# 2006 EEMC Neutral Pions: Systematics (Method of Calculation)

There are multiple ways to calculate AN. I prefer the cross-ratio method as it takes relative luminosity as well as detector acceptance out of the picture. However, given some of the odd behavior we're seeing it is probably a good idea to examine the asymmetries using another procedure. This could provide a nice sanity check of the methodology, as well as perhaps illuminate some methodology-dependent systematics.

For this study, I have chose to examine the traditional way to calculate AN. Namely, I look at

AN = (1/Pbeam)×(N-N)/(N+N).

All this requires is to separate the yields into spin states as a function of φγγ (where φγγ = π/2 - φS). The sum over the difference should reveal, in this case, a cos(φγγ)-dependence. By leaving the histograms in terms of φγγ it also proves a check on my calculation of φS.

## Figure 1: AN vs. xF (xF > 0)

Cross-ratio Method
xF Bin 1
xF Bin 2
xF Bin 3

In Fig. 1, I show a comparison of the traditional method to the cross-ratio method for the asymmetries as a function of xF for xF > 0. One can see that the methods provide completely consistent answers. Additionally, one sees that the p0's from the traditional method are equivalent to the luminosity asymmetries from the cross-ratio method. This is indeed as it should be and provides another vote of confidence for our methodology.

## Figure 2: AN vs. xF (xF < 0)

Cross-ratio Method
xF Bin 1
xF Bin 2
xF Bin 3

In Fig. 2, I show a comparison of the traditional method to the cross-ratio method for the asymmetries as a function of xF for xF < 0. Again the methods provide completely consistent answers for both the physics and luminosity asymmetries.

At this point, I will revert back to calculating the asymmetries in terms of φS. As I stated earlier, they are simply offset from each other by π/2. Having established that we have calculated φS robustly, it should be less confusing, now, to have consistent notation.

## Figure 3: AN vs. mγγ (xF > 0)

Cross-ratio Method
0.0 < mγγ < 0.1 GeV/c2
0.1 < mγγ < 0.2 GeV/c2
0.2 < mγγ < 0.3 GeV/c2
0.3 < mγγ < 0.4 GeV/c2
0.4 < mγγ < 0.5 GeV/c2
0.5 < mγγ < 0.6 GeV/c2

Figure 3 shows the asymmetries as a function of invariant mass for xF > 0. Again, the methods are consistent. Note that in the mass bin corresponding to the π0 the methods are entirely equivalent within statistics and differ numerically by only 3.1% (0.035σ).