This is a continuation of my track correction studies.

In my previous studies I tried to recreate the Full Field (bad) spectra by adding an offset to the track sagitta of the RFF tracks to change track pt. The constant offset did a reasonably good job at reproducing the FF positive track spectra, but not such a good job at reproducing the FF negative track spectra. In addition, we see that the spectra distortion is much greater for tracks from the high pt jet than it is for tracks from the low pt jet. The constant sagitta offset model does not appear to reproduce this behavior. I also checked to see if there was some difference between tracks from the high and low pt jets which could cause some tracks to be affected and others to be unaffected, I saw no difference between high and low jet tracks.

Since there does not appear to be a difference between tracks from the high and low jets I have to assume that whatever is causing the distortion is affecting all tracks. If all tracks are being affected, I have to find a scheme in which some tracks have a small correction. The scheme I am testing shifts the track sagitta by a random amount where the random number distribution is given by a gaussian. The figures below compare the FF spectra to the RFF spectra which have been modified according to gaussian distributions with different mean and sigma values.

Figures 1-3 below show various spectra for 15 different gaussian sagitta shifts. The mean and sigma of the gaussian for each pannel is given by the chart below:

1. Mean = 0.5: Sigma = 0.3, 0.4, 0.5, 0.6, 0.7
2. Mean = 0.6: Sigma = 0.4, 0.5, 0.6, 0.7, 0.8
3. Mean = 0.7: Sigma = 0.5, 0.6, 0.7, 0.8, 0.9

Figure 1: This figure shows the FF positive spectra (Red) and the modified RFF positive spectra (Black).

Figure 2: This figure shows the modified RFF negative spectra (Blue) and the spectra of tracks which started as positive and had their sagitta shifted enough to change charge sign (Red).

Figure 3: This figure shows the negative FF spectra (Red) and the negative modified RFF spectra (Black). The modified RFF spectra is the sum of the blue and red spectra shown in figure 2.

Plots showing the ratios of the FF to the modified RFF spectra can be seen here (positive tracks) and here (negative tracks).

Looking at figures 1 and 3 and looking at the ratio plots we see that as the sigma of the gaussian increases, the negative track agreement gets worse. For the next iteration, I will keep the same mean values but look at smaller sigma values. The mean and sigma for figures 4 and 5 below are given in the chart:

1. Mean = 0.5: Sigma = 0.1, 0.2, 0.3, 0.4, 0.5
2. Mean = 0.6: Sigma = 0.2, 0.3, 0.4, 0.5, 0.6
3. Mean = 0.7: Sigma = 0.3, 0.4, 0.5, 0.6, 0.7

Figure 4: This figure shows the FF positive spectra (Red) and the modified RFF positive spectra (Black).

Figure 5: This figure shows the negative FF spectra (Red) and the negative modified RFF spectra (Black). Again, the modified RFF spectrum is the combination of the modified negative spectrum and the positive tracks which change sign.

The positive track ratio plot can be seen here and the negative track ratio plot can be seen here. The plot showing the modified negative spectrum and the spectrum of tracks which change charge sign can be seen here.

From figures 4 and 5, it appears that agreement gets better at lower sigma values and larger mean values. I want to see how the agreement looks for higher mean values and lower sigma values, so for figures 6 and 7 below I use the following values:

1. Mean = 0.6: Sigma = 0.0, 0.1, 0.2, 0.3, 0.4
2. Mean = 0.7: Sigma = 0.1, 0.2, 0.3, 0.4, 0.5
3. Mean = 0.8: Sigma = 0.2, 0.3, 0.4, 0.5, 0.6

Figure 6: This figure shows the FF positive spectra (Red) and the modified RFF positive spectra (Black).

Figure 7: This figure shows the negative FF spectra (Red) and the negative modified RFF spectra (Black). Again, the modified RFF spectrum is the combination of the modified negative spectrum and the positive tracks which change sign.

 The positive track ratio plot can be seen here and the negative track ratio plot can be seen here. The plot showing the modified negative spectrum and the spectrum of tracks which change charge sign can be seen here.

In figure 7, we see that the spectra with mean = 0.8 do not match up well. For the next iteration I will look at means between 0.6 and 0.7 and sigmas between 0.2 and 0.4.

1. Mean = 0.63: Sigma = 0.20, 0.25, 0.30, 0.35, 0.40
2. Mean = 0.66: Sigma = 0.20, 0.25, 0.30, 0.35, 0.40
3. Mean = 0.69: Sigma = 0.20, 0.25, 0.30, 0.35, 0.40

Figure 8: This figure shows the FF positive spectra (Red) and the modified RFF positive spectra (Black).

Figure 9: This figure shows the negative FF spectra (Red) and the negative modified RFF spectra (Black). Again, the modified RFF spectrum is the combination of the modified negative spectrum and the positive tracks which change sign.

The positive track ratio plot can be seen here and the negative track ratio plot can be seen here. The plot showing the modified negative spectrum and the spectrum of tracks which change charge sign can be seen here.