1. 1 Vertex fitting Zeus student seminar May 9, 2003 Erik Maddox NIKHEF/UvA

2. 2 Outline What is vertexing? –K 0 s in new data ( example ) –The least squares vertex fit –A 2-dimensional example –Using a beam constraint More on vertexing –Kalman filtering Do-it-your-self-interactive-vertexing!

3. 3 A Zeus Event Hits are in the CTD and MVD Tracks are fitted in CTD and MVD Is a track primary or secondary?

4. 4 Introduction Tracks are measured with parameter vector p and covariance matrix V p –The precision of the parameters can be improved by the constraint that they all come from the same vertex. (vertex refitted) –Tracks not coming from the primary vertex Secondary decay (examples K 0 s, D* ±, b -> µ µ c) Scattering in the detector material (secondary interaction) Multiple events per bunch crossing expected at LHC. –Well enough measured tracks needed.

5. 5 K 0 s mass signal K 0 decays to  +  - –c  is 2.68 cm Method –Select secondary vertices consisting of a opposite charged track pair –Assume  mass, plot invariant mass of K 0 –Improve selection by requiring that the K 0 comes from primary vertex -> Primary vertex

6. 6 Mass spectrum Expected mass: 0.498 GeV Width depends on the resolution of the detector, a perfect detector would give the ‘natural width’ (  ) of the particle Background processes: - Photon conversion   e + e - - Random combinations

7. 7 Decay length -K0s using CTD only tracks -K0s using CTD and MVD tracks With the MVD more secondary K0s are found! correct for the boost of the particle: c  = l / 

8. 8 5 helix parameters W = q/R  0 D 0 Z 0 T=tan(  dip ) These describe the charged particle trajectory in a uniform magnetic field Used in 2D example

9. 9 The (2D) vertex problem Tracks (p) are now ‘measurements’ –Parameters are: Find best estimate for x (vertex) and  i (refitted track) use LSM

10. 10  2 equation 2*n measured values Error matrix

11. 11 Linearize h near x 0,  0,i –With -> (h-h0) describes how the ‘measurements’ change if the vertex parameters change

12. 12 Different notation = H  p vertex n+2 parameters to fit

13. 13 LSM estimation of the vertex parameters Iterative procedure to find the minimum  2 1.Start with initial ‘guess’ for vertex parameters: p 0,vtx 2.Calculate the track parameters h 0 ( p 0,vtx ) and the derivative matrix H( p 0,vtx ) V vtx = (H T V y -1 H) -1 p vtx = p 0,vtx + V vtx H T (y- h 0 ) calculate the new  2 3.Do step 2 again with p 0,vtx = p vtx until the change in  2 is small enough.  Error propagation  New vertex parameters

14. 14 2d detector model - Generated track - Fitted track Track 1 D = -0.127,  = 1.623 Cov = (0.690 0.0416 0.0416 0.00294 ) Track 2 D = -1.118,  = 3.395 Cov = (0.582 0.0350 0.0350 0.00253 ) 1 2

15. 15 After the vertex fit - Generated track - Fitted track - Vertex refitted track - Vertex Vertex x = -0.0410041, y = -1.6349 Refitted tracks  1 = 1.623,  2 = 3.935  x = 0.869,  y = 1.302 Cov = (0.755 0.716 0.044 -0.0023 0.716 1.696 0.045 0.0433 0.044 0.045 0.0029 -4.6e-08 -0.0023 0.0433 -4.6e-08 0.0025 ) Later we will improve the fit, by using a beam constraint

16. 16 3 tracks The vertex refitted tracks all intersect the vertex ZOOM - Generated track - Fitted track - Vertex refitted track - Vertex

17. 17 Primary vertex in new data Mean x and y position of primary vertex for selected runs.  Input for beam constraint vertex fit

18. 18 Using a beam constraint Information about the beam position and profile can be put into the vertex fit. –The beam position is v x, v y with covariance V 0 for the width. 2*n + 2 Measured values Error matrix

19. 19 Derivative matrix H and first extimate h 0 The procedure to find the vertex parameters stays for the rest the same.

20. 20 Vertex constraint: (0,0) with error of 0.25 compare (slide 15)

21. 21 Without beam constraint: 2*n – (n+2) = n-2 degrees of freedom ‘need at least two tracks to fit a vertex’ With beam constraint 2*n+2 – (n+2) = n degrees of freedom ‘a vertex fit with 0 tracks gives back the beam constraint’

22. 22 Kalman filter vertex fit In high multiplicity events  have to invert large (n*n) matrices, cpu time ~ n 3 LSM is not very flexible to find secondary vertices. –All tracks are evaluated in the same algorithm Better to evaluate the vertex track for track –Small matrices –Remove outliers (secondary tracks) –Start with high quality tracks Kalman filter fitting is then very useful –Kalman filter is used to estimate a state of a dynamic system in time –Consider the vertex parameters and covariance as a ‘state vector’ –Evaluate the vertex for a single track, use the  2 of the step to decide. –If the  2 do a fitting step, add the information of the current track. (update vertex and covariance) Smoothing –Update the vertex refitted tracks for the latest vertex position.