# New PID Asymmetries

**Update**: I coded up the statistical uncertainty calculation incorrectly in this post. I forgot to divide by the purity! The optimal cuts changed by a lot when I fixed that error. I also generalized the formula to account for the presence of signal in the sidebands, and I included the lowest p_{T} bin in the analysis. For more details see New PID Asymmetries II. *Bottom Line — the results on this page are wrong!*

In an earlier post I explained the new method for calculating identified particle yields that I’m using in my A_{LL} analysis. I began that study because I planned to calculate A_{LL} differently than I had been in the past. Specifically, I wanted to incorporate the proton/kaon/electron backgrounds into the statistical uncertainty instead of assigning a separate (statistics-limited) systematic uncertainty to account for their presence. I’m using the following formulas for A_{LL} and its statistical uncertainty:

where the p_{T}-dependent background fractions are defined as:

I wrote a small function to estimate the statistical precision on A_{LL} given the p_{T} bin, pion acceptance window, and sideband acceptance windows. I didn’t care about the absolute statistical precision, so I just used 1/sqrt(N) for the uncertainty on each A_{LL}. I used Minuit2 to minimize this function and extract the optimal acceptance windows, with the constraint that the purity in each sideband is never below 90%. In principle, this approach would yield four momentum-dependent cuts. In practice, the p+K sideband cut and the left side of the pion acceptance window only had a small momentum dependence, so for the sake of simplicity I keep them fixed. I also choose to fix the other two cuts in each p_{T} bin instead of letting them vary with momentum. In the end I employ the following cuts

pT bin | π window | max p+K | min electron |
---|---|---|---|

3.18 - 4.56 | (-1.90, 2.40) | -1.90 | 2.40 |

4.56 - 6.32 | (-1.90, 2.25) | -1.90 | 2.50 |

6.32 - 8.80 | (-1.90, 2.00) | -1.90 | 2.60 |

8.80 - 12.84 | (-1.90, 1.50) | -1.90 | 2.60 |

These cuts are significantly wider than the (-1.0, 2.0) acceptance window I had been using in the past. Apparently the reduction in purity is more than offset by the extra efficiency.

The electron side of the acceptance window is interesting. As momentum increases the pion band moves closer to the electron band. As a result, we need to move the electron sideband cut further out to maintain the 90% purity. This cuts down on the electron background A_{LL} statistics. The minimizer compensates for that uncertainty by restricting the right side of the pion acceptance window and thus reducing the electron background fraction.

I compared the uncertainties obtained by the minimizer with the uncertainties from my old method (a flat (-1.0, 2.0) cut that does not subtract out the background asymmetries). It turns out that the uncertainties from the new method are actually smaller in every p_{T} bin. I haven’t calculated a systematic uncertainty for this method, but if there is one it will be far smaller than the systematic from the old method (~ background fraction * sigma of background A_{LL}). In other words, using the new method is a no-brainer.

Oh, and one plot, just because I think it’s pretty:

In an earlier post I explained the new method for calculating identified particle yields that I’m using in my A_{LL} analysis. I began that study because I planned to calculate A_{LL} differently than I had been in the past. Specifically, I wanted to incorporate the proton/kaon/electron backgrounds into the statistical uncertainty instead of assigning a separate (statistics-limited) systematic uncertainty to account for their presence. I’m using the following formulas for A_{LL} and its statistical uncertainty:

where the p_{T}-dependent background fractions are defined as:

I wrote a small function to estimate the statistical precision on A_{LL} given the p_{T} bin, pion acceptance window, and sideband acceptance windows. I didn’t care about the absolute statistical precision, so I just used 1/sqrt(N) for the uncertainty on each A_{LL}. I used Minuit2 to minimize this function and extract the optimal acceptance windows, with the constraint that the purity in each sideband is never below 90%. In principle, this approach would yield four momentum-dependent cuts. In practice, the p+K sideband cut and the left side of the pion acceptance window only had a small momentum dependence, so for the sake of simplicity I keep them fixed. I also choose to fix the other two cuts in each p_{T} bin instead of letting them vary with momentum. In the end I employ the following cuts

pT bin | π window | max p+K | min electron |
---|---|---|---|

3.18 - 4.56 | (-1.90, 2.40) | -1.90 | 2.40 |

4.56 - 6.32 | (-1.90, 2.25) | -1.90 | 2.50 |

6.32 - 8.80 | (-1.90, 2.00) | -1.90 | 2.60 |

8.80 - 12.84 | (-1.90, 1.50) | -1.90 | 2.60 |

These cuts are significantly wider than the (-1.0, 2.0) acceptance window I had been using in the past. Apparently the reduction in purity is more than offset by the extra efficiency.

The electron side of the acceptance window is interesting. As momentum increases the pion band moves closer to the electron band. As a result, we need to move the electron sideband cut further out to maintain the 90% purity. This cuts down on the electron background A_{LL} statistics. The minimizer compensates for that uncertainty by restricting the right side of the pion acceptance window and thus reducing the electron background fraction.

I compared the uncertainties obtained by the minimizer with the uncertainties from my old method (a flat (-1.0, 2.0) cut that does not subtract out the background asymmetries). It turns out that the uncertainties from the new method are actually smaller in every p_{T} bin. I haven’t calculated a systematic uncertainty for this method, but if there is one it will be far smaller than the systematic from the old method (~ background fraction * sigma of background A_{LL}). In other words, using the new method is a no-brainer.

Oh, and one plot, just because I think it’s pretty:

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