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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

Vertex fitting

Zeus student seminar

May 9, 2003

Erik Maddox

NIKHEF/UvA

outline
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!
a zeus event
A Zeus Event
  • Hits are in the CTD and MVD
  • Tracks are fitted in CTD and MVD
  • Is a track primary or secondary?
introduction
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.
k 0 s mass signal

-> 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
mass spectrum
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
decay length

-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!

5 helix parameters
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

the 2d vertex problem
The (2D) vertex problem
  • Tracks (p) are now ‘measurements’
    • Parameters are:
  • Find best estimate for x (vertex) and i (refitted track)
  • use LSM
slide10
2 equation

Error matrix

2*n measured values

slide11
Linearize h near x0 , 0,i
    • With

-> (h-h0) describes how the ‘measurements’ change if the vertex parameters change

slide12
Different notation

n+2 parameters

to fit

= H pvertex

lsm estimation of the vertex parameters
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

2d detector model

- 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

after the vertex fit
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

3 tracks
3 tracks

- Generated track

ZOOM

- Fitted track

- Vertex refitted track

- Vertex

The vertex refitted tracks all intersect the vertex

primary vertex in new data
Primary vertex in new data
  • Mean x and y position of primary vertex for selected runs.

 Input for beam constraint vertex fit

using a beam constraint
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

slide19
Derivative matrix H and first extimate h0
  • The procedure to find the vertex parameters stays for the rest the same.
slide21
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’

kalman filter vertex fit
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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