, it is allowed to be rather big;  Refit the track. If refit is failed, it is not a primary. The problems:  After refit all chi-squares increased but one in vertex node becomes very small. What is the quality of the assignment? May be total chi-square of track? Not clear.  During refit, some nodes dropped and in result fit is good. But is it our track now? How many drops could be allowed? So not everything is clear yet.

14. STAR Victor Perevoztchikov, BNL StarSoft local Hits dropped in refit. During refit some nodes are dropped. What to do with according hits? Before we still kept them assigned to this track. That means we do not allow to use them for the other tracks. But very probably, this hit belongs to other track. Why our track, like “dog on the hay stock” does not use itself and does not allow to use others? But may be this hit on the crossing of two tracks, and it is hard to say which is it. So looking on the pictures presented above, some hit assignments is so weird, that I decided to allows for dropped hits to become free. So they could be used by others. We will see.

15. STAR Victor Perevoztchikov, BNL StarSoft local Tips for new detector.  High accuracy of new detector means nothing if accuracy of prediction is not enough to select best hit. By other words: l Either new detector must be not far from the previous one; l Or track density must be low enough;  The proper alignment is extremely important;  Realistic simulation of the detector, with keeping simulation information after reconstruction(idTruth), is very important. No any calculations on the back of an envelope, could replace the good simulation.  Remember, that even good simulation always gives results better than reality.

16. STAR Victor Perevoztchikov, BNL StarSoft local Results The following features were implemented and used:  Smoother used in refit  Global helix approximation ;  Primary track fitting to the common vertex point;  The failed hits are reused;  Still some unclear things remain