More track correction stuff ...

In the previous post I showed my initial attempt to correct the FF spectra. Up to this point I have been using the full magnitude of the exit vector for my sagitta calculations. After some discussion with the local group, I was convinced that it would be better to use the transverse magnitude of the exit vector so that all tracks have the same length (238.6 cm).

Since I am using a different track length, I will need to refind the mean and sigma for my random sagitta shift which transform the RFF spectra into the FF spectra. I follow the same procedure (described here) I used to get the original mean and sigma values. Here are the relevant plots: Positive Track Spectra, Positive Track Ratio, Negative Spectrum Decomposition, Negative Track Spectra, and Negative Track Ratio. For all the preceding plots, the top row has mean = 0.53, the middle row has mean = 0.56, and the bottom row has mean = 0.59. Each column is a different sigma value starting with 0.2 on the left and increasing by 0.05 increments until 0.4 is reached in the right column.

Based on the plots linked above, I have chosen the values: Mean = 0.59 and Sigma = 0.30.

Figure 1: This figure compares the unmodified RFF spectrum, the unmodified FF spectrum, and the RFF spectrum modified using the parameters above. The top plot is for positive charge sign tracks and the bottom plot is for negative charge sign tracks.

As before, we chan construct a scatter plot which relates the modified RFF values (which we then take to be equivalent to the unmodified FF values) to the unmodified RFF values. In this way we can relate the FF values we see to what we believe are the 'real' track pts which gave rise to them.

Figure 2: This figure shows the relationship between the modified RFF pt on the vertical axis to the unmodified RFF pt on the horizontal axis. The top plot shows the relationship for positive charge sign tracks and the bottom plot shows  the relationship for negative charge sign tracks.

As we see, for a given 'bad track' pt, there can be a number of 'good track' pts which could have given rise to it. Depending on how the correction from bad to good pt is done, the mapping may not be 1 to 1.

The first method I used to correct the FF spectra was to take the average value of the distributions above. So for each bin along the vertical axis, I find the average value of the distribution along the horizontal axis. This gives me one value of the 'correct' pt for each value of the 'incorrect' pt.

Figure 3: This figure shows the full correlation matrix as well and the distribution average I use to make corrections. The top four plots are for positive charge sign tracks and the bottom four plots are for negative charge sign tracks. The second and fourth rows are just zoom-ins of the first and third rows.

Using the uncorrected to corrected relationships shown in the right-hand column above, I can correct the FF spectra back to the RFF spectra. The result is shown below.

Figure 4: This figure compares the RFF (blue), unmodified FF (black), and corrected FF (red) spectra. The FF spectra was corrected by using the corrected vs original plots in figure 3. The top plot shows positive charge sign tracks and the bottom plot shows negative charge sign tracks.

The second method I used to correct the FF spectra was to take the most probable value of the distributions shown in figure 2. So for each bin along the vertical axis, I find the most probable value of the distribution along the horizontal axis. This gives me one value of the 'correct' pt for each value of the 'incorrect' pt.

Figure 5: This figure shows the full correlation matrix as well and the distribution MPV I use to make corrections. The top four plots are for positive charge sign tracks and the bottom four plots are for negative charge sign tracks. The second and fourth rows are just zoom-ins of the first and third rows.

Again, by using the uncorrected to corrected relationships shown in the right-hand column of figure 5, I can correct the FF spectra back to the RFF spectra. The result is shown below.

Figure 6: This figure compares the RFF (blue), unmodified FF (black), and corrected FF (red) spectra. The FF spectra was corrected by using the corrected vs original plots in figure 5. The top plot shows positive charge sign tracks and the bottom plot shows negative charge sign tracks.

Both methods for correcting the FF spectra back to the RFF spectra shown above do well below ~10GeV for the positive charge sign tracks and ~15GeV for the negative charge sign tracks. Above these cutoffs, the corrected FF spectra underestimate the RFF spectra by a good amount. This isn't too supprising, when you look at figures 3 and 5, it is evident that the average value and MPV lines I use to do the corrections leave out a lot of high pt tail tracks as you move higher in pt. In an attempt to get the contributions from these tail events and better reproduce the RFF spectra, I have implemented a third correction scheme. I again start with the correlation plots shown in figure 2, but now instead of taking the average or most probable 'good' value for each 'bad' value I look at the full distribution of 'good' values for each 'bad' value. For each 'bad' bin, I look at the distribution of 'good' values, I then find the mean and RMS of this distribution and then use those values to define gaussian distribution from which I can pull random numbers. So for a given FF track pt value, I can get a range of corrected track pt values based on the spread shown in figure 2. The 1-D, 'good' pt profiles for the first 200 'bad' track pt bins can be found here. 

Figure 7: This figure compares the RFF (blue), unmodified FF (black), and corrected FF (red) spectra. The FF spectra was corrected by using the method described in the preceding paragraph. The top plot shows positive charge sign tracks and the bottom plot shows negative charge sign tracks.