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## PowerPoint Slideshow about ' Vertex fitting' - kenton

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Presentation Transcript

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

- Hits are in the CTD and MVD
- Tracks are fitted in CTD and MVD
- Is a track primary or secondary?

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 -> µµc)
- Scattering in the detector material (secondary interaction)
- Multiple events per bunch crossing expected at LHC.
- Well enough measured tracks needed.

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

- 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

-K0s using CTD and MVD tracks

Decay lengthcorrect for the boost of the particle: c = l /

With the MVD more secondary K0s are found!

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

- Tracks (p) are now ‘measurements’
- Parameters are:
- Find best estimate for x (vertex) and i (refitted track)
- use LSM

Linearize h near x0 , 0,i

- With

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

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 new2

- Do step 2 again withp0,vtx = pvtxuntil the change in 2is small enough.

Error propagation

New vertex parameters

- Fitted track

2d detector modelTrack 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

- 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

- Generated track

ZOOM

- Fitted track

- Vertex refitted track

- Vertex

The vertex refitted tracks all intersect the vertex

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

- 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:

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

- 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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