1. ECAL Energy Correction for Dead Cells_August
Jie Feng, Corrine Goy, Li Tao
LAPP, Annecy

2. 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)

3. 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.

4. 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.

5. 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].

6. 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.

7. 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:

8. 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.

9. 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

10. Parameterization
Example of the Cell_34 and Layer_7:

11. Parameterization Idea:
These 4 parameters is correlated with the Layer number L and the shower energy E. Thus,

12. Parameterization
The fitting parameters change while the Layer number changes. For the 100GeV Test Beam electron, we get:

13. Parameterization
Results:

14. 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.

15. 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:

16. 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

17. 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.