Invariant Mass Distribution

The two-photon invariant mass distribution can be roughly broken up into four pieces, seen below^*.

Fig. 1

The black histogram is the invariant mass of all pion candidates (photon pairs) with pt in the given range. I simulate each of the four pieces in a slightly different way. My goal is to understand each individual piece of the puzzle and then to add all of them together to recreate the mass distribution. This will ensure that I properly understand the backgrounds and other systematic errors associated with each piece. To understand how I simulate each piece, click on the links below.

1. Pion Peak

2. Eta Peak

3. Low Mass Background

4. Combinatoric Background.

Once all of the four pieces are properly simulated they are combined to best fit the data. The individual shapes of are not changed but the overall amplitude of each piece is varied until the chisquared for the fit is minimized. Below are plots for the individual bins. Each plot contains four subplots that show, clockwise from upper left, the four individual peices properly normalized but not added; the four pieces added together and compared to data (in black); the ratio of data/simulatio histogramed for the bin; and a data simulation comparison with error bars on both plots.

Bin 1: 5.2 - 6.75 GeV/c

Bin 2: 6.75 - 8.25 GeV/c

Bin 3: 8.25 - 10.5

Bin 4: 10.5 - 16.0

Below there is a table containing the normalization factors for each of the pieces for each of the bins as well as the total integrated counts from each of the four pieces (rounded to the nearest whole count.)

Table 1: Normalization factors and total counts

bin	low norm.	low integral	pion norm.	pion integral	eta norm.	eta integral	mixed norm.	mixed integral
1	121.3	9727	146.4	75103	20.91	5290	0.723	44580
2	77.34	4467	77.81	51783	20.86	6175	0.658	34471
3	40.13	3899	29.41	23687	12.93	6581	1.02	18630
4	5.373	1464	5.225	8532	2.693	3054	0.521	6276

Table 2 below contains, for each source of background, the total number of counts in the mass window and the background fraction [background/(signal+background)].

Table 2: Background counts and Background Fraction

Bin	Low Counts	Low B.F (%)	Eta Counts	Eta B.F (%)	Mixed Counts	Mixed B.F (%)
1	2899	3.60	1212	1.50	4708	5.84
2	2205	3.96	917	1.65	3318	5.96
3	2661	9.29	633	2.21	1507	5.26
4	858	8.56	170	1.70	591	5.89

_{* Note: An astute observer might notice that the histogram in the top figure, for hPtBin2, does not exactly match the hPtBin2 histogram from the middle of the page (Bin 2.) The histogram from the middle of the page (Bin 2) is the correct one. Fig. 1 includes eta from [-1,1] and thus there are more total counts; it is shown only for modeling purposes.}

Combinatoric Background

The last piece of the invariant mass distribution is the combinatoric background. This is the result of combining two non-daughter photons into a pion candidate. Since each photon in an event is mixed with each other photon in an attempt to find true pions, we will find many of these combinatoric candidates. Below is a slide from a recent presentation describing the source of this background and how it is modeled.

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As it says above we take photons from different events, rotate them so as to line up the jet axes from the two events, and then combine them to pion candidates. We can then run the regular pion finder over the events and plot out the mass distribution from these candidates. The result can be seen below.

These distributions will be later combined with the other pieces and normalized to the data. For those who are interested in the errors on these plots please see below.

Eta Peak

I treat the eta peak in a similar way as the pion peak. I throw single etas, flat in Pt from 2 - 25, and reconstruct the two-photon invariant mass distribution for the results. The thrown etas are weighed according to the PHENIX cross-section as outlined here. The mass distributions for the four pt bins can be seen below. (I apologize for the poor labeling, the x-axis is Mass [GeV/c^2] and the y-axis is counts.) Don't worry about the scale (y-axis.) That is a consequence of the weighting. The absolute scale will later be set by normalizing to the data.

These plots will later be combined with other simulations and normalized to the data. The shape will not change. For those interested in the errors, that can be seen below.

Low Mass Background

The low mass background is the result of single photons being artifically split by the detector (specifically the SMD.) The SMD fails in it's clustering algorithm and one photon is reconstructed as two, which, by definition, comprises a pion candidate. These will show up with smaller invariant masses than true pions. Below is a slide from a recent presentation that explaines this in more detail and with pictures.

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We can reproduce this background by looking at singly thrown photons and trying to find those that are artificially split by the clustering algorithm. Indeed when we do this, we find that a small fraction (sub 1%) do indeed get split. We can then plot the invariant mass of these pion candidats. The results can be seen below. (x-axis is mass in GeV/c^2.)

These mass distributions will later be combined with other pieces and normalized to the data. For those interested in the errors on these histograms please see below.

Pion Peak

To study the pion peak section of the invariant mass distribution I looked at single pion simulations. The pions were thrown with pt from 2 - 25 GeV/c flat and were reconstructed using the cuts and parameters described in the cuts, etc. page. The mass of each reconstructed pion is corrected by adding a small amount of energy to each photon (as outlined here.) After this correction the peak of the reconstructed mass distribution is aligned with the peak of the data. The mass distributions from the four bins can be seen below.

Later, these peaks will be normalized, along with the other pieces, to the data. However, the shape will not change.

If you are interested in seeing the errors on the above plots, I reproduce those below.