March 6th, 2006 CALICE meeting, UCL 1

Position and angular resolution studies with ECAL TB prototype

IntroductionLinear fit method

Results with 1, 2, 3, and 5 GeV electronsconclusions

Anne-Marie Magnan, IC London.

March 6th, 2006 CALICE meeting, UCL 2

Introduction

o Complete test beam prototype : 30 layers, 1 cm2 cells, 9 wafers per layer.

o Objective : determine position and angular resolution in test beam data, compared with the one obtained in MC simulation.

o Method : linear fit take into account correlations between layers.

o For this study, only 1, 2, 3 and 5 GeV single electrons

(DESY test beam).

o Own generation with Mokka05.05.

Beam position and RMS : (0 ±10, 0 ± 10, -220 ± 0) (in mm).

Current LCIO output does not allow to have the “truth” position in 1st ECAL layer after scattering in air/trackers materials.

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Linear fit method : definition of variables

o Definition of x and y position per layer :

ii

iii

E

xEx

ii

iii

E

yEy

i = hits in layer L.

zVz=-220 mm

x,y

*Vx, Vy

-197.6, 1st Si layer

pxMC,yMC

Variable of interest : reconstructed position compared to the expected one :

Dx = x – xMC , Dy = y – yMC

,z

xMC p

pVzzVxx

)(

z

yMC p

pVzzVyy

)(

_ _

For this simulation :

Redefine z = 0 1st Si layer

zpp

Layer

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Linear fit method : definition of the χ2

o Estimator of how accurate the prediction of the measurement is :• Without correlations between

variables :

• With correlations between variables :

• Wij is the inverse of the error matrix Eij :

2

22 )(

ltheoreticameasured xx

zppx xxltheoretica 10

jthmeasijiji

thmeas xxWxx )()(,

2

y

xx

i,j = 1,.....,30 for x 31,.....,60 for y

jijijijiij DxDxDxDxDxDxDxDxE ),cov(

= 0

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Error matrix(= covariance matrix)

Error matrix normalized on the diagonal values(= correlation matrix)

Linear fit method : error matrix

o Wij is calculated thanks to the whole sample (25,000 electrons)

x

y

x y

x

y

x y

No significant correlations

between x and y

1 GeV

Funny pattern : due to non-overlapping layers in x ???

Layer

Diagonal termsDiagonal terms

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Error matrix for higher energies

2 GeV 3 GeV 5 GeV

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Linear fit method : minimisation of the χ2

o X and y are uncorrelated : we consider 2 (30,30) matrices

2 independent fits : one for x, the other for y.

we can then look for the parameters (p0x,p1x) of the linear fit which minimize the χ2 :

00

2

xp

0

1

2

xp

x = x or y

Reminder : jthmeasijiji

thmeas xxWxx )()(,

2

zppx xxltheoretica 10

-1

=

2

2

110

100

xxx

xxx

ppp

ppp

jiij

iij

x

x

jiijiij

iijij

xzW

xW

p

p

zzWzW

zWW

1

0

o This gives the following equation :

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Linear fit method : expected resolution

σp0x (mm) σp0y (mm) σp1x (mrad) σp1y (mrad)

1 GeV 2.4 2.8 58 60

2 GeV 2.4 2.8 48 50

3 GeV 2.3 2.7 41 44

5 GeV 2.2 2.5 35 37

z-220 mm

x,y

*

-197.6, 1st Si layer

p

p0

p1σp0

σp1

o Angular resolution decrease when E increase :more layers on the back better constraintso Position resolution decrease as well when E increase : more hits better estimate of the average positiono Position resolution is higher in y , why ???????

Best case : if all layers

!!+ tracker resolution!!!! in data !!

Best case : if all layers

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Position and angular resolutionobtained on an event by event basis

Therefore have to solve it event by event.

jiij

iij

x

x

jiijiij

iijij

xzW

xW

p

p

zzWzW

zWW

1

0

To solve this, need to take into account only layers i and j with hits remove layers with no hit from error matrix, then invert to have W matrix.

Equation to solve :

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Results event by event for parameter resolution matrices

xp1

yp1

xp0

yp0

=

2

2

110

100

xxx

xxx

ppp

ppp

-1

jiij

iij

x

x

jiijiij

iijij

xzW

xW

p

p

zzWzW

zWW

1

0

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Result event by eventfor (p0 – p0MC)x,y

x y

Energy σp0x (mm) if all layers σp0y (mm) if all layers

1 GeV 2.6 2.4 3.1 2.8

2 GeV 2.5 2.4 2.9 2.8

3 GeV 2.4 2.3 2.8 2.7

5 GeV 2.2 2.2 2.6 2.5

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Result event by eventfor (p1 – p1MC)x,y

x y

Energy σp1x (mrad) if all layers σp1y (mrad) if all layers

1 GeV 71 58 74 60

2 GeV 54 48 56 50

3 GeV 45 41 48 44

5 GeV 36 35 39 37

March 6th, 2006 CALICE meeting, UCL 13

Consistency checks

o Pull of the distributions for Δp0 (=p0 – p0MC) and Δ p1

xp

xp

0

0

yp

yp

0

0

xp

xp

1

1

yp

yp

1

1

0.88-0.96 0.96-0.99

0.95-0.97 0.97-0.98

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With material in front of ECAL

o Beam position : (4,7,10000) mm

o Expected effect of air scattering in 10 m ~ 13 mm spread.

o Observed <x> : ~ 16 mm spread.

o Expected resolution :• The “true” position is now given by

hits in last DC layer

• σp0x = 5.2 mm, σp0y = 5.3 mm

• σp1x = 70 mrad, σp1y = 69 mrad

still correlations. Need to have the “true” position of MC particle at front ECAL face.

1 GeV

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Future work

o Study with missing layers : better to have front, middle, back ? First layers needed for position resolution, and last ones for angular resolution… but depend on energy.

Hakan + AM

o Redo everything with material in front, and truth entry point.

o Study of reconstructed tracking resolution to separate the 2 sources and allow to compare with data.

o Redo everything when realistic digitisation is available.

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Thank you for your attention