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

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].