Discussion

Testing the single shower PSF fitter we realized that we are facing several difficulties in determining the correct shape of the shower:

• - by nature, conversion of the EM energy in the calorimeter is a stochastic process: we can deduce only the average shower shape. Then we can compare fit-data difference with the "noise", and accept or reject the result.
• - finite width of the block: we sample our function at the center of the block to compare it with the block energy. This is OK for a function that is changing slowly across the block, but might be misleading for rapidly changing functions. We can take care of this by replacing the value of the function at the center of the block by its integral over the block area. If the function is not analytically integrable we can use 2-nd, or a higher even-order expansion of the integral (in any integral expansion around the center of the bin only even terms survive). We found out that these corrections are very important for the Gaussian with small width compared to the block size. If a function is not analytically integrable, or if it is too complicated to find higher order corrections, numerical integration might be necessary.
• - the shape of the shower is not radially symmetric with respect to the shower center, but rather distorted in the direction defined by impact angle $(\theta, \phi)$. This can be taken into account by introducing an elliptical PSF function. However, this again complicates the fit if numerical integration is required.

This single Gaussian study shows that a single shower fitter is stable when $R$ is fixed. The fitted shower energy is biased by more than $\approx 10\%$ for higher energy showers. The bias results from the fact that we fit a sharply peaked distribution with a single Gaussian. The bias remains even when we integrate the fitting function across the block area. The bias itself should not be a problem; it can be removed by proper calibration of the fitter. A bigger problem is the 2-block dominance in the shower energy distribution of MC showers. It might arise as a consequence of low light attenuation along the LGD blocks. We expect that real data will have a broader distribution because of higher attenuation. The next step is to look at real data before we fine-tune PSF.