The asymmetry for the observable, Pzc

1. Equation

2. Asymmetry

Black(Topology 1) and Red(Average)

Black(Topology 2) and Red(Average)

EBin[03]: 0.7GeV - 0.8GeV
EBin[04]: 0.8GeV - 0.9GeV
EBin[05]: 0.9GeV - 1.0GeV
EBin[06]: 1.0GeV - 1.1GeV
EBin[07]: 1.1GeV - 1.2GeV
EBin[08]: 1.2GeV - 1.3GeV
EBin[09]: 1.3GeV - 1.4GeV
EBin[10]: 1.4GeV - 1.5GeeV
EBin[11]: 1.5GeV - 1.6GeV
EBin[12]: 1.6GeV - 1.7GeV
EBin[13]: 1.7GeV - 1.8GeV
EBin[14]: 1.8GeV - 1.9GeV
EBin[15]: 1.9GeV - 2.0GeV
EBin[16]: 2.0GeV - 2.1GeV
EBin[17]: 2.1GeV - 2.2GeV
EBin[18]: 2.2GeV - 2.3GeV
EBin[3]: 0.7GeV - 0.8GeV
EBin[4]: 0.8GeV - 0.9GeV
EBin[5]: 0.9GeV - 1.0GeV
EBin[6]: 1.0GeV - 1.1GeV
EBin[7]: 1.1GeV - 1.2GeV
EBin[8]: 1.2GeV - 1.3GeV
EBin[9]: 1.3GeV - 1.4GeV
EBin[10]: 1.4GeV - 1.5GeV
EBin[11]: 1.5GeV - 1.6GeV
EBin[12]: 1.6GeV - 1.7GeV
EBin[13]: 1.7GeV - 1.8GeV
EBin[14]: 1.8GeV - 1.9GeV
EBin[15]: 1.9GeV - 2.0GeV
EBin[16]: 2.0GeV - 2.1GeV
EBin[17]: 2.1GeV - 2.2GeV
EBin[18]: 2.2GeV - 2.3GeV

Black(Topology 3) and Red(Average)

Black(Topology 4) and Red(Average)

EBin[03]: 0.7GeV - 0.8GeV
EBin[04]: 0.8GeV - 0.9GeV
EBin[05]: 0.9GeV - 1.0GeV
EBin[06]: 1.0GeV - 1.1GeV
EBin[07]: 1.1GeV - 1.2GeV
EBin[08]: 1.2GeV - 1.3GeV
EBin[09]: 1.3GeV - 1.4GeV
EBin[10]: 1.4GeV - 1.5GeeV
EBin[11]: 1.5GeV - 1.6GeV
EBin[12]: 1.6GeV - 1.7GeV
EBin[13]: 1.7GeV - 1.8GeV
EBin[14]: 1.8GeV - 1.9GeV
EBin[15]: 1.9GeV - 2.0GeV
EBin[16]: 2.0GeV - 2.1GeV
EBin[17]: 2.1GeV - 2.2GeV
EBin[18]: 2.2GeV - 2.3GeV
EBin[3]: 0.7GeV - 0.8GeV
EBin[4]: 0.8GeV - 0.9GeV
EBin[5]: 0.9GeV - 1.0GeV
EBin[6]: 1.0GeV - 1.1GeV
EBin[7]: 1.1GeV - 1.2GeV
EBin[8]: 1.2GeV - 1.3GeV
EBin[9]: 1.3GeV - 1.4GeV
EBin[10]: 1.4GeV - 1.5GeV
EBin[11]: 1.5GeV - 1.6GeV
EBin[12]: 1.6GeV - 1.7GeV
EBin[13]: 1.7GeV - 1.8GeV
EBin[14]: 1.8GeV - 1.9GeV
EBin[15]: 1.9GeV - 2.0GeV
EBin[16]: 2.0GeV - 2.1GeV
EBin[17]: 2.1GeV - 2.2GeV
EBin[18]: 2.2GeV - 2.3GeV

3. Check the symmetry (EBin=20, ABin=9, PBin=10)

A) compare front observables and back observables

The upper asymmetry plots are made on the φ angle as the x-axis. The φ angles are divided to 10 bins. We give numbers from the left observable.We compare the front numbers(1,2,3,4,5) to the back numbers(6,7,8,9,10). That is, we compare observable 1 and observable 10. observable 2 and observable 9. In the following histograms, we found the ratio and difference between front observales and back observables.

w/ weight

w/o weight

w/ weight

w/o weight

B) compare observables of topologies

We compare observables between topologies in the same bins. For example, PzcDiff12 is the observables difference betweeb topology 1 and topology 2.

Difference btw top. 1 and 1

Difference btw top. 1 and 2

Difference btw top. 1 and 3

Difference btw top. 1 and 4

Difference btw top. 2 and 3

Difference btw top. 2 and 4

Difference btw top. 3 and 3

Difference btw top. 3 and 4

Ratio btw top. 1 and 1

Ratio btw top. 1 and 2

Ratio btw top. 1 and 3

Ratio btw top. 1 and 4

Ratio btw top. 3 and 3

Ratio btw top. 3 and 4

The difference of topologies has the gaussian shape with mean 0. We think the ratio also may have the gaussian with mean 1. However, their shapes are not gaussian and do not have mean 1. That is so intersting.

C) simulation

We have a simple test. We make the difference and ratio plots of gaussian histograms.

The gaissian(h1, h2, h3, and h4) with const=10, mean=0, and sigma=0.5

Difference btw h1 and h1

Difference btw h1 and h2

Difference btw h1 and h3

Difference btw h1 and h4

Difference btw h2 and h2

Difference btw h2 and h3

Difference btw h2 and h4

Difference btw h3 and h3

Difference btw h3 and h4

Difference btw h4 and h4

Ratio btw h1 and h1

Ratio btw h1 and h2

Ratio btw h1 and h3

Ratio btw h1 and h4

Ratio btw h2 and h2

Ratio btw h2 and h3

Ratio btw h2 and h4

Ratio btw h3 and h3

Ratio btw h3 and h4

Ratio btw h4 and h4

The difference plots have the gaussian with mean 0, however, the ratio plots is not the gaussian and their peaks are around zero.

4. Check the symmetry (EBin=20, ABin=20, PBin=10)

A) compare front observables and back observables

Averaged Topology
Topology 1
Topology 2
Topology 3
Topology 4
Averaged Topology
Topology 1
Topology 2
Topology 3
Topology 4

B) compare observables of topologies

We compare observables between topologies in the same bins. For example, PzcDiff12 is the observables difference betweeb topology 1 and topology 2.

Difference btw top. 1 and 1

Difference btw top. 1 and 2

Difference btw top. 1 and 3

Difference btw top. 1 and 4

Difference btw top. 2 and 2

Difference btw top. 2 and 3

Difference btw top. 2 and 4

Difference btw top. 3 and 3

Difference btw top. 3 and 4

Difference btw top. 4 and 4

Ratio btw top. 1 and 1

Ratio btw top. 1 and 2

Ratio btw top. 1 and 3

Ratio btw top. 1 and 4

Ratio btw top. 2 and 2

Ratio btw top. 2 and 3

Ratio btw top. 2 and 4

Ratio btw top. 3 and 3

Ratio btw top. 3 and 4

Ratio btw top. 4 and 4