Shower Function Tuned to Geant Simulation with Incident Angle Effect

 

Previously, we found that the relatively small incident angle in the FPD at middle position (~0.05 radian) affects the X-position resolution enough that it becomes an issue in calculating the two photon separation. This is one of the main obstacles in calibrating the detector within a couple of percents using Pi0 mass.

Link to previous discussions 1 

Link to previous discussions 2 

The default shower function that we use seems to be a compromise between X and Y, with FPD at some position that is probably not too different from where it was in run6. With the default parameters, we see structures in both X and Y, but fiddling with the parameters tells us that while it's easy to improve the Y resolution, that tends to make the X even worse.

Consequently, we set out to fix the shower function in a way that improves both X and Y. The most direct way to correct the shower function is to make a distribution of energy deposition in a cell as a function of the distance from the cell to the photon. This is the basic distribution that is actually measured, and used for the fitting. By tuning the parameters against this distribution, we can constrain the shower function uniquely. 

In the Y-direction, the shape of the distribution is well described using the existing shower function with somewhat different parameters. Unfortunately, the shape of this distribution in the X direction turned out to be asymmetrical in a rather nasty way. I tried to use various scaling functions to take the Y-shape and transform it to X-shape, but after two days of trying I was not able to find such a function that is analytic and sensible. 

I've also tried a very simple "South Park" projection (a term coined by Steve) where I take multiple copies of the shower function and add one on top of the other with varying weight and X-offset. This would work if X-Y profile of the shower is largely de-coupled to the Z-profile. So not surprisingly, it did not work either.  

As a result, it became clear to us that we needed to understand the longitudinal profile of the shower, as well as how it couples to the transverse profile. While there are certainly a better, more legit way to look into this issue, I decided to try the simplest thing I can do with my very limited understanding of the GSTAR programming language, as following. 

1. Turn the ES FPD inward to make the center of it projective.

2. Vary the length (depth) of the pb glass cells from 5cm to the full 45cm in 5cm intervals. The radiation length for our glasses is about 2.5cm, so this allows me to see how the shower develops in 2 radiation lengths interval.

3. Throw single photons of energy 40GeV at the central region of the FPD. 100,000 photons were thrown for each length of the glass.

4. The result is the shower developed from the face of the glass to a certain point. This is not nearly as nice as having an actually Z-segmented detectors, as I lose all information on the correlation in longitudinal shower development, but at a distribution level, I get to see what happens in Z. The hope was that this would be useful enough. (And it pretty much was)

 

Fig. 1. Example of a data set with pb glass length set at an intermediate value. 

 

The "integrated" shower development shows that, as expected, the transverse profile gets increasingly wider as we go deeper into the detector. Also noteworthy is the fact that at the beginning, the "head" is quite flat meaning that the energy deposition doesn't change much just off the shower axis. In all cases, the existing shower function can be made to work reasonably well with new parameters. The shower max is estimated (based on simulation) at ~17cm from the face at this energy.

 

Fig. 2. Energy Deposited In a Cell vs. Distance from Cell to Photon (cm), from the detector face to a point in 5 cm interval.  

 

Fig. 2a. Profile Plot of Energy Deposited In a Cell vs. Distance from Cell to Photon (cm), from the detector face to a point in 5 cm interval. 

 

The default shower shape uses three copies of the shower function, two with large positive weights and one with small negative weight. It seems that the slices before the shower max does require this third negative piece to fit properly, while after the shower max the two positive pieces are enough.

Next, I take these profile plots, and simply take the difference between the two consecutive distributions. This is a crude way to see how much energy is deposited on average in each 5cm interval of pb glass. I was not sure how to calculate the errors, as these plots are in some sense correlated, so I simply added the errors. (This isn't that crucial since I'm just doing shape fitting)

 

Fig. 3. Average Energy Deposited In a Cell vs. Distance from Cell to Photon (cm) within a 5cm thick segments, from 5 to 45cm in 5cm interval

 

Again, the existing shower function works fine, with pre-shower-max slices requiring the third negative piece. 

Once the parameters are decided for each slice, we can look at the Z-profile given by these sets of functions. Here, I assume that the photon hit the center of a cell, and only look at the energy that would be in the high tower and the 4 first neighbors. Using the functions in figure 3,

Etot = E(0cm) + 4 * E(3.81cm)

Fig. 4. Z-profile of the shower

 

From this, we do the more intelligent version of the South Park projection again. Except 9 slices are a bit too much, so I trimmed it to 4. This was a first guess, but it seems to work pretty well. I used the left hand plot of figure 4 to calculate the average Z-offset for each slice. The slices are,

Slice 1: 0~10cm, first 4 radiation length, at 6.73cm from the face.

Slice 2: 10~20cm, radiation length 4~8, including the shower max at ~6.5 radiation length, at 15.1cm from the face.

Slice 3. 20~30cm, radiation length 8~12, at 24.55cm from the face.

Slice 4. 30~45cm, radiation length 12~18, at 35.6cm from the face.

 

Fig. 5. Average Energy Deposited In a Cell vs. Distance from Cell to Photon (cm) for the 4 slices used for projection

 

With these, we make a new shower function that includes X and Y incident angles as its parameters. Each slice receives different X and/or Y offset calculated from the incident angle, (which can be set photon by photon) and the Z-offset listed above. The slices are then added together to make up the final shape upon which the Minuit fitting is done. 

 

Fig. 6. Average Energy Deposited In a Cell vs. Distance from Cell to Photon (cm) in Y direction, with very little incident angle. Red is the new shower function.

 

 

Fig. 7. Average Energy Deposited In a Cell vs. Distance from Cell to Photon (cm) in X direction, at the center of the FPD at its run6 position.

 

 

Fig. 8. Average Energy Deposited In a Cell vs. Distance from Cell to Photon (cm) in X direction, with the FPD pushed back against the wall at ~70cm from the beam. The illuminated cells were about 80cm from the beam, with shoer max at Z=815cm

We see that at this level, the 4 slice projection is starting to show its limits. But even so, it should still be a big improvement over the default shape. This doesn't quite cover the edge of the FMS, but close. 

 

Now the following are the results from single photon, flat distribution GSTAR only simulations.

 

Fig. 9. Reconstructed vs. Generated Position, with the default shower function

 

 

Fig. 10. Reconstructed vs. Generated Position, with the new shower function

 

 

Fig. 11. Reconstructed vs. Generated Position in 1D, with the default shower function

 

 

Fig. 12. Reconstructed vs. Generated Position in 1D, with the new shower function