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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


Error matrix

2*n measured values


  • Linearize h near x0 , 0,i

    • With

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


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


  • Derivative matrix H and first extimate h0

  • The procedure to find the vertex parameters stays for the rest the same.



  • Without beam constraint: 15)

    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 15)

  • 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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