1

Vertex fitting

Zeus student seminarMay 9, 2003

Erik MaddoxNIKHEF/UvA

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Outline

• What is vertexing?– K0

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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A Zeus Event

• Hits are in the CTD and MVD

• Tracks are fitted in CTD and MVD

• Is a track primary or secondary?

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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 K0

s , D*±, b -> µµc)

• Scattering in the detector material (secondary interaction)• Multiple events per bunch crossing expected at LHC.

– Well enough measured tracks needed.

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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 the K0 comes from primary vertex

-> Primary vertex

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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

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Decay length

-K0s using CTD only tracks

-K0s using CTD and MVD tracks

With the MVD more secondary K0s are found!

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

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5 helix parameters

• W = q/R0

• D0

• Z0

• T=tan(dip)These describe the charged particle trajectory in a uniform magnetic field

Used in 2D example

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The (2D) vertex problem

• Tracks (p) are now ‘measurements’– Parameters are:

• Find best estimate for x (vertex) and i (refitted track)

• use LSM

000020

200 cossin yx

qx

qxyxD

0)sin(

)cos(

0

0

y

x

q

q

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2 equation

n

nD

D

y

1

1

)],h([)],h([ 12iy

Ti xyVxy

np

p

p

y

V

V

V

V

,

2,

1,

00

00

00

2*n measured valuesError matrix

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• Linearize h near x0 , 0,i

– With

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

)()(),(h),h( ,00,000 iiiiii BxxAxx

00

00

yx

y

D

x

D

Aii

ii

i

i

i

i

i

i

D

B

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• Different notation

nnn

ii

x

BA

BA

BA

xx

122

11

,000

00

00

00

),(h),h(

= H pvertex

n+2 parameters

to fit

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LSM estimation of the vertex parameters

• Iterative procedure to find the minimum 2

1. Start with initial ‘guess’ for vertex parameters: p0,vtx

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

3. Do step 2 again with p0,vtx = pvtx until the change in 2 is small enough.

Error propagation

New vertex parameters

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2d detector model

- Generated track

- Fitted track

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

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After the vertex fit

- Generated track

- Fitted track

- Vertex refitted track

- Vertex

Vertex

x = -0.0410041, y = -1.6349

Refitted tracks

1= 1.623, 2 = 3.935

x = 0.869, y = 1.302

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

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3 tracks

The vertex refitted tracks all intersect the vertex

ZOOM- Generated track

- Fitted track

- Vertex refitted track

- Vertex

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Primary vertex in new data

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

Input for beam constraint vertex fit

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Using a beam constraint

• Information about the beam position and profile can be put into the vertex fit.– The beam position is vx, vy

with covariance V0 for the width.

n

n

y

x

D

D

v

v

y

1

1

np

p

p

y

V

V

V

V

V

,

2,

1,

0

0000

0

000

000

000

2*n + 2 Measured values

Error matrix

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• Derivative matrix H and first extimate h0

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

vtxpn

n

y

x

nn D

D

v

v

h

BA

BA

BA

I

H

,0

1

1

022

11

,

00

0

00

00

000

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

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• Without beam constraint:2*n – (n+2) = n-2 degrees of freedom

‘need at least two tracks to fit a vertex’

• With beam constraint2*n+2 – (n+2) = n degrees of freedom

‘a vertex fit with 0 tracks gives back the beam constraint’

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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.