74 CHAPTER 5: Estimat<strong>in</strong>g <strong>of</strong> <strong>the</strong> <strong>Jet</strong> <strong>Energy</strong> <strong>Scale</strong> Calibration Factor• Θ(t h , b h ) which is <strong>the</strong> space angle between hadronic top quark and hadronic b,• Θ(t h , b l ) which is <strong>the</strong> space angle between hadronic top quark and leptonic b,• Θ(t h , e) which is <strong>the</strong> space angle between hadronic top quark and electron,• Θ(W h , b h ) which is <strong>the</strong> space angle between hadronic W boson and hadronic b,• Θ(W h , b l ) which is <strong>the</strong> space angle between hadronic W boson and leptonic b,• Θ(W h , e) which is <strong>the</strong> space angle between hadronic W boson and electron,• Θ(b h , b l ) which is <strong>the</strong> space angle between hadronic b and leptonic b,• Θ(b h , e) which is <strong>the</strong> space angle between hadronic b and electron,• Θ(b l , e) which is <strong>the</strong> space angle between leptonic b and electron,• Θ(q, ¯q) which is <strong>the</strong> space angle between light quarks appear<strong>in</strong>g <strong>in</strong> <strong>the</strong> decay <strong>of</strong>hadronic W boson.Most <strong>of</strong> <strong>the</strong> above mentioned variables are not <strong>in</strong>dependent and conta<strong>in</strong> <strong>the</strong> same<strong>in</strong>formation. This can be seen by look<strong>in</strong>g at <strong>the</strong>ir l<strong>in</strong>ear correlations which are shown<strong>in</strong> Table 5.1. S<strong>in</strong>ce <strong>the</strong> masses <strong>of</strong> <strong>the</strong> W boson and <strong>the</strong> top quark are go<strong>in</strong>g to be usedas constra<strong>in</strong>ts <strong>in</strong> this analysis, as will be expla<strong>in</strong>ed <strong>in</strong> <strong>the</strong> follow<strong>in</strong>g sections, it is alsorequired that <strong>the</strong> selected variables are not highly correlated with <strong>the</strong> W boson andtop quark masses. The last two rows <strong>of</strong> Table 5.1 conta<strong>in</strong> <strong>the</strong> correlation factors amongall considered space angle variables with <strong>the</strong> masses <strong>of</strong> <strong>the</strong> W boson and top quark.The space angle variables are required to have a mutual correlation factor among<strong>the</strong>mselves less than 40%. They are also asked to be correlated with <strong>the</strong> W boson and<strong>the</strong> top quark masses less than 20%. S<strong>in</strong>ce <strong>the</strong> separation powers <strong>of</strong> each <strong>in</strong>dividualvariable are <strong>of</strong> <strong>the</strong> same order, tak<strong>in</strong>g <strong>in</strong>to account <strong>the</strong> requirement on <strong>the</strong> correlationfactor, only 5 variables among <strong>the</strong> above mentioned space angle variables are selectedto be used <strong>in</strong> <strong>the</strong> tra<strong>in</strong><strong>in</strong>g <strong>of</strong> <strong>the</strong> MVA method. In addition, <strong>the</strong>re are two morevariables which are selected for tra<strong>in</strong><strong>in</strong>g. A first one, which is def<strong>in</strong>ed as <strong>the</strong> transversemomentum <strong>of</strong> <strong>the</strong> hadronic top quark candidate relative to <strong>the</strong> sum <strong>of</strong> <strong>the</strong> transversemomenta <strong>of</strong> all hadronic top quark candidates pth TΣp T. S<strong>in</strong>ce with <strong>the</strong> four lead<strong>in</strong>g jets,<strong>the</strong>re are four different ways to comb<strong>in</strong>e three <strong>of</strong> <strong>the</strong>m to reconstruct a top quarkcandidate, <strong>the</strong>n <strong>the</strong> sum <strong>in</strong> <strong>the</strong> denom<strong>in</strong>ator is performed over <strong>the</strong> transverse momenta
CHAPTER 5: Estimat<strong>in</strong>g <strong>of</strong> <strong>the</strong> <strong>Jet</strong> <strong>Energy</strong> <strong>Scale</strong> Calibration Factor 75Θ(t h , W h ) Θ(t h , b h ) Θ(t h , b l ) Θ(t h , e) Θ(W h , b h ) Θ(W h , b l ) Θ(W h , e) Θ(b h , b l ) Θ(b h , e) Θ(b l , e) Θ(q, ¯q)Θ(t h , W h ) +100.0 - - - - - - - - - -Θ(t h , b h ) -10.0 +100.0 - - - - - - - - -Θ(t h , b l ) +5.6 +6.9 +100.0 - - - - - - - -Θ(t h , e) +5.2 +5.8 +40.8 +100.0 - - - - - - -Θ(W h , b h ) +32.4 +67.9 +7.9 +8.7 +100.0 - - - - - -Θ(W h , b l ) +2.0 +7.3 +60.1 +36.5 +8.0 +100.0 - - - - -Θ(W h , e) +0.5 +4.7 +24.0 +61.1 +4.4 +35.2 +100.0 - - - -Θ(b h , b l ) +3.6 -4.4 +43.2 +32.1 -1.4 +32.7 +33.6 +100.0 - - -Θ(b h , e) +3.8 -3.7 +18.3 +43.5 -0.1 +25.6 +31.5 +31.1 +100.0 - -Θ(b l , e) +15.0 +11.8 +1.2 +0.5 +15.9 +1.3 -1.3 -1.0 +3.1 +100.0 -Θ(q, ¯q) +16.8 -1.8 +2.2 +4.5 +23.6 +0.7 +1.3 +4.3 +5.8 +4.4 +100.0m W -0.7 +2.0 +0.9 -0.1 +1.4 -1.3 -0.5 +3.2 +1.2 -2.5 +21.3m top +3.6 +2.0 +1.7 +2.1 +7.1 -0.4 +1.6 +0.9 +1.2 -1.5 +10.7Table 5.1: The mutual correlation factors <strong>of</strong> some space angle variables and <strong>the</strong>ir correlation coefficients with <strong>the</strong> masses <strong>of</strong> <strong>the</strong>W boson and <strong>the</strong> top quark. All numbers are quoted <strong>in</strong> %.