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Vertex fitting

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

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Vertex fitting

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

  2. Outline • What is vertexing? • K0s 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. A Zeus Event • Hits are in the CTD and MVD • Tracks are fitted in CTD and MVD • Is a track primary or secondary?

  4. Introduction • Tracks are measured with parameter vector p and covariance matrix Vp • 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 K0s , D*±, b -> µµc) • Scattering in the detector material (secondary interaction) • Multiple events per bunch crossing expected at LHC. • Well enough measured tracks needed.

  5. -> Primary vertex K0s mass signal • K0 decays to +- • c is 2.68 cm • Method • Select secondary vertices consisting of a opposite charged track pair • Assume  mass, plot invariant mass of K0 • Improve selection by requiring that theK0 comes from primary vertex

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

  8. 5 helix parameters • W = q/R • 0 • D0 • Z0 • T=tan(dip) Used in 2D example These describe the charged particle trajectory in a uniform magnetic field

  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. 2 equation Error matrix 2*n measured values

  11. Linearize h near x0 , 0,i • With -> (h-h0) describes how the ‘measurements’ change if the vertex parameters change

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

  13. LSM estimation of the vertex parameters • Iterative procedure to find the minimum 2 • Start with initial ‘guess’ for vertex parameters:p0,vtx • Calculate the track parameters h0( p0,vtx ) and the derivative matrix H( p0,vtx ) Vvtx =(HTVy-1H)-1 pvtx =p0,vtx +Vvtx HT (y- h0 ) calculate the new2 • Do step 2 again withp0,vtx = pvtxuntil the change in 2is small enough.  Error propagation  New vertex parameters

  14. - Generated track - Fitted track 2d detector model 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. After the vertex fit • Vertex • x = -0.0410041, y = -1.6349 • Refitted tracks • 1= 1.623, 2= 3.935 x= 0.869, y = 1.302 - Generated track - Fitted track - Vertex refitted track - Vertex 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. 3 tracks - Generated track ZOOM - Fitted track - Vertex refitted track - Vertex The vertex refitted tracks all intersect the vertex

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

  18. Using a beam constraint • Information about the beam position and profile can be put into the vertex fit. • The beam position is vx, vywith covariance V0 for the width. 2*n + 2 Measured values Error matrix

  19. Derivative matrix H and first extimate h0 • The procedure to find the vertex parameters stays for the rest the same.

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

  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. Kalman filter vertex fit • In high multiplicity events  have to invert large (n*n) matrices , cpu time ~ n3 • 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.

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