Calibration software for the HADES electromagnetic calorimeter (EMC)

Calibration software for the HADES electromagnetic calorimeter (EMC)

Dimitar MihaylovExcellence Cluster ‘Universe’, TU Munich

HADES collaboration meeting XXVGSI, November 2012

2

Overview

The proposed electromagnetic calorimeter (EMC)

Data analysis

Motivation for a calibration procedure

Calibration procedure – Mathematical model

Calibration procedure – Realization

Summary

Ongoing improvements and outlook

3

Overview of EMC

Parameters and benefits:• Identification of light neutral pseudoscalar mesons (π0 and η).

• Good e/π-separation at high momenta.

• Six sectors, each containing 163 lead-glass blocks.

• High refractive index of the lead-glass blocks (n = 1.708).

• Radiation length (X0) of 2.51 cm.

• Almost full azimuthal coverage.

• The polar angle is covered between 12˚ and 45˚.

• The energy resolution is 𝜎𝐸 𝐸Τ = 6% ඥ𝐸 𝐺𝑒𝑉ΤΤ . Czyzycki et al., arXiv:1109.5550v2

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Data analysis

Simulation

Reconstruction Output data(reconstr. γ)

Invariant massspectrum (IMS)

Input data(initial γ)

Experiment

• Unaccounted energy losses.

• Undetected photons.

• Reconstruction of a non-existing photon (false signals).

• Large combinatorial background in the IMS (polynomial part of the fit).

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Overview of the calibration procedure

Fitting the IMS

Calibration of single γ

Final IMS𝑴𝝅 = 𝒎𝝅 ? ?

𝑝Ԧ𝐶 = 𝑀𝑝Ԧ, The general form of the calibration procedure is:

where M is a calibration matrix.

A calibration procedure capable of compensating for those inaccuracies is needed.The calibration procedure will be performed on individual photons.

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Is only energy calibration enough?

• The error in the energy is significantly larger compared to the error in the polar and azimuthal angles.

Distribution of ES/E Distribution of θS/θ Distribution of ϕS/ϕ

• Only spherical coordinates will be used.

• Comparison between the initially simulated and reconstructed values of the components of the photon momentum.

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Choice of calibration function

The general form of the calibration function is:

The exact form of the expressions Fi can be determined by reconstructing simulated data and analyzing the resulting deviations.

𝑓ሺ𝐸,𝜃,𝜑ሻ= 𝑒𝑥𝑝൝ 𝐴𝑖𝐹𝑖(𝐸,𝜃,𝜑)𝑖 ൡ

D.J. Tanner, MSc thesis, University of Manchester, Oct 1998

The general form of the calibration procedure is:

𝐸𝐶 = 𝐸𝑓ሺ𝐸,𝜃,𝜑ሻ

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Mathematical model

𝝀𝒊 ≔ 𝟏𝟐ሺ𝑭𝟏𝒊 + 𝑭𝟐𝒊ሻ • Definition of some variables for photon pairs

𝝆≔ 𝒍𝒏𝑴𝜸𝜸 − 𝒍𝒏𝒎𝝅 = 𝒍𝒏𝑴𝜸𝜸𝒎𝝅

• The calibration is performed by minimizing the function

𝓛≔ ൣ�𝒍𝒏𝑴𝜸𝜸𝒋− 𝒍𝒏𝒎𝝅൧𝟐𝑵

𝒋=𝟏 = 𝝆𝒋𝟐𝑵

𝒋=𝟏

𝑹𝒊 ≔ 𝟏𝟐 𝝏𝓛𝝏𝑨𝒊 = 𝝆𝒋𝝀𝒊𝒋𝑵

𝒋=𝟏 = 𝟎

𝑀𝛾𝛾 = ඥ2𝐸𝐶1𝐸𝐶2(1− 𝑐𝑜𝑠𝜂)

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Energy dependence

𝐸𝑓𝑖𝑡1 = 𝑒𝑥𝑝ሺ𝑃0 + 𝑃1𝑙𝑛𝐸+ 𝑃2𝑙𝑛2𝐸ሻ with 𝜒2 𝑁𝐷𝐹Τ = 5.34

𝐸𝑓𝑖𝑡2 = 𝑒𝑥𝑝ሺ𝑃0 + 𝑃1𝑙𝑛𝐸+ 𝑃2𝑙𝑛2𝐸+ 𝑃3𝑙𝑛3𝐸ሻ with 𝜒2 𝑁𝐷𝐹Τ = 2.96

The energy dependence of the calibration function can be well described with function of the type:

𝑓𝐸ሺ𝐸ሻ= 𝑒𝑥𝑝൝ 𝑃𝑖𝑙𝑛𝑖𝐸)𝑖 ൡ

𝑓ሺ𝐸,𝜃,𝜑ሻ= 𝑒𝑥𝑝൝ 𝐴𝑖𝐹𝑖(𝐸,𝜃,𝜑)𝑖 ൡ

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Result of the calibration procedure

The results below are obtained with the following calibration functions: 𝑓1 = 𝑒𝑥𝑝ሺ𝐴0 + 𝐴1𝑙𝑛𝐸+ 𝐴2𝑙𝑛2𝐸ሻ, 𝑓2 = 𝑒𝑥𝑝൛𝐴0 + 𝐴1𝑙𝑛𝐸+ 𝐴2𝑙𝑛2𝐸+ 𝐴3𝑙𝑛3𝐸+ 𝐴4𝑐𝑜𝑠൫𝜃෨𝜔𝜃𝑐𝑜𝑠𝜑 − 𝜃𝑠ℎ𝑖𝑓𝑡൯+ 𝐴5𝑐𝑜𝑠൫𝜑𝜔𝜑𝑠𝑖𝑛𝜃− 𝜑𝑠ℎ𝑖𝑓𝑡൯ൟ, 𝑓3 = 𝑒𝑥𝑝ሺ𝐴0 + 𝐴1𝑙𝑛𝐸+ 𝐴2𝑙𝑛2𝐸+ 𝐴3𝑙𝑛3𝐸ሻ.

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Accuracy

Example of the calibration functions f1, f2 and f3 as a function of the energy.

Error in the energy of single photons before and after calibration.

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Accuracy

Distribution of the error in the energy before and after calibration.

Position of the η peak after calibration.

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Implementation in C++

• The described calibration procedure has been implemented in a standalone C++ program called IMS-expert.

• Only standard C++ and ROOT libraries have been used.

• IMS-expert allows the user to perform the calibration with different types of functions.

• After a calibration has been performed, the calibration parameters are saved and can be used for further calibrations of different data.

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Summary

• The presented calibration procedure is able to accurately correct the IMS for invariant masses in the range of the mass of π0 (135 MeV/c2) and of η (548 MeV/c2).

• The systematic shift of the IMS is compensated by the calibration procedure but the statistical error, which is mostly related to the energy resolution, remains approximately the same.

• The error in the energy of individual photons decreases, but it still remains significant.

• The energy dependence of the calibration function has the biggest influence on the final result.

• The calibration procedure is implemented in a standalone program called IMS-expert and can be used for any EMC.

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Ongoing improvements

𝐸𝐶 = 𝐸𝑓𝐸ሺ𝐸,𝜃,𝜑ሻ 𝜃𝐶 = 𝜃+𝑓𝜃ሺ𝐸,𝜃,𝜑ሻ 𝜑𝐶 = 𝜑+𝑓𝜑ሺ𝐸,𝜃,𝜑ሻ

Look-up table approach

Analytical angle calibration

Thank You For Your Attention

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Backup slide

If not all Ri=0, then:𝑹𝒊 + 𝜹𝑹𝒊 = 𝟎

For small δRi𝜹𝑹𝒊𝜹𝑨𝒍 ≈ 𝝏𝑹𝒊𝝏𝑨𝒍 = 𝑮𝒊𝒍

𝑮𝒊𝒍 ≔ 𝝏𝑹𝒊𝝏𝑨𝒍 = 𝟏𝟐 𝝏𝟐𝓛𝝏𝑨𝒊𝝏𝑨𝒍 = 𝝀𝒍𝒋𝝀𝒊𝒋𝑵

𝒋=𝟏 𝑹𝒊 ≔ 𝟏𝟐 𝝏𝓛𝝏𝑨𝒊 = 𝝆𝒋𝝀𝒊𝒋𝑵

𝒋=𝟏 = 𝟎

𝑮𝒊𝒍𝜹𝑨𝒍 = 𝜹𝑹𝒊 −𝑮𝒊𝒍𝜹𝑨𝒍 = 𝑹𝒊 𝜹𝑨𝒍 = −𝑮𝒊𝒍−𝟏𝑹𝒊

𝑓ሺ𝐸,𝜃,𝜑ሻ= 𝑒𝑥𝑝൝ 𝐴𝑖𝐹𝑖(𝐸,𝜃,𝜑)𝑖 ൡ