Chapter 7 Jet algorithm - FastJet 25
8.4 Choice of the appropriate variables
8.4.1 Jet Radius
The most important variable, that we have take into accound is the jet radius. Our aim is to choose the most efficient compination of the radius Rj1 andRj2, in order to achive the maximization of the number of Gravitons that created. We try different combinations of the R and Rj2 for the signal process, they are the following:
1) Rj1=1 and Rj2=0.4 2) Rj1=1 and Rj2=0.3 3) Rj1=0.8 and Rj2=0.4 4) Rj1=0.6 and Rj2=0.4 5) Rj1=0.6 and Rj2=0.3
From all these cases it is worth to see in detail the two most efficient choices, which are the 1) and the 4). In the following plots we can see the signal process in all the possible decays of Z boson into quark and antiquark.
h_mj1 Entries 99995 Mean 83.3 Std Dev 27.77
20 40 60 80 100 120 140
0 1000 2000 3000 4000 5000 6000 7000
8000 h_mj1
Entries 99995 Mean 83.3 Std Dev 27.77
h_mj1
Figure 8: The distribution of the mass of the Fatjet withRj1=1 for the quarks, d,s and u.
h_mj1 Entries 99986 Mean 81.14 Std Dev 27.42
20 40 60 80 100 120 140
0 1000 2000 3000 4000 5000 6000
h_mj1 Entries 99986 Mean 81.14 Std Dev 27.42
h_mj1
Figure 9: The distribution of the mass of the Fatjet withRj1=1 for the quarks, c and b.
Histo_mj2 Entries 30687 Mean 97.71 Std Dev 19.76
20 40 60 80 100 120 140
0 100 200 300 400 500
Histo_mj2 Entries 30687 Mean 97.71 Std Dev 19.76
Histo_mj2
Figure 10: The distribution of the mass of the two small jets with Rj1=1 for the quarks,d,s and u.
Histo_mj2 Entries 30895 Mean 95.82 Std Dev 21.88
20 40 60 80 100 120 140
0 50 100 150 200 250 300 350 400
Histo_mj2 Entries 30895 Mean 95.82 Std Dev 21.88
Histo_mj2
Figure 11: The distribution of the mass of the two small jets with Rj1=1 for the quarks, c and b.
h_mlj Entries 99995 Mean 970.1 Std Dev 126.3
500 600 700 800 900 1000 1100 1200 1300
0 2000 4000 6000 8000 10000
h_mlj Entries 99995 Mean 970.1 Std Dev 126.3
h_mlj
Figure 12: The distribution of the mass of the graviton from the two muons and the Fatjet with Rj1=1 for the quarks d,s and u.
h_mlj Entries 99986 Mean 956.6 Std Dev 129.3
500 600 700 800 900 1000 1100 1200 1300
0 1000 2000 3000 4000 5000 6000 7000
h_mlj Entries 99986 Mean 956.6 Std Dev 129.3
h_mlj
Figure 13: The distribution of the mass of the graviton from the two muons and the Fatjet with Rj1=1 for the quarks c and b.
Histo_ml2j2 Entries 6144 Mean 968.7 Std Dev 116.9
500 600 700 800 900 1000 1100 1200 1300
0 50 100 150 200 250 300 350 400 450
Histo_ml2j2 Entries 6144 Mean 968.7 Std Dev 116.9
Histo_ml2j2
Figure 14: The distribution of the mass of the graviton from the two muons and the two small jets withRj1=1 for the quarks d,s and u.
Histo_ml2j2 Entries 30895 Mean 1007 Std Dev 164.3
500 600 700 800 900 1000 1100 1200 1300
0 100 200 300 400 500
600 Histo_ml2j2
Entries 30895 Mean 1007 Std Dev 164.3
Histo_ml2j2
Figure 15: The distribution of the mass of the graviton from the two muons and the two small jets withRj1=1 for the quarks c and b.
h_mj1 Entries 99987 Mean 60.55 Std Dev 32.55
20 40 60 80 100 120 140
0 1000 2000 3000 4000 5000 6000 7000 8000
h_mj1 Entries 99987 Mean 60.55 Std Dev 32.55
h_mj1
Figure 16: The distribution of the mass of the Fatjet with Rj1=0.6 for the quarks, d,s and u.
h_mj1 Entries 99979 Mean 58.94 Std Dev 31.52
20 40 60 80 100 120 140
0 1000 2000 3000 4000 5000 6000
h_mj1 Entries 99979 Mean 58.94 Std Dev 31.52
h_mj1
Figure 17: The distribution of the mass of the Fatjet withRj1=0.6 for the quarks, c and b.
Histo_mj2 Entries 48303 Mean 91.85 Std Dev 16.49
20 40 60 80 100 120 140
0 200 400 600 800 1000 1200 1400 1600
1800 Histo_mj2
Entries 48303 Mean 91.85 Std Dev 16.49
Histo_mj2
Figure 18: The distribution of the mass of the two small jets with Rj1=0.6 for the quarks,d,s and u.
Histo_mj2 Entries 49229 Mean 89.5 Std Dev 18.27
20 40 60 80 100 120 140
0 200 400 600 800 1000 1200
Histo_mj2 Entries 49229 Mean 89.5 Std Dev 18.27
Histo_mj2
Figure 19: The distribution of the mass of the two small jets with Rj1=0.6 for the quarks, c and b
h_mlj Entries 46912 Mean 980.4 Std Dev 95.58
500 600 700 800 900 1000 1100 1200 1300
0 1000 2000 3000 4000 5000 6000 7000
h_mlj Entries 46912 Mean 980.4 Std Dev 95.58
h_mlj
Figure 20: The distribution of the mass of the graviton from the two muons and the Fatjet with Rj1=0.6 for the quarks d,s and u.
h_mlj Entries 45805 Mean 968.3 Std Dev 98.1
500 600 700 800 900 1000 1100 1200 1300
0 1000 2000 3000 4000 5000
h_mlj Entries 45805 Mean 968.3 Std Dev 98.1
h_mlj
Figure 21: The distribution of the mass of the graviton from the two muons and the Fatjet with Rj1=0.6 for the quarks c and b.
Histo_ml2j2 Entries 13313 Mean 962.6 Std Dev 114.9
500 600 700 800 900 1000 1100 1200 1300
0 200 400 600 800 1000 1200 1400
1600 Histo_ml2j2
Entries 13313 Mean 962.6 Std Dev 114.9
Histo_ml2j2
Figure 22: The distribution of the mass of the graviton from the two muons and the two small jets withRj1=0.6 for the quarks d,s and u.
Histo_ml2j2 Entries 12661 Mean 951.8 Std Dev 116
500 600 700 800 900 1000 1100 1200 1300
0 200 400 600 800 1000
Histo_ml2j2 Entries 12661 Mean 951.8 Std Dev 116
Histo_ml2j2
Figure 23: The distribution of the mass of the graviton from the two muons and the two small jets withRj1=0.6 for the quarks c and b
The following table shows the sigma of the mass distribution in all the previous cases with Rj1=1 and Rj1=0.6 and Rj2=0.4.
Sigma Z boson and Graviton for d,u and s quark Width Rj1=1 Rj1=0.6 Rj2=0.4
and Rj1=1
Rj2=0.4 and Rj1=0.6
Sigma Z 3.4 2.78 4.1 3.34
sigma G 11.1 9.028 30.5 13.16
Sigma Z boson and Graviton for b and c quark Width Rj1=1 Rj1=0.6 Rj2=0.4
and Rj1=1
Rj2=0.4 and Rj1=0.6
Sigma Z 3.8 3.43 5.7 4.6
sigma G 15.8 17.5 32 22.37
According to the previous plots and table it is obvious that we have better distribution in the case of the combination Rj1=0.6 and Rj2=0.4.
Nevertheless, that is not our aim, so in order to be completly clear which combination is the most efficient we have to calculate the number of the gravitons are ”reconstructed” in each of these cases. Collecting all the nec- essary information for each quarks, I represent the results in the following table.
Percentages of graviton creation
Quarks Rj1=1 andRj2=0.4 Rj1=0.6 and Rj2=0.4
u,d,s 48 40
b,c 40 31
According to the results we can see that in the case ofRj1=1 andRj2=0.4 we have almost ten percent more production of Gravitons. This result is into agreement with the combination of jet radius that is in use in data analysis.
Having choose the combination of jet radius we can continuou our anal- ysis.The following histogramms show the invariant mass distrubution of the one FatJet and the two Small jets and the mass of the graviton in these two
cases for both signal and background processes.
Figure 24: Distribution of the invariant mass of the FatJet.
Figure 25: Distribution of the invariant mass of the two small jets.
Figure 26: Distribution of the invariant mass of the Graviton from muons and the FatJet.
Figure 27: Distribution of the invariant mass of the Graviton from muons and the two small jets.
Then I apply certain cuts in the invariant mass of the jets (both FatJet and small jets). These are:
• ∆m= 10 GeV around the Z nominal mass.
• ∆m= 20 GeV around the Z nominal mass.
• ∆m= 25 GeV around the Z nominal mass.
• ∆m= 30 GeV around the Z nominal mass.
• ∆m= 40 GeV around the Z nominal mass.
All these cuts help us to understand which value is appropriate in order to maximize our signal and reduce our background. Our aim is to find the value that can give us both good discrimination of the signal process and good rejection for the background process close to the Graviton region. The following figures show the efficiency for the signal and the rejection for the background as function of the cuts that we have applied.
Cuts
0 5 10 15 20 25 30 35 40 45 50
Efficiency (%)
0 10 20 30 40 50 60 70 80 90
100 Signal
WZ ZZ ttbar Z+Jets /10
Rejection
0 20 40 60 80 100
Figure 28: Efficiency-Rejection plot for the mass of the one Fatjet.
Cuts
0 5 10 15 20 25 30 35 40 45 50
Efficiency (%)
0 2 4 6 8 10 12 14 16 18
20 Signal
WZ ZZ ttbar Z+Jets
Rejection
0 5 10 15 20 25 30 35
Figure 29: Efficiency-Rejection plot for the mass of the two small jets.
Cuts
0 5 10 15 20 25 30 35 40 45 50
Efficiency (%)
0 10 20 30 40 50 60 70 80 90
100 Signal
WZ ZZ ttbar Z+Jets
Rejection
0 5 10 15 20 25 30 35
Figure 30: Efficiency-Rejection plot for the mass of the addition of the Fatjet and the two small jets.
Cuts
0 5 10 15 20 25 30 35 40 45 50
Efficiency (%)
0 10 20 30 40 50 60 70 80 90
100 Signal
WZ ZZ ttbar Z+Jets
Rejection
0 500 1000 1500 2000 2500 3000
Figure 31: Efficiency-Rejection plot for the mass of the Graviton.
From these plots is obvious that a good mass cut will be either the
∆m = 25GeV or ∆m = 30GeV. To conclude, having see all the previous plots we came to this ”decision”, we use Rj1=1 and Rj2=0.4 jet radius in order to select the maximum number of Gravitons that we can, and we apply jet mass cut equal to 25 GeV and that because this cut give us the maximun efficiency for our signal process and simultaneously the maximun rejection for the background processes.