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# Vertex fitting - PowerPoint PPT Presentation

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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## PowerPoint Slideshow about 'Vertex fitting' - kenton

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### Vertex fitting

Zeus student seminar

May 9, 2003

NIKHEF/UvA

• 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!

• Hits are in the CTD and MVD

• Tracks are fitted in CTD and MVD

• Is a track primary or secondary?

• 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.

-> 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

• 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 length

correct for the boost of the particle: c = l / 

With the MVD more secondary K0s are found!

• W = q/R

• 0

• D0

• Z0

• T=tan(dip)

Used in 2D example

These describe the charged particle trajectory in a uniform magnetic field

• 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

• Iterative procedure to find the minimum 2

• 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

- 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

• 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

- Generated track

ZOOM

- Fitted track

- Vertex refitted track

- Vertex

The vertex refitted tracks all intersect the vertex

• Mean x and y position of primary vertex for selected runs.

 Input for beam constraint vertex fit

• 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’

• 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)

• 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.