This page details my attempts to come up with a way to correct the bad spectra.

The basic idea is that the sector by sector pt spectra from the RFF part of the run are correct. I want to see if there is a way to transform the FF spectra to match the RFF spectra. For this study, I assume that the problems in the calibration in the FF part of the run cause a constant shift in the sagitta of the track and thus change the track pt.

Figure 1: Comparison of track pt spectra from RFF part of run (blue) and the FF part of the run (red) for TPC sector 20. The left column is for positive tracks and the right column is for negative tracks. The first row shows tracks which come from the high pt jet, the second row shows tracks which come from the low pt jet, and the last row shows the tracks from both jets together. The red curve is normalized to the blue curve in each pannel by the ratio of the number of counts in each spectra.

The plots for all 24 TPC sectors can be found in this pdf. The page number corresponds to the sector number.

There are several interesting / troubling features shown in figure 1:

• The difference between the RFF and FF spectra is much greater for the positive tracks than it is for the negative tracks. Sector 20 is the most pathological, but in several other sectors where there is a shift to higher pt in the positive track spectrum, the shift for negative tracks is much smaller.
  
• The difference between the RFF and FF spectra is much more pronounced for tracks which come from the high pt jet than it is for tracks from the low pt jet. It is still unclear to me why the tracks in the low pt jet seem to be unaffected.
  
• For sector 20, the negative high pt (above ~25) FF (red) counts may be positive tracks that have been distorted enough to change charge sign.

Of the issues I've listed, the second bullet point is the most troubling to me. If this really is a sector effect, then I don't understand why some tracks going into the sector are affected while others are not. This will need to be understood if we want to make a correction at the jet finder level.

As I said above, the model I'm using to transform the spectra is a constant shift in track sagitta. Given a track length and pt, the sagitta can be calculated by using the formula: S = R - Sqrt(R^2-0.25*L^2) where S is the sagitta, R is the radius of curvature which is proportional to pt, and L is the track length. So for each track, I calculate the sagitta and then add or subtract a set constant. I then take the new sagitta and back calculate the track pt.

NOTE: Because I am using the jet finder, I do not have access to all the TPC quantities. Instead of finding the track length from the first and last point, I get the track length by looking at the magnitude of the vector which points to where the track intersects a calorimeter. The point where the track intersects the calorimeter is taken to be at a depth near the SMD. In addition, the track length is calculated incorrectly for the endcap in the jet finder. Given these points, I DO NOT calculate a modified pt for tracks which: 1) Do not point at the barrel calorimeter and 2) which have a negative value of R^2 - 0.25*L^2.

I show modified spectra below. For all comparisons I am using the high and low jet combined track spectra. This corresponds to the last row in figure 1.

Figure 2: This figure shows the effect that reducing the sagitta has on positive tracks from the RFF part of the run. In each pannel, the blue line is the RFF pt spectrum, the red line is the FF pt spectrum and the black line is the RFF spectrum with the sagitta reduced a certain amount. Each pannel shows a different value for the change in sagitta, starting with 0.1 in the upper left and increasing by 0.1 in each pannel until 1.0 is reached in the bottom right.

We can see that as the ammount by which the sagitta is reduced increases, the RFF spectrum becomes more like the FF spectrum. It appears that pannels 7 and 8 show the best agreement. To get a better idea of the agreement, I have also ploted the ratios of FF spectra to the modified RFF spectra.

Figure 3: This figure shows the ratio of the FF spectra (red line in figure 2) to the modified RFF spectra (black line in figure 2) for each of the pannels seen in figure 2.

Another way to gauge the similarity of the two spectra is to look at the chi2 probability.

Figure 4: This figure shows the chi2 probability between the FF spectra and modified RFF spectra for each value of shift in sagitta. The bin number corresponds to the pannel number from figures 2 and 3.

So far I have only shown the results for the positive tracks. Figures 5 and 6 below are the negative track complements to figures 2 and 3.

Figure 5: This figure shows the effect that increasing the sagitta has on negative tracks from the RFF part of the run. In each pannel, the blue line is the RFF pt spectrum, the red line is the FF pt spectrum and the black line is the RFF spectrum with the sagitta reduced a certain amount. Each pannel shows a different value for the change in sagitta, starting with 0.1 in the upper left and increasing by 0.1 in each pannel until 1.0 is reached in the bottom right.

Figure 6:  This figure shows the ratio of the FF spectra (red line in figure 5) to the modified RFF spectra (black line in figure 5) for each of the pannels seen in figure 5.

We see that for the positive tracks, shifting the sagitta does a pretty good job at recreating the FF spectrum whereas it does a less satisfactory job for the negative tracks.