leun's blog

In the last analysis meeting, I reported that the biggest uncertainty in photon measurement comes form the uncertainty in the relative energy scale between one and two photon events. After a more careful investigation, I found the following.

1. In the simulation, the single photon energy is always a few percent more overestimated compared to the pi0 events at the deposited energy level. THe degree of overestimation seems more or less constant as a function of energy, and is about 3~4%.

Fig. 1. Deposited energy / thrown energy, 1 photon vs 2 photon events

2. The 1 photon vs 2 photon DIFFERENCE in ratio of deposited and reconstructed energy is consistent between data and simulation, suggesting that the shower shape discrepancy is not causing a huge problem in terms of reconstructing 1 vs. 2 photon events.

Fig. 2. Reconstructed energy / deposited energy, simulation vs. data.

First, we look at the cause of overestimation for the single photon events. We have found that there is a real energy dependent gain shift due to the Cherenkov photon propagation in am attenuated medium, and we have corrected for this effect by using a polynomial gain correction factor as a function of energy.

But function of what energy? For the pi0/eta analysis, it was a function of di-photon summed energy, which is really not quite right as the attenuation depends on the energy of individual photons. But since Zgg distribution is well understood and averaged out, that was good enough.

This is clearly not the case for the photon anlaysis, and in fact we would expect a 50GeV photon to have more gain shift (to higher gain) than an average 50GeV pi0, since the shower max for the photon would be deeper into the detector. By how much? To estimate this, I ran a toy simulation where I compare pi0s with flat Zgg distribution up to Zgg<0.85, which is our Zgg cut, and single photons. I use 12% energy smearing, and three different functions for the energy dependent gain shift. I simply calculate the estimated observed energy based on the given energy smearing and gain shift for both pi0 and gamma, and compare them.

Fig. 3. Result of toy simulation. LEFT: Three different functions for energy dependent gain shift, RIGHT: estimated observed energy for pi0 / gamma.

The black function is the estimated gain shift based on the simulation, and it is the function used for the correction in both data and simulation for pi0/eta analysis. The blue and red are chosen arbitrarily to estimate the effect of mis-simulating the energy dependent gain correrction. Given the data-MC match on other aspects, those seem to be good, conservative estimates.

So we see that first, the observed few percent difference in gain between 1 and 2 photon events can be mostly understood in terms of the energy dependent gain shift. Secondly, having somewhat more or less energy dependent gain shift in data compared to the simulation would result in over/under-estimating 1 photon vs. 2 photon gain by around ~1%.

Based on the black curve, I applied constant 3% correction to the 1 photon gain relative to the 2 photon gain by brining 1 photon energy down by 3% regardless of the photon energy. After this, I compared the ratio between the peak location of reconstructed/thrown energy distribution in energy bins, for pi0 and gamma.

Fig. 4. Reconstructed / thrown energy distribution for 1 photon after correction, 40~70GeV, 5GeV bins.

Fig. 5. Reconstructed / thrown energy distribution for 2 photon, 40~70GeV, 5GeV bins.

Fig. 6. Pi0 / gamma ratio

CONCLUSION: Reducing 1 photon gain by 3% and using the same shape for the energy dependent gain shift is good enough to match the relative energy between 1 and 2 photon events in the simulation. From here, we can only assume that this will work for the data as well, and I add 1% scale uncertainty (based on figure 3) between 1 and 2 photon events for the data into the systematics for both cross-section and AN measurements.

NOTE: I have also verified that the above result is not sensitive to the small mismatch in energy slope between data and simulation. None of this is very sensitive on the bin rolling down effect.

Now that I can fix the energy scale between pi0/eta and gamma, I now estimate the background fraction for the photon sample coming from pi0 and eta.

This is done based on the previous measurement of pi0 and eta X-section. I use the energy weighted full simulation, the same ones used for the pi0 and eta analysis, to obtain the "pi0 in 8GeV true energy bin to pi0-originated photon background in 8GeV true energy bin". By applying this to the efficiency corrected number distribution of pi0 and eta, I get the pi0 and eta background.

The caveat is that while this can most easily be obtained in 8GeV true energy binning, since that is what's used for the X-section measurement, I also need it in 5GeV measured energy binning for the AN measurement. What I did was to calculate the "8GeV true energy bins to 5GeV measured energy bins" smearing matrix based on the same full simulation, and use it to smear the 8GeV result into 5GeV bins. This obviously adds more systematic errors, but I think this is better than redoing the cross-section measurement in 5GeV binning from scratch. (We went with 8GeV binning instead of 5GeV due to the size of energy smearing in the first place.)

Fig. 7. Smearing matrix for 8GeV to 5GeV bin conversion. Each element has error associated with it.

All of thess results have an added systematic error coming from 1% scale uncertainty between 1 and 2 photon events.

Fig.8. Pi0 and eta background contribution in 8GeV true energy bins

Fig.9. Pi0 and eta background contribution in 5GeV measured energy bins

As can be seen, the pi0 background is oddly flat as a function of energy. While the smoothness is largely the efffect of unfolding, the fact that it is so constant, as far as I can see, is a coincident. I cannot think of anything that forces it this way, and the relative energy scale fix between 1 and 2 photon events happened to result in such flat distribution.

Fig. 9. Cross-section for direct photon, and directo photon / pi0 ratio

Again, the photon to pi0 ratio is very flat. This means that essentially, our single photon sample has a very similar energy slope compared to the di-photon sample, after we correct for the energy scale difference between the two. What does this mean?

Fig. 10. Direct photon AN using data-driven background correction