Transverse Momentum Resolution

One question of importance is by how much to shift the reconstructed jet p_T to obtain the particle or parton-jet p_T. One way to obtain this estimate is to look at the difference in reconstructed and thrown p_T in embedding. Another is to examine the true p_T distribution of jets reconstructed in a particular p_T bin. In the end the latter is probably the path I will pursue.

In the present case, I have matched reconstructed jets to "parton" jets. "Thrown" p_T is the p_T of the "parton" jet. The thrown p_T distributions are quite asymmetric. To estimate the mean and σ I have fit the distributions with a skew normal distribution:

The mean and width are then calculated by

and

To estimate the errors on the mean and width I have only taken the uncertainties on the first fitted parameter in each expression, as there is undoubtedly some correlation between the parameters which would be difficult to isolate for proper error propagation.

For the two of the three highest p_T bins I have fixed the α parameter to be zero. The fitting was returning α's within 1σ of zero for these bins; and to get a sensible error it was simpler to fix the α parameters in the three relevant bins.

As a reminder, I use p_T bins defined as

const double xbins[nbins+1] = { 5.0, 6.0, 7.1, 8.4, 9.9, 11.7, 13.8, 16.3, 19.2, 22.7, 26.8, 31.6, 37.3, 45., 55., 65., 80., 100., 250. };

Figure 1

The fitted means are plotted in black circles, while the simple arithmetic mean from the histograms is plotted in blue triangles. For lower p_T, the values are quite similar, deviating slightly at the three highest p_T bins. The resolutions dip from bin-7 to bin-9, likely, a reflection of the lack of lower partonic-p_T bins. I expect all of the means will shift lower from the "rolling downhill" effect when the low-partonic-p_T embedding is included. The two highest p_T bins appear to be rather thoroughly dominated by misreconstructed jets. It may, thus, make sense to consider excluding these bins from subsequent analysis. For reference, attached to this blog are plots from the individual fits.

Figure 2

It is also interesting to consider the p_T shifts event-by-event. For a reference, Fig. 2 shows the δp_T vs. p_{T, reco} with the arithmetic means of the bins plotted as black circles. One can see that the means fall below 0 until rather high p_T.

Figure 3

Figure 3 shows δp_T vs. p_{T, reco}-bin using the skew normal fitting and related definitions of mean and width, now for δp_T rather than p_{T, thrown}. One sees the resolution degrading at higher p_T. Additionally, one sees δp_T trend more positive with higher p_T.

Azimuthal Resolution in η-Bins

In a Sivers measurement it is crucial to have reliable azimuthal resolution. Our detectors tend to have good resolution in azimuth, but it is good to confirm in embedding that there are no surprises, here. Additional checks can be made in data, e.g. using weighted events to verify the absence of significant distortion from finite binning coupled to non-uniform acceptance.

To estimate the resolution in φ I have separated matched jets into bins of reconstructed η, φ_S, and p_T. For a first look, I integrate over φ_S and examine δφ_S as a function of p_T, reco. Within bins of p_T I fit the distribution of δφ_S with a Gaussian. The figures show the extracted centroids and σ's. I have attached PDF files of the individual fits. The χ² of the fits is not very good, but in general the distributions do appear to be fairly Gaussian in shape.

One concern in investigating resolutions in this method is whether or not the sigma; of the fit sufficiently accounts for the resolution degredation, e.g. at the base of the peak, where the distribution is no longer Gaussian. In my dissertation I simply used the RMS of the distribution. Given that, now, I have substantially more data, perhaps the RMS approach is not the most efficient.

φ_S-Resolution vs. p_T-Bins

Figure 4: -1 < η < -0.5

Figure 5: -0.5 < η < 0

Figure 6: 0 < η < 0.5

Figure 7: 0.5 < η < 1

The figures show centroids fluctuating around zero across all p_T-bins. The χ² values tend to indicate that the uncertainties are under-estimated, however, the fitted values are quite small with flucutations within 0.01 radians. The σ's tend to start out around 0.04 at the lowest p_T bin and decrease steadily with high p_T. This is not so surprising as the tracks tend to become straight-lines at high p_T. For high-p_T jets, the limiting factor is thus the track p_T resolution rather than the φ_S resolution.

φ_S-Resolution vs. φ_S-Bins

It is also worthwhile to examine the azimuthal resolution in bins of the Sivers angle. Large fluctuations could indicate an acceptance effect. Such a study could complement and inform the data-driven analysis of weighting histogram yields to observe finite binning coupled to non-uniform distribution within the bins. As an overview I post figures of the azimuthal resolution integrated over p_T.

Figure 8: -1 < η < -0.5

Figure 9: -0.5 < η < 0

Figure 10: 0 < η < 0.5

Figure 11: 0.5 < η < 1

The figures show centroids fluctuating around zero across all p_T-bins. The χ² values, again, indicate that the uncertainties are under-estimated. Again, however, the fitted values are quite small with flucutations within 0.01 radians. The σ's tend to be flat around 0.03. This is undoubtedly correlated to the average p_T, e.g. cross-reference with earlier figures. A quick eye-ball scan of the various p_T bins suggests the resolution is constant in φ_S within the various p_T bins.

If I take the RMS of the distributions within the range -π/4 < δφ_S < π/4 I find a resolution that begins around 0.06 and decreases with increasing p_T. In one case, I find a resolution that peaks around 0.08. To get an idea of the effect on the Sivers asymmetry, I convoluted a sine function with a Gaussian of width equal to the resolution of 0.08. The resulting distribution represents a sine function damped by a factor of 0.9968.