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

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84 CHAPTER 5: Estimating of the Jet Energy Scale Calibration FactorAll of the partial derviatives of the constraints, in each step of the iteration, needto be obtained at the value of the parameters at the previous iteration X ∗ . Since theconstraints depend directly on the four-vector components P ⃗ = (⃗p, E), the chain ruleis used to obtain the partial derivatives ∂ f ⃗ = ∂ f ⃗∂⃗y ∂ ⃗P .∂ P ⃗ . As four-momenta of the particles∂⃗ycan be parametrized in different ways, the definition of the ∂ P ⃗ is not unique. Various∂⃗yparametrization for the reconstructed objects exist which are described in more detailin the next section.In the new notation, one has to minimize the likelihood expression which is written asL = ∆⃗y T V −1 ∆⃗y + 2λ T (A∆⃗a + B∆⃗y −⃗c).The conditions for a local minimum are obtained, after differentiating the above expressionwith respect to ⃗y, ⃗a and ⃗ λ, as listed belowV −1 ∆⃗y + B T ⃗ λ = 0,A T ⃗ λ = 0,B∆⃗y + A∆⃗a = c.Collecting the above three equations in a compact form with partitioned matrices⎛ ⎞ ⎛ ⎞ ⎛ ⎞V −1 0 B T ∆⃗y 0⎝ 0 0 A T ⎠ ⎝∆⃗a⎠ = ⎝0⎠B A 0 λ cThis matrix equation will be solved iteratively for the unknown values ∆⃗y, ∆⃗a and∆ ⃗ λ.The iteration procedure is repeated until some predefined convergence criteria are fulfilled.The first one requires that the change in χ 2 expression between the currentiteration, n, and the previous iteration, n-1, is smaller than a given value ǫ S , and iswritten asS(n − 1) − S(n)< ǫ S ,ndfwhere ndf is the difference between the number of constraints and the number ofunmeasured quantities, beingndf = m − p.The second convergency criterion which is checked in each iteration, requires that theconstraints are fulfilled better than a given value ǫ F , and is expressed asF = Σ m k=1 f(n) k(⃗y,⃗a) < ǫ F .A more complete description of the kinematic fit technique can be found in [103].
CHAPTER 5: Estimating of the Jet Energy Scale Calibration Factor 855.2.2 Four-Vector ParametrizationSeveral parametrizations in the kinematic fit package are implemented, two of whichare commonly used in hadron colliders and are introduced briefly here. A first one,called TFitParticleEtThetaPhi, uses E T , θ and φ variables for parametrization. Themomentum vector and the energy can be written in this parametrization as⎛ ⎞E T cosφ⃗p f = ⎝E T sin φ⎠ ,E T cot θE = E Tsin θ .In a second parametrization, called TFitParticleEtEtaPhi, where E T , η and φ variablesare used as parameters, the four-vector can be expressed as⎛ ⎞E T cosφ⃗p f = ⎝ E T sin φ ⎠,E T sinh ηE = E T cosh η.In the two parametrizations explained above, the objects are considered as massless.In this study, TFitParticleEtThetaPhi parametrization is used with an additionalrequirement. During the fit procedure, it is crucial to keep the ratio of the measuredenergy of a jet to its measured momentum constant. Hence, in the kinematic fit methodwhich is used in the current study, this additional requirement is applied which isexpressed as followsE fitted|⃗p fitted | = E measured|⃗p measured | .5.2.3 Resolutions on the Jet KinematicsThe uncertainties on the measured parameters enter the kinematic fit package via thecovariance matrix, as mentioned before. In this analysis where the measured objects arereconstructed jets, the resolutions on the jet properties are required to be calculated.These resolutions are obtained from simulated e+jets t¯t events in which a correct jetpartoncombination, according to the generator level information, is found. Accordingto the jet-parton matching algorithm described in Section 5.1.1, the reconstructedjets are matched to the partons if ∆R difference between them is less than 0.3. Thereconstructed jets are grouped in either light jets, which are matched to the partonsarising from the decay of the W boson, or b jets, which are matched to the b flavoredpartons originating from the decay of the top quarks. Since the obtained resolutionsare found to be dependent on the transverse momentum and the pseudorapidity of thereconstructed jets, jets are further categorized in various bins of p T and η. In thisanalysis, eleven bins of transverse momentum and twelve bins of pseudorapidity areconsidered. For those jets which populate a certain p T -bin and η-bin, the resolutionsare calculated as explained in the following. The resolution on the X property of a
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Vrije Universiteit BrusselFaculteit
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IntroductionThe Standard Model of p
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Chapter 1The Standard Model of Part
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Chapter 2The CMS experiment at the
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SummaryJets appear in the final sta
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Measurement of the Jet Energy Scale in the CMS experiment ... - IIHE

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