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

There are multiple ways to calculate A_{N}. 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 A_{N}. Namely, I look at

A_{N} = (1/P_{beam})×(N^{↑}-N^{↓})/(N^{↑}+N^{↓}).

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

## Figure 1: A_{N} vs. x_{F} (x_{F} > 0)

Traditional Method |
---|

Cross-ratio Method |

x_{F} Bin 1 |

x_{F} Bin 2 |

x_{F} 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 x_{F} for x_{F} > 0. One can see that the methods provide completely consistent answers. Additionally, one sees that the p_{0}'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: A_{N} vs. x_{F} (x_{F} < 0)

Traditional Method |
---|

Cross-ratio Method |

x_{F} Bin 1 |

x_{F} Bin 2 |

x_{F} 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 x_{F} for x_{F} < 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 φ_{S}. As I stated earlier, they are simply offset from each other by π/2. Having established that we have calculated φ_{S} robustly, it should be less confusing, now, to have consistent notation.

## Figure 3: A_{N} vs. m_{γγ} (x_{F} > 0)

Traditional Method |
---|

Cross-ratio Method |

0.0 < m_{γγ} < 0.1 GeV/c^{2} |

0.1 < m_{γγ} < 0.2 GeV/c^{2} |

0.2 < m_{γγ} < 0.3 GeV/c^{2} |

0.3 < m_{γγ} < 0.4 GeV/c^{2} |

0.4 < m_{γγ} < 0.5 GeV/c^{2} |

0.5 < m_{γγ} < 0.6 GeV/c^{2} |

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

## Figure 4: A_{N} vs. m_{γγ} (x_{F} < 0)

Traditional Method |
---|

Cross-ratio Method |

0.0 < m_{γγ} < 0.1 GeV/c^{2} |

0.1 < m_{γγ} < 0.2 GeV/c^{2} |

0.2 < m_{γγ} < 0.3 GeV/c^{2} |

0.3 < m_{γγ} < 0.4 GeV/c^{2} |

0.4 < m_{γγ} < 0.5 GeV/c^{2} |

0.5 < m_{γγ} < 0.6 GeV/c^{2} |

In Figure 4 I show the asymmetry as a function of invariant mass for x_{F} < 0. Again, the methods yield consistent results, even for the lowest mass bin. Thus, I cannot conclude at this point that the odd behavior in that bin is due to the methodology. It still may be the result of something stupid on my part, however, I can't conclude that based on this study.

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