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

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