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

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82 CHAPTER 5: Estimating of the Jet Energy Scale Calibration Factorprotons can not be known. However, the usage of the kinematic fit method is still feasibleby applying the mass constraints in the specific event topologies. For example, inthe processes where a W boson is produced and subsequently decays to a pair of quarksW → q¯q, applying the mass constraint of the W boson and forcing the four-momentaof its decay products to fulfill the mass equation, beingm 2 W = (E jet 1+ E jet2 ) 2 − ( ⃗ P jet1 + ⃗ P jet2 ) 2 ,would improve significantly the resolution on the reconstructed jets. In the currentstudy, the mass constraints of both the W boson and top quark are applied in thehadronic branch of e+jets t¯t system, namely t → Wb → q¯qb. The reconstructedobjects in the final state, being the light jets arrising from the W boson decay and theb quark jet arrising from the top quark decay, are asked to fulfill the mass constraintswhich are expressed asm 2 W = (E l 1+ E l2 ) 2 − ( ⃗ P l1 + ⃗ P l2 ) 2 ,m 2 top = (E l 1+ E l2 + E b ) 2 − ( ⃗ P l1 + ⃗ P l2 + ⃗ P b ) 2 ,where index l i refers to one of the two light jets originating from light quarks, namelyup, down, charm and strange. Also index b in the above formula, refers to the b jetarrising from the heavy quark, namely the bottom quark.The kinematic fit results are used to estimate the residual jet energy correction factorsfor both light as well as b quark jets as will be described in detail in Section 5.4. Themathematical concept of the kinematic fit technique is reviewed in the next section.5.2.1 General Mathematical ConceptA physics problem consists of measured quantities, ⃗y = (y 1 , y 2 , . . .,y n ), and unmeasuredvalues, ⃗a = (a 1 , a 2 , . . .,a p ). Also a certain hypothesis originating from the physics principalsuch as conservation of the energy and momentum may be introduced. Thereforethe observed events, containing n measured and p unmeasured values, are constrainedto fulfill the hypothesis. Due to the presence of uncertainties on the measured values,usually the hypothesis is not respected in the physics problem which is under consideration.The hypothesis, which is expressed via constraint equations ⃗ f = (f 1 , f 2 , . . .,f m ),is satisfied for only the true parameters, namely ȳ and ā as written below.f 1 (ā 1 , ā 2 , . . .,ā p , ȳ 1 , ȳ 2 , . . ., ȳ n ) = 0,f 2 (ā 1 , ā 2 , . . .,ā p , ȳ 1 , ȳ 2 , . . ., ȳ n ) = 0,.f m (ā 1 , ā 2 , . . .,ā p , ȳ 1 , ȳ 2 , . . .,ȳ n ) = 0.In order to obtain the correction to the measured values, ∆⃗y, which yields the constraintsto be fulfilled for the corrected measurements, the extrema of the above equationsshould be found. Hence the Lagrange Multipliers method is used to determinethe true values, ⃗y ′ = ⃗y + ∆⃗y, for which the constraints are fulfilled. At the same
CHAPTER 5: Estimating of the Jet Energy Scale Calibration Factor 83time the calculated corrections to the measured values should be minimized which isgoverened with the use of a χ 2 term. Together with the Lagrange term, one could writea combined likelihood expression, L(⃗y,⃗a, ⃗ λ), as followsL(⃗y,⃗a, ⃗ λ) = S(⃗y) + 2Σ m k=1 λ kf k (⃗y,⃗a),where ⃗ λ are the Lagrange Multipliers and S(⃗y) is the χ 2 term and is expressed explicitlyasS(⃗y) = ∆⃗y T V −1 ∆⃗y,where V is the covariance matrix of the measured parameters.In order to find a local minimum for the likelihood equation, one has to minimize S(⃗y)under the constraints f k (⃗y,⃗a) = 0. Due to the non-linearity of the physics constraintswhich are used, one has to solve the constraints iteratively.Since the constraints are fulfilled in the limit of true values, (⃗y ′ ,⃗a ′ ), then f k (⃗y,⃗a) canbe expanded arount a desired value, for example could be the value obtained duringthe previous iteration represented by (⃗y ∗ ,⃗a ∗ ), as followsf k (⃗y ′ ,⃗a ′ ) ≈ f k (⃗y ∗ ,⃗a ∗ ) + Σ p ∂f kj=1 .(∆a j − ∆a ∗∂aj) + Σ n ∂f ki=1 .(∆y i − ∆yi ∗ ) ≈ 0,j ∂y iwhere ⃗ X, ⃗ X ∗ and ⃗ X ′ , with X could be either y or a, are respectively the start value,the value after the previous iteration and the value after the current iteration. Alsothe ∆ ⃗ X and ∆ ⃗ X ∗ are defined as ∆ ⃗ X = ⃗ X ′ − ⃗ X and ∆ ⃗ X ∗ = ⃗ X ∗ − ⃗ X. One can writethe kinematic constraints in vector notation⃗f ∗ + A(∆⃗a − ∆⃗a ∗ ) + B(∆⃗y − ∆⃗y ∗ ) ≈ 0,or equivalentlywithA∆⃗a + B∆⃗y −⃗c = 0,⎛ ⎞f 1 (⃗a ∗ ,⃗y ∗ )⃗c = A∆⃗a ∗ + B∆⃗y ∗ − f ⃗ ∗ ; f ⃗ ∗ f 2 (⃗a ∗ ,⃗y ∗ )= ⎜ ⎟⎝ . ⎠f m (⃗a ∗ ,⃗y ∗ )In the vector notaion, the matrices A and B are defined as A = ∂ f ⃗ and B = ∂ f ⃗∂⃗awhich their components are expressed explicitly below.⎛⎞∂f 1 ∂f 1 ∂f∂a 1 ∂a 2. . . 1∂a p∂f 2 ∂f 2 ∂fA =∂a 1 ∂a 2. . . 2∂a p⎜⎟⎝ .⎠∂f m ∂f∂a 2. . . m∂a p⎛B = ⎜⎝∂f m∂a 1∂f 1 ∂f 1∂y 1∂f 2 ∂f 2∂y 1.∂f m∂y 1∂y 2. . .∂y 2. . .∂f m∂y 2. . .∂f 1⎞∂y n∂f 2∂y n⎟⎠∂f m∂y n∂⃗y for
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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 1: The Standard Model of Pa
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Chapter 2The CMS experiment at the
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CHAPTER 2: The CMS experiment at th
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Chapter 3Object ReconstructionThe e
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BIBLIOGRAPHY 133[40] The ALICE Coll
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BIBLIOGRAPHY 135[76] The CMS Collab
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SummaryJets appear in the final sta
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