Kinematic Fitting. Mark M. Ito. July 11, Jefferson Lab. Mark Ito (JLab) Kinematic Fitting July 11, / 25

Transcription

1 Kinematic Fitting Mark M. Ito Jefferson Lab July 11, 2013 Mark Ito (JLab) Kinematic Fitting July 11, / 25

2 Introduction Motivation Particles measured independently Often: known relationships between their kinematics energy and momentum conservation daughters of common parent particle of known mass particles from a common vertex Measured kinematics do not respect relationships How to incoporate knowledge of relationships? Why? better measurements hypothesis testing Mark Ito (JLab) Kinematic Fitting July 11, / 25

3 PrimEx Experiment in Hall B PrimEx Detector Mark Ito (JLab) Kinematic Fitting July 11, / 25

4 PrimEx Experiment in Hall B Hybrid Calorimeter Mark Ito (JLab) Kinematic Fitting July 11, / 25

5 PrimEx Experiment in Hall B Event Display Mark Ito (JLab) Kinematic Fitting July 11, / 25

6 PrimEx Experiment in Hall B Two-photon Mass Distribution Mark Ito (JLab) Kinematic Fitting July 11, / 25

7 Example Example: Decay of Neutral Pion into Two Photons π 0 γγ both photons detected assume photon directions are known precisely energies have relative uncertainty σ E /E = 5%/ E for simplicity: look at 500 MeV/c π 0 moving in z-direction but, will not assume 500 MeV/c nor z-direction for pion Mark Ito (JLab) Kinematic Fitting July 11, / 25

8 Example two-photon invariant mass Mark Ito (JLab) Kinematic Fitting July 11, / 25

9 Example relative error on single photon energy Mark Ito (JLab) Kinematic Fitting July 11, / 25

10 Example two-photon measured momentum Mark Ito (JLab) Kinematic Fitting July 11, / 25

11 Example What is the problem? want to improve resolution assume that photons came from π 0, so mass is known can we use this information? could adjust one photon (why just one?) could scale them both (high energy γ: better E measurement) could minimize ( ) χ 2 E1,fit E 2 ( 1,meas E2,fit E 2,meas = + σ 1 but minimum is clear: E fit = E meas (something is missing!) must introduce constraint: (k 1 + k 2 ) 2 = m 2 π gives 2E 1 E 2 (1 cos θ) = m 2 π Problem: minimize χ 2 while simultaneously satisfying constraint Mark Ito (JLab) Kinematic Fitting July 11, / 25 σ 2 ) 2

12 Example Minimization Strategy multi-variable minimization with constraints: Lagrange multipliers instead of minimizing over two variables, minimize over three E 1,fit, E 2,fit, and λ ( ) χ 2 E1,fit E 2 ( 1,meas E2,fit E 2,meas = + σ 1 σ 2 +2λ [ 2E 1,fit E 2,fit (1 cos θ) mπ 2 ] ) 2 Mark Ito (JLab) Kinematic Fitting July 11, / 25

13 Results fit two-photon invariant mass Mark Ito (JLab) Kinematic Fitting July 11, / 25

14 Results fit relative error on single photon energy Mark Ito (JLab) Kinematic Fitting July 11, / 25

15 Results fit two-photon measured momentum Mark Ito (JLab) Kinematic Fitting July 11, / 25

16 General Formalism Variable Definitions measured variables N number of measured variables (input) y vector of measurements, N-dimensional (input) V covariance matrix, N N (input) η vector of fit values of measured variables, N-dimensional χ 2 = (y η) T V 1 (y η) unmeasured variables J number of unmeasured variables (input) ξ vector of unmeasured variable values, J-dimensional Mark Ito (JLab) Kinematic Fitting July 11, / 25

17 General Formalism Variable Definitions 2 constraints K number of constraints (input) f vector of constraint functions, K dimensional (input) Each constraint a function of measured and unmeasured variables. When constraint is satisfied f k (η 1,..., η N, ξ 1,..., ξ J ) = 0 for k = 1,..., K Lagrange multipliers λ vector of mulipliers, K-dimensional Extended χ 2 to be minimized: χ 2 (η, ξ, λ) = (y η) T V 1 (y η) + 2λ T f (η, ξ) Mark Ito (JLab) Kinematic Fitting July 11, / 25

18 General Formalism Minimization Condition Set all partial derivatives to zero: where χ 2 [ ] = 2V 1 (y η) + 2Fη T λ η = 0, n n χ 2 ] = [2Fξ T ξ λ = 0, j j χ 2 λ k = [2f ] k = 0, j = 1,..., J k = 1,..., K (F η ) kn = f k η n and (F ξ ) kj = f k ξ j n = 1,..., N All equations manifestly linear, except for the multiplier derivatives. Mark Ito (JLab) Kinematic Fitting July 11, / 25

19 General Formalism Linearize the Constraints; Iterate the Solutions Go to a linear form of the constraints: 1 Change variables: η = η 0 + δη, ξ = ξ 0 + δξ, and λ = λ 0 + δλ 2 Take first order in Taylor expansion: f (δη, δξ, δλ) = f (η 0, ξ 0, λ 0 ) + (F η )(δη) + (F ξ )(δξ) Now have a linear system of N + J + K equations with N + J + K unknowns, i. e. δη, δξ, δλ But an approximation was made: need to iterate Also: must initialize variables Mark Ito (JLab) Kinematic Fitting July 11, / 25

20 General Formalism Errors covariance matrices: fit variables V η = V [ ] I (G HUH T )V unmeasured variables V ξ = U covariances between fit and unmeasured variables V η,ξ = VHU where G = Fη T S 1 F η, H = Fη T S 1 F ξ, U 1 = Fξ T S 1 F ξ and S = Fη T VF η Mark Ito (JLab) Kinematic Fitting July 11, / 25

21 Diagnostics Stretch Functions or Pulls How to tell if the thing is working? look at use these N quantities: z n = y n η n σ 2 (y n ) σ 2 (η n ) n = 1,..., N Gaussian with mean at 0, σ of 1 If not there are problems: offset mean: measurements biased wrong width: errors not correct tails: non-gaussian tails in measurements, background in sample Mark Ito (JLab) Kinematic Fitting July 11, / 25

22 Diagnostics Unmeasured Variables, Number of Constraints go back to π 0 decay could have introduced unmeasured variables: p π,x, p π,y, p π,z but then would have to apply 3-momentum conservation now have 4 constraints with 3 measured variables used to have 1 constraint with 0 measured variables same problem recast: 1-C fit C = K J Mark Ito (JLab) Kinematic Fitting July 11, / 25

23 Diagnostics Check χ 2 χ 2 should have a standard probablility density distribution: f (χ 2 ) Convenient to check χ 2 probability: P(χ 2 0) = P runs from 0 to 1 χ 2 0 f (χ 2 )dχ 2 for nominal χ 2 distribution P will be uniform non-uniformity: problem with errors, check the pulls often see peaks near 0: bad χ 2, background in sample Mark Ito (JLab) Kinematic Fitting July 11, / 25

24 Diagnostics Example Probability Distribution Mark Ito (JLab) Kinematic Fitting July 11, / 25

25 Summary Summary measured variables, with or without statistical correlation, may have physical relationships kinematic fit varies values of measured quantities to satisfy relationships minimize χ 2 with constraints improved measurements diagnostics about bias and errors in measurements are generated goodness of fit a handle on correctness of physical relationships assumed Reference: A. G. Frodesen, O. Skjeggestad, H. Tøfte. Probability and Statistics in Particle Physics. Universitetsforlaget, ISBN Mark Ito (JLab) Kinematic Fitting July 11, / 25