Measurement of the Jet Energy Scale in the CMS experiment ... - IIHE

CHAPTER 5: Estimating of the Jet Energy Scale Calibration Factor 73information of all input variables ⃗x = (x 1 . . .x n ) as explained in Section 5.1.2, is introducedin an n-dimensional phase space.By definition, L can take values in a range of [0,1]. Therefore, in the language of LikelihoodRatio, an event is assigned as signal if the measurement of its X property wouldhappen in the bin whose Likelihood Ratio value is greater than 0.5. Then the event ismore signal-like compared to background-like. Otherwise, it is labeled as background.In case of finding the correct jet combination, where eleven backgrounds, being wrongcombinations, versus one signal, being the true combination, exist, the chosen combinationis defined as the combination whose Likelihood Ratio value is the maximumamong the other combinations.5.1.4 Input Variables for TrainingVarious observable variables can be used to train the Likelihood Ratio method. Also acombination of two or three variables is allowed. Among so many candidate variableswhich can be used in the training, the most discriminating variables that differentiatemaximally between signal and backgrond, are desired. The discriminating powerof a generic variable can be defined using different methods. As an example, the“separation” S, which can be a measure of discriminating power between signal andbackground, is defined asS = 1 ∫ (PS (X) − P B (X)) 22 (P S (X) + P B (X)) dX,where P S (X) and P B (X) are the PDFs of the observable X for signal and background,respectively. By definition, for identical signal and background shapes, the separationis zero. If the signal and background PDFs do not overlap, then the separation takesa value equal to one. Therefore higher values of the separation, indicate the variablehas a higher probability to be a good discriminant candidate.Since the four vector of the neutrino is not known, hence the leptonic W boson andconsequently the leptonic top quark can not be fully reconstructed. Therefore the inputvariables which can be chosen to train the MVA method with, should be independentfrom the reconstructed neutrino. This makes the input candidates to be limited to asmaller collection. Different kinematics of the objects such as transverse momentump T , pseudorapidity η, polar angle θ and azimuthal angle φ can be used to comparethe candidates in t¯t system. In this analysis, the Θ variable, which is defined as thespace angle between two vectors in a three-dimensional phase space is used which isexpressed asΘ(⃗v 1 ,⃗v 2 ) = cos −1 ( ⃗v1 .⃗v 2|⃗v 1 ||⃗v 2 |)= cos −1 (sin θ 1 sin θ 2 (cos(φ 1 − φ 2 )) + cosθ 1 cos θ 2 ).The two-component variables in the t¯t system, excluding those variables related to thekinematics of neutrino, which are considered in this analysis, are listed below.• Θ(t h , W h ) which is the space angle between hadronic top quark and hadronic Wboson,