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ECAL Energy Correction for Dead Cells_August
Jie Feng, Corrine Goy, Li Tao LAPP, Annecy
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MissingEnergy= F(x, L, E)
Principle We assume that the missing energy of the dead cell is a function of the distance of the center of gravity from the dead cell(x), the Layer number(L) and the Energy of the shower(E), as shown in the following expression: MissingEnergy= F(x, L, E) Our purpose is to find the formula of F(x, L, E)
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Current Method Currently, the correction of the ECAL dead cell energy is: b=1/2(a+c) Where b is the energy of the dead cell and a and c are the energies of the neighboring cells. This formula works well except at the cell of the center of gravity.
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Current Method If the center of gravity of 1D Cluster happens to be at the dead cell, the missing energy will be about 75% of the layer, as shown in the Figure below. The correction formula above can correct 12% of the energy so about 60% of that will still be missing.
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Simulation of a dead cell
The TProfile of deposited energy without dead cells can be found on the left, while that with a dead cell can be found on the right. We simulate that the dead cell is located at the cell[34] of the Layer[7].
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Subtract the profile with a dead cell from that without any dead cell and we get the distribution of the missing energy, as shown in the figure below.
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A Direct Correction(Path 2)
I study the Radial shower profile in the article by Grindhammer and Peters ( , formula (23)) : I integrate it along y and get:
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A Direct Correction(Path 2)
I use the above function to fit the Missing Energy and get: It seems to be better than the one fitted by the Double Gaussian.
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Parameterization For the function:
Where x is the distance between the dead cell the and shower center, which is computed by the cell ratio method, 2p is the normalization of the core, R_C is the RMS of the core, 2(1-p) is the normalization of the tail and R_T is the RMS of the tail. core tail
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Parameterization Example of the Cell_34 and Layer_7:
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Parameterization Idea:
These 4 parameters is correlated with the Layer number L and the shower energy E. Thus,
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Parameterization The fitting parameters change while the Layer number changes. For the 100GeV Test Beam electron, we get:
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Parameterization Results:
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Parameterization I also analyze those parameters with different Energies. 20GeV positrons, 80GeV positrons, 100GeV electrons, 120GeV electrons and 180GeV electrons. The graph below is the parameter N_1(L) for different energies.
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Parameterization Comparing the 5 curves, we can analyze the relation between the parameter and the Energy. For example, the graph below shows the analysis of p0. Thus, we can roughly get:
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Conclusion As far as I can do, the deposited energy of a dead cell should be a function of distance from the Shower Center x, the Layer number and the energy of the particle E. Thus, where
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Comparison with the old correction
The TProfile with the old correction(ADC[i]=(ADC[i-1]+ADC[i+1])/2) is shown in the left, while that with the new correction is shown in the right.
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