1. Vertex and Kinematic fitting with Pandaroot
Vishwajeet Jha
PANDA Meeting : GSI December 09 1

2. Outline:
1. Introduction
2. PandaRoot Vertex Fitter and New Developments
3. Preliminary performance tests with New Vertex fitter
4. Tests of kinematic fitters in pandaroot
5. Summary and outlook

3. Introduction:
Vertex fitters are implemented at the end of analysis chain
To find the crossing point compatible with reconstructed tracks
Output is the vertex position and its errors with fit probability
Improvement in the resolution of track parameters
In Pandaroot vertex fitters with Rho track candidates

4. New Vertex Fitter I: χ2 = (α – α0)T Vα0 -1 (α – α0)
α0 Track parameters from track fit , Vα0 Covariance Matrix from track fit
α New parameters , Iterations are needed to converge to the real vertex
Algorithm same as BABAR Chi2Vertexer
New Vertex Fitter II:
Algorithm same as PndVtxFitter (Based on Belle Kfitter, a new implementation)
A good start point is needed specially for displaced vertices.
Start point : Vertex finder, ( crossing point , point of minimum distance)
Propagation of the track parameters and covariance matrix to the start point

5. New Vertex Fitter II :
Least Square minimization with lagrange multiplier for constraints.
χ2 = (α – α0)T Vα0 -1 (α – α0)+ λ(D (α – αa )+ E(x – xa ) + d)
The constraint equation H(α) =0 is linearized around suitable point (αa ,xa )
D is derivative w.r.t α, E is derivative w.r.t x, d is H (αa ,xa )
Minimize χ2 with respect to α , x and λ to determine track parameters and vertex
Constraint : Tracks pass through an unknown point, For a solenoid field this leads to two constraints

6. Λ Reconstruction and Vertex Fitting:
ppbar -> Λ Λbar 1.5 GeV/c,
Λ-> pπ-
Reconstructed Λ
proper decay length (L*M/P) =
8.43 ± cm
PDG Value = 7.89 cm

7. Λ Reconstruction and Vertex Fitting:
New Vertex Fitter
PndVtxFitter

8. Ds Reconstruction and Vertex Fitting:
ppbar -> Ds Ds*(2317) , Ds -> K+ K- π
Reconstructed Ds (proper) Decay length = 185 ± 29 μm
PDG Value =147 μm

9. Ds Reconstruction and Vertex Fitting:

10. Kinematic Fitting with other constraints:
Same procedure can be used to apply additional constraints
Global : All constraints are applied simultaneously
Sequential : Constraints applied sequentially to build the decay chain
Kinematic constraints:
i) Invariant mass constraint
ii) Total energy or momentum constraint
iii) 4 vector constraint
iv) Equal mass constraint
Geometric Constraints :
i) Pointing constraint
ii) Back to back constraint

11. Kinematic Fitting Mass Constraint :
PndMassFitter

12. Kinematic 4 Momentum Fitting:
Pnd4CFitter
ppbar -> J/ψ π+π-
Full MC Simulation
and reconstruction
4 Momentum fit
for the ppbar system )

13. Summary and Outlook: New Vertex Fitters have been implemented
Preliminary tests of its performance have been made.
More detailed tests and improvements are in progress.
Kinematic fitters with more constraints are planned to be included. 13

14. New Vertex Fitter I: χ2 = (α – α0)T Vα0 -1 (α – α0)
α0 Track parameters from track fit , Vα0 Covariance Matrix from track fit
α New parameters ,
Starting from parameters in cartesian coordinates α ( px,py,pz,E ,x,y,z)
Gxx = Σ Txxi , Txxi = Wxxi –Di , Di = Wxpi * Wppi * Wxpi T
x = GxxTx , Tx = Σ Txxi
p = Σ pi =poi – Di T(x-x0i)
Based on BABAR Chi2MinVertexer
Iterations are needed to converge to the real vertex